Detecting crosstalk errors in quantum information processors

Crosstalk occurs in most quantum computing systems with more than one qubit. It can cause a variety of correlated and nonlocal errors, which we call crosstalk errors. They can be especially harmful to fault-tolerant quantum error correction, which generally relies on errors being local and relatively predictable. Mitigating crosstalk errors requires understanding, modeling, and detecting them. In this paper, we introduce a comprehensive framework for crosstalk errors and a protocol for detecting all kinds of crosstalk errors. We begin by giving a rigorous definition of crosstalk errors that captures a wide range of disparate physical phenomena that have been called"crosstalk". The heart of this definition is a concrete error model for crosstalk-free quantum processors. Errors that violate this model are crosstalk errors. Next, we give an equivalent but purely operational (model-independent) definition of crosstalk errors. Finally, using this definition, we construct a protocol for detecting crosstalk errors in a multi-qubit processor. It detects crosstalk errors by evaluating conditional dependencies between observed experimental probabilities, and it is highly efficient in the sense that the number of unique experiments required scales linearly with the number of qubits. We demonstrate the protocol using simulations of 2-qubit and 6-qubit processors.


I. INTRODUCTION
Quantum computing has grown from a theoretical concept into a nascent technology.Cloud-accessible quantum information processors (QIPs) with 20+ qubits exist today, and ones with around 100 qubits may appear in the next few years [1].Fundamental operations -gates, state preparation and measurements (SPAM) -are approaching the demanding error rates required by the theory of fault-tolerance on a number of physical platforms, including superconducting qubits and trapped ions [2].However, as experimentalists and engineers have begun to build systems of 10-20 qubits, it is becoming clear that emergent failure modes may be an even bigger problem than errors in elementary operations.The most obvious failure mode that emerges at scale is crosstalk.
"Crosstalk" describes a wide range of physical phenomena that vary significantly across physical platforms used for quantum computing.We will focus, instead, on the visible effects of crosstalk on the quantum logical behavior of a physical system that is used and treated like a quantum computer.We refer to these hardware-agnostic effects as crosstalk errors -deviations from the ideal behavior of quantum gates and circuits, which can be formalized and captured in an architecture-independent way.Crosstalk errors violate either of two key assumptions that go into any well-behaved model of QIP dynamics: spatial locality, and independence of operations.Gates and other operations are supposed to act nontrivially only in a specific "target" region of the QIP, and their action on that region is supposed to be independent of the context in which they are applied.These assumptions enable tractable models for quantum computing, and crosstalk errors violate them.Here, we give a rigorous definition of crosstalk errors that captures the effects of crosstalk, while avoiding the need to engage deeply with the physical phenomena themselves.
We begin in Sec.II and Sec.III by defining what it means * mnsarov@sandia.govfor a quantum processor to be "crosstalk-free" at the logical level.In Sec.IV, we construct an explicit error model for Markovian crosstalk-free behavior.Markovian dynamics that are not consistent with that model constitute crosstalk errors.Then, in Sec.V and Sec.VI, we take an operational approach to crosstalk and show how to detect it using only correlations between experimental variables -the settings and the outcomes of experiments.The protocol we develop specifies a set of O(n) experiments for detecting crosstalk on an n-qubit QIP.The analysis of the data from these experiments is also efficient and uses techniques adapted from causal inference on probabilistic graphical models [3,4].Much recent work has been published on detecting, quantifying, and modeling crosstalk and crosstalk errors in quantum computing devices [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].Variants of Ramsey sequences have been used to detect and quantify coherent coupling between qubits [17].This technique is very hardware-specific and typically limited to detecting crosstalk in the form of unwanted Hamiltonian couplings of known form.Several groups have also demonstrated mitigation of crosstalk in readout lines by detailed characterization and compensation [12,13] (see also Supplementary Information in Refs.[5,6,11,18,19]).A very different approach, which is platformindependent and model-free like the work we present here, is the simultaneous randomized benchmarking (SRB) technique for detecting and quantifying crosstalk between pairs of qubits [7,9].The crosstalk detection protocol we present here is similar in motivation to SRB, and is meant to be used as a light-weight diagnostic for the presence of crosstalk.It is specifically designed to be run efficiently on many-qubit QIPs, whereas we are not aware of an efficient application of SRB that reveals the crosstalk structure in a many-qubit QIPs.Moreover, our protocol is designed to detect all kinds of crosstalk errors and is more flexible in terms of the experiments that are performed, allowing it to be tailored towards detection of certain types of crosstalk errors.However, SRB has at least one clear advantage over our protocol; it measures the quantitative rate of certain crosstalk errors, whereas our protocol is just designed to detect them, and has limited quantitative ability.Finally, we note that in a previous paper [14] arXiv:1908.09855v1[quant-ph] 26 Aug 2019 we gave a protocol for detecting context dependence, including crosstalk, that can be seen as a precursor to the protocol given here.

II. CROSSTALK AND CROSSTALK ERRORS
Before we embark on defining things precisely, a brief discussion of exactly what we are defining is apropos.In particular, the distinction between "crosstalk" and "crosstalk errors" needs further explanation.
Crosstalk is an imprecise but widely used term that appears primarily in electrical engineering and communication theory, and generally refers to "unwanted coupling between signal paths" [20].In experimental quantum computing, the word has been adapted to describe a range of physical phenomena in which some subsystem of an experimental device -a qubit, field, control line, resonator, photodetector, etc. -unintentionally affects another subsystem.
A specific quantum computing device will generally display more than one such effect.For example, a transmonbased quantum processor might experience • Residual coherent couplings between transmons that should be uncoupled, • Traditional electromagnetic (EM) crosstalk between microwave lines, • Stray on-chip EM fields due to imperfect microwave hygiene, • Coupling between readout resonators attached to distinct qubits, • 60Hz line noise that influences all the qubits.
Any and all of these phenomena could legitimately be termed crosstalk.All of them are architecture-specific; a trapped-ion processor would have its own endemic crosstalk effects, some analogous to these and some not.
Our goal is to understand and address crosstalk in a platform-independent way that facilitates comparisons between quantum processors without reference to the underlying physics.This is clearly inconsistent with the established use of the term crosstalk to describe specific physics phenomena.There is no reasonable direct comparison between an unwanted 2-transmon coupling (measured in MHz) and the intensity of a control laser spillover in a trapped-ion setup (measured in W/m 2 ).But we can legitimately compare their effects at the logical level of abstraction, where each device is required to behave like a quantum computer, performing logic gates and quantum circuits.
We introduce the new term "crosstalk errors" for this purpose.It means any observable effect at the logical level (qubits, gates, quantum circuits, and their associated probabilities) that stems uniquely from some form of physical crosstalk.Some forms of physical crosstalk may result in purely local errorse.g., independent bit flips -at the logical level; these are not crosstalk errors (despite their source) because they could have been produced by local noise.Similarly, if physical crosstalk exists but has no effect (perhaps because of intentional mitigation) then we say that the system is "crosstalk-free" at the logical level.

III. DEFINITION OF CROSSTALK ERRORS
Crosstalk errors are undesired dynamics that violate either (or both) of two principles: locality, and independence.In an ideal QIP, each qubit is completely isolated from the rest of the universe, and evolves independently of it, except when an operation is applied.Operations, including gates and measurements, couple qubits to other systems, such as external control fields and/or other qubits.This coupling is supposed to be precise and limited in scope.
Unfortunately, real QIPs are not ideal.They experience all manner of noise and errors.Of course, not all errors constitute crosstalk errors.Errors can cause deviations from ideal behavior yet still respect locality and independence.Unwanted dynamics that do violate locality or independence constitute crosstalk errors.We now make this precise by defining locality and independence.

Locality of operations:
The physical implementation of a quantum circuit does not create correlation between any qubits, or disjoint subsets of qubits, unless that circuit contains multiqubit operations that intentionally couple them.
If a processor obeys locality, then it makes sense to talk about the action of operations on their target qubits, and we can go further and define independence.If locality is violated, then operations do not necessarily have well-defined actions on their targets, and independence may not be well-defined.
Independence of local operations: When an operation (gate, measurement, etc) appears in a quantum circuit acting on target qubits q at time t, the dynamical evolution of q at time t does not depend on what other operations (acting on disjoint qubits) appear in the circuit at the same time t.A QIP's behavior is crosstalk-free if its behavior, when implementing arbitrary circuits, satisfies locality and independence.

IV. AN EXPLICIT ERROR MODEL FOR CROSSTALK-FREE PROCESSORS
The definitions in the previous section are abstract.They neither rely upon nor define a concrete model for crosstalk errors or for crosstalk-free processors.In this section, we specialize to Markovian processors and construct an explicit model for crosstalk-free Markovian processors.By assuming Markovianity we are able to rule out many conceivable failures and define a model in which only finitely many things can go wrong.By defining crosstalk-free within this framework, we get a division of Markovian errors into crosstalk-free or local errors, and everything else (i.e., crosstalk errors).The dynamics of a stable QIP are Markovian if each layer in the circuit, including state preparation and measurement, can be represented by a CPTP map that depends only on the operations that comprise that layer, and not on any external context.For example, the two shaded circuit layers are identical and therefore must be represented by the same CPTP map.(c) The dynamics are Markovian and crosstalk-free if the gate operations are modular: the CPTP map describing a given circuit layer can be written as a tensor product of CPTP maps describing each of the component gates (locality), and these component maps do not depend on the other gates in the layer (independence).For example, each appearance of the shaded X, CNOT, or H gates must be represented by the same CPTP maps.

A. Defining crosstalk-free for Markovian QIPs
We place Markovianity in context within a hierarchy of models for quantum hardware, based on increasing levels of modularity (see Fig. 1).A processor is stable if its implementation of any fixed quantum circuit can be described by a fixed distribution over outputs.It is Markovian if each circuit layer is stable and can be described by an n-qubit CPTP map.And a stable, Markovian processor is crosstalk-free, if its implementation of quantum operations satisfies locality and independence.

Stable QIPs
We call a QIP stable if every circuit's outcome probability distribution (over n-bit strings) is independent of external contexts [21].Contexts on which these probabilities might depend include the time at which the circuit is run, the identity of the circuit that was run before it, or even the phase of the moon.Stability is the weakest notion of modularity: a stable QIP is modular only in the sense that its output distribution is independent of any external contexts, so that each circuit run on the QIP forms a "module".If a QIP is not stable, then modeling or probing its behavior becomes much more difficult.Importantly for this work, protocols for detecting crosstalk will likely be corrupted by this instability, and there is no obvious way to draw conclusions about crosstalk.The amount of instability can be bounded by using protocols proposed in Refs.[14,22].

