A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits)

Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.

more fundamental question. In previous work, the saturation was shown through a handful of examples. Now, we provide a complete descriptive answer to: What is the set of mechanisms responsible for the discrepancy between the best and the worst fidelity of a circuit?
This would not be much of a fundamental question if the answer didn't also unravel an important dichotomy in classifying quantum errors. Indeed, given the intricate geometry of quantum states [3], the answer could have included some obscure blend of non-intuitive mechanisms, leaving us with yet another resignation in the attempt to intuitively reason about quantum dynamics. Although, for once, this is not the case: the discrepancy between the best and worst fidelity is, to high precision, entirely taken into account by unitary dynamics 2 . Even more surprisingly, the unitary dynamics itself is precisely the product of the "unitary factors" of individual circuit components. Yes, as we later demonstrate, every non-catastrophic 3 channel can be decomposed as a physical unitary followed (or preceded) be a decoherent channel. For realistic errors 4 , the unitary is unique and is referred to as the coherent factor. This factorization is analog to the so-called matrix polar decomposition and, as we will show, directly stems from it. The uniqueness of the coherent factor might puzzle the skeptical reader. Indeed, for example, how should we unambiguously define such factor in the case of an error which consists of a mixture of near-identity unitaries (i.e. A(ρ) = i p i U i ρU † i , where U i ≈ I d )? Should it be the unitary operation with the highest weight? Should it relate with some kind of ensemble average over the associated Hamiltonians? To systematically answer this type of question, we introduce the leading Kraus (LK) approximation, a sub-parameterization of quantum channels which, among other things, exposes a natural definition for the coherent and decoherent factors of a channel.
What allows us to really profit from the channel polar decomposition is the surprising property that the LK approximation, despite its seemingly bare structure, closely captures the evolution of the fidelity and unitarity in circuits. That is, we can mathematically replace all the channels in a circuit by their respective LK approximation and still expect to accurately bound its fidelity and unitarity. Working with the uncluttered structure offered by the LK approximation helped us identify and rule out pathological error scenarios, which we refer to as "extremal". For all realistic noisy channel, we derive the following observations (they hold to high precision): i. The infidelity (the counterpart to the fidelity) of a channel can be split into two terms: (a) a coherent infidelity, which corresponds to the infidelity of the coherent factor to the target channel; (b) a decoherent infidelity, which corresponds to the infidelity of the decoherent factor to the identity.
ii. The decoherent infidelity of a channel is in one-to-one correspondence with its unitarity. Moreover, the decoherent infidelity corresponds to the minimum infidelity of the channel after the application of a unitary (the coherent infidelity is correctable through a composition with a unitary).
iii. The unitarity of a composite channel is a decay function expressed in terms of individual channels' unitarity.
iv. The fidelity of the composition of decoherent channels is a decay function expressed in terms of individual channels' fidelity.
v. The fidelity of a general composition is upper bounded by a decay dictated by the decoherent factors (hence by the unitarity of individual components).
vi. The discrepancy between the upper and the lower bound of the fidelity is captured by the fidelity of the composition of the coherent factors (to the target circuit).
These realizations are directly applicable to the analysis and development of process characterization methods. The fidelity of various error processes can be robustly and efficiently estimated through a scalable experimental protocol known as randomized benchmarking (RB) [4][5][6][7] and a family of generalizations thereof [2,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. To remain efficient as quantum devices grow larger, RB experiments only extract partial information about specific sets of components. A known challenge is to leverage this limited view to formulate a more rounded understanding of the device. By looking at the fidelity of well-designed compositions, it should be possible to extract other figures of merit attached to quantum processes. The idea is that since maps dictate the evolution of the fidelity, conversely, the evolution of the fidelity can tell us information about the maps. The relationship between the maps and the fidelity is generally obscure, but the LK approximation allows seeing through it more clearly.
We structure the paper as follows. In section II, we introduce important characterization figures of merit -the average process fidelity and the unitarity -and relate them with the Kraus operator formalism. In section III, we define the LK approximation and present its aptitude in capturing important characteristics of evolving quantum circuits. In section IV, based on the emergent mathematical structure of LK approximated channels, we show the existence of a channel polar unitary-decoherent decomposition. In section V, we make use of the approximation to demonstrate key behavioral aspects of quantum circuits based on partial knowledge of their components.
For the sake of conciseness, most demonstrations are pushed to the appendix. Moreover, in the main text, certain results have been abridged by gathering higher order terms under the acronym "H.O.T.". The complete expressions -which are not any more insightful than their abbreviated analog -are provided in the appendix.

