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arXiv:1812.03936 · Physical review. B./Physical review. B

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

David Poulin, Roger G. Melko, Matthew B. Hastings

Published Mar 8, 2019• 9 citations

Abstract

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 two-dimensional Ising model, but that systems with local, i.e., gauge symmetries, have the required properties. Previous work identified a three-dimensional (3D) quantum system which realizes a self-correcting classical memory [E. Dennis et al., J. Math. Phys. 43, 4452 (2002)]. Here, we show that a purely classical 3D system, Wegner's Ising lattice gauge model [F. J. Wegner, J. Math. Phys. 12, 2259 (1971)], 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 [I. S. Tupitsyn et al., Phys. Rev. B 82, 085114 (2010)], except that self-correction extends beyond the topological phase, past the lower critical temperature.

DOI: 10.1103/physrevb.99.094103

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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 two-dimensional Ising model, but that systems with local, i.e., gauge symmetries, have the required properties. Previous work identified a three-dimensional (3D) quantum system which realizes a self-correcting classical memory [E. Dennis et al., J. Math. Phys. 43, 4452 (2002)]. Here, we show that a purely classical 3D system, Wegner's Ising lattice gauge model [F. J. Wegner, J. Math. Phys. 12, 2259 (1971)], 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 [I. S. Tupitsyn et al., Phys. Rev. B 82, 085114 (2010)], except that self-correction extends beyond the topological phase, past the lower critical temperature.

We show that the self-correcting memory in 3d Wegner gauge theory is stable to arbitrary small enough perturbations of the Hamiltonian. Our proof relies on a new method we dub ``gauge averaging'', which gives conditions under which explicitly broken gauge symmetries are effectively restored by fluctuations. These conditions show that the self-correcting memory phase is fluctuation stabilized, with its robustness to perturbations increasing with increasing temperature, up to some

T_c > 0

.

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 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.

The existence of self-correcting quantum memories in three dimensions is a long-standing open question at the interface between quantum computing and many-body physics. We take the perspective that large contributions to the entropy arising from fine-tuned spatial symmetries, including the assumption of an underlying regular lattice, are responsible for fundamental challenges to realizing self-correction. Accordingly, we introduce a class of disordered quantum codes, which we call "cored product codes". These codes are derived from classical factors via the hypergraph product but undergo a coring procedure which allows them to be embedded in a lower number of spatial dimensions while preserving code properties. As a specific example, we focus on a fractal code based on the aperiodic pinwheel tiling as the classical factor and perform finite temperature numerical simulations on the resulting three-dimensional quantum memory. We provide evidence that, below a critical temperature, the memory lifetime increases with system size for codes up to 60000 qubits.

After fifty years of lattice gauge theories (LGTs), the nature of the transition between their topological phases (confinement or deconfinement) remains challenging due to the absence of a local order parameter. In this work, we conduct a percolation analysis of Wegner's three-dimensional Z_{2} lattice gauge model using intensive Monte Carlo simulations and finite-size scaling, offering fresh insights into the topological phase transitions of gauge-invariant systems. We demonstrate that, regardless of the connection rules, geometrical loops, constructed by piercing excited plaquettes percolate precisely at the thermal critical point T_{c}, with critical exponents coinciding with those of the loop representation of the dual 3D Ising model. Further, we construct Fortuin-Kasteleyn (FK) clusters in a random-cluster representation, showing that they also percolate at T_{c}, enabling access to all thermal critical exponents. Strikingly, the Binder cumulants of the percolation order parameters for both loops and FK clusters reveal a pseudo-first-order transition. This work sheds new light on the critical behavior of pure LGTs, with potential implications for condensed matter systems and quantum error correction.

We construct self-correcting 3D surface codes over

\mathbb Z_q

with local dimension

q

growing with system size. The model preserves the local geometry and stabilizer support of the ordinary 3D surface code and remains translation invariant in the bulk, while modifying only the stabilizer Pauli powers. We use the Lee metric to characterize error weight and consider local Metropolis dynamics with single-qudit

X^{\pm1}

and

Z^{\pm1}

updates, each changing the energy by at most a constant independent of

q

and system size

N

. We prove that, below the critical temperature, when the local dimension scales as

q=Θ(N)

, the memory times in both error sectors grow at least as

\exp\left[Ω(N^{1/3})\right]

. Our numerical results further support these analytical results. Together, these results establish growing local Hilbert-space dimension as a new resource for designing self-correcting quantum memories while retaining simple local geometry and local thermal dynamics.