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arXiv:2402.01532

The closed-branch decoder for quantum LDPC codes

Antonio deMarti iOlius, Josu Etxezarreta Martínez

Published Feb 2, 2024• 0 citations
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Abstract

Quantum error correction is the building block for constructing fault-tolerant quantum processors that can operate reliably even if its constituting elements are corrupted by decoherence. In this context, real-time decoding is a necessity for implementing arbitrary quantum computations on the logical level. In this work, we present a new decoder for Quantum Low Density Parity Check (QLDPC) codes, named the closed-branch decoder, with a worst-case complexity loosely upper bounded by $\mathcal{O}(n\text{max}_{\text{gr}}\text{max}_{\text{br}})$, where $\text{max}_{\text{gr}}$ and $\text{max}_{\text{br}}$ are tunable parameters that pose the accuracy versus speed trade-off of decoding algorithms. For the best precision, the $\text{max}_{\text{gr}}\text{max}_{\text{br}}$ product increases exponentially as $\propto dj^d$, where $d$ indicates the distance of the code and $j$ indicates the average row weight of its parity check matrix. Nevertheless, we numerically show that considering small values that are polynomials of the code distance are enough for good error correction performance. The decoder is described to great extent and compared with the Belief Propagation Ordered Statistics Decoder (BPOSD) operating over data qubit, phenomenological and circuit-level noise models for the class of Bivariate Bicycle (BB) codes. The results showcase a promising performance of the decoder, obtaining similar results with much lower complexity than BPOSD when considering the smallest distance codes, but experiencing some logical error probability degradation for the larger ones. Ultimately, the performance and complexity of the decoder depends on the product $\text{max}_{\text{gr}}\text{max}_{\text{br}}$, which can be considered taking into account benefiting one of the two aspects at the expense of the other.