Markovian QIPs
We need a stronger notion of modularity to predict how a QIP will perform on new quantum circuits that have not been run before.Circuits have a well-defined notion of time, which usually defines a natural partitioning into consecutive layers [23] of parallel operations (gates, state preparations or measurements).See Fig. 1 for an example circuit with 9 layers that we notate L 0 , ..., L 8 .Operations within a single layer are effectively simultaneous.A layer is uniquely defined by the list of operations applied to each qubit during that layer, where "operations" can include idles, measurements, and initialiation/reset operations as well as logic gates.Figure 1(b) shows a circuit partitioned into layers.
We call the QIP Markovian if we can describe and model each unique layer by a CPTP map acting on all n qubits in the system.We use a broad definition of CPTP map here, in which the input and output spaces need not be the same, and can include classical systems.Typically, an initialization operation is represented by a density matrix, which is a CPTP map from a trivial (1-dimensional) state space to a d 2 -dimensional vector in quantum state space (Hilbert-Schmidt space); here d = 2 n .Logic gates are represented by "square" CPTP maps from a quantum state space to itself.Terminating measurements are represented by POVMs, which are CPTP maps that map a d 2 -dimensional quantum state to a d-outcome classical distribution.Intermediate measurements are represented by quantum instruments [24], which are CPTP maps that map a d 2 -dimensional quantum state to a d 2 -dimensional quantum state and a d-dimensional classical distribution.Layers involving multiple kinds of operations are represented by CPTP maps whose input and output spaces correspond to the tensor product of the input/output spaces of all the component operations.
There are many ways to violate the Markovian condition.For example, a layer might appear multiple times in the circuit, and act differently each time.But if a QIP is Markovian, then the CPTP map representing each layer depends only on the identity of the layer, not on any external context (e.g., the time, or which layers occurred previously).Hereafter, we will only consider Markovian QIPs.The abstract definitions of locality and independence can presumably be instantiated in a non-Markovian model, but since no general model for non-Markovian processors is known we leave this for future work.
We use L to denote a CPTP map describing a given circuit layer.To specify which layer L describes, we will either index its position in the circuit or specify its component operations explicitly.For example, in Fig. 1(b), layers 1 and 3 (highlighted) are identical; each involves an X gate on qubit 0, a CNOT gate from qubit 2 to qubit 1, and a Hadamard gate on qubit 3.So we denote the CPTP map for this layer by: The probabilities of the measurement outcomes for a quantum circuit are determined entirely by the CPTP maps describing the circuit's layers.For a depth-N circuit that begins with an initialization layer ρ, ends with a POVM measurement layer {M i }, and includes N − 2 gate layers in the middle, the probability of the ith possible result is Here i is an n-bit string denoting the measurement result.Markovianity ensures that the QIP's behavior is modular in time.It is the layers that are modular; each layer's effect on the QIP's state must be well-defined, stable, and independent of context.This is a powerful assumption.It makes modeling possible -we can now predict the results of new circuits as long as they are composed of layers that we have characterized already.
But efficient modeling of n-qubit circuits requires a stronger modularity condition.Representing every possible layer by an n-qubit CPTP map is neither compact nor tractable.Exponentially many layers need to be described, and each one requires O(16 n ) real numbers.Even storing that model is impractical for large n, and learning it from data becomes infeasible for as few as three qubits.Stronger modularity assumptions, like the absence of crosstalk errors, enable efficient models like the one we present below.
Although general n-qubit Markovian models are intractable to reconstruct, Markovianity (like stability) can be tested.Published protocols include those in Refs.[25][26][27][28][29][30].Violations of the Markovian model -generally termed non-Markovianity -may result from a number of underlying causes, including time-dependence, persistent bath memories, or even serial context dependence, where the performance of a layer operation is influenced by the layers that immediately proceed (or even follow) it due to the finite bandwidth of control pulses.
We expect that all QIPs are at least a little bit non-Markovian, but we also expect that our Markovian model for crosstalk errors will (like the Markovian CPTP map model itself) fail gracefully, and continue to work well for slightly non-Markovian QIPs.However, our experience is that crosstalk detection protocols (including the one we develop in the second half of the paper) can confuse violations of Markovianity for crosstalk.So, in practice, it is important to test for non-Markovian effects before (or simultaneously with) testing for crosstalk.

Crosstalk-free Markovian QIPs
Whereas Markovianity allows modularity in time, a processor without crosstalk is modular in spacei.e., across qubits and regions.Layer operations can reliably be composed by combining even smaller operations, that act locally and independently.Earlier, we said that a processor is crosstalk-free if it obeys locality and independence.If it is also Markovian, then the conditions for locality and independence can be stated as explicit conditions on the CPTP maps describing circuit layers.
Each ideal circuit layer defines a locality (tensor product) structure that divides the qubits into disjoint and uncoupled target subsets.A Markovian QIP satisfies locality if and only if the CPTP map describing each layer obeys that structure.(The proof is trivial: for any bipartite system AB and operation G AB , there exists an initial product state If a Markovian model satisfies both of these conditions, then we say it is crosstalk-free.Its behavior is consistent with the absence of physical crosstalk, and its dynamics contain no crosstalk errors.Conversely, any violation of these conditions constitutes a crosstalk error. If a QIP satisfies Condition 1 (locality), then each layer's CPTP map is a tensor product over the target subsets implied by that layer.The CPTP map for the layer described above, in the example of Markovianity, would be where G, indexed by the gate operation and qubit target, represents a component CPTP map for that gate.
To satisfy Condition 2 (independence), a gate that appears in multiple layers must act identically in each of them.For example, in Fig. 1, layers 1, 3, 5, and 7 all contain a Hadamard gate acting on the fourth qubit.So the CPTP map describing layer 5 then must take the form where G(H 3 ) is the same local map that appeared in the other layers (although this map does not have to be the same as the same gate on another qubit, e.g., G(H 4 )).
Initialization and measurement operations must obey the same structure.For the four-qubit QIP shown in Fig. 1, this means that: where M j,i j is the POVM effect operator for outcome i j on qubit j.If the initial state is correlated, or the output bit on one qubit depends on another qubit's state, then the QIP is not crosstalk-free.
B. Discussion of the crosstalk-free QIP model In classical systems, crosstalk usually refers to a signal in one channel influencing the signal on another channel.For example, inductive coupling between adjacent copper telephone wires may cause a conversation on one line to be heard on another.Analogous effects occur in QIPs -laser beams have finite width and may illuminate neighboring ions, superconducting transmission line resonators may capacitively couple to each other, or qubits themselves may interact directly.These interactions can be modeled by coupling Hamiltonians.So it is tempting to say that "crosstalk" is nothing more than a coupling Hamiltonian, and the complex abstraction that we have introduced is unnecessary.But this misses three key points.First, those Hamiltonians appear in low-level device modeling, and are specific to particular physical implementations.Second, like all low-level device Hamiltonians, they fluctuate in time and with the state of the environment.Third, the systems that they couple are often ancillary ones -control wires, ambient fields, etc -that would not normally appear in an effective description of the processor and its qubits.Defining, detecting, and modeling crosstalk at this low level is possible -and even desirable for device physicists -but not portable across many devices.
We have presented a high-level, hardware-agnostic effective model.This approach is common.It is present when qubits are described as 2-dimensional Hilbert spaces, when logic gates are described by CPTP maps, and when errors are modeled as depolarization or T 1 processes.Our model, like all of those techniques, trades the conceptual simplicity of Hamiltonian dynamics on very large system-specific Hilbert spaces for the practical tractability of an effective model on n qubits.The CPTP map formalism strikes a good balance between rigorous, low-level device models and cross-platform, high-level abstraction -but as a picture of the underlying physics, it is coarse-grained and can sometimes be counterintuitive.
For example, consider two qubits in a magnetic field along the Z axis whose strength varies slowly in time, see Fig. 2. The field causes both qubit states to rotate around the Z axis.Clearly, there is neither coupling nor communication between the qubits.So, if we include the magnetic field in our model, then it seems that there should be no crosstalk between the qubits.But if we only model the two qubits, and integrate out the field, then the CPTP map describing the effective dynamics of the two qubits violates the crosstalk free model -they experience correlated Z errors, which violate locality.This may appear counterintuitive, since the qubits are not coupled, and   If the field's state is modeled and tracked, then there is no crosstalk between the two qubits the correlations between their states and errors are fully explained by the field and its coupling to them.
(b) But if we do not track the field, focusing on the two qubits only then there is crosstalk between the qubits, in the form of correlated stochastic errors mediated by the (untracked) magnetic field.
neither has any causal effect on the other.But it reflects the fact that there is crosstalk in the system, between each qubit and the magnetic field.Even when the field is eliminated from the model, it still mediates an effect that creates unexpected correlations between the qubits.Crosstalk errors can occur at the coarse-grained level even between two qubits that are not directly coupled by the underlying physics.
The stable/Markovian/crosstalk-free hierarchy of models given above is based on strict criteria that, as stated, are either true or false.One might object that these conditions are practically useless -no processor is perfectly Markovian or crosstalk-free, and could not be proven so even if it were.While this objection is strictly speaking true, it dismisses the utility of idealized models.No operation is perfectly unitary, yet unitary dynamics is both well-defined and highly useful as an ideal.In the same way, what matters is not whether a QIP is perfectly crosstalk-free, but how close it is to the ideal.The definitions given above lay the groundwork for metrics that quantify that closeness, and thus for measuring how much crosstalk is present.
Similarly, perfect Markovianity is not required.In a real and slightly non-Markovian QIP, we can confidently detect crosstalk as long as the violations of Markovianity (or stability) are small compared to the violations of the crosstalk-free conditions.An experiment to detect crosstalk has a certain duration and a certain statistical power.If it detects crosstalk, that conclusion is reliable as long as the QIP's instability and non-Markovianity do not rise above the experiment's level of sensitivity over its duration.
Finally, note that the CPTP maps describing experimental operations are only unique up to a gauge freedom [31][32][33].In multiqubit QIPs, this gauge freedom is non-local.Gauge transformations -which simply change the description of the QIP, and have no observable consequences -can change the tensor product structure of operations, transforming a CPTP map that respects a tensor product structure to one that does not, and vice versa.This raises the question of whether the "crosstalk-freeness" of a model is real and experimentally testable, since it appears to be not gauge-invariant.
Fortunately, there is a simple resolution: a stable, Markovian QIP is crosstalk-free if there exists some gauge in which Conditions 1-2 hold.This is directly analogous to the definition of a perfectly error-free gate set.An ideal target set of operations can be written down in many gauges.In all but one of them, the CPTP maps appear to be different from the original "ideal" ones.But this is the nature of gauge theories.What matters are the observable probabilities predicted from the theory.Those are identical in all gauges.So if there exists any gauge in which a gate set coincides with its ideal target, then no experiment (with this gate set) will ever detect any error.Similarly, if there exists any gauge in which a set of n-qubit operations is crosstalk-free, then no experiment (with this gate set) will detect evidence of crosstalk.A processor is crosstalk-free if and only if it admits some crosstalk-free model.

C. Examples
We now consider some examples of crosstalk phenomena, and the crosstalk errors they induce.All the examples in this section involve a QIP with just two qubits, which we label A and B. The examples can be generalized easily to more qubits.
1. Pulse spillover: Quantum gates should act only on their target qubits, but control pulses may spill over onto neighboring qubits and affect them.This is the most widely discussed form of crosstalk, e.g., [8,[34][35][36][37].For example, consider two qubits that experience no errors when both are idle.But whenever an X π gate is applied to qubit A, the control field spills over onto qubit B and induces a small X rotation.Each layer still respects the tensor product structure of the two qubits, so locality is not violated.However, the effect of the idle operation on qubit B depends on whether an idle or an X π gate was applied to qubit A at the same time, so this scenario violates independence.
2. Always-on Hamiltonian: Suppose that when both qubits are idle, they experience an unwanted XX Hamiltonian.Thus, if A is in the |+ (respectively, |− ) state, B undergoes a slow rotation around the +X (respectively, −X) axis.Each qubit is influenced by the state of the other.This example violates locality, because the map describing the global idle is an entangling unitary operation, which is not a tensor product of two single-qubit CPTP maps.
3. Correlated stochastic errors from common causes: Correlated dynamics caused by a common influence can violate locality.For example (see Fig. 2), suppose both qubits interact with a common magnetic field along the quantization axis, and that field undergoes white-noise fluctuations.This produces correlated (weight-2) dephasing or ZZ errors while the qubits are idle.This is not a tensor product map, and violates locality.Note that a constant field would only cause local unitary rotations, which respect the tensor product structure and does note result in crosstalk errors.
4. Detection crosstalk: Measurements of a qubit's state may be influenced by the state of neighboring qubits.As an example, consider measuring trapped-ion qubits A and B simultaneously using resonance fluorescence.If light scattered from qubit B spills over onto the detector for qubit A, then the result of measuring qubit A will depend on the state of qubit B. We refer to this type of crosstalk error as detection (or readout) crosstalk, because it specifically affects measurement results.This example violates locality -the POVM describing the measurement does not respect the QIP's tensor product structure, because the marginal effects corresponding to "0" and "1" on qubit A act nontrivially on qubit B.
5. Correlated state preparation: Correlated errors in the controls used to prepare the qubits can create correlated, or even entangled, initial states.This violates locality.For example, consider initializing qubits A and B to the |0 state using a common control field.Occasionally, some noise in the common control field may increase the state preparation error for both qubits.For any single trial, the resulting state would be a product state, but when averaged over many initializations the density matrix describing the initial state can no longer be factorized, so locality is violated.
This list of examples is not exhaustive, but we hope it helps to connect common notions of crosstalk to the conditions that define the crosstalk-free model.

