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arXiv:2110.05710 · PRX Quantum

Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes

Julia Wildeboer, Thomas Iadecola, Dominic J. Williamson

Published May 10, 2022• 20 citations

Abstract

We study a mechanism whereby quantum information present in the initial state of a quantum manybody system can be protected for arbitrary times due to a combination of symmetry and spatial locality.Remarkably, the mechanism is sufficiently generic that the dynamics can be fully ergodic upon resolving the protecting symmetry and fixing the encoded quantum state, resulting in a quantum memory that persists up to infinite temperature.After exemplifying the mechanism in a strongly nonintegrable two-dimensional (2D) spin model inspired by the surface code, we find it has a natural interpretation in the language of noiseless subsystems and stabilizer subsystem codes.This interpretation yields a number of further examples, including a nonintegrable Hamiltonian with quantum memory based on the Bacon-Shor code.The lifetime of the encoded quantum information in these models is infinite provided that the dynamics respect the stabilizer symmetry of the underlying subsystem code.In the presence of symmetry-violating perturbations, we make contact with previous work leveraging the concept of prethermalization to show that the encoded quantum information can acquire a parametrically long lifetime under dynamics with an enlarged continuous symmetry group.The prethermalization mechanism hinges on the application of external fields that are much larger than the perturbations themselves.We identify conditions on the underlying subsystem code that enable such a prethermal enhancement of the memory lifetime.

DOI: 10.1103/prxquantum.3.020330

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Whether self-correcting quantum memories can exist at nonzero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [S. Roberts and S. D. Bartlett, Phys. Rev. X 10, 031041 (2020)] that symmetry-protected topological (SPT) systems in three space dimensions subject to a strong constraint---that the quantum dynamics respect a 1-form symmetry---realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a subvolume of the system. This strongly suggests that the SPT character of the state is not essential. We confirm this by constructing an explicit example with a trivial (paramagnetic) bulk that realizes a self-correcting quantum memory. We therefore show that the enforcement of a 1-form symmetry on a measure-zero subvolume of a three-dimensional system can be sufficient to stabilize a self-correcting quantum memory at nonzero temperature.

We show how combining a discrete symmetry with topological Hilbert space fragmentation can give rise to exponentially many topologically stable qubits protected by a single discrete symmetry. We illustrate this explicitly with the example of the

\mathsf{CZ}_p

model, where the encoded qubits are stable to arbitrary symmetry-respecting perturbations for parametrically long times, substantially enhancing the robustness of a recently proposed construction based on nontopological fragmentation. In this model, the encoded qubits naturally come in pairs for which a universal set of transversal logical gates can be performed, ruling out (by the Eastin-Knill theorem) the possibility of using them for quantum error correction. We also comment on the combination of symmetry enrichment and topological fragmentation more generally, and the implications for use of systems exhibiting Hilbert space fragmentation as quantum memories.

We derive new tradeoffs for reliable quantum information storage in a 2D local architecture based on subsystem quantum codes. Our results apply to stabilizer subsystem codes, that is, stabilizer codes in which part of the logical qubits does not encode any information. A stabilizer subsystem code can be specified by its gauge group - a subgroup of the Pauli group that includes the stabilizers and the logical operators on the unused logical qubits. We assume that the physical qubits are arranged on a two-dimensional grid and the gauge group has spatially local generators such that each generator acts only on a few qubits located close to each other. Our main result is an upper bound kd = O(n), where k is the number of encoded qubits, d is the minimal distance, and n is the number of physical qubits. In the special case when both gauge group and the stabilizer group have spatially local generators, we derive a stronger bound kd2= O(n) which is tight up to a constant factor.