Improved belief propagation is sufficient for real-time decoding of quantum memory

Message passing

• Check-to-error messages encode syndrome-consistent beliefs using a sign determined by κi,j(t) (−1)σi and a magnitude set by the least confident contributing message.The message µi→j(t) represents check i’s belief that error j occurred based on messages from its other neighboring error nodes.Source: DMem-BP— · p. 2
• Error-to-check messages combine beliefs from other check nodes with a time-dependent bias term Λj(t).The bias term shifts messages received by error node j and is updated in DMem-BP rather than remaining fixed as in standard BP.Source: DMem-BP— · p. 3

Initialization and stopping

• DMem-BP initializes beliefs and biases from error-prior log-likelihoods, while initial marginals may be supplied separately.After each message-passing iteration, the algorithm computes updated marginals and a hard decision for each error node.Source: DMem-BP— · p. 3
• The algorithm returns a hard decision when it satisfies Hê(t) = σ; otherwise it continues until the maximum iteration count T, after which it is unsuccessful.This stopping rule makes convergence depend on finding a parity-check-consistent error estimate within the available iterations.Source: DMem-BP— · p. 3

Memory strengths

• DMem-BP parameterizes decoding by real-valued, node-specific memory strengths Γ = {γj}j∈[N].Setting all γj to zero recovers standard BP, while a constant value between zero and one recovers Mem-BP.Source: DMem-BP— · p. 3

Relay integration

• DMem-BP instances can run in parallel, but sequentially connecting them into a relay ensemble typically improves performance.Each DMem-BP instance is treated as one leg of Relay-BP.Source: Relay-BP-S— · p. 3

Relay specification

• Relay-BP-S is specified by the number of solutions sought S, maximum relay legs R, per-leg iteration limits Tr, and memory strengths Γr.These parameters define both the relay budget and the memory-strength schedule across legs.Source: Relay-BP-S— · p. 3

Relay initialization

• The first relay leg applies DMem-BP using initial marginals and memory strengths derived from the error priors.The initialization is written as Mj(0) = log 1−pj in the supplied description.Source: Relay-BP-S— · p. 3

Relay execution

• Each later leg initializes its marginals with the previous leg’s final marginals, passing information forward through the relay.A leg stops after finding a solution or reaching its iteration limit Tr.Source: Relay-BP-S— · p. 3

Selection and stopping

• The algorithm stops after R legs or S solutions and returns the lowest-weight solution found among all legs.The solution weight is defined as w(ê) = P pj in the supplied passage.Source: Relay-BP-S— · p. 3

Evaluation

• The experiments evaluate Relay-BP under circuit-level noise on two bivariate-bicycle codes and a distance-11 rotated surface code.The examples use CSS-type circuits and noisy QEC cycles followed by one perfect cycle.Source: Decoding examples— · p. 3

Decoding strategies

• XYZ-decoding directly computes a correction from H and σ, whereas XZ-decoding independently decodes σX and σZ before combining the partial corrections.The decomposed strategy uses derived check matrices HX and HZ.Source: Decoding examples— · p. 3
• XZ-decoding simplifies computation with smaller decoding objects but may degrade performance by modeling X and Z errors as independent.The supplied evaluation description states that XZ-decoding is used throughout unless otherwise indicated.Source: Decoding examples— · p. 3

Simulation configuration

• Relay-BP uses Tr = 60 iterations per leg except the first, which uses T0 = 80 because initial convergence is slower.The iteration settings are shared across the reported simulations.Source: Decoding examples— · p. 3
• The first leg uses γ = 0.125 for the gross and two-gross codes and γ = 0.35 for the surface code.These values were selected from preliminary sweep results.Source: Decoding examples— · p. 3
• Subsequent-leg memory strengths are sampled independently and uniformly from a range centered at γcenter with width γwidth.This produces randomized memory-strength instances after the initial leg.Source: Decoding examples— · p. 3

Memory-strength landscape

• The gross and two-gross code heatmaps show performance hotspots for intervals that include negative memory strengths.A similar but less pronounced hotspot appears for the surface code.Source: Memory strength selection— · p. 3
• The selected memory-strength intervals were obtained through gradient-free optimization and used for subsequent simulations.The heatmaps mark these optimized intervals with circles.Source: Memory strength selection— · p. 3

Decoding performance

• Relay-BP achieves low logical error rates across all three decoding examples.The result is reported for the two bivariate-bicycle codes and the rotated surface-code example evaluated in the study.Source: Flexible decoding— · p. 3

• Relay-BP compares circuit-noise decoding on two BB codes and a distance-11 rotated surface code using circuit-level noise simulations.Source: Flexible decoding— · p. 4
• At p = 3 × 10−3, Relay-BP-5 and XYZ-Relay-BP-5 outperform BP+OSD+CS-10 on the gross code by approximately one and two orders of magnitude, respectively.Source: Flexible decoding— · p. 4
• For the two-gross code at p = 3 × 10−3, Relay-BP-5 shows even larger improvements over BP+OSD+CS-10.Source: Flexible decoding— · p. 4
• XYZ-Relay-BP-5 outperforms Relay-BP-5 and performs comparably to Matching on the surface code.Source: Flexible decoding— · p. 4
• Relay ensembling improves both convergence rate and solution quality compared with independently restarting each leg from the original priors.On the gross code, the relay version reaches (7 ± 1) × 10−6 in 330.8 ± 0.5 iterations, versus (1.4 ± 0.2) × 10−5 in 578 ± 2 iterations for independent ensembling.Source: Flexible decoding— · p. 4