D. Useful terminology for crosstalk errors
Any violation of the crosstalk-free model results in crosstalk errors, but there are many ways to violate the model.Some of them are quite distinct from others, both in the physical phenomena that typically produce them, and in their consequences and behavior.It is useful to identify the most common categories and give them names, if only to facilitate answering the question "What kind of crosstalk do you see?" We suggest some useful categories here, based on our experience examining data and modeling noise.
First, we observe a fundamental difference between errors that violate locality, and those that only violate independence.Any violation of locality can be traced to at least one specific layer operation that creates unexpected correlations.We call these crosstalk errors absolute.In contrast, violations of independence cannot be isolated to a specific layer operation.Some local operation just behaves differently in different layers, and no one layer defines the correct behavior of that operation.We call these crosstalk errors relative.
In addition to these terms, which are relatively rigorous, we have found the following less-precise categories to be useful.These categories are not intended to be exhaustive, and may not prove over time to be the most useful classification.For example, the "correlated state preparation" example given in the previous section does not fall into any of these categories (it could define another category, but it is not clear that it is sufficiently common or important).Other violations of the crosstalk-free model can be invented that fall into none of these three categories, or bridge them.Furthermore, we do not yet have specific protocols for rigorously distinguishing these categories.Nonetheless, we have found them useful, and so we propose them to the research community.
Idle crosstalk is any violation of locality when all qubits are idle.The unique layer in which no nontrivial operations are performed corresponds to a CPTP map that we call the global idle, and if the global idle is not a tensor product of 1-qubit CPTP maps, then we say there is idle crosstalk.Any error occurring during the global idle that produces correlation between qubits (an error of weight 2 or higher) is an idle crosstalk error.Examples 2 and 3 in the previous section are examples of idle crosstalk errors.The same physical phenomena (always-on Hamiltonians, correlated decoherence, etc.) can also cause high-weight errors during nontrivial gates, but their effects are usually strongest and easiest to detect during the global idle.
Operation crosstalk refers to violations of independence caused by particular logic operations.A QIP displays operation crosstalk if the act of performing an operation on qubits in region A changes the dynamics of qubits in a disjoint region B. It is not always possible to unambiguously ascribe a crosstalk error to an operation (i.e., to define operation crosstalk orthogonally to idle crosstalk), but we have found it useful to have terminology for crosstalk errors that change as (non-idle) operations are applied to a QIP.Operation crosstalk is a special case of relative crosstalk, corresponding to cases where the change in region B's dynamics can confidently be blamed on a particular operation.
Detection crosstalk refers to violations of locality in the outcomes or results of measurement operations.If the result of a measurement on one qubit depends on the premeasurement state of another qubit, that is detection crosstalk.We avoid the term "measurement crosstalk" because it is ambiguous; it could also refer to errors on spectator qubits that are caused by measuring a target qubit in the middle of a circuit, which would be an instance of operation crosstalk instead of detection crosstalk.Example 4 in the previous section is an instance of detection crosstalk.

V. AN OPERATIONAL PROTOCOL FOR DETECTING CROSSTALK ERRORS
In this section, we return to the abstract definitions of locality and independence presented in Sec.III to build a protocol for detecting crosstalk errors based on the fact that violations of these conditions can be observed directly from operational phenomena.
In Sec.V A we present the model-free and operational definition of crosstalk-free QIPs that forms the basis of the protocol.Then in Sec.V B we discuss the types of experiments that could detect crosstalk in this model-free framework, and design a lightweight protocol that requires only O(n) circuits for an n-qubit QIP.In the last subsection, Sec.V C, we describe how the data from this (and related) protocols can be efficiently analyzed using statistical tools originally developed for inference on probabilistic graphical models.

A. Model-free framework and definitions
Consider a QIP comprising n qubits.Let r be a partition of the n qubits into M < n disjoint subsets, r i ⊂ {0, ...n − 1}, which we call regions, and let n(r i ) be the number of qubits in region r i .We assume no model, only that for each region r i we

FIG.
3. Illustration of the type of circuits used in our protocol.A 4qubit QIP is partitioned into three regions, labeled r 0 , r 1 , r 2 , and the goal is to detect crosstalk errors between these regions.To do so, we perform circuits that only apply coupling operations between qubits within a region, never between regions (across the red lines in the figure).The random variable outcome from measuring the qubits in region r i is denoted R r i .In this example R r 0 and R r 2 are 1-bit-valued while R r 1 is 2-bit-valued.
(1) apply operations that ideally should only affect qubits in r i and should not affect qubits in any other region, and (2) make measurements whose results should only depend on the state of qubits in r i .We will define crosstalk errors in terms of the settings that denote the operations applied to a region, and the results of measurements on qubits in a region.An experiment is defined by a tuple Ω ≡ (S r 0 , S r 1 , ..., S r M−1 , R r 0 , R r 1 , ..., R r M−1 ), where S r i are the settings assigned to the qubits in region r i and R r i are the measurement results from the qubits in region r i .We treat each member of this tuple as a random variable drawn from some sample space, S r i ∈ S r i , R r i ∈ R r i .It is clear that the results are random variables; they are the results of measurements on quantum systems, which are always random variables.We also treat the settings as random variables, but for a different reason.In a large QIP, it is not feasible to perform an exhaustive set of experiments that enumerates all the possible experimental settings.So, in practice, observed data constitute a sample over all the possible settings.As we shall see, a random sampling over settings often yields good results.The random variable R r i takes values that are bit strings of length n(r i ), obtained by measuring all qubits in region r i in some basis.More complicated scenarios, e.g., involving detection of leakage, are possible but we restrict ourselves to the simplest case here.Fig. 3 illustrates these definitions.
The settings S r i are random variables that describe (i) what state is prepared natively on the qubits in r i , (ii) what gates are applied to the qubits in r i , and (iii) what basis the qubits in r i are measured in.So S r i labels a quantum circuit for that region (defined here as the state preparation, applied gates and measurement basis choice for a region).We note that most quantum computing architectures have only one qubit state that is natively prepared (e.g., the ground state) and only one measurement basis (e.g., the Z basis).Therefore the only setting that can be varied is the gates applied to the qubits in between state preparation and measurement.Hence in most quantum computing architectures, the settings will be synonymous with "gates applied to qubits in r i ".
We say that a region r i is free of crosstalk errors to/from other regions if conditional distributions over the measure-ment results on this region satisfy: This means that the distribution of measurement results on region r i depends only on the settings for r i ; conditioned on those settings, it is independent of all the other random variables in Ω.Any violation of these conditions is a witness to some kind of crosstalk error.
It is preferable to define the crosstalk-free condition in terms of conditional independence as opposed to marginal independencei.e., P(R r i , S r j ) = P(R r i )P(S r j ) -because it is more robust to confounding by hidden (or intentional) correlations in settings, which can become an issue when detecting crosstalk errors in large QIPs.Appendix A discusses this further.
This model-free definition of crosstalk errors is equivalent to our model-based definition of crosstalk errors stated in Sec.III; see Appendix B for proof.The two definitions capture the same notions of locality and independence of quantum operations -the model-based definition does so in terms of conditions on models of quantum operations (i.e., CPTP maps), while the model-free definition does so in terms of conditions on operational random variables that arise naturally in a QIP.
The model-free definition given by Eq. ( 7) leads directly to practical tests for crosstalk, because if we draw a circuit at random from the distribution defined by P(S r 0 , S r 1 , ..., S r M−1 ) and perform it on the QIP, the result is a sample from the joint probability distribution P(Ω) ≡ P(S r 0 , S r 1 , ..., S r M−1 , R r 0 , R r 1 , ..., R r M−1 ).These samples can be used to statistically test the conditions implied by Eq. (7).In Sec.V C we define a protocol built on such statistical tests that does exactly this.First, we present an example to illustrate the notation introduced above.
Example.We wish to detect crosstalk errors induced by single qubit operations on a QIP with 3 qubits, partitioned into two regions r 0 = {0} and r 1 = r0 = {1, 2}.The following elementary single-qubit operations can be performed: initialization in |0 ; initialization in |+ ; idle gate (i.e., do nothing for one clock cycle); X π/2 gate; Z π/2 gate; and measurement in the computational basis.Circuits can be performed that comprise (1) parallel initialization of all 3 qubits, (2) a sequence of k layers built from arbitrary single-qubit gates on each qubit in parallel, and (3) measurement of all qubits in the computational basis.Then the sample space of settings on region r 0which includes only qubit 0 -is where we have distinguished prep settings (S p ) and gate settings (S g ).Only one measurement layer is allowed and the only measurement basis accessible is the computational basis, so there are no measurement settings.The space of settings for region r 1 is isomorphic to two copies of the settings for r 0 : S r 1 = S r 0 × S r 0 .The spaces of possible results for each of the two regions are simply R r 0 = {0, 1} and R r 1 = {0, 1} 2 .
In this example, each experiment is labeled by the following tuple of nine random variables, where P i ∈ S p , G i ∈ S g and R i ∈ {0, 1} label (respectively) the preparation, sequence of gates, and measurements results for qubit i.

B. Lightweight experiment design
When circuits are performed that ideally do not couple regions (Fig. 3), no correlations should exist or be created between (a) the settings on region r i and the results on a distinct region r j , or (b) the results on two distinct regions r i and r j ( j i in both cases).Crosstalk errors produce such correlations, and they can (in principle) be detected from data (samples from P(Ω)).But to generate data, we need to run circuits, and so we need to pick particular circuits.In Sec.V C we outline a rather general analysis method that can be used to detect signatures of crosstalk errors in the data produced by many sets of circuits.But first, in this section, we construct a specific experiment -set of circuits -that is well-suited for detecting crosstalk errors.
We only consider circuits that do not (intentionally) couple regions, which means that for each region there is a welldefined subcircuit comprising all operations applied to it.We also assume, for the sake of simplicity, that the QIP has unique initialization and measurement operations (in |0 ⊗n and the { 0| , 1|} ⊗n basis.Thus, the settings for a region correspond precisely to the gates in the subcircuit on that region. Our goal here is to detect any kind of crosstalk, between any regions.Each possible circuit on the QIP is composed of the parallel application of multiple subcircuits, one on each region.The simplest approach is to choose a collection of N circ subcircuits for each region, and then perform all combinations of those subcircuits.We refer to this collection of subcircuits as the "bag" of circuits applied to a region.(We postpone the question of what subcircuits to place in the bag to the end of this subsection.)We call this the exhaustive experiment, in which each subcircuit on region r i gets performed in an exhaustive variety of different contextsi.e., in parallel with all N circ circuits on other regions -and so violations of independence are easy to detect in the data.Unfortunately, this experiment defines a hypercube containing N M circ distinct circuits, which grows too rapidly with M (the number of regions) to be feasible.
However, we observe that in the exhaustive experiment, each subcircuit on every region r i is performed in exponentially many distinct contexts (defined by the settings on the other regions r j r i ).This is arduous but powerful -it can detect contrived and unlikely forms of crosstalk, like a situation where the Mth region experiences extra errors if and only if a specific subcircuit is performed on all of the other regions (e.g., if and only if an X gate is performed on M − 1 qubits, the Mth qubit experiences additional depolarization).
It is easy to see that detecting arbitrary conditional dynamics of this type demands performing every possible combination of settings.But it is also overkill; real-world crosstalk usually has pairwise manifestations, meaning that if the dynamics of region r i are correlated with the settings on several regions, they are also correlated with each of those regions' settings individually.So we will choose a sparse subset of the experiments in the hypercube defining the exhaustive experiment, with the goal of defining a small set of experiments that allow the realistic crosstalk errors to be detected.

An explicit construction
The sparse sampling of the hypercube should maintain two important properties of the exhaustive experiment.First, each subcircuit in the bag for each region r i must appear in multiple contexts (but not exponentially many).Second, that set of contexts in which each subcircuit gets performed must vary on each of the other regions.These properties ensure that -whatever subcircuits we select for each region's bag -the data will reveal whether the local results of those subcircuits are significantly influenced by the settings (choice of subcircuit) on any other region.
The construction we outline now ensures that these properties are preserved, even with much fewer experiments.It is defined by three adjustable integer parameters: • L is the length or depth of all the subcircuits.Subcircuits on different regions are applied in parallel, so they must all be the same length, so L must be chosen and fixed.
• N circ is the number of circuits in the bag for each region.
• N con is the number of random contexts in which each subcircuit will be tested.
First, we choose a bag of N circ depth-L subcircuits for each of the M regions (see below for their construction).Now, for each region m ∈ [0 . . .M − 1] and each of the subcircuits ν m in that region's bag, we define N con different circuits that perform ν m in different contexts, by choosing a subcircuit for each of the other regions at random from the corresponding bag, and performing all those subcircuits (including ν m ) in parallel.This circuit selection procedure is illustrated in Fig. 4.This design ensures that (1) for each region, approximately N circ different subcircuits are studied in detail, (2) each of these subcircuits is performed in N con different contexts, and (3) those contexts vary independently across all the other regions.
We have found that a small refinement improves the protocol in practice.Often, the idle gate is less noisy than others, and the depth-L idle circuit is the least noisy depth-L circuit and most sensitive to crosstalk.We have found it useful to artificially boost the probability of sampling all-idle circuits when the random contexts are defined (not when ν m is drawn).To do this, we sample context subcircuits normally, but replace each sample by the depth-L idle circuit with probability n F E 7 a e w 6 h z t 9 q J H v d 2 3 j 7 t 7 z 4. Illustration of the circuits performed in the lightweight crosstalk error detection experiment design.The QIP is divided into M non-overlapping regions, and subcircuits of fixed length L are applied to each region.This diagram represents one "epoch", during which the N circs subcircuits in the bag for region 0 (denoted ν i ) are iterated over.Each ν i is repeated N con times on region 0, and in each instance (each line in the diagram) the subcircuits on the other M − 1 regions are randomly sampled (with replacement) from the subcircuits in the bag for that region.In the diagram ν m i, j denotes the subcircuit applied to region m in the jth context when ν i is applied to region 0. The experiment design prescribes M such epochs; in epoch m the N circs subcircuits in the bag for region m are iterated over while the subcircuits for all the other regions are randomly sampled.It is assumed that all qubits are initialized in their ground states, and the measurements after each of the prescribed circuits are performed simultaneously on all qubits in the computational basis.Finally, each of the experiments is repeated N rep times in order to collect statistics.
p idle .This experiment design is also described by pseudocode in Appendix C.
Two additional variations are useful in some circumstances.First, in certain regimes, we find that crosstalk can be comprehensively detected without testing all N circ subcircuits in each region's bag.Iterating ν m over a randomly chosen subset is sufficient.Second, it is sometimes easier to sample the subcircuits ν m at random (with replacement) -just as the contexts are sampled randomly -than to iterate over them.

