Self-correcting Quantum Memory Research Articles

A cellular automaton decoder for a noise-bias tailored color code

Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.

Thermalization in Kitaev’s quantum double models via tensor network techniques

Abstract We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.

Quantum XYZ Product Codes

We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3). While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

Symmetry-protected self-correcting quantum memory in three space dimensions

Whether self correcting quantum memories can exist at non-zero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [1] 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 sub-volume of the system which is measure zero in the thermodynamic limit. 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 sub-volume of a three dimensional system can be sufficient to stabilize a self correcting quantum memory at non-zero temperature.

Symmetry-Protected Self-Correcting Quantum Memories

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.

Self-correction in Wegner's three-dimensional Ising lattice gauge theory

Motivated by the growing interest in self-correcting quantum memories, we study the feasibility of self-correction in classical lattice systems composed of bounded degrees of freedom with local interactions. We argue that self-correction, including a requirement of stability against external perturbation, cannot be realized in system with broken global symmetries such as the 2d Ising model, but that systems with local, i.e. gauge, symmetries have the required properties. Previous work gave a three-dimensional quantum system which realized a self-correcting classical memory. Here we show that a purely classical three dimensional system, Wegner's 3D Ising lattice gauge model, can also realize this self-correction despite having an extensive ground state degeneracy. We give a detailed numerical study to support the existence of a self-correcting phase in this system, even when the gauge symmetry is explicitly broken. More generally, our results obtained by studying the memory lifetime of the system are in quantitative agreement with the phase diagram obtained from conventional analysis of the system's specific heat, except that self-correction extends beyond the topological phase, past the lower critical temperature.

Topological order and memory time in marginally-self-correcting quantum memory

We examine two proposals for marginally self-correcting quantum memory, the cubic code by Haah and the welded code by Michnicki. In particular, we prove explicitly that they are absent of topological order above zero temperature, as their Gibbs ensembles can be prepared via a short-depth quantum circuit from classical ensembles. Our proof technique naturally gives rise to the notion of free energy associated with excitations. Further, we develop a framework for an ergodic decomposition of Davies generators in CSS codes which enables formal reduction to simpler classical memory problems. We then show that memory time in the welded code is doubly exponential in inverse temperature via the Peierls argument. These results introduce further connections between thermal topological order and self-correction from the viewpoint of free energy and quantum circuit depth.

Quantum memories at finite temperature

To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the state-of-the-art developments in this field in an informative and pedagogical way. We give the main principles of self-correcting quantum memories and we analyze several milestone examples from the literature of two-, three- and higher-dimensional quantum memories.

Thermalization Time Bounds for Pauli Stabilizer Hamiltonians

We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of N-qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life time of such a quantum memory and can be used to estimate the time until the system relaxes to thermal equilibrium when brought into contact with a thermal heat bath. The bound can be shown to behave as λ≥O(N^(−1) exp(−2βϵ)), where ϵ is a generalization of the well known energy barrier for logical operators. Particularly in the low temperature regime we expect this bound to provide the correct asymptotic scaling of the gap with the system size up to a factor of N^(−1). Furthermore, we discuss conditions and provide scenarios where this factor can be removed and a constant lower bound can be proven.

Thermalization of topological entropy after a quantum quench

Topologically ordered quantum phases are robust in the sense that perturbations in the Hamiltonian of the system will not change the topological nature of the ground-state wave function. However, in order to exploit topological order for applications such as self-correcting quantum memories and information processing, these states need to be also robust both dynamically and at finite temperature in the presence of an environment. It is well known that systems like the toric code in two spatial dimensions are fragile in temperature. In this paper, we show a completely analytic treatment of the toric code away from equilibrium, after a quantum quench of the system Hamiltonian. We show that, despite being subject to unitary evolution (and at zero temperature), the long-time behavior of the topological entropy is thermal, therefore vanishing. If the quench preserves a local gauge structure, there is a residual long-lived topological entropy. This also is the thermal behavior in presence of such gauge constraints. The result is obtained by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way.

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

We propose a family of local CSS stabilizer codes as possible candidates for self-correcting quantum memories in 3D. The construction is inspired by the classical Ising model on a Sierpinski carpet fractal, which acts as a classical self-correcting memory. Our models are naturally defined on fractal subsets of a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does not imply that these models can be realized with local interactions in , we also discuss this possibility. The X and Z sectors of the code are dual to one another, and we show that there exists a finite temperature phase transition associated with each of these sectors, providing evidence that the system may robustly store quantum information at finite temperature.

Quantum error correction for quantum memories

Active quantum error correction using qubit stabilizer codes has emerged as a promising, but experimentally challenging, engineering program for building a universal quantum computer. In this review we consider the formalism of qubit stabilizer and subsystem stabilizer codes and their possible use in protecting quantum information in a quantum memory. We review the theory of fault-tolerance and quantum error-correction, discuss examples of various codes and code constructions, the general quantum error correction conditions, the noise threshold, the special role played by Clifford gates and the route towards fault-tolerant universal quantum computation. The second part of the review is focused on providing an overview of quantum error correction using two-dimensional (topological) codes, in particular the surface code architecture. We discuss the complexity of decoding and the notion of passive or self-correcting quantum memories. The review does not focus on a particular technology but discusses topics that will be relevant for various quantum technologies.

