Low-weight quantum syndrome errors in belief propagation decoding

Abstract

We describe an empirical approach to identify low-weight combinations of columns of the decoding matrices of a quantum circuit-level noise model, for which belief-propagation (BP) algorithms converge possibly very slowly. Focusing on the logical-idle syndrome cycle of the low-density parity check gross code, we identify criteria providing a characterization of the Tanner subgraph of such low-weight error syndromes. We analyze the dynamics of iterations when BP is used to decode weight-four and weight-five errors, finding statistics akin to exponential activation in the presence of noise or escape from chaotic phase-space domains. We study how BP convergence improves when adding to the decoding matrix relevant combinations of fault columns, and show that the suggested decoder amendment can result in the reduction of both logical errors and decoding time.

Figures (14)

• Figure 1: The "hard decision" result bit of part of the faults (variable nodes) at the end of part of the iterations in the middle of a Relay-BP execution. Each row corresponds to a single fault column in the decoding matrix, and the decoder attempts to decide whether it contributed to the measured syndrome, where a black (white) pixel indicates yes (no). In this example, a low-weight error syndrome does not converge to a correct result even with 200 relay legs (and over 12,000 BP iterations). There are 60 iterations per relay leg, with a bit more than four legs shown. The faults are ordered by their total (integrated) brightness along all iterations, and only the first 121 from the top are shown. For comparison, a quickly converging weight-four error is presented in Fig. \ref{['fig:success']}.
• Figure 2: The number of fault syndrome columns shared by each detector pair, $n_s(c_i, c_j)$ measured at the end of the first syndrome cycle, in the circuit-level decoding matrix $H_Z$ of $Z$-type errors ($X$-type stabilizers) for the gross code. The logical idle circuit analyzed here has $n/2=72$ check qubits measured at the end of each syndrome cycle, partitioning the detectors into groups of 72, and only the pairs of checks from the first group are shown. The count of shared columns of the pairs ranges from $n_s=0$ to $n_s=8$. The structure of $H_Z$ beyond the first cycle is shown in Fig. \ref{['fig:z.pairs.4']} (and $H_X$ with $X$-type errors in Fig. \ref{['fig:x.pairs.4']}), and the distribution of $n_s$ in Fig. \ref{['fig:shared']}.
• Figure 3: Frequency of $n_s(c_1, c_i)$ among the 71 checks $c_i\neq c_1$ in the same syndrome cycle group with which one (arbitrary) check $c_1$ shares between zero to eight fault columns. The data corresponds to the circuit-level noise decoding matrices $H_Z$ (first group of checks, as in Fig. \ref{['fig:z.pairs']}) and $H_X$ (Fig. \ref{['fig:x.pairs.4']}, second group of checks) for the gross code with the idle logic cycle.
• Figure 4: Distribution of the number of total canceled checks ($n_c$) in the syndromes of all weight-four errors constructed by choosing any combination of two of the shared columns of the fixed pair $p_0=(c_1,c_2)$ together with any two of the shared columns of every other pair $p_i=(c_3,c_4)$ of the 71 maximal $n_s=8$ pairs in one cycle of $H_Z$ (first group of checks, as in Fig. \ref{['fig:z.pairs']}) and $H_X$ (second group of checks, shown in Fig. \ref{['fig:x.pairs.4']}).
• Figure 5: A possible structure of part of the Tanner subgraph of weight-four errors that are slow to converge with BP iterations. Checks are drawn as squares and faults as circles, lines connect checks to their faults, and only the eight canceled checks are explicitly shown, similar to the structure in Tab. \ref{['structure_table']}.