Discussion
The experiment defined above can be seen as a sparse filling of the hypercube defined by the exhaustive experiment, as long as N con does not grow exponentially with M. Ideally, it should not grow with M at all.In practice, we find a constant N con (with respect to M) to be sufficient to detect crosstalk.The protocol also requires specifying both N circ and how to construct the subcircuits, which we discuss at the end of this section.The total number of experimental configurations to be performed for any fixed length L is N exp ≈ M × N circ × N con [38], which scales linearly in the number of regions, M.
We emphasize, again, that this lightweight experiment is not designed to detect every possible crosstalk error.It is easy to imagine pathological or adversarial crosstalk error models that it almost certainly would not detect.This follows from a simple counting argument; if there are g native gates that can be applied to each region in an M-region QIP, this results in g M possible layers that can be executed in this QIP.If only a few layers in this exponentially large set manifest crosstalk errors we have negligible probability of sampling that layer in the above lightweight experiment and hence would not detect these crosstalk errors.However, such phenomena are not expected in most well-behaved quantum computing architectures.
Each of the circuits prescribed for this protocol should be repeated N rep times to collect statistics.Each repetition yields a single datum, comprising a label for the circuit applied to each region (S r i ) and a bit string describing the measurement results from each region (R r i ).This is a single sample from the distribution over settings and results P(Ω) that we seek to test for correlations that signal crosstalk errors.
As much as possible, the repetitions of the various circuits should be distributed uniformly over the entire time of the experimentnot performed all at once in a single chunk.They may be rasterized (each circuit is performed once, in succession, and this is repeated N rep times), or randomized (all the circuit repetitions are shuffled and performed in completely random order).This minimizes the probability of systematic false positives caused by drift.If behavior of the device (e.g., error rates) is correlated with timei.e., it drifts -then if the settings are also correlated with time, this will produce spurious evidence of correlation between settings and results.Time is an unobserved, or latent, variable; e.g., in the simplest case an unobserved classical degree of freedom (e.g., a twostate fluctuator) may cause drift by providing a fluctuating local potential.When a variable that is a common cause for multiple other variables is not observed, it can create a fictitious conditional dependence between these variables [39].Randomization and rasterization reduce or destroy correlations between the settings and time, reducing the risk of conflation; see also related discussion about conflation in Sec.V C 2. Finally, we note that rasterizing also facilitates concurrent drift detection with the same data [22].

Choosing the subcircuits for each region's bag
Exactly what subcircuits to choose or define for each region is a critical component.We have left it open for now for a simple reason: there are many reasonable, yet very different, possible choices.For the sake of concreteness, we specify particular circuits here.But we also expect that new, creative, and perhaps objectively better choices can be usefully explored.
The subcircuits run on each region have two purposes: to manifest crosstalk, and to detect crosstalk.It may be that only certain circuits, when run on r i , cause or amplify errors on r j .And it may also be that certain circuits on r j are more sensitive to these effects.Our goal is to detect whatever crosstalk errors exist.Therefore, in principle, the subcircuits chosen for each region's bag should be those that (1) cause the greatest effects on other regions, and (2) are the most sensitive to effects caused by other regions.
Given a specific physical model of crosstalk, it is possible to design subcircuits with these properties.Or, given a parameterized model of the sorts of crosstalk that might occur, it is possible to design a rather larger set of subcircuits that collectively amplify all the effects appearing in that model.(This is how the circuits for gate set tomography (GST) are chosen).
An entirely different approach is to switch from trying to detect all crosstalk to focusing on the crosstalk errors that impact a specific application.This motivates choosing subcircuits that are representative of the subroutines that appear in specific algorithms, and would emphasize detection of crosstalk errors that impact execution of those particular algorithms.
But we have intentionally assumed no specific model (except for the general assumption that certain highly nonlinear and/or adversarial errors are unlikely) and no specific application.In the absence of any other guidance, random circuitslike those used in randomized benchmarking -are a sensible choice.These have certain known drawbacks; they are less sensitive than periodic circuits (e.g., those used in GST or robust phase estimation) to some forms of noise because they twirl it [31,40,41].But random circuits are both common and hard to fool -their sensitivity to noise is not always high, but it is reliable.Therefore, we propose that the bag for each region be constructed by choosing N circ subcircuits uniformly at random from an ensemble of random sequences of the processor's native gates.The simulations presented in Sec.VI use this construction.

Configuring the set of circuits in the protocol
This protocol has several user-adjustable parameters.Their values can be chosen, but not arbitrarily -they control the reliability and power of the experiment.Here, we provide some heuristic guidance on how to choose them.
1. N rep is the number of repetitions of each experiment.
Increasing N rep reduces statistical noise, at the cost of requiring more time to take data.We suggest that this should be as large as possible, and no less than 1000.
2. L is the the length or depth of the circuits, and two useful rules of thumb suggest what L should be.The first is that longer circuits (all else being equal) amplify crosstalk effects and therefore permit more sensitive detection.However, once L becomes greater than 1/ , where is the rate of stochastic errors or decoherence, generic noise tends to swamp the effect sought.Therefore, L should be as large as feasible, but no greater than O(1/ ).
3. N circs is the number of circuits in each region's bag, which in turn are randomly selected from the population of all depth-L subcircuits on each region.This parameter can be chosen to be a constant, independent of M, of order 10 − 30.At a minimum, it needs to be large enough to guarantee that all possible native gates that can be performed in a region appear in at least one of the subcircuits chosen for that region.
4. N con is the number of random contexts in which each subcircuit is intentionally performed.Empirically, we find that it should be O(N circs ) -but the best value for this parameter depends on the relative strength of crosstalk errors and local errors, which we refer to as signal-to-noise ratio.When crosstalk errors are comparable to local errors (low signal-to-noise), we require N con ∼ N circs /2.But if crosstalk errors dominate (high signal-to-noise), we find that N con ∼ N circs /4 is sufficient.
5. p idle is the probability of sampling the length L idle circuit on any of the M − 1 regions when constructing a context.The recommended value of p idle depends on whether the idle operation has a significantly lower local error rate than other operations.If it does, then we recommend choosing p idle ∼ 1/M, so that there is probability ∼ (M − 1)/M ≈ 1 for large M that the idle circuit is among the contexts provided.Otherwise, p idle should be smaller -but even in this case, we find that p idle > 0 is often advantageous, although we do not have a good rule of thumb for how the optimal value varies with the idle error rate.

C. Analyzing data to detect crosstalk errors
The experiment described in Sec.V B generates data, which can be analyzed to detect and quantify crosstalk errors in a QIP.In this section we explain how to do this.
Running the circuits described above generates samples from a joint probability distribution of settings and results over the M regions, P(S r 0 , S r 1 , ..., S r M−1 , R r 0 , R r 1 , ..., R r M−1 ).Testing this joint distribution for violations of the conditions in Eq. ( 7) enables detecting whether where are crosstalk errors between any of the regions in the QIP.But we can go further, by determining the structure of the crosstalk errorsi.e., which pairs of regions experience crosstalk errors.This can be achieved using techniques from causal inference that discover conditional dependence relationships between the 2M variables in this distribution.Specifically, we show how to adapt techniques developed to learn causal structure in Bayesian networks [3,42], to efficiently detect the structure of crosstalk errors.
A Bayesian network is a directed graph where each node represents a random variable and the edges represent joint probabilistic relationships between the variables.It is a concise representation of the joint distribution over the variables, with an edge indicating a conditional dependence between the variables -an edge from node i to node j indicates that variable j is dependent on variable i, when conditioned on the other variables in the graph.This is notated (i j) | A, where A is a set representing all other nodes/variables in the graph.
Identifying causal network structure from data is an active and rapidly evolving area of research in the field of causal inference, and there are many algorithms available to do such causal network discovery [43].These algorithms fall into two broad classes.The first, termed search-and-score methods, enumerate or search through graph structures (each of which corresponds to a particular form of the joint distribution over the variables) and evaluates the how well each fits the data according to a score (which is often its likelihood [44]).The second class of algorithms, referred to as constraint-based methods, operate by reconstructing a graph that is consistent with the conditional dependencies seen in the data by performing a series of hypothesis tests.Search-and-score methods are typically very computationally expensive, especially for datasets with a large number of variables, so we focus on constraintbased algorithms in this work.
Any constraint-based algorithm for causal network structure discovery can be split into two parts: (i) a statistical test that tests for conditional independence (CI) of some sets of random variables, given samples and at a level of statistical significance, and (ii) a network discovery algorithm that repeatedly applies this CI test to determine the edges in the network.The keys to developing a "good" network learning algorithm are to formulate a CI test that is efficient and powerful, and to formulate a network discovery algorithm that is efficient, in the sense of needing to applying as few CI tests as possible.Given the mature body of research in this field, we seek to apply a previously developed network discovery algorithm to reveal the conditional dependence structure between the random variables we have in the context of crosstalk error detection.In the following subsections we present specific choices for the CI test and network discovery algorithm.
We emphasize that although we are using tools traditionally used in causal inference, we are not making claims about causality.Specifically, an edge between nodes S r i and R r j (or R r i and R r j ) for i j does not imply a direct causal relationship between the regions r i and r j , just that there is some crosstalk error between these regions.This is an important caveat.Even in the context of classical physics, it is well known that statistical causal discovery algorithms are only heuristics for revealing causal relationships (especially in the presence of latent, or unobserved, variables) [3,43,45].In quantum theory, even defining causality and a definite causal order between random variables is thorny [46,47].So we emphasize that we are simply using causal inference tools to efficiently assess conditional independence relationships that form the basis of our model-free definition of crosstalk errors.

Statistical tests for conditional independence
There are many statistical tests for conditional independence.In the protocol described in Sec.V B the random variables of interest represent experimental settings and measurement outcomes.Both are drawn from a finite set.Therefore, all random variables in a data set resulting from such experiments will be categorical.For such variables, a well-motivated test for conditional independence is the loglikelihood ratio test, or G 2 test [48].
To describe the test statistic associated with this test, let us first describe the data.The dataset consists of samples from K random variables X = {X k } K−1 k=0 some of which represent experimental settings (S r i ) and some of which represent measurement outcomes (the R r i ).We assume that each X k takes values from a finite set X k of size |X k |.Then the G 2 test statistic that tests for the conditional dependence between variable X i and X j , conditioned on the variables in the set A ⊂ X is defined as [48] G 2 (i, where n i jA (x i , x j , x A ) is the frequency of the random variables (X i , X j , X A ) taking on the values (x i , x j , x A ) in the dataset, and similarly for the other quantities.Note that X A is a composite random variable since one may want to condition on several variables, i.e., |A| > 1.Under the null hypothesis, where (X i X j ) | A, this test statistic is asymptotically distributed as chisquared with degrees of freedom This test statistic is a scaled version of the empirical estimate of the conditional mutual information between variables X i and X j , given A. Thus this quantity also has a convenient information theoretic interpretation [48].Finally, we note that in the simplest case where the conditioning set is null, A = ∅, this statistical test is often referred to as a homogeneity or independence test (with d f = (|X i | − 1)(|X j | − 1)) since it tests whether the distribution of variable i is the same (homogeneous) regardless of the value of the variable j.