Fault-tolerant logical gates in quantum error-correcting codes

Recently, Bravyi and K\"onig have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and applications of their characterization in various directions. First, we present a new no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. Second, we prove that the code distance of a D-dimensional local stabilizer code with non-trivial locality-preserving m-th level Clifford logical gate is upper bounded by $O(L^{D+1-m})$. For codes with non-Clifford gates (m>2), this improves the previous best bound by Bravyi and Terhal. Third we prove that a qubit loss threshold of codes with non-trivial transversal m-th level Clifford logical gate is upper bounded by 1/m. As such, no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes, and is not restricted to geometrically-local codes. Fourth we extend the result of Bravyi and K\"onig to subsystem codes. A technical difficulty is that, unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig is recovered.

3D topological quantum memory with a power-law energy barrier.

We discuss energy barriers and their relationship to self-correcting quantum memories. We introduce the solid code, a 3D version of Kitaev's surface code, and then combine several solid codes using a technique called welding. The resulting code is a [[O(L³),1,O(L(4/3))]] stabilizer code with an energy barrier of O(L(2/3)), which is an exponential improvement over the previous highest energy barrier in 3D. No-go results are avoided by breaking microscopic translation invariance.

Effective quantum-memory Hamiltonian from local two-body interactions

In [Phys. Rev. A 88, 062313 (2013)] we proposed and studied a model for a self-correcting quantum memory in which the energetic cost for introducing a defect in the memory grows without bounds as a function of system size. This positive behavior is due to attractive long-range interactions mediated by a bosonic field to which the memory is coupled. The crucial ingredients for the implementation of such a memory are the physical realization of the bosonic field as well as local five-body interactions between the stabilizer operators of the memory and the bosonic field. Here, we show that both of these ingredients appear in a low-energy effective theory of a Hamiltonian that involves only two-body interactions between neighboring spins. In particular, we consider the low-energy, long-wavelength excitations of an ordered Heisenberg ferromagnet (magnons) as a realization of the bosonic field. Furthermore, we present perturbative gadgets for generating the required five-spin operators. Our Hamiltonian involving only local two-body interactions is thus expected to exhibit self-correcting properties as long as the noise affecting it is in the regime where the effective low-energy description remains valid.

Self-Correcting Quantum Memories Beyond the Percolation Threshold

We analyze several high dimensional generalizations of the toric code at a nonzero temperature. We find that in a large enough dimension, there can be a distinct separation between the critical temperature Tc, given by thermodynamic singularities, and the percolation temperature Tp, given by the percolation of defects. We argue that the regime Tp<T<Tc is a range of temperatures where a self-correcting quantum memory can operate despite having percolating defects. We present analytic arguments and numerical evidence in support of this scenario, including a mean-field treatment and Monte Carlo simulations. Near Tc, simulations observe a large hysteretic behavior, which may have applications by allowing the self-correcting phase to survive in a "superheated" regime.

Topological phases and self-correcting memories in interacting anyon systems

Recent studies have shown that topological models with interacting anyonic quasiparticles can be used as self-correcting quantum memories. Here we study the behavior of these models at thermal equilibrium. It is found that the interactions allow topological order to exist at finite temperature, not only in an extension of the ground-state phase, but also in a novel form of topologically ordered phase. Both phases are found to support self-correction in all models considered, and the transition between them corresponds to a change in the scaling of memory lifetime with system size.

Quantum Self-Correction in the 3D Cubic Code Model

A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least L(cβ), where L is the lattice size, β is the inverse temperature of the bath, and c>0 is a constant coefficient. However, this bound applies only if the lattice size L does not exceed a critical value which grows exponentially with β. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.

Area laws in a many-body localized state and its implications for topological order

The question whether Anderson insulators can persist to finite-strength interactions—a scenario dubbed many-body localization—has recently received a great deal of interest. The origin of such a many-body localized phase has been described as localization in Fock space, a picture we examine numerically. We then formulate a precise sense in which a single energy eigenstate of a Hamiltonian can be adiabatically connected to a state of a non-interacting Anderson insulator. We call such a state a many-body localized state and define a many-body localized phase as one in which almost all states are many-body localized states. We explore the possible consequences of this; the most striking is an area law for the entanglement entropy of almost all excited states in a many-body localized phase. We present the results of numerical calculations for a one-dimensional system of spinless fermions. Our results are consistent with an area law and, by implication, many-body localization for almost all states and almost all regions for weak enough interactions and strong disorder. However, there are rare regions and rare states with much larger entanglement entropies. Furthermore, we study the implications that many-body localization may have for topological phases and self-correcting quantum memories. We find that there are scenarios in which many-body localization can help to stabilize topological order at non-zero energy density, and we propose potentially useful criteria to confirm these scenarios.

Self-correcting quantum computers

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.