Circuits Of Space And Time Quantum Channels

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Pavel Kos and Georgios Styliaris

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany

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Abstract

Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting of unitary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to consider the impact of noise to dual-unitary dynamics and its exact solvability.
In this work we generalise the ideas of dual-unitarity to obtain exact solutions in noisy quantum circuits, where each unitary gate is substituted by a local quantum channel. Exact solutions are obtained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolutions along one or both of the spatial directions and possibly backwards in time. This gives rise to new families of models that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solutions for the spatio-temporal correlation functions, spatial correlations after a quantum quench, and the structure of steady states for these families of models. We show that noise unbiased around the dual-unitary family leads to exactly solvable models, even if dual-unitarity is strongly violated. We prove that any channel unital in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Finally, we extend the definition of solvable initial states to matrix-product density operators. We completely classify them when their tensor admits a local purification.

Featured image: In this work, we consider dynamics given by a circuit of local quantum channels. These channels may have additional sideways unitality conditions, which enable exact calculations.

Popular summary

Understanding how quantum systems of many spins evolve in time is a challenging task. In most cases, the relevant aspects of the complicated evolution can be extracted by examining the correlation functions. However, the problem of computing correlation functions for models exhibiting chaos is in general hard, so providing examples where they can be analyzed is crucial for our understanding.

In our work, we generalize one such example –dual-unitary circuits– to systems beyond unitary dynamics, called space-time channels. Here coupling with the environment results in quantum dynamics consisting of local quantum channels, that is, open-system evolution. These space-time quantum channels are characterized by the property that the evolution is still physical upon changing the roles of space and time, exactly as in the case of dual-unitary circuits. This property defines different rich families of models with tractable dynamics.

Our work opens new doors to exactly solvable open quantum circuits. As quantum evolution, simulation, or computation is never entirely isolated from the environment this knowledge is much needed. Moreover, our work also explains why the signature of dual-unitarity (vanishing correlations inside the light cone), which was already witnessed in the experiment, is preserved under typical noise.

► BibTeX data

@article{Kos2023circuitsofspacetime, doi = {10.22331/q-2023-05-24-1020}, url = {https://doi.org/10.22331/q-2023-05-24-1020}, title = {Circuits of space and time quantum channels}, author = {Kos, Pavel and Styliaris, Georgios}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1020}, month = may, year = {2023}
}

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Cited by

[1] Alessandro Foligno and Bruno Bertini, "Growth of entanglement of generic states under dual-unitary dynamics", arXiv:2208.00030, (2022).

[2] Katja Klobas, Cecilia De Fazio, and Juan P. Garrahan, "Exact "hydrophobicity" in deterministic circuits: dynamical fluctuations in the Floquet-East model", arXiv:2305.07423, (2023).

[3] Richard M. Milbradt, Lisa Scheller, Christopher Aßmus, and Christian B. Mendl, "Ternary Unitary Quantum Lattice Models and Circuits in 2 +1 Dimensions", Physical Review Letters 130 9, 090601 (2023).

[4] Pieter W. Claeys, Austen Lamacraft, and Jamie Vicary, "From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics", arXiv:2302.07280, (2023).

[5] Michael A. Rampp, Roderich Moessner, and Pieter W. Claeys, "From Dual Unitarity to Generic Quantum Operator Spreading", Physical Review Letters 130 13, 130402 (2023).

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