II. CHANNEL PROPERTIES CAPTURED BY THE LEADING KRAUS OPERATOR
A quantum channel is a completely-positive (CP), trace-preserving (TP) map acting on M d (C). Given a quantum channel A : and e i are canonical orthonormal vectors. The Choi matrix is positive semi-definite iff A is CP, and has trace d if A is TP or unital 5 . Since Choi(A) ≥ 0, it has a spectral decomposition of the form where col(A) ∈ C d 2 denotes the column vectorization of a matrix A ∈ M d (C) 6 , and · p denotes the Schatten p-norm. The eigenvectors col(Ā i ) are orthonormal, an without loss of generality the eigenvalues are ordered with respect to the Frobenius norm (Schatten 2-norm): Given a spectral decomposition like eq. (2), we can express the channel's action on states ρ ∈ M d (C) as [26]: where the usual Hilbert-Schmidt inner product is used. Notice that the TP condition implies that i ( A i 2 2 /d) = 1. The matrices A i ∈ M d (C) are referred to as (ordered) canonical Kraus operators. In this work, A 1 (which is associated with the highest Choi matrix eigenvalue A 1 2 2 ) will deserve special attention, and is attributed the title of "leading Kraus (LK) operator". In general, A 1 might be non-unique when the spectrum of the Choi matrix is degenerate. However, in this work we focus on non-catastrophic channels (definition 1), for which A 1 is unique.
Given an operation A and a target unitary channel U(ρ) = U ρU † 7 , we can compare the overlap of their outputs given specific inputs M ∈ M d (C) through the M -fidelity: The so-called average gate fidelity is obtained by averaging the M -fidelities uniformly 8 over all physical pure states |ψ ψ|: The average infidelity r is simply a shorthand for 1 − F . Instead of averaging over quantum states, we could also average uniformly over all operators M ∈ M d (C). More precisely, given any orthogonal operator basis {B i } for M d (C), we can uniformly average over the M -fidelities f Bi , which yields the average process fidelity 9 Compared to Φ, F puts a slightly higher weight over the identity component I d . The TP condition enforces this special component to take a fixed value, f I d = 1. Hence the two quantities are closely related via [27]: F (A, U) is the overlap between the output state A(ρ) of an implemented channel A and its ideal output U(ρ), averaged over all physical pure input states |ψ ψ|. While F (A, U) conveys a more graspable interpretation, it will remain easier here to work with Φ(A, U) since it ties with the Kraus operators through Since {Ā i } forms an orthonormal basis and U/ √ d 2 = 1, it follows that If A i 2 2 /d can be thought as the "weights" of the Kraus operators, can be thought as normalized overlaps with the target U .
To quantify the coherence of a quantum channel, one could wonder how much the Bloch vectors (the traceless component of quantum states [28]) are contracted. For instance, consider the unitarity, which is the squared length ratio of the Bloch vectors before and after the action of the channel A, averaged over all physical Bloch vector inputs corresponding to pure states |ψ ψ| − I d /d [2]: Let's extend the domain of Φ to include a new function of A: Straightforward calculations closely relate the unitarity to Υ via (Notice that the notation alludes to the connection between greek and latin alphabets; it relates "phi" to "F" and "upsilon" to "u".) We are ready to express a first result:

Lemma 1
Consider a CPTP map A with ordered canonical Kraus decomposition Then, , which completes the proof.
The LK operator alone provides a very accurate approximation of 1 − Φ and 1 − Υ. This only begins a list of realizations regarding the role of LK operators in quantum dynamics. As we will see, they also contain most of the information necessary to describe the evolution of Φ and Υ.