Network discovery algorithms
The second part of a constraint-based causal network structure learning algorithm applies a CI test on data to reconstruct a network consistent with the data.The PC algorithm by Spirtes and Glymour [49] is a popular network discovery algorithm that has been widely implemented.Chapter 5 of Ref. [45] has a comprehensive description of the algorithm, but here we outline its basic steps.The PC algorithm starts with a complete undirected graph with edges between all nodes (each of which represents a variable in the dataset).Then each edge is tested for conditional independence, given some conditioning set A comprising neighbors of the nodes connected by the edge, for conditioning sets of increasing size (starting from an empty set).The resulting undirected graph is called the skeleton, and the last step applies certain edge orientation rules in order to estimate a directed acyclic graph (DAG) representing the causal relations in the data.
For crosstalk error detection, we will omit the last, edge orientation, step of the PC algorithm and will focus on the graph skeleton.We do this because we are not interested in identifying causal relationships (for reasons mentioned at the beginning of the section) and simply wish to detect conditional dependence relationships that signal violation of the crosstalkfree model.
In the worst case, the runtime of the PC algorithm grows exponentially with the number of variables.However, graph sparsity greatly reduces computational cost, and the algorithm has been demonstrated on data with hundreds and thousands of variables [50].Furthermore, detecting crosstalk errors is simpler than general causal network learning, because we can exploit sparsity by encoding physically motivated information into the graph from the start.For example, the edge between any two experimental settings can be removed if they are randomized according to the experiment design outlined in Sec.V B.
The PC algorithm performs multiple hypothesis tests to determine conditional independence relationships between random variables.In such multiple hypothesis testing scenarios one typically applies a significance adjustment, such as the Bonferroni correction, to control the number of false positives (type-I errors).These corrections are not done in the standard PC algorithm, because controlling the family-wise error rate is complicated by the structure of the PC algorithm: one does not know how many hypothesis tests will be performed a priori.However, we note that there have been recent attempts to incorporate statistical methods for controlling the false discovery rate by modifying the PC algorithm [51,52].Implementing this more complex algorithm may increase the reliability and statistical rigor of the crosstalk error detection protocol.Alternatively, α-significance weak control of the family-wise error rate [53] can be maintained by setting the input significance of the standard algorithm to α/K where K is the number of edges in the initial graph.
Using the PC algorithm to identify crosstalk structure in a QIP implies some subtle assumptions about the crosstalk errors.To clarify these, we first note that the PC algorithm is known to fail to detect causal network structure when the probability distribution being sampled from is not faithful to the underlying causal graph [43,45,54].In our context, faithfulness means that if there exists crosstalk between regions r i and r j , then there exist at least some random variables in r i that exhibit dependence to some random variables r j , vice versa, or both.The classic example [54,55] where the faithfulness assumption is violated and the PC algorithm fails is with three random variables X 1 , X 2 , X 3 , that are pairwise independent; e.g., if X 1 , X 2 , X 3 are binary, and This means that X i X j , for any i, j, but (X i X j ) | X k (for i j k).So each pair X i and X j are conditionally dependent, but marginally independent.The PC algorithm's first step tests each pair of variables for marginal independence [45].This step would indicate that all pairs are marginally independent, and therefore all edges would be removed and the algorithm would terminate.Therefore the PC algorithm evaluated on samples from this distribution (even in the infinite sample size limit) would yield a graph with three nodes and no edges, despite the fact that these variables are clearly dependent.An analogue of this example in the context of crosstalk detection in QIPs is the following: suppose one is trying to detect crosstalk caused by single qubit gates in an n-qubit QIP.The regions are composed of single qubits, and suppose that the crosstalk errors are such that with the circuits that are tested, one ends up preparing an entangled state of the n qubits with any two-local marginal density matrix that is completely mixed (e.g., multiparty data hiding states [56]).Then the results of measuring any qubit will be uncorrelated with the results from any other qubit (if all other measurement results are ignored) and the PC algorithm would not indicate any crosstalk between regions.The basic problem is that the marginal/local states do not produce distributions over measurement outcomes that are faithful to the underlying dependence (and correlation) between local subsystems.Testing the dependence (or correlation) between a large number of subsystems would reveal strong dependence.But the PC algorithm orders its tests by increasing number of variables (increasing size of conditioning set) for efficiency, and declares two variables to be independent as soon as it fails to detect a dependence.Therefore, it would never perform the necessary tests to reveal the dependence, which is also the root cause of the failure in the pairwise independent, three variable example given above.
Fortunately, producing unfaithful distributions over the random variables in the crosstalk error detection setting appears to be extremely artificial.Every case where we have been able to manufacture such distributions requires either, (i) highweight error generators acting non-trivially on several regions, (ii) extremely large crosstalk errors (e.g., errors causing π 2rotations), or (iii) fine-tuned crosstalk that cancels or adds up in precise ways.Moreover, we have not encountered this issue in any of the physically-relevant crosstalk error models that we have simulated .Therefore, we note it as an issue to be aware of when using the PC algorithm, but something that does not seem to practically affect the performance of the crosstalk detection protocol developed here.

Quantifying crosstalk errors
Applying the PC algorithm to a dataset reporting the experimental settings and measurement outcomes for regions will reconstruct a graph whose edges can be used to detect crosstalk errors at a specified significance level.However, we can also use this analysis to statistically quantify the amount of crosstalk error across any edge that represents crosstalk.
Let the edge that represents crosstalk in a reconstructed graph be between variables X and Y, i.e., X → Y. Recall that X takes values in the set {x 0 , ...x |X|−1 } and Y takes values in {y 0 , ...y |Y|−1 }.We compute the following total variation distance (TVD) estimates for 0 ≤ i, j ≤ |X| − 1.This quantity is a measure of the difference between the distribution of Y when X = x i and when X = x j .We quantify the amount of crosstalk error across the edge X → Y as the maximum over all i, j, since this represents the maximum deviation in the distribution of Y when X is varied: Often, we also calculate the median over these TVDs to understand how much of an outlier the maximum is.One has to be a little careful with this definition when Y is a result random variable and X is a setting random variable.To see this, suppose X = S 1 and Y = R 2 .Then, the most sensible thing is to compare the distributions of R 2 generated by the same setting S 2 , as S 1 is varied.This requires that S 2 take on a value s when S 1 = i and S 2 = j in the above definition.In this case, we calculate the above TVD for every such common setting S 2 for a pair S 1 = i and S 2 = j, and maximize over these.If no such common settings exist (which can happen for example, in the experiment specified in Sec.V B since it is a randomized design), then we fail to compute a TVD for that edge.

VI. SIMULATIONS
In this section we illustrate the crosstalk error detection and quantification procedure developed above by simulation.The analysis of simulated data is performed using crosstalk error detection routines in the pyGSTi package [57] that implement the PC algorithm as described above.

A. Two-qubit simulations
We first simulate several kinds of crosstalk error on a twoqubit system, with qubits (which form the regions in this case) labeled 0 and 1.The settings for each qubit enumerate the subcircuits applied (which are just gate sequences in this case since each region is composed of a single qubit), and the experiments simulated correspond to the experiment design outlined in Sec.V B. In addition to the crosstalk error models, we also simulate local errors through a local depolarization channel (after every gate, including the idle) with depolarization rate p local .To illustrate the efficacy of the technique, in all the simulations below, we operate in the low signal-to-noise regime where the local error rates are comparable or larger than the crosstalk error rates.This is where we expect that it is most challenging to detect the crosstalk errors.
The native one-qubit gates are assumed to be X π/2 , Y π/2 , I, where I is an idle or identity gate that takes the same time as the other gates.The native state preparation is always ideally the |0 state for both qubits, and the measurements are in the computational basis.The gate sequences are determined according to the experiment design outlined in Sec.V B. In all of the simulated experiments, we follow the guidance in Sec.V B and use the suggested value N con = N circs /2 (since the parameters chosen are in the low signal-to-noise regime).The values of N circs , N rep , L and p idle vary and are specified below for each case.
Finally, since the regions in this case are composed of single qubits, in this section we simplify notation and dispense with the additional r subscript when denoting results and settings; i.e., S r i → S i and R r i → R i .

Weight-1 operation crosstalk error
The first error model we simulate is what we refer to as operation crosstalk error, and is also sometimes termed classical, or control line, crosstalk in literature.An X π/2 gate on qubit 0 induces a depolarization channel on qubit 1 with depolarization rate p, i.e., where X π/2 (ρ) = e −i π 4 σ x ρe i π 4 σ x denotes a superoperator representation of a X π/2 unitary rotation, I denotes an identity superoperator, and D p (ρ) = (1 − p)ρ + pI denotes a depolarization channel with depolarization probability p. = 100.Finally, the significance level of the hypothesis tests used to test for conditional independence was set to α = 0.01.The red edge between settings in region 0 and results in region 1 in the graph signals the crosstalk between the qubits.

Weight-2 operation crosstalk error
The next error model we simulate is an example of what is sometimes called coherent, or Hamiltonian, crosstalk.We model an X π/2 gate on qubit 0 as inducing the desired rotation on qubit 0, but with an additional small two-qubit Z ⊗ Z Hamiltonian rotation as well, i.e., (therefore, N exp = 100), and the significance level of the hypothesis tests was set to α = 0.01.Note that the coherent crosstalk error shows up at ∼ 2 in the measurement probabilities since we are using random gate sequences, and this is why more samples are required to detect this crosstalk error.
The red edges in the crosstalk graph indicate crosstalk errors between the qubits.In this case there are conditional dependencies between settings and results in different regions, and also between the results in different regions.There is no clear causal direction for this type of crosstalk error (and one can show using a model of this kind of crosstalk error and calculations such as in Appendix B that conditional dependencies between results are expected for this kind of crosstalk error).

Detection crosstalk error
The final error model we simulate is a model of crosstalk during the qubit measurement process.The measurement effects, indexed by the outcome values, are: In other words, if the measured value for the first qubit is 1, there is a p m probability that the measured value of the second qubit is flipped.This could, for example, model detection crosstalk due to scattered photons that flip the neighboring qubit state.Fig. 5(c) shows the reconstructed crosstalk graph for this error model.The error model parameters used are: p m = 10 −2 , p local = 10 −2 .The parameters defining the simulated experiment are L = 10, N circs = 20, p idle = 0, N rep = 10 5 (therefore, N exp = 400), and the significance level of the hypothesis tests was set to α = 0.01.Unlike the previous crosstalk error models, the effects of this error do not potentially build up over a gate sequence, and thus only impact the outcome probabilities weakly.Moreover, its effect is reduced in the experiments where the first qubit's outcome is 0 with high probability.For these reasons, we found that a larger N rep and N circs are required to detect this crosstalk error.Reconstructed graphs for various crosstalk error models in a systems of two qubits; see Sec.VI A for details of error models.The regions in this case are composed on one qubit each.R i represents the measurement result on qubit i and S (0) i represents the setting on qubit i (the superscript (0) indexes the settings for a region -in all our examples there is only one setting per region since only the applied gate sequence is varied).The blue edges indicate conditional dependencies between variables that are expected (i.e., both variables belong to the same region).The red edges indicate conditional dependencies between variables in different regions, and these represent crosstalk.The red edges are labeled with the maximum TVD (and median TVD in parentheses) for that conditional dependence (see main text for definitions of these quantities).
The reconstructed graph in this case shows a red edge between the results on the two qubits indicating a conditional dependence that should not exist without some form of crosstalk error.

Crosstalk error quantification
It is important to keep in mind that the weights on the edges of a crosstalk graph are estimated maximum TVDs of outcome distributions, and not necessarily physical quantities like error rates.To illustrate this, in Fig. 6 we return to the weight-2 operation crosstalk error model in Sec.VI A 2 and plot the weight of the edge from R 0 to R 1 and S 1 to R 0 as the amount of crosstalk, i.e., the magnitude of the coherent Z ⊗ Z coupling term, is varied.The experiment sampling and physical parameters are the same as in Sec.VI A 2, except that we use N circs = 5.We see that while the max TVD varies monotonically with increasing for one edge, it does not for the other.Even if the max TVD should vary monotonically with some crosstalk parameter when computed over all random circuits, in practice it is a function of the experiments that are sampled and therefore sensitive to finite sampling variations in these experiments.
Therefore, the maximum TVD quantification should not be thought of as a direct measure of the physical degree of crosstalk.It should instead be used as a way to identify the regions of a multiqubit device that require the most attention in terms of needing crosstalk mitigation.

B. Six-qubit simulations
In this section we illustrate the scalability of the crosstalk error detection protocol by simulations on a 6-qubit device.The hypothetical device has a ladder layout, as shown in Fig. 7(a) and we are interested in detecting the crosstalk errors caused by single qubit gates.So we divide the QIP into six regions with a single qubit in each.The settings for each qubit enumerate the gate sequences applied, and the experiments simulated correspond to the experiment design outlined in Sec.V B.
The crosstalk error model is similar to the weight-1 operation crosstalk error model detailed in Sec.VI A 1; all single qubit gates on any of the qubits in the bottom line (qubits 3,4,5) result in a depolarizing channel with depolarization rate p on the vertical neighboring qubit.In addition to these crosstalk errors we also simulate local errors through a depolarization channel with depolarization rate p local after every gate and rate p idle after every idle clock cycle.All other details (native gate set, state preparation and measurement, and form of experimental gate sequences used) are the same as in the two-qubit simulations.The red edges in the crosstalk graph indicate crosstalk between the qubits that are vertical neighbors, as expected.We emphasize that this accurate crosstalk detection is achieved with just 300 distinct experiments, which highlights the benefits of using a technique with experimental burden that scales linearly with the number of qubits.

