High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory

TL;DR

The paper introduces quantum radial codes (QRCs) built from the lifted product of classical radial codes, achieving high encoding rates with distance scaling and enabling single-shot decoding under circuit-level noise. It develops both the classical radial code construction and a general quantum lifted-product framework, establishing stabilizer weight, logical operator structure, and a tunable parameter set with a theoretical distance bound and empirical distance scaling. Through circuit-level simulations, the authors demonstrate competitive error suppression relative to surface codes of similar distance while using far fewer physical qubits, and they propose an efficient, parallelizable stabilizer measurement schedule enabling single-shot-like decoding with an overlapping-window BP+OSD decoder. The work highlights practical potential for near-term quantum devices and outlines open questions on decoding, circuit design, and broader applicability of lifted-product constructions.

Abstract

We present a new family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasi-cyclic codes. The codes are defined using a pair of integers $(r,s)$ and have parameters $[\![2r^2s,2(r-1)^2,\leq2s]\!]$, with numerical studies suggesting average-case distance linear in $s$. In simulations of circuit-level noise, we observe comparable error suppression to surface codes of similar distance while using approximately five times fewer physical qubits. This is true even when radial codes are decoded using a single-shot approach, which can allow for faster logical clock speeds and reduced decoding complexity. We describe an intuitive visual representation, canonical basis of logical operators and optimal-length stabiliser measurement circuits for these codes, and argue that their error correction capabilities, tunable parameters and small size make them promising candidates for implementation on near-term quantum devices.

Figures (10)

• Figure 1: Radial layouts for two examples of $(r,s)=(2,3)$ classical codes with parity check matrices given in \ref{['eq:classical_example_a']} and \ref{['eq:classical_example_b']} respectively. Bits/checks are shown as circles/squares. Rings (and ring indices) are shown in red. Spokes (and spoke indices) are shown in blue.
• Figure 2: Structure of the QRC with PCMs \ref{['eq:example_hx']} and \ref{['eq:example_hz']}. The code is made of four $(r,s) = (2,3)$ classical radial codes, with two (red and blue) providing $Z$ stabilisers and two (orange and green) providing $X$ stabilisers. An example of a $X$ stabiliser from ring $1$ of the orange code (supported on a qubit from each ring of the orange code and a ring $1$ qubit in each of the red and blue codes) is also shown.
• Figure 3: Embedding of the radial code described by \ref{['eq:example_hx']} and \ref{['eq:example_hz']} onto a twisted $2$-torus, where it is equivalent to a surface code. Qubits are circles with colouring showing which of the classical radial codes in \ref{['fig:small_stack']} they belong to. $X$ and $Z$ stabilisers are square faces.
• Figure 4: Representation of the structure of a quantum radial code. The code is formed from $r$ copies of a classical radial code $H_1$ (the $X$ codes) and $r$ copies of a classical radial code $H_2^*$ (the $Z$ codes). Each stabiliser is associated with a ring $u$ check in one of these classical codes and has support on a qubit from each ring of that code and a ring $u$ qubit from each code of the other type ($u=0$ in the example above)
• Figure 5: Numerical estimates of average distances for QRCs of different $s$ for (a) $r=3$ and (b) $r=4$. 100 codes were generated for each pair $(r,s)$. Error bars are the standard error on the mean and are typically smaller than the marker.