III. THE LK APPROXIMATION AND TWO EVOLUTION THEOREMS
The last section naturally suggests the following channel approximation as a mean to partially characterize noncatastrophic quantum dynamics: We define its leading Kraus (LK) approximation as: Notice that A is always CP (Choi(A ) ≥ 0), but is TP iff A is unitary. Hence, A fails to be generally physical. However, as we will see, it closely describes the dynamics of certain physical quantities, so one may qualify this map as "quasi-dynamical". The general specification of a map acting on a d-dimensional quantum system requires roughly d 4 parameters, and due to the intricate geometry of quantum states, the parameterization of its range of action is quite convoluted. In contrast, the LK approximation is remarkably transparent: it is fully parameterized by d × d matrices with spectral radius smaller than 1 (contractions) and Frobenius norm greater than d/ √ 2 11 . If the noise is non-catastrophic, every quantum map has a corresponding LK approximation, and every d × d linear contraction corresponds to at least one quantum operator.
Given m channels A i , we denote the composition A m • A m−1 • · · · • A 2 • A 1 as A m:1 . Replacing every element of the composition by its LK approximation, A m • A m−1 • · · · • A 2 • A 1 , is noted as A m:1 . In general, the composition operation doesn't commute with the LK approximation, that is A m:1 = (A m:1 ) . To put it in other words, the LK operator of a circuit is generally not the multiplication of the LK operators of its elements. However, while A m:1 provides an incomplete description of A m:1 , they still might share some comparable characteristics. That is, there might exist some function f : CP maps → R for which f (A m:1 ) ≈ f (A m:1 ). As we show, not only there exist such functions, but some correspond to important experimental figures of merit. From the previous section, we know that Φ(A, U) ≈ Φ(A , U) and Υ(A) ≈ Υ(A ). What may be more surprising are the following two theorems: Theorem 1: The unitarity of a circuit barely changes after LK approximating its elements Consider m non-catastrophic channels A i with respective unitary targets U i and suppose that the composition A m:1 is also non-catastrophic. Then, Theorem 2: The process fidelity of a circuit barely changes after LK approximating its elements Consider m non-catastrophic channels A i with respective unitary targets U i and suppose that the composition A m:1 is also non-catastrophic. Then, and A differs from the veritable channel A in many ways as shown by comparing various M -fidelities f M (A m:1 , U m:1 ) with f M (A m:1 , U m:1 ) (see fig. 1). Of course, some kind of discrepancy is expected since the LK approximation contains only d 2 parameters instead of ∼ d 4 . Essentially, the LK operators closely dictate the evolution of the average of Mfidelities Φ = Ef M (see eq. (8)), while the other Kraus operators add or subtract to specific M -fidelities f M in such a way that the sum of those variations almost exactly cancels.
The bar plot on the right illustrates the evolution of different M -infidelities (the four Pauli unitaries), together with the evolution of 1 − Φ. In a), the noise model consists in amplitude damping towards the |0 state. In b), each A i is sampled from a distribution of noisy operations with high coherence level (the notion of coherence level is defined in section V. The evolution theorems presented in this section will greatly help classify different types of errors 12 . Indeed, they allow tying behavioral signatures in the evolution of Υ and Φ to more digestible error profiles. In particular, the two theorems further motivate, as shown in section V, the definition of a natural dichotomy in quantum channels (itself introduced in section IV).

IV. A POLAR DECOMPOSITION FOR QUANTUM CHANNELS
Due to the intricate geometry of d-dimensional quantum states [3], quantum processes can be delicate to dissect. One of the main reasons the single qubit Bloch sphere is frequently invoked stems from the simple picture it offers: i. There is a clear bijection between quantum states and the Bloch ball [28].
ii. The action on the Bloch vectors can be decomposed into a positive semi-definite contraction |M | ≤ I 3 , followed by orthogonal matrix R ∈ O(3), which corresponds to a physical unitary U ∈ SU (2), added to a translational vector t (the non-unital vector) [29][30][31]: where |M | denotes (M † M ) Not every contraction |M | is physical; for instance, transforming the Bloch sphere into a disk violates CP-ness (the "no pancake" theorem). A thorough analysis of CPTP maps acting on M 2 (C) is provided in [30]. For higher dimensions, the Bloch sphere imagery falls apart in many ways: i. The generalized Bloch space is not a (d 2 − 1)-ball (with respect to the 2-norm on R d 2 −1 ) [3].
ii. If we express the action on the Bloch vector as in eq. (30) where R ∈ O(d 2 − 1) and |M | ≥ 0, we realize that To see this, consider the following canonical Kraus decomposition: The spectrum of the associated unital part M is a subset of the spectrum of A * 13 . By expanding up to order α 4 , it is straightforward to show that the phase factors of M are all ≈ 1 except for a single conjugate pair φ ± ≈ exp(±i3α 3 /2). This single pair can't be factored into any unitary process since any non-trivial U * ⊗ U contains at least two conjugate pairs. Hence, trying to cancel the rotating component of the spiraling action (see fig. 2) induced on v ± by φ ± would merely relocate the spiraling motion on an other pair of eigenvectors v ± (or on multiple other pairs). To put it simply, spiraling is inherent to some decoherent processes. To explicitly show this, we constructed an example in which the rotation factors in the spirals couldn't be accounted for by any physical unitary (without creating more spirals).
Separating a quantum channel A into a composition of a physical unitary U and a decoherent operation D (i.e. A = U • D or A = D • U) demands a more careful surgery. Allocate too much rotation factors to the process U 12 An error channel simply refers to a channel with identity target I. 13 A * 1 ⊗A 1 +A * 2 ⊗A 2 is the matrix acting on the column-vectorized density matrices, and has an extra eigenvalue of 1 due the TP condition. Here the star * denotes the complex conjugation, which is not to be confused with the star used for the LK approximation. and it fails to remain physical; allocate too little and the allegedly decoherent constituent D may still contain some physically reversible motion. In fact, depending on the definition of decoherence, it is not even clear if such surgery is even possible. Here, we propose a definition of decoherence that allows to easily decompose any non-catastrophic quantum channel into a composition of a unitary channel with a decoherent one.
Consider a channel A.
This polar decomposition provides a geometric understanding of the range of action of LK approximated channels on the space of quantum states. The absence of phase factors in the spectrum of |A 1 | motivates the following definition:

Definition 3: decoherent channel
A non-catastrophic channel A is said to be decoherent if its LK operator is positive semi-definite: From this definition immediately follows a unitary-decoherent decomposition for quantum channels: Theorem 3: Unitary-decoherent polar decomposition for quantum channels Any non-catastrophic quantum channel A can be express as a composition of a unitary channel U with an In terms of LK approximation, we have: Proof. Under the composition U † • A, the canonical Kraus operators {A i } of A are mapped to {U † A i }, since it preserves their orthonormality. Given the polar decomposition A 1 = U |A 1 |, it follows that the LK operator of U † • A is positive semi-definite.
FIG. 2: Representation of the spiraling action of a normal matrix acting on a 2 × 2 subspace. The polar decomposition, in this case, separates the azimuthal and radial components of the action. Quantum dynamics on d > 2 can generate spiraling actions on the Bloch space for which the rotation factor can't be interpreted as a physical unitary operation. In this sense, spiraling, despite generating some rotating action, is inherent to some decoherent dynamics.
While the proof nearly trivially follows from definition 3, it remains to show that decoherent channels as we defined them deserve such an appellation. An interesting angle to initially justify our definition of decoherence is to observe its contribution in the Gorini-Kossakowski-Sudarshan-Lindblad equation [33,34]. Consider a time evolution dictated by instantaneous CPTP channels 14 with (possibly time-dependent) canonical Kraus operators {A k (t, dt)}: Since dt is infinitesimal, the instantaneous LK operator A 1 (t, dt) must be close to I, and can be expressed as where H(t) is Hermitian and P (t) is positive semi-definite. The TP condition can be expressed as which combined with eq. (36) yields This enforces the remaining instantaneous Kraus operators A k =1 (t, dt) to scale as √ dt, and leaves us with where and [A, B] := AB −BA, {A, B} := AB +BA are respectively the so-called commutator and anticommutator. The fact that {A k (t, dt)} are canonical (hence orthogonal) at every moment in time implies that A 1 (t, dt), A k =1 (t, dt) = 0, which by using eq. (36) results in This together with eq. (40) implies that Notice that the Lindblad operators featuring in a master equation generally do not have a zero trace, but since the master eq. (39) is derived from instantaneous canonical Kraus operators, they do. That is, for every Lindblad master equation, there exists an alternate one, giving rise to the same dynamics, for which the Lindblad operators have a zero trace. This is an important feature for what follows. Let's re-express eq. (39) as a differential equation acting on the column-vectorized states, col(ρ). Using the property col A quick calculation suffices to show that the three indicated terms are mutually orthogonal. This means that their respective actions have no overlap. The first term should be familiar as it corresponds to the generator of unitary evolution. The remaining two terms are often referred to as the relaxation or decoherent part of the Lindbladian [35,36]. This integrates well with our notion of decoherence since the instantaneous channels are decoherent if and only if the Hamiltonian is null at every moment in time: To formulate it otherwise, the Lindbladian consists solely of a decoherent part orthogonal to any commutator if and only if the instantaneous channels are decoherent. An additional interesting remark is that the LK approximation applied to the instantaneous channels essentially eliminates the term iii, leaving only the commutator (term i) and the anticommutator (term ii). In particular, the master equation with LK approximated instantaneous decoherent channels consists of an anticommutator only: When considered as infinitesimal perturbations from the identity, the channels that we refer to as "decoherent" correspond to the generators of the familiar class of decoherent master equations. While our notion of decoherence connects with previous physics literature in the infinitesimal case, it remains to show that our definition is also appropriate without taking such limit. As we will see in the next section, not only the definition is appropriate, but the polar decomposition that results from it (i.e. theorem 3) unravels a series of interesting realizations about Φ(A m:1 , U m:1 ) and Υ(A m:1 ).