VII. CONCLUSIONS
We make two contributions in this paper.First, we provide a universal and hardware-agnostic definition of crosstalk errors in terms of a model for QIP dynamics based on representations of gates, state preparations and measurements on the device.Second, we provide a model-free definition of crosstalk in terms of operational variables (QIP settings and measurement results), and develop a protocol for detecting crosstalk errors based on it.
The protocol is based on testing conditional independence relations between a potentially large number of random variables.We have tested the protocol and associated data processing on simulated experiments on QIPs with up to six qubits.The technique shows promise for crosstalk error detection on medium-scale QIPs since it requires a number of experiments that is linear in the number of qubits.
An avenue for future research is to explore the utility of alternatives to the PC algorithm for discovering the crosstalk structure in a QIP.The PC algorithm is arguably the most established constraint-based algorithm for causal network structure discovery, but there is an active field of study developing new approaches to causal network structure discovery, e.g., the new kernel-based learning methods in Refs.[58,59], and it would be interesting to study whether any of these present any advantages when post-processing lightweight experimental data for crosstalk error detection.
Of course, detecting crosstalk is just the first step.One would ideally like to also characterize crosstalk errors once detected in order to learn their form and possibly also their origin.In future work we will utilize the model-based definition of crosstalk developed here to construct efficient protocols for characterizing crosstalk errors. in which case the equality would hold if the initial state is invariant under the error map.However, note that these CPTP maps represent the action of gate sequences, and the error map E 1 (s b 2 ) is the effective error on qubit 1 when some sequence M 2 (s b 2 ) is performed on qubit 2, after the desired sequence on qubit 1, M 1 (s a 1 ) has been factored out.These sequences are composed of elementary gates, some subset of which violate the independence condition, which leads the whole map to violate the independence condition.However, for a sufficiently rich set of sequences if one sequence violates independence, then it is likely that others do as well (in other words, there are a set of (a, b) satisfying Eq. (B11).And the probability that the measurement effects are invariant under all the effective error maps E 1 (s b 2 ) is extremely unlikely.Therefore we conclude that for a sufficiently rich set of settings, violation of independence results in violation of model-free definition of a crosstalk-free QIP.The above subsections prove the two directions of implication required to establish equivalence between the model-based definition (Definition 1) and the model-free definition (Definition 2) of crosstalk-free QIPs.for 0 < n < N circs − 1 do For each region, iterate over the N circs circuits sampled for that region return Expts

FIG. 1 .
FIG.1.A hierarchy of modularity for QIPs.The dotted lines indicate the modular components in each layer of the hierarchy.(a) A quantum circuit is specified by a schedule of quantum gates on target qubits.It is stable if the associated measurement outcome distribution does not depend on any external context.(b) The dynamics of a stable QIP are Markovian if each layer in the circuit, including state preparation and measurement, can be represented by a CPTP map that depends only on the operations that comprise that layer, and not on any external context.For example, the two shaded circuit layers are identical and therefore must be represented by the same CPTP map.(c) The dynamics are Markovian and crosstalk-free if the gate operations are modular: the CPTP map describing a given circuit layer can be written as a tensor product of CPTP maps describing each of the component gates (locality), and these component maps do not depend on the other gates in the layer (independence).For example, each appearance of the shaded X, CNOT, or H gates must be represented by the same CPTP maps.
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FIG. 2 .
FIG.2.Two qubits influenced by the same fluctuating magnetic field ( B).(a) If the field's state is modeled and tracked, then there is no crosstalk between the two qubits the correlations between their states and errors are fully explained by the field and its coupling to them.(b) But if we do not track the field, focusing on the two qubits only then there is crosstalk between the qubits, in the form of correlated stochastic errors mediated by the (untracked) magnetic field.

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s 8 1 d r 9 X 2 5 6 8 A L 1 F 3 r L X x L I 7 r S 8 n i V r K / 0 q C M A 7 W 8 5 3 n w 8 f a z fV / C 7 4 4 J 0 v d X b L G 4 F a a J H E f T 9 2 g T t 1 G 8 h x k H 9 S y 7 l C V B a M W k 7 C O H o E 1 u F o N v b 2 K 5 g g 2 W a / L P s t F r O + 7 x x H 3 n 8 K m c / x o G D 8 e R u + e h A f P V 8 9 k F 9 1 D 9 9 F D F K O n 6 A C 9 Q o d o j C g q 0 G f 0 B X 3 1 P n n f v O / e j y V 1 s L P S 3 E X n z P v 5 B 4 P z Z V Y = < / l a t e x i t > ⌫ M l a t e x i t s h a 1 _ b a s e 6 4 = " O 5 0 i 3 z t 9 S / d A 3 3 x y 7 E E E D n H 1 O o E = " > A A A E E H i c j V P N j t M w E P Y 2 / C z h Z 7 t w 5 B J R I X G A K g E k O K 6 A A x x W u 6 z a 3 Z X q b u U 4 k 9 S q 4 0 S 2 A 1 S W X 4 I r V 3 g H b m i v v A G P w F v g p B X q p l v E S H Z G 8 3 2 f Z z z x x C V n S o f h r 6 2 O d + X q t e v b N/ y b t 2 7 f 2 e n u 3 j 1 W R S U p D G n B C 3 k a E w W c C R h q p j m c l h J I H n M 4 i W e v a / z k A 0 j F C j H Q 8 x L G O c k E S x k l 2 o U m 3 R 0 s q j O z / y S y E x M 9 d n u 3 F / b D x o J 1 J 1 o 6 P b S 0 w 8 l u 5 z d O C l r l I D T l R K l R F J Z 6 b I j U j H K w P q 4 U l I T O S A Y j 5 w q S g x q b p n I b P H r S Y F B j E l b R I 7 A G F 8 u m 1 0 / R H M E 6 6 1 3 e Z r m I 9 X 0 3 H F F 7 F N a d 4 6 f 9 6 F k / f P + 8 t / d q O S b b 6 D 5 6 g B 6 h CL 1 A e + g t O k R D R F G F v q C v 6 J v 3 2 f v u / f D O F 9 T O 1 l J z D 1 0 w 7 + c f I m V o N g = = < / l a t e x i t > . . .⌫ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " A o O 9 R E y C i g Q u U N / E E a n d 4 k s b 5 k k = " > A A A E B n i c j V P L j t M w F P U 0 P I b w m o E l m 4 g I i V W V A B I s R 8 A C F o h h 1 E 5 H q j s j x 7 l J r T h O Z D t A Z X n P l i 3 8 A z v E l t / g E / g L n L R C n X S K u J L j q 3 v O 8 X 3 E T mr O l I 6 i X z s D 7 9 L l K 1 d 3 r / n X b 9 y 8 d X t v / 8 6 x q h p J Y U w r X s m T h C j g T M B Y M 8 3 h p J Z A y o T D J C l e t P j k P U j F K j H S i x p m J c k F y x g l 2 o U m W D S n J r J n e 2 E 0 j D o L N p 1 4 5 Y R o Z Y d n + 4 s 8 1 d r 9 X 2 5 6 8 A L 1 F 3 r L X x L I 7 r S 8 n i V r K / 0 q C M A 7 W 8 5 3 n w 8 f a z fV / C 7 4 4 J 0 v d X b L G 4 F a a J H E f T 9 2 g T t 1 G 8 h x k H 9 S y 7 l C V B a M W k 7 C O H o E 1 u F o N v b 2 K 5 g g 2 W a / L P s t F r O + 7 x x H 3 n 8 K m c / x o G D 8 e R u + e h A f P V 8 9 k F 9 1 D 9 9 F D F K O n 6 A C 9 Q o d o j C g q 0 G f 0 B X 3 1 P n n f v O / e j y V 1 s L P S 3 E X n z P v 5 B 4 B W Z V U = < / l a t e x i t > ⌫ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " A o O 9 R E y C i g Q u U N / E E a n d 4 k s b 5 k k = " > A A A E B n i c j V P L j t M w F P U 0 P I b w m o E l m 4 g I i V W V A B I s R 8 A C F o h h 1 E 5 H q j s j x 7 l J r T h O Z D t A Z X n P l i 3 8 A z v E l t / g E / g L n L R C n X S K u J L j q 3 v O 8 X 3 E T mr O l I 6 i X z s D 7 9 L l K 1 d 3 r / n X b 9 y 8 d X t v / 8 6 x q h p J Y U w r X s m T h C j g T M B Y M 8 3 h p J Z A y o T D J C l e t P j k P U j F K j H S i x p m J c k F y x g l 2 o U m W D S n J r J n e 2 E 0 j D o L N p 1 4 5 Y R o Z Y d n + 4

