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.
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.
Haah's 3D cubic code was introduced as a candidate for a self-correcting quantum memory in three dimensions, using the absence of string logical operators to obstruct thermal errors. It has been a long-standing open question whether it is a self-correcting quantum memory. Even though the memory time was explored numerically, no analytical upper bound was known. Here, we settle this question rigorously and show that the cubic code thermalizes rapidly at every positive temperature. Our proof relies on the framework of [Commun. Math. Phys. 407, 195 (2026)] to turn a decay-of-correlations condition into a modified logarithmic Sobolev inequality for the Davies generator. To the best of our knowledge, this is the first analytic proof of rapid mixing for all positive temperatures for any fracton code.
We construct self-correcting 3D surface codes over Zq 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±1 and Z±1 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[Ω(N1/3)]. 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.
The storage of large-scale quantum information at finite temperature requires an autonomous and reliable quantum hard drive, also known as a self-correcting quantum memory. It is a long-standing open problem to find a self-correcting quantum memory in three dimensions. The recently introduced layer codes achieve the best possible scaling of code parameters and logical energy barrier in three dimensions; these are tantalizing features for the purposes of self-correction. In this Letter, we show that a family of layer codes based on good quantum Tanner codes exhibit partial self-correction. Their memory time grows exponentially with linear system size, up to a length scale that is exponential in the inverse temperature. At this length scale, the memory time scales as a double exponential of inverse temperature. To establish this result, we introduce a concatenated matching decoder that combines three rounds of parallelized minimum-weight perfect matching with a decoder for good quantum Tanner codes. We show that our decoder corrects errors up to a constant fraction of the energy barrier, and a constant fraction of the code distance, for a family of layer codes. Our results position layer codes as the leading candidate for a partially self-correcting memory in three dimensions in the asymptotically large system size regime. While they fall short of achieving strict self-correction in the thermodynamic limit, our Letter highlights the potential of these local codes in three dimensions, with fast distance and logical qubit growth, fast decoders, and a long memory time over a wide range of parameters. In particular, recent work by Gu et al. reports promising numerical results for the memory time of small explicit instances of layer codes based on random input codes.
The storage of large-scale quantum information at finite temperature requires an autonomous and reliable quantum hard drive, also known as a self-correcting quantum memory. It is a long-standing open problem to find a self-correcting quantum memory in three dimensions. The recently introduced Layer Codes achieve the best possible scaling of code parameters and logical energy barrier in three dimensions, these are tantalizing features for the purposes of self-correction. In this work we show that a family of Layer Codes, based on good Quantum Tanner Codes, exhibit partial self-correction. Their memory time grows exponentially with linear system size, up to a length scale that is exponential in the inverse temperature. At this length scale, the memory time scales as a double exponential of inverse temperature. To establish this result we introduce a concatenated matching decoder that combines three rounds of parallelized minimum-weight perfect-matching with a decoder for good Quantum Tanner Codes. We show that our decoder corrects errors up to a constant fraction of the energy barrier, and a constant fraction of the code distance, for a family of Layer Codes. Our results position Layer Codes as the leading candidate for a partially self-correcting memory in three dimensions. While they fall short of achieving strict self-correction in the thermodynamic limit, our work highlights the potential of these local codes in three dimensions, with fast distance and logical qubit growth, fast decoders, and a long memory time over a wide range of parameters.
Abstract Self-correcting quantum memories would store logical information without a real-time decoder, and Haah's cubic code is the best studied three-dimensional candidate, but how its memory time depends on the environment beyond a flat Markovian bath has not been mapped. Here we simulate the thermal syndrome dynamics of the cubic code at β=4.5--5 for up to 13718 qubits under detailed-balance rate filters on each of its three energy channels, under an Ornstein--Uhlenbeck modulation that gives the bath a correlation time, and at rescaled code energies. The zero-energy diffusion channel controls the memory time: suppressing it by 103 lengthens the memory time 80- to 120-fold for L=7--13, the gain enters as a prefactor rather than a steeper size law, and it outlives the flat-bath optimum, where the memory time peaks between L=13 and 17 and falls by 39% at L=19 while the filtered bath still grows to L=17 and gains a factor 45 at L=19. The pair-creation channel is invisible over three decades; a bath correlation time adds a factor of two to eleven, consistent with annealed and quenched hazard averages; and a uniform 12% energy rescaling raises the size exponent from 2.7 to 4.3 along the Bravyi--Haah line, which no gapped pseudomode we solve provides. These results form a channel-resolved finite-size benchmark for bath engineering of a passive three-dimensional memory; they establish neither asymptotic scaling nor an advantage over active decoding.
Cubic codes were proposed by Haah as candidates for self-error correction in three dimensions (3D). While these codes are not self-correcting, Bravyi and Haah showed that they are partially self-correcting. In this paper we are interested in generalizations of the cubic code to prime alphabet. Kim initiated the study of such codes over prime alphabet. Haah also proposed a framework based on modules that enables the study of cubic codes over higher alphabet. However, there are many open questions remaining, especially those pertaining to the performance of nonbinary cubic codes. Building on Bravyi and Haah's decoder, we study the performance of nonbinary cubic codes over the quantum erasure and depolarizing channels. This is the first such study of nonbinary cubic codes.