V. BEHAVIORAL SIGNATURES OF COHERENCE AND DECOHERENCE
The introduction in the previous section of the dichotomy between coherence and decoherence, together with the demonstration of a polar decomposition for quantum channels wasn't void of ulterior motives. In this section, we leverage the intrinsic differences between coherent and decoherent channels to explore the behavior of the average process fidelity and the unitarity as circuits grow in depth. Before we begin such investigation, however, let's first make a side step to define various classes of operations which will harmonize with our notion of decoherence.
A. Extremal dephasers, extremal unitaries, and equable error channels The non-catastrophic condition still leaves room for pathological noise scenarios. We highlight two extreme (unrealistic) types of channel; the first is of decoherent nature, and the second is purely unitary.

Extremal dephasers
For a channel A to be non-catastrophic, the singular values of its LK operator σ i (A 1 ) must nearly average to 1, but nothing else constrains their distribution. Consider a 10-qubit error A that essentially acts as identity on all operators in M d (C), but cancels any phase between |0 and |i for i = 0 (that is, |0 i|, |i 0| → 0 for i = 0). It is easily shown that the LK operator is A 1 = i =0 |i i|; this is an instance of what we call an "extremal dephaser". An extremal dephaser is defined as a channel for which there exists a singular value σ j ∈ {σ i (A 1 )} (in our example, it is σ 0 = 0) that deviates from the average by much more than the average perturbation: To obey eq. (46), channels must involve excessively strong 15 dephasing mechanisms between a small number of states and the rest of the system 16 . Let's come back to our example: a quick calculation shows that A has an infidelity of around O(2 −10 ) = O(10 −3 ): extremal dephasers can have a high average fidelity; they are not ruled out by the non-catastrophic assumption. However, based on realistic grounds, one might discard such scenarios by assuming that the perturbations of the singular values Indeed, most physically motivated noise mechanisms -such as unitary, amplitude damping and stochastic channels 17 -perturb the singular values of A 1 in a rather homogeneous way (see table I).

Extremal unitaries
The same argument that was made about the singular values of A 1 = U |A 1 | , which are the eigenvalues of its positive semidefinite factor |A 1 |, can be made for the eigenvalues of the unitary factor U . To mimic our previous example, consider a 10-qubit unitary error U that essentially acts as identity on operators in M d (C), but maps |0 i| → −|0 i| |i 0| → −|i 0| for i = 0. It is easily shown that the LK operator is U = −|0 0| + i =0 |i i|; this is an instance of what we call an "extremal unitary". An extremal unitary is defined as a unitary error U for which there exists an eigenvalue λ j ∈ {λ i (U )} (in our example, it is λ 0 = −1) that deviates from the average by much more than the average perturbation. An easy way to make this precise is to fix the phase of U such that Tr U ∈ R + , and project the eigenvalues on the real axis (this is easy to picture on an Argand diagram):

Equable error channels
In this paper, we qualify as "equable" the non-catastrophic error channels A = U • D for which the factors D and U are not extremal. Notice that the equability assumption ensures a unique polar decomposition since the LK operator is guaranteed to be full rank.
While ruling out extremal error channels seems reasonable, we also define a weaker condition based on the variance of the perturbations:

Definition 4: Wide sense equable (WSE) error channels
Consider a non-catastrophic error channel A = U • D with LK operator A 1 = U |A 1 |. Let {σ i } be the singular values of A 1 and {λ i } be the eigenvalues of U for which the phase is fixed such that Tr U ∈ R + . We define the WSE decoherence constant γ D as: We define the WSE coherence constant γ U as: A non-catastrophic error channel is said to be equable in the wide sense if First notice that ruling out extremal errors, which can be imposed by for γ D , γ U obeying eqs. (50) and (51), trivially implies equability in the wide sense. The WSE condition is weaker: for instance, there could be some singular values σ i (A 1 ) that widely differ from the average, but not enough of them to increase the variance in a substantial manner. Let's modify our previous extremal dephaser example by adding some depolarizing component to A. Let A 1 = i =0 (1 − 10 −1.5 )|i i| . A quick calculation confirms that despite being a extremal dephaser, this channel is still equable in the wide sense. The reason to invoke the WSE condition rather than simply ruling out extremal errors is twofold: i. Wide sense equability is sufficient for deriving our results.
ii. Since the definitions of the WSE constants γ D , γ U rely on simple statistics (i.e. variances and means), it seems conceivable to efficiently bound them experimentally (it's always desirable to not discard verifiability if possible).  Consider m non-catastrophic channels A i . Then Υ(A m:1 ) has the following properties: (Quasi-subadditive property) Those inequalities are almost saturated by extremal channels. If we introduce the WSE decoherence constants γ D (A i ) ≤ γ D ), we obtain: which guarantees quasi-multiplicativity of Υ when the errors are WSE.
Of course, those results can be immediately translated in terms of unitarity by using eq. (14). Without using the LK approximation, showing the monotonicity of the unitarity can be difficult, since quantum channels aren't contractive maps; going to the LK picture fixes this issue since Kraus operators are contractions. Quasi-multiplicativity is another way of stating that the unitarity of a composition essentially behaves as a multiplicative decay involving the unitarity of individual components: Equation (54) should be seen as a staple of wide sense equability; deviations from this behavior indicates the presence of extremal dephasers. Another signature of wide sense equability can be seen through the following result: Moreover, if we introduce the WSE decoherence constant γ D ), we obtain: If the error attached to A is WSE, then the interval virtually collapses and max

A quasi-maximal choice of unitary correction consists in
Essentially, wide sense equability ensures a quasi-one-to-one correspondence between the maximal average gate fidelity (through a unitary correction) and the unitarity: (57)

C. Justifying our notion of decoherence
Let's now return to our allegedly decoherent channels. Typically, quantum error channels are said to enact decoherently if they exhibit a non-reversible deterioration. In turn, coherent error channels correspond to a mishandling of information -which can in principle be reverted -rather than a loss of information. An additional expected property of decoherent operations is that they shouldn't allow for coherent buildups such as in the case accumulating over-rotations. Given m non-catastrophic unitary channels U i ≈ I with the infidelity grows faster than linearly (let the composition U m:1 be non-catastrophic so that mθ ≤ π/4) [1]: As an intuitive pair of properties of our so-called decoherent channels, we show that i. The average process fidelity of decoherent error channels cannot be substantially recovered by any unitary (quasimonotonicity).
ii. The evolution of the infidelity of a circuit composed of decoherent operations is (approximately) at most additive in the individual infidelities. There is no substantial coherent buildup.
If the channels are WSE, Φ(D m:1 , I) is essentially multiplicative.
Using the simple relation between F and Φ (eq. (9)) we come to this observation: the average gate fidelity of a composition of non-catastrophic decoherent WSE channels behaves almost exactly as a multiplicative decay in the average process fidelity of individual components, that is The decay becomes exact with the depolarizing channel P p (ρ) = pρ + (1 − p)(Tr ρ)I d /d, which is a celebrated example of decoherent operation.

D. The coherence level
Let's extend theorem 7 by appending a coherent operation to the decoherent composition: The channel average infidelity of a channel can be split into a sum of a coherent and decoherent terms (given WSE errors). r decoh is not substantially correctable through any composition, and can be obtained from the unitarity alone: r coh can be corrected through a composition with a unitary (see theorem 5). Equation (65) motivates the definition of coherence level as the fraction of the infidelity that is associated to coherence. It can be obtained by combining the infidelity and the unitarity through: Similarly, the decoherence level is defined as r decoh /r. Equation (65) strengthen the insight behind notion of coherence level introduced (under different appellations) in [37,38]. In those previous works, the RHS of eq. (65) is generally depicted as a lower bound on the infidelity, which can be reduced to r decoh through a unitary correction. The (approximate) equality -which is much more valuable since it provides an upper bound on r -is shown for single qubit case in [37] using the polar decomposition of the action on Bloch sphere. Here, we have shown the (approximate) equality (in the WSE scenario) for all dimensions using the polar decomposition of LK operators.