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s 8 1 d r 9 X 2 5 6 8 A L 1 F 3 r L X x L I 7 r S 8 n i V r K / 0 q C M A 7 W 8 5 3 n w 8 f a z f V / C 7 4 4 J 0 v d X b L G 4 F a a J H E f T 9 2 g T t 1 G 8 h x k H 9 S y 7 l C V B a M W k 7 C O H o E 1 u F o N v b 2 K 5 g g 2 W a / L P s t F r O + 7 x x H 3 n 8 K m c / x o G D 8 e R u + e h A f P V 8 9 k F 9 1 D 9 9 F D F K O n 6 A C 9 Q o d o j C g q 0 G f 0 B X 3 1 P n n f v O / e j y V 1 s L P S 3 E X n z P v 5 B 4 B W Z V U = < / l a t e x i t > ⌫ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " A o O 9 R E y C i g Q u U N / E E a n d 4 k s b 5 k k = " > A A A E B n i c j V P L j t M w F P U 0 P I b w m o E l m 4 gI i V W V A B I s R 8 A C F o h h 1 E5 H q j s j x 7 l J r T h O Z D t A Z X n P l i 3 8 A z v E l t / g E / g L n L R C n X S K u J L j q 3 v O 8 X 3 E T m r O l I 6 i X z s D 7 9 L l K 1 d 3 r / n X b 9 y 8 d X t v / 8 6 x q h p J Y U w r X s m T h C j g T M B Y M 8 3 h p J Z A y o T D J C l e t P j k P U j F K j H S i x p m J c k F y x g l 2 o U m W D S n J r J n e 2 E 0 jD o L N p 1 4 5 Y R o Z Y d n + 4 P f O K 1 o U 4 L Q l B O l p n F U 6 5 k h U j P K w f q 4 U V A T W p A c p s 4 V p A Q 1 M 1 2 9 N n j g I m m Q V d I t o Y M u u q 4 w p F R q U S a O W R I 9 V 3 2 s D V 6 E T R u d P Z s Z J u p G g 6 D L R F n D A 1 0 F b f N B y i R Q z R f O I V Q y V 2 t A 5 0 Q S q t 2 I f B + / B N e M h D f u 4 L c 1 S K I r a b C W 1 r i 1 B R 3 9 E 5 W F N Z K I Y h v u 6 s i t 6 b 4 + F v C B V m V J R G p w I o m d x j O D Q a h G Q t u q w R w y j T k R O Q c T x h Z L l s 8 1 d r 9 X 2 5 6 8 A L 1 F 3 r L X x L I 7 r S 8 n i V r K / 0 q C M A 7 W 8 5 3 n w 8 f a z f V / C 7 4 4 J 0 v d X b L G 4 F a a J H E f T 9 2 g T t 1 G 8 h x k H 9 S y 7 l C V B a M W k 7 C O H o E 1 u F o N v b 2 K 5 gg 2 W a / L P s t F r O + 7 x x H 3 n 8 K m c / x o G D 8 e R u + e h A f P V 8 9 k F 9 1 D 9 9 F D F K O n 6 A C 9 Q o d o j C g q 0 G f 0 B X 3 1 P n n f v O / e j y V 1 s L P S 3 E X n z P v 5 B 4 B W Z V U = < / l a t e x i t > ⌫ l a t e x i t s h a 1 _ b a s e 6 4 = " g X m O o X s B h E F p y s T N f a z x g v 7 8 K 1 Y = " > A A A E D H i c j V P N b t Q w E H Y 3 / J T w 1 8 K R S 8 Q K i Q N a J Y A E x w o 4 w A F R q t 2 2 0 j q t H G e S t e I 4 w X a A l e V X 4 M o V 3 o E b 4 s o 7 8 A i 8 B U 5 2 h b b Z L m I k O 6 P 5 v s 8 z n n i S m j O l w / D X 1 s C 7 c P H S 5 e 0 r / t V r 1 2 / c 3 N m 9 d a i q R l K Y 0 I p X 8 j g h C j g T M N F M c z i u J Z A y 4 X C U F M 9 b / O g 9 S M U q M d b z G u K S 5 I J l j B L t Q j E W z Y m J 7 K k J H 7 h 9 Z x i O w s 6 C d S d a O k O 0 t P 3 T 3 c F v n F a 0 K U F o y o l S 0 y i s d W y I 1 I x y s D 5 u F N S EF i S H q X M F K U H F p q v a B v d c J A 2 y S r o l d N B F V x W G l E r N y 8 Q x S 6 J n q o + 1 w f O w a a O z p 7 F h o m 4 0 C L p I l D U 8 0 F X Q t i B I m Q S q + d w 5 h E r m a g 3 o j E h C t W u U 7 + M X 4 C 4 j 4 b U 7 + E 0 N k u h K G q y l N W 5 t Q M f / R G V h j S S i 2 I S 7 O n J r u t 3 H A j 7 Q q i y J S A 1 O J L H T K D Y Y h G o k t F c 1 m E O m M S c i 5 2 C G k c W S 5 T O N 3 U / W t i c v Q G + Q t + w Vs e x O 6 8 t J o h b y v 5 J g G A W r + c 7 y 4 W P t + v q / B Z + f k 6 X u L V l j c C t N k q i P p 6 5 R J + 5 D 8 h x k H 9 S y 7 l C V B e M W k 7 C K H o A 1 u F o 2 v X 2 K 5 g D W W a / K P s t F r O + 7 4 Y j 6 o 7 D u H D 4 c R Y 9 G 4 d v H w 7 1 n y z H Z R n f Q X X Q f R e g J 2 k M v 0 T 6 a I I r e o c / o C / r q f f K + e d + 9 H w v q Y G u p u Y 3 O m P f z D 1 c t Z 3 Y = < / l a t e x i t > Exp.Ncircs×"#$% Exp."#$% ⌫ Ncircs 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " I o g X e h K B s S N I 2 a y q u L G g f Y n e / y k = " > A A A E F n i c j V P N j t M w E P Z u + F n C X x f E i Y t F h c S F q l m Q 4 L g C D n A A l l W 7 u 1 L d r R x n k l p 1 n M h 2 g M r y e 3 D l C u / A D X H l y i P w F j h p h b r p F j G S 4 9 F 8 3 + c Z T z x x K b g 2 / f 6 v r e 3 g w s V L l 3 e u h F e v X b 9 x s 7 N 7 6 0 g X l W I w Z I U o 1 E l M N Q g u Y W i 4 E X B S K q B 5 L O A 4 n j 2 v 8 e P 3 o D Q v 5 M D M S x j n N J M 8 5 Y w a H 5 p 0 7 h B Z n d o 3 E 0 C b D d y B H F s 6 k h / n c b 1 5 L P w G y Q 1 + w V s W p O a 8 t p r B f y v x L c j f B q v r N 8 + F j 6 v v 5 v w e f n 5 I l / S 8 5 a U k v j O G r j i W / U q d 9 o l o F q g 0 a V D a p T P K g x B a v o I T h L i m X T 6 6 d o D 2 G d 9 S p v s 3 z E h a E f j q g 9 C u v O 0 V 4 v e t T b e / e 4 u / 9 s O S Y 7 6 C 6 6 h x 6 g C D 1 B + + g l O k B D x J B F n 9 E X 9 D X 4 F H w L v g c / F t T t r a X m N j p j w c 8 / o x R r V g = = < / l a t e x i t > ⌫ Ncircs 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " I o g X e h K B s S N I 2 a y q u L G g f Y n e / y k = " > A A A E F n i c j V P N j t M w E P Z u + F n C X x f E i Y t F h c S F q l m Q 4 L g C D n A A l l W 7 u 1 L d r R x n k l p 1 n M h 2 g M r y e 3 D l C u / A D X H l y i P w F j h p h b r p F j G S 4 9 F 8 3 + c Z T z x x K b g 2 / f 6 v r e 3 g w s V L l 3 e u h F e v X b 9 x s 7 N 7 6 0 g X l W I w Z I U o 1 E l M N Q g u Y W i 4 E X B S K q B 5 L O A 4 n j 2 v 8 e P 3 o D Q v 5 M D M S x j n N J M 8 5 Y w a H 5 p 0 7 h B Z n d o 3 E 0 C b D d y B H F s 6 k h / n c b 1 5 L P w G y Q 1 + w V s W p O a 8 t p r B f y v x L c j f B q v r N 8 + F j 6 v v 5 v w e f n 5 I l / S 8 5 a U k v j O G r j i W / U q d 9 o l o F q g 0 a V D a p T P K g x B a v o I T h L i m X T 6 6 d o D 2 G d 9 S p v s 3 z E h a E f j q g 9 C u v O 0 V 4 v e t T b e / e 4 u / 9 s O S Y 7 6 C 6 6 h x 6 g C D 1 B + + g l O k B D x J B F n 9 E X 9 D X 4 F H w L v g c / F t T t r a X m N j p j w c 8 / o x R r V g = = < / l a t e x i t > ⌫ Ncircs 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " I o g X e h K B s S N I 2 a y q uL G g f Y n e / y k = " > A A A E F n i c j V P N j t M w E P Z u + F n C X x f E i Y t F h c S F q l m Q 4 L g C D n A A l l W 7 u 1 L d r R x n k l p 1 n M h 2 g Mr y e 3 D l C u / A D X H l y i P w F j h p h b r p F j G S 4 9 F 8 3 + c Z T z x x K b g 2 / f 6 v r e 3 g w s V L l 3 e u h F e v X b 9 x s 7 N 7 6 0 g X l W I w Z I U o 1 E l M N Q g u Y W i 4 E X B S K q B 5 L O A 4 n j 2 v 8 e P 3 o D Q v 5 M D M S x j n N J M 8 5 Y w a H 5 p 0 7 h B Z n d o 3 E 0

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C b D d y B H F s 6 k h / n c b 1 5 L P w G y Q 1 + w V s W p O a 8 t p r B f y v x L c j f B q v r N 8 + F j 6 v v 5 v w e f n 5 I l / S 8 5 a U k v j O G r j i W / U q d 9 o l o F q g 0 a V D a p T P K g x B a v o I T h L i m X T 6 6 d o D 2 G d 9 S p v s 3 z E h a E f j q g 9 C u v O 0 V 4 v e t T b e / e 4 u / 9 s O S Y 7 6 C 6 6 h x 6 g C D 1B + + g l O k B D x J B F n 9 E X 9 D X 4 F H w L v g c / F t T t r a X m N j p j w c 8 / o x R r V g = = < / l a t e x i t > ⌫ l a t e x i t s h a 1 _ b a s e 6 4 = " z H i F z S 3 Z 7 Z q 3 m F / p m 9 1 8 5 L W R i o g = " > A A A E C n i c j V P N j t M w E P Y 2 / C z h b x e O X C I q J A 6 o q h c k O K 4 W D n B A L K t 2 d 6 W 6 W z n O J L X q O J H t L F S W 3 4 A r V 3 g H b o g r L 8 E j 8 B Y 4 a Y W 6 6 R Y x k u P R f N / n G U 8 8 c S m 4 N v 3 + r 6 1 O c O X q t e v b N 8 K b t 2 7 f u b u z e + 9 Y F 5 V i M G S F K N R p T D U I L m F o u B F w W i q g e S zg J J 6 9 r P G T c 1 C a F 3 J g 5 i W M c 5 p J n n J G j Q + N i K z O 8 M T i J 9 h N d r r 9 X r + x a N 3 B S 6 e L l n Y 4 2 e 3 8 J k n B q h y k Y Y J q P c L 9 0 o w t V Y Y z A S 4 k l Y a S s h n N Y O R d S X P Q Y 9 v U 7 K J H P p J E a a H 8 k i Z q o q s K S 3 O t 5 3 n s m T k 1 U 9 3 G 6 u B l 2 K g y 6 Y u x 5 b K s D 7 x m r 4 h V c 1 p b T m O 9 k P + V R F 0 c r e a 7 y I e P p e / r / x Z 8 e U 6 e + L f k r C W 1 N I 5 x G 0 9 8 o 8 7 8 R r M M V B s 0 q m x Q n U a D G l O w i h 6 B s 6 R Y N r 1 + i v Y I 1 l l v 8 j b L R 1 w Y + u H A 7 V F Y d 4 7 3 e v h p b + / 9 s + 7 + w X J M t t E D 9 B A 9 R h g 9 R / v o N T p E Q 8 R Q g T 6 j L + h r 8 C n 4 F n w P f i y o n a 2 l 5 j 6 6 Y M H P P 4 P B Z m 0 = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " 7 f J w o U A W h Y 4 6 D T 1 / w N w i I o l m R g A = " > A A A E C n i c j V P N j t M w E P Z u + F n C 3 y 4 c u U R U S B x Q l S x I c F w B B z g g l l W 7 u 1 L d r R x n k l p 1 7 M h 2 g M r y G 3 D l C u / A D X H l J X g E 3 g I n r V A 3 3 S J G c j y a 7 / s 8 4 4 k n r T j T J o 5 / b W 0 H l y 5 f u b p z L b x + 4 + a t 2 7 t 7 d 4 6 1 r B WF I Z V c q t O U a O B M w N A w w + G 0 U k D K l M N J O n v R 4 C f v Q W k m x c D M K x i X p B A s Z 5 Q Y H x p h U Z 8 l E x s / i t 1 k t x f3 4 9 a i d S d Z O j 2 0 t M P J 3 v Z v n E l a l y A M 5 U T r U R J X Z m y J M o x y c C G u N V S E z k g B I + 8 K U o I e 2 7 Z m F z 3 w k S z K p f J L m K i N r i o s K b W e l 6 l n l s R M d R d r g h d h o 9 r k z 8 a W i a o 2 I O E b B P u 6 y i c b b 8 h F v C B y r I k I r M 4 V c S N k r H F I H S t o L m q x R x y g z k R B Q f b S x x W r J g a 7 H + x c R 3 5 D M w G e c N e E a v 2 t K 6 c p H o h / y u J e k m 0 m u 8 8 H z 5 W v q / / W / D F O V n m 3 5 K z F j f S N E 2 6 e O Y b d e Y 3 U h S g u q B R V Y v q P B o 0 m I J V 9 A i c x X L Z 9 O Y p 2 i N Y Z 7 0 u u y w f c W H o h y P p j s K 6 c 7 z f T x 7 3 9 9 8 9 6 R 0 8 X 4 7 J D r q H7 q O H K E F P 0 Q F 6 h Q 7 R E F E k 0 W f 0 B X 0 N P g X f g u / B j w V1 e 2 u p u Y v O W f D z D 3 y F Z m s = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " u k 2 F 9 U D A / v d W T Z g 0 9 8 P y o 4 0 k l 8 I = " > A A A E C n i c j V P N j t M w E P Z u + F n C 3 y 4 c u U R U S B x Q l S x I c F w B B z g g l l W 7 u 1 L d r R x n k l p 1 7 M h 2 g M r y G 3 D l C u / A D X H l J X g E 3 g I n r V A 3 3 S J G c j y a 7 / s 8 4 4 k n r T j T J o 5 / b W 0 H l y 5 f u b p z L b x + 4 + a t 2 7 t 7 d 4 6 1 r B W E b B P u 6 y i c b b 8 h F v C B y r I k I r M 4 V c S N k r H F I H S t o L m q x R x y g z k R B Q f b S x x W r J g a 7 H + x c R 3 5 D M w G e c N e E a v 2 t K 6 c p H o h / y u J e k m 0 m u 8 8 H z 5 W v q / / W / D F O V n m 3 5 K z F j f S N E 2 6 e O Y b d e Y 3 U h S g u q B R V Y v q P B o 0 m I J V 9 A i c x X L Z 9 O Y p 2 i N Y Z 7 0 u u y w f c W H o h y P p j s K 6 c 7 z f T x 7 3 9 9 8 9 6 R 0 8 X 4 7 J D r q H

