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Channel-resolved memory time of Haah's cubic code under reversible rate filtering and temporally correlated baths

Hikaru Wakaura

Published Oct 12, 2026• 0 citations
Publisher

Abstract

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 $\beta=4.5$--$5$ 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 $10^{3}$ 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.

DOI
10.21203/rs.3.rs-11241681/v1

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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\beta=4.5β=4.5--555 for up to 137181371813718 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 10310^{3}103 lengthens the memory time 808080- to 120120120-fold for L=7L=7L=7--131313, 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=13L=13L=13 and 171717 and falls by 39%39\%39% at L=19L=19L=19 while the filtered bath still grows to L=17L=17L=17 and gains a factor 454545 at L=19L=19L=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%12\%12% energy rescaling raises the size exponent from 2.72.72.7 to 4.34.34.3 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.
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.
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.
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.
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.
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.
Unrecorded or ``forgetful'' measurements -- physically, a local dephasing channel -- can arise naturally as syndrome-measurement errors in fault-tolerant protocols and as uncontrolled decoherence in open systems. We numerically study random Clifford circuits in which recorded projective ZZZ measurements compete with unrecorded (forgotten) ZZZ measurements, focusing on the dynamical, finite-depth regime relevant to decoding and quantum memory. In the forget-only limit the global entropy density thermalizes at a rate independent of system size, and the forgetting rate at which thermalization sets in decays with circuit depth as a power law pf∗∼dvp_f^{\ast}\sim d^{v}pf∗​∼dv, consistent with the critical-depth scaling inversely with the noise rate found for related noisy-circuit quantities. With recorded measurements present, the mutual information I(A:B)I(A{:}B)I(A:B) -- which quantifies the recoverable correlation between the two halves -- develops a finite-depth peak whose height grows sub-extensively with NNN: its effective scaling exponent α(pf)α(p_f)α(pf​) decreases monotonically, so that forgetting drives the recoverable correlation from near-volume-law toward area-law scaling. We map the recoverability landscape in the (pm,pf)(p_m,p_f)(pm​,pf​) plane and show that forgetting destroys the measurement-induced purification transition. These results complement earlier stat-mech analyses of noisy monitored circuits and quantify the competition between measurement and forgetting relevant to quantum error correction.