Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes

TL;DR

<3-5 sentences high-level summary describing: the problem of decoding qLDPC codes for fault-tolerant quantum computing; the Ambiguity Clustering (AC) approach which uses Belief Propagation to inform staged, cluster-based decoding; the key results showing AC achieves up to $27\times$ speedups over BP-OSD with matched accuracy (under circuit-level depolarising noise of $0.3\%$) and decodes the $144$-qubit Gross code in $135\,\mu\text{s}$ per syndrome round on a single M2 CPU; the significance is real-time decoding practicality for non-surface qLDPC codes on near-term hardware.

Abstract

Error correction allows a quantum computer to preserve states long beyond the decoherence time of its physical qubits. Key to any scheme of error correction is the decoding algorithm, which estimates the error state of qubits from the results of syndrome measurements. The leading proposal for quantum error correction, the surface code, has fast and accurate decoders, but several recently proposed quantum low-density parity check (qLDPC) codes allow more logical information to be encoded in significantly fewer physical qubits. The state-of-the-art decoder for general qLDPC codes, BP-OSD, has a cheap Belief Propagation stage, followed by linear algebra and search stages which can each be slow in practice. We introduce the Ambiguity Clustering decoder (AC) which, after the Belief Propagation stage, divides the measurement data into clusters that can be decoded independently. We benchmark AC on the recently proposed bivariate bicycle qLDPC codes and find that, with 0.3% circuit-level depolarising noise, AC is up to 27x faster than BP-OSD with matched accuracy. Our implementation of AC decodes the 144-qubit Gross code in 135us per round of syndrome extraction on an M2 CPU, already fast enough to keep up with neutral atom and trapped ion systems.

Figures (7)

• Figure 1: The decoding problem and main innovation of AC.a. A Tanner graph for the 5-bit repetition code. Check nodes (squares) are each connected to an adjacent pair of error nodes (circles), representing the condition that adjacent bits of the codewords $00000$ and $11111$ agree. An unknown error pattern (shaded circles) has an observable syndrome (shaded squares) consisting of the checks adjacent to an odd number of errors. Below is the associated parity check matrix $H$. The syndrome $\sigma$ is the mod 2 sum of the columns corresponding to the unknown error. Any $e$ satisfying $He = \sigma$ is an explanation of the observed syndrome. b, c. A different choice of Tanner graph and parity check matrix. The checks represent the conditions that each bit of a codeword should agree with the final bit. It is particularly easy to read off the sets of columns summing to the observed syndrome: columns 2 and 3 (b), or column 5 together with columns 1 and 4 (c), corresponding to $e=(0,1,1,0,0)$ and $e=(1,0,0,1,1)$. d. An arbitrary parity check matrix (left) can be converted to a similar special form (right) by Gaussian elimination. There are typically a great many combinations of columns to search through. BP-OSD streamlines this search by reordering the columns according to the output of BP, placing columns more likely to represent an error further to the left, before performing Gaussian elimination. High accuracy can then be obtained by searching over only small sets of columns from the right-hand side of the matrix. e. A partial block structure, covering the rows where the syndrome is non-zero, which would be useful for decoding. If we are confident that errors corresponding to the rightmost block of columns did not occur, then we can search over explanations for each section of the syndrome separately. The space explored by this search can be exponentially larger, in the number of blocks, than a search which does the same amount of work without regard to the block structure. f. Blocks correspond to isolated clusters in an updated Tanner graph. The key idea of AC is that, by careful manipulation of the parity check matrix, this block structure can usually be obtained.
• Figure 2: The split belief problem. The syndrome marked in red has two explanations a, b. If all four errors (circular nodes) have the same prior probability, then their posterior probabilities are identical (in fact, equal to $1/2$) (c). It is impossible to retrieve a solution to \ref{['eqn:error-to-syndrome']} by applying a uniform threshold to the posteriors.
• Figure 3: Comparison of BP-OSD and AC. Top. Logical error rates for quantum memory using the bivariate bicycle codes, labelled by code parameters $[[n,k,d]]$ (the number of physical qubits, logical qubits, and the distance of the code). Syndromes were extracted for $d$ rounds; logical error and timing data are per round of syndrome extraction. Dashed lines are the BP-OSD-CS(7) (see Section \ref{['sec:BPOSD']}) data from IBMCodes. For each data point, the AC parameter $K$ (Section \ref{['sec:ac-stage-2']}) has been set to match or exceed the accuracy of BP-OSD. The shaded region indicates one (binomial) standard deviation of error. Only experiments for which we observed at least 5 failures are shown. The [[108,8,10]] code is omitted for clarity as it has similar performance to the [[90,8,10]] code. Bottom. Decoding time per round of syndrome extraction at $p = 0.3\%$ ($p=0.35\%$ for the [[288,12,18]] code).
• Figure 4: A pivot operation.a. A pivot $ij$ = $11$ (yellow) is chosen. b. The pivot row is added to every other row with a $1$ in the pivot column. c. The pivot column is now in reduced form. Further pivot operations in other rows will no longer change this column, so it will remain in reduced form. The syndrome vector on the right is modified in the same way to ensure the solutions (one of which is shown in grey) to the linear system are unchanged.
• Figure 5: Evolution of the decoding problem through AC stages 1 and 2. a. A general, unstructured, parity check matrix $H$. The syndrome $\sigma$ is shown as a column vector on its right. b. The associated Tanner graph. Checks with a $1$ in the syndrome are marked in red. c. After stage 1, $H$ is reduced with respect to the syndrome (Definition \ref{['def:pcm-solved-form']}): an identity block covers the rows in which the syndrome has a $1$. In general, the rows and columns of the identity 'block' may be scattered throughout the matrix. Because a pivot row can still be involved in later pivot operations, there might (as in this example) be rows in the identity block without a $1$ in the syndrome. d. In the Tanner graph, being reduced with respect to $\sigma$ means that every marked check has an associated error node adjacent only to that check. e. After stage 2, an expanded set of blocks covers the syndrome. The blocks $C_i$ are shown as coloured rectangles. Each consists of an $m_i \times m_i$ identity matrix $I_i$, and optionally some linearly dependent columns in the $m_i \times (n_i - m_i)$ matrix $B_i$. Again, in practice these rows and columns may be scattered throughout the matrix. f. In the Tanner graph, each cluster consists of at least one check node--error node pair, indicated with an arrow, and optionally some additional errors adjacent only to those checks 'gluing them together' into a cluster. Every marked check is part of some cluster.

Theorems & Definitions (6)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Definition 3.1
• Theorem B.1
• proof