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arXiv:0907.2807 · New Journal of Physics

Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes

Stefano Chesi, Daniel Loss, Sergey Bravyi, Barbara M. Terhal

Published Feb 26, 2010• 50 citations
arXivHTMLPDF

Abstract

We discuss several thermodynamic criteria that have been introduced to characterize the thermal stability of a self-correcting quantum memory. We first examine the use of symmetry-breaking fields in analyzing the properties of self-correcting quantum memories in the thermodynamic limit; we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results of Alicki et al to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.

DOI
10.1088/1367-2630/12/2/025013

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We discuss several thermodynamic criteria that have been introduced to characterize the thermal stability of a self-correcting quantum memory. We first examine the use of symmetry-breaking fields in analyzing the properties of self-correcting quantum memories in the thermodynamic limit; we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results of Alicki et al to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.
We study properties of stabilizer codes that permit a local description on a regular D -dimensional lattice. Specifically, we assume that the stabilizer group of a code (the gauge group for subsystem codes) can be generated by local Pauli operators such that the support of any generator is bounded by a hypercube of size O (1). Our first result concerns the optimal scaling of the distance d with the linear size of the lattice L . We prove an upper bound d = O ( L D -1 ) which is tight for D =1, 2. This bound applies to both subspace and subsystem stabilizer codes. Secondly, we analyze the suitability of stabilizer codes for building a self-correcting quantum memory. Any stabilizer code with geometrically local generators can be naturally transformed to a local Hamiltonian penalizing states that violate the stabilizer condition. A degenerate ground state of this Hamiltonian corresponds to the logical subspace of the code. We prove that for D =1, 2, different logical states can be mapped into each other by a sequence of single-qubit Pauli errors such that the energy of all intermediate states is upper bounded by a constant independent of the lattice size L . The same result holds if there are unused logical qubits that are treated as 'gauge qubits'. It demonstrates that a self-correcting quantum memory cannot be built using stabilizer codes in dimensions D =1, 2. This result is in sharp contrast with the existence of a classical self-correcting memory in the form of a two-dimensional (2D) ferromagnet. Our results leave open the possibility for a self-correcting quantum memory based on 2D subsystem codes or on 3D subspace or subsystem codes.
Is the notion of a quantum computer (QC) resilient to thermal noise unphysical? We address this question from a constructive perspective and show that local quantum Hamiltonian models provide self-correcting QCs. To this end, we first give a sufficient condition on the connectedness of excitations for a stabilizer code model to be a self-correcting quantum memory. We then study the two main examples of topological stabilizer codes in arbitrary dimensions and establish their self-correcting capabilities. Also, we address the transversality properties of topological color codes, showing that six-dimensional color codes provide a self-correcting model that allows the transversal and local implementation of a universal set of operations in seven spatial dimensions. Finally, we give a procedure for initializing such quantum memories at finite temperature.
We present rigorous bounds on the thermalization time of the family of quantum mechanical spin systems known as stabilizer Hamiltonians. The thermalizing dynamics are modeled by a Davies master equation that arises from a weak local coupling of the system to a large thermal bath. Two temperature regimes are considered. First we clarify how in the low temperature regime, the thermalization time is governed by a generalization of the energy barrier between orthogonal ground states. When no energy barrier is present the Hamiltonian thermalizes in a time that is at most quadratic in the system size. Secondly, we show that above a universal critical temperature, every stabilizer Hamiltonian relaxes to its unique thermal state in a time which scales at most linearly in the size of the system. We provide an explicit lower bound on the critical temperature. Finally, we discuss the implications of these result for the problem of self-correcting quantum memories with stabilizer Hamiltonians.
The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error-correcting codes, one requires the ability to apply a procedure which exactly reverses on the error-correcting subspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform corrections only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error-correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error-correction procedures.
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.
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.