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 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.

Physical Review LettersJul 29, 20260 citations

Layer Codes as Partially Self-Correcting Quantum Memories

Shouzhen Gu, Libor Caha, Shin Ho Choe +3 more

We investigate layer codes, a family of three-dimensional stabilizer codes that achieves optimal scaling of code parameters and energy barrier, as candidates for self-correcting quantum memories. First, we introduce two decoding algorithms for layer codes with provable guarantees for local stochastic and adversarial noise, respectively. We then prove that layer codes constitute partially self-correcting quantum memories which outperform previously analyzed models such as the cubic and welded solid codes. Notably, we argue that partial self-correction without the requirement of efficient decoding is more common than expected, as it arises solely from a diverging energy barrier. This draws a sharp distinction between partially self-correcting systems and partially self-correcting memories. Another novel aspect of our work is an analysis of layer codes constructed from random Calderbank-Shor-Steane codes. We show that these random layer codes have optimal scaling (up to logarithmic corrections) of code parameters and energy barrier. Finally, we present numerical studies of their memory times and report behavior consistent with partial self-correction.We investigate layer codes, a family of three-dimensional stabilizer codes that achieves optimal scaling of code parameters and energy barrier, as candidates for self-correcting quantum memories. First, we introduce two decoding algorithms for layer codes with provable guarantees for local stochastic and adversarial noise, respectively. We then prove that layer codes constitute partially self-correcting quantum memories which outperform previously analyzed models such as the cubic and welded solid codes. Notably, we argue that partial self-correction without the requirement of efficient decoding is more common than expected, as it arises solely from a diverging energy barrier. This draws a sharp distinction between partially self-correcting systems and partially self-correcting memories. Another novel aspect of our work is an analysis of layer codes constructed from random Calderbank-Shor-Steane codes. We show that these random layer codes have optimal scaling (up to logarithmic corrections) of code parameters and energy barrier. Finally, we present numerical studies of their memory times and report behavior consistent with partial self-correction.

arXiv:2510.06659Physical Review LettersApr 9, 20261 citation

Layer codes

Dominic J. Williamson, Nouédyn Baspin

Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions. Quantum error correcting codes are inefficient when implemented on platforms with local connectivity in two dimensions, which motivates platforms with 3D connectivity and codes applicable in these conditions. Here, the authors show how to transform any CSS stabilizer code into a 3D topological code, leading to codes with optimal asymptotic scaling in 3D.Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions. Quantum error correcting codes are inefficient when implemented on platforms with local connectivity in two dimensions, which motivates platforms with 3D connectivity and codes applicable in these conditions. Here, the authors show how to transform any CSS stabilizer code into a 3D topological code, leading to codes with optimal asymptotic scaling in 3D.

arXiv:2309.16503Nature CommunicationsNov 4, 20249 citations

Self-Correcting 3D Surface Code with Growing Local Dimension

Jinkang Guo

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.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.

quant-pharXiv:2610.35099Oct 27, 20260 citations

Quantum Self-Correction in the 3D Cubic Code Model

Sergey B. Bravyi, Jeongwan Haah

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.

arXiv:1112.3252Physical Review LettersNov 12, 2013292 citations

Layer codes

Dominic J. Williamson, Nouédyn Baspin

Abstract Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.Abstract Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.

Jan 1, 20240 citations

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.