E. Bounding the worst and best case fidelity of a circuit
Now, let's revisit theorem 8 for general circuit depth m. This will allow us to identify the worst and best case fidelity of a circuit. Consider m channels A i with target U i and polar decomposition D i • V i . The circuit A m:1 can be re-expressed as where D k := (V k:1 ) † • D k • V k:1 are decoherent channels with the same fidelity as D k . This means that: In this last expression, we clearly see that the evolution of Φ is factored into a decoherent decay multiplied by a function Φ(V m:1 , U m:1 ) which captures the fidelity of a purely coherent process. This is already an interesting realization: since the decoherent decay is fixed, all the freedom in the evolution of the fidelity is contained in the coherent factors. An assessment concerning the circuit's average process fidelity must rely on a characterization of coherent effects. Since we know that such effects are correctable through composition, we first get: For WSE errors, the maximal unitary correction of the composition A m:1 is essentially In short, the average gate fidelity of a composite circuit is upper bounded by a decaying envelope which is closely prescribed by the decoherent factors of its individual components: This unforgiving behavior harmonizes well with the more typical comprehension of decoherence as a limiting process. To find the worst possible Φ(A m:1 , U m:1 ), it suffices to use a lower bound for the coherent factor Φ(V m:1 , U m:1 ). This is partially done in [1], where the inequality is shown to be saturated in even dimensions. For odd dimensions, we find the following saturated bound: Proof. The generalization to odd dimensions almost immediately follows by looking at the saturation case in even dimensions, which consists of commuting unitary errors of the form In the odd dimension case, it suffices to always pick the global phase to fix the first eigenvalue of V m:1 (U m:1 ) −1 to 1. The minimization over | Tr V m:1 (U m:1 ) −1 | then falls back to the even dimensional case, since the saturation case has a real trace.
can formulate a quasi-saturated assessment about the average process fidelity of the circuit A m:1 given a partial information about its components A i (in the WSE scenario):

(Odd dimensions)
The terms in the cosine function are very close to what was defined as "coherence angles" in [1]. Their sum can be interpreted as a coherent buildup. In some sense, the coherence angle is just another way to go about the notion of coherence level: it ties r coh to an optimal rotation angle.

VI. CONCLUSION
In this work, we investigated a quasi-dynamical sub-parameterization of quantum channels that we referred to as the LK approximation. A remarkable realization is that this reduced picture still allows to closely follow the evolution of two important figures of merit, namely the average process fidelity and the unitarity.
Working with a simplified portrait sets aside superfluous subtleties and typically grants new mathematical properties to the object of consideration 19 . In our case, LK approximated mappings can be parameterized as contractions in M d (C); this set of matrices offers a much more intelligible categorization of error scenarios than the more abstruse full process matrix parameterization. Any matrix A ∈ M d (C) has a polar decomposition U |A| where |A| ≥ 0 and U is unitary. U corresponds to a purely coherent physical operation U(ρ) = U ρU † , whereas the positive contraction |A| is the LK operator belonging to what we classify as a decoherent channel. In a nutshell, the polar decomposition in M d (C) translates into a coherent-decoherent factorization for quantum channels. We leveraged this dichotomy between types of noise to derive fundamental principles of behavior concerning our two considered figures of merit. Among other properties, we demonstrated, up to high precision, the general monotonicity of the unitarity as well as the monotonicity of the average process fidelity of circuits with decoherent components.
To pursue our analysis further, we introduced the wide sense equable parameters γ D , γ U , which are defined through the LK parameterization. Wide sense equable (WSE) error channels, for which γ D , γ U are not too high, includes all realistic noise models (and more). Under the WSE condition, we make multiple interesting connections between individual channels and compositions thereof: i. The infidelity of any channel can be decomposed into a sum of two terms: a decoherent infidelity and a coherent one (respectively tied to the decoherent/coherent components of the channel).
ii. The unitarity, as well as for the fidelity of circuits with decoherent elements, obey decay laws. Both these decays are closely dictated by the unitarity of individual components alone.
iii. The decoherent decay (that is, the decay prescribed by the decoherent factors of the circuit components) forms an upper bound to the total average process fidelity. Any substantial deviation from this upper bound is due to coherent effects alone (which gives us a lower bound).
This work was primarily cast as a stepping-stone to formulate assessments about the performance of circuits based on partial knowledge of their constituents. While we do provide some assertion formulas, we want to emphasize that the more fundamental introduction of the LK approximation should also benefit the development of further characterization schemes. Indeed, the simple parameterization offered by the LK approximation facilitates the identification of specific noise signatures. Extremal dephaser (channel) -Strongly dephases a small set of states from the rest of the system. Since the set of states is small, extremal dephasers can still have high fidelity.
Extremal unitary (channel) -Strongly dephases a small set of states from the rest of the system. Since the set of states is small, extremal dephasers can still have high fidelity.
-Excludes pathological behaviors induced by extremal errors.
-Should apply to all realistic scenarios.
Wide sense equable (WSE) channel  Acknowledgments -The authors would like to thank Joel J. Wallman for his helpful discussions. This research was supported by the U.S. Army Research Office through grant W911NF-14-1-0103, TQT, CIFAR, the Government This section is dedicated to demonstrating a useful trace inequality.

