Cored product codes for quantum self-correction in three dimensions

TL;DR

This work tackles the challenge of achieving self-correcting quantum memories in three spatial dimensions by abandoning translational symmetry and introducing disordered cored product codes built from slead classical codes. The construction starts with a 4D hypergraph product of pinwheel-based slead factors and applies a coring procedure to embed the code in 3D while preserving CSS structure, distance, and a favorable energy landscape. Finite-temperature simulations indicate a finite-temperature threshold below which memory lifetime grows with system size for codes up to around 60,000 qubits, with lifetimes following a stretched-exponential scaling in code size and evidence of algebraic energy barriers that suppress entropic degradation. Together, these results support a pathway to passive, self-correcting quantum memories in three dimensions via disorder, offering a framework for further exploration of disordered LDPC codes with geometric locality and tunable properties.

Abstract

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.

Figures (21)

• Figure 1: Three dimensional self-correcting quantum memory via cored product codes.(a) Our proposed code construction procedure. A four-dimensional quantum code is produced from the hypergraph product of a pair of two-dimensional classical "slead" codes. Our main construction utilizes a pair of so-called classical pinwheel codes Tan2024Fracton. The four-dimensional quantum code is then reduced to three dimensions via a coring operation. (b) Depicts the schematic setting for a passive quantum memory. In particular, the passive operation of a self-correcting quantum memory entails storing logical information in a quantum code at a sufficiently low temperature. The code is subject to thermal noise at this temperature and decoding is performed once at readout to retrieve the logical information. (c) A self-correcting quantum memory ideally exhibits stretched exponential growth (power $\alpha > 0$) in memory lifetime with code size at sufficiently low temperatures.
• Figure 2: Example of a slead code. The code shown has local Newman--Moore check $f=1+x+y$, up to truncation. In this figure bits and checks are placed on the faces of the square lattice drawn in black, on the two-dimensional interval $[0,8]^2$. The equivalent slead is drawn in blue, with labels indicating the level $p$ of each vertex in the topological ordering. The single source vertex at $p=0$ is shaded green and the single sink vertex at $p=14$ is shaded red. The self-loop on the source vertex is removed, and the induced codeword is indicated in gray on both the physical system and the slead.
• Figure 3: Cancellation in algebraic codes. Panels (a)--(h) show a classical walk implementing the codeword in a small slead code described by a polynomial $f$ in the bulk. A filled vertex indicates a spin flip, and a red boundary indicates a violated local check. Cancellation occurs at levels $p=2^a$, $a \in \mathbb{Z}$, and despite the introduction of boundaries the walk achieves a logarithmic barrier. Note that the number of spin flips at both levels $p=1,2$ in the codeword is only $|f|-1=3$.
• Figure 4: Worldline implementations in unstructured codes. A slead code is shown, oriented such that the graph topological ordering labels $p$ correspond to the vertical axis. We perform check depletion on an indicated vertex at level $p_\mathrm{dep} = p_\mathrm{max} - \tau$, and the resulting lightcone is indicated by the dashed line. Due to geometric locality, the width of the lightcone scales as $\tau^{D-1}$. A partial worldline implementation of the codeword is indicated, with excitations generated along the thick boundary between the shaded (implemented spin flips along the worldline) and unshaded (no spin flips) regions. This is a coarse-grained counterpart to the middle panels of Fig. \ref{['fig:ti_barrier']}.
• Figure 5: Energy barriers in classical fractal codes. Numerical upper bounds on energy barrier versus code size in two-dimensional classical fractal codes, obtained by greedy search over minimal Pauli walks. As a contrasting example, the Newman-Moore code with periodic boundary conditions (red) exhibits logarithmic barrier with code size. The pinwheel codes (purple) discussed in Sec. \ref{['sec:pinwheel_slead_codes']}, and another family of fractal codes that are translation-invariant in the bulk but with open boundaries (brown), exhibit algebraic scaling of energy barrier $\mathcal{E} \sim L^\eta$ for $\eta > 0$.