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.
Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension
3−ε and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.
We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight
X- and
Z-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over
Zq for arbitrary
q, using joint and separate Lee metrics for the
X- and
Z-components of errors. For qubits, the joint metric counts
Y errors twice and can yield codes that detect and correct
X and
Z errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over
Z4 via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on
n qudits, we prove that the Lee distance cannot exceed
O(n), uniformly in
q, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-
q qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical
q-ary ``helical repetition codes'' have exponentially long memory times at fixed temperature for sufficiently large
q (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for
Z errors, but self-correction for
X errors remains an open question.
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.
We construct the first fully spatially local fault-tolerant quantum computer based on topological codes in fewer than four spatial dimensions. Our construction is a two-dimensional architecture that uses only geometrically local quantum and classical operations, bounded-speed classical communication and computation, and a constant density of quantum and classical resources. The core component is a new time-translation-invariant cellular-automaton decoder for the surface code. This decoder preserves logical information for a time stretched-exponential in the code distance and operates continuously during state injection, stabilizer-state preparation, lattice surgery, and transversal readout. We also prove that every translation-invariant topological Pauli stabilizer code is locally decodable under phenomenological noise.
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.