Lemma 3: Noteworthy trace inequality
Let A, B ∈ M d (C) be Hermitian matrices with eigenvalues of at most ρ A , ρ B respectively. Then, Proof. We first show this inequality for positive semi-definite matrices with eigenvalues of at most 1, under the condition that In such case, the inner product is minimized by the sum of eigenvalues paired in opposite order [39] (it's a matrix equivalent to the Hardy-Littlewood rearrangement inequality): This is in turn minimized when both {λ i (A)} and {λ i (B)} are maximized in terms of strong majorization. Since the eigenvalues are between zero and 1, both majorizations have a simple form: With such spectrum and the condition d < Tr A + Tr B + 2, we are ensured that which, together with eq. (A3), yields eq. (A1) in this simpler case. Now, consider the general case of Hermitian matrices A, B with eigenvalues of at most ρ A , ρ B respectively. Let A = (A + n A I) − n A I, B = (B + n B I) − n B I, for n A , n B ∈ R + , and consider the following expansion: Now, let's pick n A , n B large enough so that For i we can simply pick n A ≥ min λ(A), n B ≥ min λ(B). To see why ii is also possible, realize that lim n A →∞

Tr(
meaning that there exists a finite n A such that ii is fulfilled. Moreover, realize that the maximum eigenvalue of both since this corresponds to our initial simpler case. Substituting eq. (A9) into eq. (A7) and simplifying, we get eq. (A1) which completes the proof.
This inequality pairs well with the so-called Von-Neuman's trace inequality, as when Tr AB ≥ 0, lemma 3 provides a much better lower bound. To see this, consider the following inequality which is trivially derived from Von's Neumann's trace inequality:

Lemma 4: Flavored Von Neumann's trace inequality
Let A, B ∈ M d (C) be matrices with spectral radius of at most ρ A , ρ B respectively. Then, Recalling that A 2 2 = Tr A † A and using those two last inequalities, we get the following norm inequality:

Lemma 5: Norm inequality
Consider two matrices A, B with spectral radius of at most 1. Then, Appendix B: Proofs of the main results

Notation and remarks
Before we start proving theorems 1 and 2, let's introduce some handy notation. The i th canonical Kraus operator of a channel A j is denoted A j i . Let a ≥ b; we denote where i ∈ N a−b+1 simply contains indices i k ∈ {1, · · · , d 2 }. Finally we denote 1 = (1, · · · , 1) for which the dimension is left implicit.
Remark that the set {A m:1 i } i consist in a valid Kraus decomposition for the composite channel A m:1 , and can be used to calculate Φ(A m:1 , U m:1 ) and Υ(A m:1 ) through eqs. (10) and (13) respectively. However, these Kraus operators are generally not orthogonal one another (this is not the canonical decomposition), which prevents the same proof technique as in lemmas 1 and 2. i with i k = 1 for some k ∈ {1, · · · m}. Using the properties of contractions, we have Hence, if we suppose argmax which completes the proof.

Proof of the evolution theorem 2
Proof. We will show that the inequality eq. (28) holds for m = 2 n , ∀n ∈ N. This suffices since if N < 2 n , then we can append I N −2 n :1 to the composition A N :1 so that A N :1 • I N −2 n :1 is a composition of length 2 n . Appending I N −2 n :1 has no effect on eq. (28).
From the definition of Φ, we have that Φ(A m:1 , U m:1 ) − Φ(A m:1 , U m:1 ) ≥ 0, so it only remains to derive an upper bound on Φ(A m:1 , U m:1 ) − Φ(A m:1 , U m:1 ). Our approach will be to split the sum as follows: The double sum (last term) can be bounded via Cauchy-Schwarz inequality followed by the usage of lemma 5: With regards to the first two terms on the RHS of eq. (B6), let's split them both into three terms once again: By iterating the same subdivision technique, we end up with Bounding the first term on the RHS can be done by alternating between the AM-GM inequality and square completions. First let's perform the AM-GM inequality on the terms of the summation restricted to i = n: Then, let's add in the terms with index i = n − 1 and complete the squares (taking n = 3 as an example is recommended): (eq. (B11)) Similarly, we can then add in the terms with index i = n − 2, complete the squares and use the AM-GM inequality on the leftover summation: Repeating this procedure until i = 1, we get The last term on the RHS of eq. (B10) is upper-bounded using an alternate technique. First, we get For fixed j, {A j i } forms an orthonormal basis. Since (A m:j+1 , we have that, for any j:

(Equation (C2))
To bound m i=1 Φ(D i , I), we express it as a sum of three terms: