# Perspective on “Circuits of space and time quantum channels”

*This is a Perspective on "Circuits of space and time quantum channels" by Pavel Kos and Georgios Styliaris, published in Quantum 7, 1020 (2023).*

**By Shane Dooley (Department of Physics, Trinity College Dublin, Dublin 2, Ireland).**

Published: | 2023-07-26, volume 7, page 75 |

Doi: | https://doi.org/10.22331/qv-2023-07-26-75 |

Citation: | Quantum Views 7, 75 (2023) |

## Summary

Throughout the history of physics, exactly solvable models have provided valuable insight into various physical phenomena [1]. Conventional wisdom says that exact solutions are usually only possible for relatively simple integrable systems, and that one must resort to approximations or numerical simulations to learn about the dynamics of more complicated chaotic systems. But this is not entirely accurate. For example, in recent years it was discovered that for quantum circuits with a property called *dual-unitarity* it is possible to exactly calculate some dynamical quantities, even in quantum chaotic many-body systems.

Dual-unitarity means that the local bipartite gates that make up the circuit are unitary not only as propagators in time, but are also unitary if reinterpreted as propagators in the spatial direction [2]. In the work Circuits of space and time quantum channels, recently published in *Quantum* [3], the authors generalise the dual-unitary property to open quantum systems. Their circuits are made up of local quantum dynamical maps that are completely positive and trace preserving (i.e., they are quantum channels) not only in the forward-time directionbut also when reinterpreted as propagators in the backward-time direction, or in the left/right spatial directions. Just like for dual-unitary circuits, these special properties, dubbed *space-time unitality*, can enable the exact calculation of some dynamical quantities. This significantly extends the dual-unitary framework, and enlarges the class of exactly solvable models in many-body quantum physics.

## Diagrammatic tensor notation

Crucial to understanding this work (as well as the prior literature on dual-unitary circuits) is a diagrammatic tensor notation in which quantum states and channels are represented graphically. For example, a bipartite quantum state $\rho $ is represented as a box with two outgoing lines, or legs, corresponding to the two parts of the system:

A quantum channel $\Lambda $ acts on the state $\rho $ and is assumed to output a state $\Lambda (\rho )$ with the same subsystems. It is represented as a box with two input legs and two output legs, so that the output state $\Lambda (\rho )$ is:

In this work, the authors consider a system of many qudits in which the dynamics is generated by a circuit of local quantum channels in a brickwork pattern:

## Space-time unitality

With this graphical notation in hand, the authors define several properties that a quantum dynamical map may have:

Here, the open circle attached to an input leg corresponds to the maximally mixed input state for that subsystem, while the open circle attached to an output leg corresponds to the partial trace applied to that subsystem.

The first property, trace preservation, is satisfied for any map that is a well-defined quantum channel, i.e., a completely positive and trace-preserving map (complete positivity is already ensured by construction [3]). The other three graphical identities simply represent trace preservation of the channel, if it is reconsidered as a propagator in the backward-time direction (unitality), or as a propagator along the left or right spatial direction (left unitality or right unitality, respectively). In other words, when additional space-time unitality conditions are satisfied the quantum map is also a well-defined quantum channel along that space-time direction of propagation.

## Exact solvability

If the quantum channels satisfy some combination of the four space-time unitiality properties above, then the corresponding identities can be used to do graphical calculations that might otherwise be quite cumbersome in the usual (non-graphical) algebraic approach.

For example, consider the infinite-temperature spatio-temporal function $\langle a_\ell (t) b_{\ell ‘} \rangle = Tr[ a_\ell (t) b_{\ell ‘} 𝟙 ] / D$, where $D$ is the total dimension, and $a_\ell $ and $b_{\ell ‘}$ are observables localised at qudit $\ell $ and qudit $\ell ‘$, respectively. Such a quantity in graphical notation might be expressed as (assuming periodic boundary conditions):

If, for instance, the trace-preservation, left-unitality and right-unitality conditions are satisfied, then their corresponding graphical identities can be used repeatedly to simplify the picture to:

which can, it turns out, be evaluated efficiently by transfer-operator methods. More generally, the authors show that infinite-temperature spatio-temporal functions can be evalutated exactly if any three of the space-time unitality conditions (including trace preservation) are satisfied. This generalises a similar result for dual-unitary circuits [2].

In a similar spirit, they also find a class of solvable initial states $\rho (0)$, represented as matrix-product operators. From this class of initial states the authors show how to efficiently calculate spatial correlation functions of the form $Tr[a_\ell b_{\ell ‘} \rho (t)]$, as well as how to determine properties of the long-time steady state of the circuit dynamics. The authors also analyse some general properties of space-time unital channels. For example, they find the dimension of the subsets of channels corresponding to different combinations of the spact-time unitality properties. Interestingly, the analagous dimension for dual-unitaries acting on qudits is still unknown for local dimension $d>2$. At the same time, other results that are known for dual-unitaries appear to be more difficult to calculate for space-time unital channels (e.g., the calculation of entanglement growth).

## Outlook

Since no real system is perfectly closed, this generalisation of dual-unitarity to open systems may help to bridge the gap between experiments and the known exact results for dual-unitary circuits. Indeed, several experiments have been performed recently around the dual-unitary point of quantum circuits [4,5]. The results have been consistent with the exact calculations, indicating a certain robustness to the noise that is present in all experiments. The framework presented in this paper may help to explain this robustness.

### ► BibTeX data

### ► References

[1] Bill Sutherland. Beautiful models: 70 years of exactly solved quantum many-body problems. World Scientific, 2004.

https://doi.org/10.1142/5552

[2] Bruno Bertini, Pavel Kos, and TomažProsen. Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions. Phys. Rev. Lett., 123:210601, Nov 2019.

https://doi.org/10.1103/PhysRevLett.123.210601

[3] Pavel Kos and Georgios Styliaris. Circuits of space and time quantum channels. Quantum, 7:1020, May 2023.

https://doi.org/10.22331/q-2023-05-24-1020

[4] Eli Chertkov, Justin Bohnet, David Francois, John Gaebler, Dan Gresh, Aaron Hankin, Kenny Lee, David Hayes, Brian Neyenhuis, Russell Stutz, et al. Holographic dynamics simulations with a trapped-ion quantum computer. Nat. Phys., 18(9):1074–1079, 2022.

https://doi.org/10.1038/s41567-022-01689-7

[5] Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandra, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, et al. Information scrambling in quantum circuits. Science, 374(6574):1479–1483, 2021.

https://doi.org/10.1126/science.abg5029

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This View is published in Quantum Views under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.