DOI
10.48550/arxiv.2402.01532

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Quantum error correction is the building block for constructing fault-tolerant quantum processors that can operate reliably even if its constituting elements are corrupted by decoherence. In this context, real-time decoding is a necessity for implementing arbitrary quantum computations on the logical level. In this work, we present a new decoder for Quantum Low Density Parity Check (QLDPC) codes, named the closed-branch decoder, with a worst-case complexity loosely upper bounded by O(nmaxgrmaxbr)\mathcal{O}(n\text{max}_{\text{gr}}\text{max}_{\text{br}})O(nmaxgr​maxbr​), where maxgr\text{max}_{\text{gr}}maxgr​ and maxbr\text{max}_{\text{br}}maxbr​ are tunable parameters that pose the accuracy versus speed trade-off of decoding algorithms. For the best precision, the maxgrmaxbr\text{max}_{\text{gr}}\text{max}_{\text{br}}maxgr​maxbr​ product increases exponentially as ∝djd\propto dj^d∝djd, where ddd indicates the distance of the code and jjj indicates the average row weight of its parity check matrix. Nevertheless, we numerically show that considering small values that are polynomials of the code distance are enough for good error correction performance. The decoder is described to great extent and compared with the Belief Propagation Ordered Statistics Decoder (BPOSD) operating over data qubit, phenomenological and circuit-level noise models for the class of Bivariate Bicycle (BB) codes. The results showcase a promising performance of the decoder, obtaining similar results with much lower complexity than BPOSD when considering the smallest distance codes, but experiencing some logical error probability degradation for the larger ones. Ultimately, the performance and complexity of the decoder depends on the product maxgrmaxbr\text{max}_{\text{gr}}\text{max}_{\text{br}}maxgr​maxbr​, which can be considered taking into account benefiting one of the two aspects at the expense of the other.
In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters [[144,12,12]][[144,12,12]][[144,12,12]], our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks (w=6,8w=6,8w=6,8). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.
We propose a decoder for quantum low density parity check (LDPC) codes based on a beam search heuristic guided by belief propagation (BP). Our beam search decoder applies to all quantum LDPC codes and achieves different speed-accuracy tradeoffs by tuning its parameters such as the beam width. We perform numerical simulations under circuit level noise for the [[144,12,12]][[144, 12, 12]][[144,12,12]] bivariate bicycle (BB) code at noise rate p=10−3p=10^{-3}p=10−3 to estimate the logical error rate and the 99.9 percentile runtime and we compare with the BP-OSD decoder which has been the default quantum LDPC decoder for the past six years. A variant of our beam search decoder with a beam width of 64 achieves a 17×17\times17× reduction in logical error rate. With a beam width of 8, we reach the same logical error rate as BP-OSD with a 26.2×26.2\times26.2× reduction in the 99.9 percentile runtime. We identify the beam search decoder with beam width of 32 as a promising candidate for trapped ion architectures because it achieves a 5.6×5.6\times5.6× reduction in logical error rate with a 99.9 percentile runtime per syndrome extraction round below 1ms at p=5×10−4p=5 \times10^{-4}p=5×10−4. Remarkably, this is achieved in software on a single core, without any parallelization or specialized hardware (FPGA, ASIC), suggesting one might only need three 32-core CPUs to decode a trapped ion quantum computer with 1000 logical qubits.
We propose a decoder for quantum low-density parity-check (LDPC) codes based on a beam search heuristic guided by belief propagation (BP). Our beam search decoder applies to all quantum LDPC codes and achieves different speed-accuracy tradeoffs by tuning its parameters such as the beam width. We perform numerical simulations under circuit level noise for the [ [ 144 , 12 , 12 ] ] bivariate bicycle (BB) code at noise rate p = 10 − 3 to estimate the logical error rate and the 99.9 percentile runtime and we compare with the BP-OSD decoder which has been the default quantum LDPC decoder for the past six years. A variant of our beam search decoder with a beam width of 64 achieves a 17 × reduction in logical error rate. With a beam width of 8, we reach the same logical error rate as BP-OSD with a 26.2 × reduction in the 99.9 percentile runtime. We identify the beam search decoder with beam width of 32 as a promising candidate for trapped ion architectures because it achieves a 5.6 × reduction in logical error rate with a 99.9 percentile runtime per syndrome extraction round below 1 ms at p = 5 × 10 − 4 . Remarkably, this is achieved in software on a single core, without any parallelization or specialized hardware (FPGA, ASIC), suggesting one might only need three 32-core CPUs to decode a trapped ion quantum computer with 1000 logical qubits.
Accurate decoding of quantum low-density parity-check (qLDPC) codes often relies on expensive post-processing search, although decoding difficulty varies strongly between syndromes. We find that the benefit of deeper post-processing search is highly concentrated in a small subset of decoding instances, and that these instances can be identified directly from the decoder itself. To this end, we introduce an adaptive decoder based on belief propagation (BP) and ordered-statistics decoding (OSD), using the disagreement between the BP hard decision and the syndrome-consistent order-zero OSD solution as an internal risk signal to determine where deeper search is required. On a [[144,12,12]][[144,12,12]][[144,12,12]] bivariate bicycle code under circuit-level depolarising noise, encoding 121212 logical qubits in 144144144 data qubits with distance 121212, escalating only the highest-risk 20%20\%20% of instances recovers 85%85\%85% to 92%92\%92% of the improvement in logical error rate available from a full one-free-variable sweep, while reducing the mean serial decoding cost by a factor of 3.63.63.6 relative to applying the same truncated search to every instance. At matched budget, disagreement-guided routing also outperforms routing based on the BP syndrome residual, syndrome weight and random selection. The same behaviour reappears on a structurally distinct radial qLDPC code obtained from a lifted-product construction, where escalating 20%20\%20% of instances recovers 87%87\%87% to 94%94\%94% of the available gain. A ZZZ-memory experiment on the Quantinuum H2 trapped-ion processor, implementing both XXX-type and ZZZ-type checks, further shows that the signal remains predictive under device noise. These results show that decoder-internal disagreement can expose where additional decoding effort is valuable, allowing classical computation to be concentrated on the instances most likely to benefit from it.
We introduce Decision Tree Decoders (DTDs), which rely only on the sparsity of the binary check matrix, making them broadly applicable for decoding any quantum low-density parity-check (qLDPC) code and fault-tolerant quantum circuits. DTDs construct corrections incrementally by adding faults one-by-one, forming a path through a Decision Tree (DT). Each DTD algorithm is defined by its strategy for exploring the tree, with well-designed algorithms typically needing to explore only a small portion before finding a correction. We propose two explicit DTD algorithms that can be applied to any qLDPC code: (1) A provable decoder: Guaranteed to find a minimum-weight correction. While it can be slow in the worst case, numerical results show surprisingly fast median-case runtime, exploring only www DT nodes to find a correction for weight-www errors in notable qLDPC codes, such as bivariate bicycle and color codes. This decoder may be useful for ensemble decoding and determining provable code distances, and can be adapted to compute all minimum-weight logical operators of a code. (2) A heuristic decoder: Achieves higher accuracy and faster performance than BP-OSD on the gross code with circuit noise in realistic parameter regimes.
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.