• Relay-BP-1 achieves error rates orders of magnitude lower than BP and Mem-BP within 30 iterations, and outperforms high-resource BP+OSD+CS-10 by a factor of three.Source: Real-time decoding— · p. 4
• Within the estimated 600-iteration real-time budget, Relay-BP achieves substantially lower error rates than other decoders.Source: Figure 3 · p. 5
• At 300 iterations, Relay-BP-9 achieves a twofold improvement over Relay-BP-1 and nearly matches Relay-BP-100, indicating diminishing returns beyond S = 9 in this scenario.Source: Real-time decoding— · p. 5
• For larger codes, increased problem size may raise iteration time and exceed a single FPGA’s capacity under the fully parallel implementation strategy considered.Source: Real-time decoding— · p. 5

• Relay-BP achieves orders of magnitude better logical error rates than BP+OSD+CS-10 for LDPC codes and comparable error rates to matching-based decoders for the rotated surface code.Source: Discussions and Outlook— · p. 5
• All evidence suggests that Relay-BP is fast enough for real-time FPGA decoding while achieving very low logical error rates.Source: Discussions and Outlook— · p. 5
• The paper leaves open why negative memory strengths matter, how Relay-BP performs beyond memory experiments, how to choose memory strengths, and how a full FPGA implementation performs.Source: Discussions and Outlook— · p. 5

Graph representation

• The decoding graph represents H with error nodes as columns and check nodes as rows, while the syndrome records each check’s parity.Filled nodes indicate matrix value 1; unfilled nodes indicate 0.Source: Figure 4 · p. 7, Appendix Decoding objects for logical… · p. 7

Circuit construction

• Each QEC cycle measures fixed stabilizer generators, producing detector outcomes from parity across successive rounds.The simulations use d noisy cycles followed by one error-free final cycle.Source: Appendix Decoding objects for logical… · p. 7

Noise model

• The circuit-level noise model independently samples discrete error modes, whose detector flips and residual Paulis define H and A.Each error mode corresponds to a column in H and A, with its probability stored in p.Source: Appendix Decoding objects for logical… · p. 7

Decoding objects

• XYZ decoding uses H, A, and p directly, whereas XZ decoding extracts type-specific objects and independently decodes X- and Z-associated syndromes.The partial corrections are later combined into a full correction.Source: Appendix Decoding objects for logical… · p. 7

Column compression

• Identical columns of H are compressed by retaining one representative and replacing their probabilities with the probability of an odd number of errors in the group.The process repeats until all columns of H are unique.Source: Appendix Decoding objects for logical… · p. 7

Probability model

• The noise model assumes each bit is drawn independently according to the probability vector p.This independence assumption enables factorization of the conditional error distribution.Source: Decoding— · pp. 7–8

BP decoding

• BP computes marginals of the conditional error distribution, which are used to estimate the bit-wise most likely error rather than the maximum-likelihood error.The marginals are calculated using a suitable factorization for BP.Source: Decoding— · p. 7

XYZ-decoding analysis

• Figure 5 plots logical error rate against average BP iterations at p = 3 × 10^-3 for XYZ-Relay-BP-S, XYZ-BP, and XYZ-Mem-BP.Relay-BP-S varies the maximum number of legs R, while the other curves vary maximum iteration count.Source: Figure 5 · p. 8, XYZ-decoding real-time analysis— · p. 8

Probability model

• The factorization uses each row H_k of H and Iverson’s notation to express parity-check constraints.Iverson’s bracket is 1 when its enclosed proposition is true and 0 otherwise.Source: Decoding— · p. 8

Notation

• The pseudo-code suppresses ∀j for brevity when the intended quantification is clear.This is a notational simplification rather than an algorithmic change.Source: Pseudo-Code for Relay Algorithm— · p. 8
• The omitted universal quantifier applies to indexed operations whose scope is evident from the surrounding pseudo-code.The convention keeps the displayed algorithm shorter.Source: Pseudo-Code for Relay Algorithm— · p. 8
• Readers should restore ∀j mentally where the pseudo-code makes universal indexing implicit.The stated convention concerns presentation only.Source: Pseudo-Code for Relay Algorithm— · p. 8

Inputs and output

• Relay-BP-S accepts a target number of solutions S, relay-leg limit R, per-leg iteration limit T_r, and leg-specific memory strengths Γ_r.It returns whether a solution was found together with the estimated error.Source: Pseudo-Code for Relay Algorithm— · p. 8

Stopping conditions

• The algorithm proceeds to another leg when a leg ends without satisfying the stopping condition and breaks when enough solutions have been found.The solution count is compared with S.Source: Pseudo-Code for Relay Algorithm— · p. 8

Relay legs

• Final memory values are reused as the initial memory state for the next relay leg.This carries information across consecutive DMem-BP runs.Source: Pseudo-Code for Relay Algorithm— · p. 8

Inputs and output

• Relay-BP-S returns a Boolean indicating whether at least one solution was found, together with the estimated error vector.The return condition is s > 0.Source: Pseudo-Code for Relay Algorithm— · p. 8