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3 g e i A 4 5 h K Y z u b O o U x y V y t m U y o p 0 2 5 s v k + e g 2 t G w i t 3 8 J s S J NW F N E R L a 9 z a g A 7 + i c q Z N Z K K 2 S b c 1 Z F a 0 3 x 9 I u A 9 K / K c i t i Q S F I 7 C s e G g F C V h L p V Q z J I N M m o S D M w v d A S y d O p J u 6 X a 9 u S z 0 B v k N f s F b F s T m v L a a Q W 8r 8 S 3 A v x a r 7 T f P h Q u r n + b 8 F n 5 + S x u 0 v W G F J L o y h s 4 7 E b 1 I n b a J q C b I N a l g 2 q E j y o M Q m r 6 A F Y Q 4 r l 0 O u r a A 5 g n f U y b 7 N c x P q + e x x h + y m s O 4 e 7 / f B h f / f t o 9 7 e 0 + U z 2 U a 3 0 R 1 0 D 4 X o M d p D L 9 A + G i K G P q L P 6 A v 6 6 n 3 y v n n f v R 8 L a m d r q b m J T p n 3 8 w + x 7 2 x q < / l a t e x i t > ⌫ l a t e x i t s h a 1 _ b a s e 6 4 = " r P 9 U a b 4 i W I Y Z j + A S t W M + S w Z V X 7 E = " > A A A E G n i c j V P L j t M w F P W 0 P I b w 6 s A K s b

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r 6 6 i P Y B 1 1 s u s y f I R F w T + c Y T N p 7 D u H O 7 2 w o e 9 3 b e P u n t P l 8 9 k G 9 1 G d 9 A 9 F K L H a A + 9 Q P t o i B j 6 i D 6 j L + h r + 1 P 7 W / t 7 + 8 e C 2 t p a a m 6 i U 9 b + + Q e 1 j G x r < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " I KJ L p + q O o D z 4 h b D F H / f t B T F 9 A Z c = " > A A A E E H i c j V P N j t M w E P Y 2 / C z h Z 7 t w 5 B J R I X G A K t l F g u M K O M B h x b J q d 1 e q u 5 X j T F K r j h P Z D l B Z f g m u X O E d u C G u v A G P w F v g p B X q p l v E S I 5 H 8 3 2 f Z z z x x C V n S o f h r 6 2 O d + X q t e v b N / y b t 2 7 f 2 e n u 3 j 1 R R S U p D G n B C 3 k W E w W c C R h q p j m c l R J I H n M 4 j W c v a / z 0 P U j F C j H Q 8 x L G O c k E S x k l 2 o U m 3 R 0 s q n N z + C S y E x M + D u 2 k 2 w v 7 Y W P B u h M t n R 5 a 2 t F k t / M b J w W t c h C a c q L U K A p L P T Z E a k Y 5 W B 9 X C k p C Z y S D k X M F y U G N T V O 5 D R 6 6 S B K k h X R L 6 K C J r i o M y Z W a 5 7 F j 5 k R P V R u r g 5 d h o 0 q n z 8 e G i b L S I O g i U V r x Q B d B 3 Y Y g Y R K o 5 n P n E C q Z q z W g U y I J 1 a 5 Z v o 9 f g b u M h E N 3 8 N s S J N G F N F h L a 9 z a g A 7 + i c q Z N Z K I 2 S b c 1 Z F Z 0 3 x 9 L O A D L f K c i M T g W B I 7 i s Y G g 1 C V h P q q B n N I N e Z E Z B x M L 7 J Y s m y q s f v R 2r b k M 9 A b 5 D V 7 R S y b 0 9 p y E q u F / K 8 k 6 E X B a r 6 L f P h Y u r 7 + b 8 G X 5 2 S J e 0 v W G F x L 4 z h q 4 4 l r 1 L n b S J a B b I N a l g 2 q 0 m B Q Y x J W 0 W O w B h f L p t d P 0 R z D O u t N 3 m a 5 i P V 9 N x x R e x T W n Z O 9 f r T f 3 3 v 3 t H f w Y j k m 2 + g + e o A e o Q g 9 Q w f o N T p C Q 0 R R h T 6 j L + i r 9 8 n 7 5 n 3 3 f i y o n a 2 l 5 h 6 6 Y N 7 P P x v N a D Y = < / l a t e x i t > ⌫ M l a t e x i t s h a 1 _ b a s e 6 4 = " O X n 3 I d 0 n 6 w x E v 5 X w q M Q H Q I M B N t g = " > A A A E E H i c j V P N j t M w E P Y 2 / C z h Z 7 t w 5 B J R I X G A K t l F g u M K O M B h x b J q d 1 e q u 5 X j T F K r j h P Z D l B Z f g m u X O E d u C G u v A G P w F v g p B X q p l v E S H Z G 8 3 2 f Z z z x x C V n S o f h r 6 2 O d + X q t e v b N / y b t 2 7 f 2 e n u 3 j 1 R R S U p D G n B C 3 k W E w W c C R h q p j m c l R J I H n M 4 j W c v a / z 0 P U j F C j H Q 8 x L G O c k E S x k l 2 o U m 3 R 0 s q n N z + C S y E x M + d n u 3 F / b D x o J 1 J 1 o 6 P b S 0 o 8 l u 5 z d O C l r l I D T l R K l R F J Z 6 b I j U j H K w P q 4 U l I T O S A Y j 5 w q S g x q b p n I b P H

1 Ncircs 1 , 1 < 1
r b k M 9 A b 5 D V 7 R S y b 0 9 p y E q u F / K 8 k 6 E X B a r 6 L f P h Y u r 7 + b 8 G X 5 2 S J e 0 v W G F x L 4 z h q 4 4 l r 1 L n b S J a B b I N a l g 2 q 0 m B Q Y x J W 0 W O w B h f L p t d P 0 R z D O u t N 3 m a 5 i P V 9 N x x R e x T W n Z O 9 f r T f 3 3 v 3 t H f w Y j k m 2 + g + e o A e o Q g 9 Q w f o N T p C Q 0 R R h T 6 j L + i r 9 8 n 7 5 n 3 3 f i y o n a 2 l 5 h 6 6 Y N 7 P P x 9 s a D c = < / l a t e x i t > ⌫ M l a t e x i t s h a 1 _ b a s e 6 4 = " m X I M W a y g / 5 3 6 JY L R q W 3 s w Z m 6 n Y M = " > A A A E H n i c j V P L b h M x F H U 7 P M r w S m G J k C w i J B Y 0 y h Q k W F b A A h a F U i V t p T i N P J 4 7 E y s e z 8 j 2 A J H l F T / C l i 3 8 A z v E F j 6 B v 8 C T R C i d N I g r e X x 1 z z m + j 7 H j U n B t u t 1 f G 5 v B h Y u X L m 9 d C a 9 e u 3 7 j Z m v 7 1 p E u K s W g z w p R q J O Y a h B c Q t 9 w I + C k V E D z W M B x P H l e 4 8 f v Q G l e y J 6 Z l j D M a S Z 5 y h k 1 P j R q 3 S W y O r X 7 O 5 E b 2 d c j S 1 S O G V d M u 5 3 o o Y + 1 2 t 1 O d 2 Z 4 1 Y k W T h s t 7 G C 0 v f m b J A W r c p C G C a r 1 I O q W Z m i p M p w J c C G p N J S U T W g G A + 9 K m o M e 2 l k f D t / 3 k Q S n h f J L G j y L L is s z b W e 5 r F n 5 t S M d R O r g + d h g 8 q k T 4 e W y 7 I y I N k 8 U V o J b A p c D w U n X A E z Y u o d y h T 3 t W I 2 p o o y 4 0 c X h u Q F + G Y U 7 P u D 3 5 S g q C m U J U Y 5 6 9 c a t P d PV E 2 c V V R O 1 u G + j s z Z 2 T c k E t 6 z I s + p T C y J F X W D a G g J S F 0 p q F u 1 R E B q i K A y E 2 D b k S O K Z 2 N D / G 8 3 r i G f g F k j r 9 l L Y j U 7 r S m n s Z 7 L / 0 p w O 8 L L + c 7 y 4 U P p 5 / q / B Z + f k y f + L j l r S S 2 N 4 6 i J J 3 5 Q p 3 6 j W Q a q C R p V z l C d 4 l 6 N K V h G D 8 F Z U i y G X l 9 F e w i r r F d 5 k + U j L g z 9 4 4 i a T 2 H V O d r t R I 8 6 u 2 8 f t / e e L Z 7 J F r q D 7 q E H K E J P 0 B 5 6 i Q 5 Q H z H 0 E X 1 G X9 D X 4 F P w L f g e / J h T N z c W m t v o j A U / / w C x U G 4 F < / l a t e x i t > ⌫ M Ncircs 1,0 < l a t e x i t s h a 1 _ b a s e 6 4 = " x 0 E v q C F T B a 7 d x I s R z p T Q 0 c B 8 F 4 c = " > A A A E H n i c j V N N b 9 M w G P Y W P k b 4 6 u C I k C I q J A 6 s S g Y S H C f g A I f B m N p t U t 1 V j v M m t e o 4 k e 0 A l e U T f 4 Q r V / g P 3 B B X + A n 8 C 5 y 0 Q l 2 6 I l 7 J 8 a v 3 e R 6 / H 7 H j k j O l w / D X x q Z 3 4 e K l y 1 t X / K v X r t + 4 2 d m + d a S K S l I Y 0 I I X 8 i Q m C j g T M N B M c z g p J Z A 8 5 n A c T 5 / X + P E 7 k I

Fig. 5 (
Fig. 5(b) shows the reconstructed crosstalk graph for this error model.The error model parameters used are: = 2 • 10 −2 , p local = 10 −2 .The parameters defining the simulated experiment are L = 30, N circs = 10, p idle = 0, N rep = 10 5(therefore, N exp = 100), and the significance level of the hypothesis tests was set to α = 0.01.Note that the coherent crosstalk error shows up at ∼ 2 in the measurement probabilities since we are using random gate sequences, and this is why more samples are required to detect this crosstalk error.The red edges in the crosstalk graph indicate crosstalk errors between the qubits.In this case there are conditional dependencies between settings and results in different regions, and also between the results in different regions.There is no clear causal direction for this type of crosstalk error (and one can show using a model of this kind of crosstalk error and calculations such as in Appendix B that conditional dependencies between results are expected for this kind of crosstalk error).
FIG.5.Reconstructed graphs for various crosstalk error models in a systems of two qubits; see Sec.VI A for details of error models.The regions in this case are composed on one qubit each.R i represents the measurement result on qubit i and S(0)  i represents the setting on qubit i (the superscript (0) indexes the settings for a region -in all our examples there is only one setting per region since only the applied gate sequence is varied).The blue edges indicate conditional dependencies between variables that are expected (i.e., both variables belong to the same region).The red edges indicate conditional dependencies between variables in different regions, and these represent crosstalk.The red edges are labeled with the maximum TVD (and median TVD in parentheses) for that conditional dependence (see main text for definitions of these quantities).

FIG. 6 .
FIG. 6.The weight of the edge from R 0 to R 1 and S 1 to R 0 versus the the physical crosstalk error magnitude, , for the weight-2 operation crosstalk error model detailed in Sec.VI A 2.

FIG. 7 .
FIG. 7. Reconstructed graphs for 6 qubit QIP with weight-1 operation crosstalk errors along vertical lines; see Sec.VI B for details of error models.The regions in this case are composed on one qubit each.R i represents the measurement results on qubit i and S (0)i represents the setting on qubit i (the superscript (0) indexes the settings for a region -in all our examples there is only one setting per region since only the gate sequence applied is varied).The blue edges indicate conditional dependencies between variables that are expected (i.e., both variables belong to the same region).The red edges indicate conditional dependencies between variables in different regions, and this represent crosstalk.The red edges are labeled with the maximum TVD (and median TVD in parentheses) for that conditional dependence (see main text for definitions of these quantities).

4 .
Defintion 1 ⇐⇒ Defintion 2 Appendix C: Pseudocode for lightweight experiment design Algorithm 1 Lightweight crosstalk detection experiment generation.The output is a set of roughly M × N circs × N con experiments on an M region QIP, with each experiment consisting of length L circuits on each region.

7 : s m ← Circuit number n from bag m 8 : 9 :if k m then 11 :← Sample a circuit from bag k 15 :
for 0 < c < N con do Generate N con experiment with s m circuit on region mfor 0 < k < M − 1 do 10:if Unif(0,1) < p idle then With probability p idle region k gets idle circuit 12: s k ← the length L idle circuit on region k Append to Expts the experiment defined by parallel application of s n (for 0 < n < M − 1) to the M regions 16: Expts ← RemoveDuplicates(Expts) Remove duplicate experiments 17: not a product state -if and only if G AB is not a tensor product of operations).Therefore, a Markovian model obeys locality if and only if each layer can be represented by a tensor product of local operations.For such a model, independence is well-defined.A local, Markovian QIP satisfies independence of operations if and only if each local operation (gate, initialization, measurement) is represented by the same local CPTP map in every layer where it appears.(No proof is needed -this is just a restatement of the definition of independence above in terms of CPTP maps).