Numerical and analytical bounds on threshold error rates for hypergraph-product codes

We study analytically and numerically decoding properties of finite-rate hypergraph-product quantum low density parity-check codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent $X$ and $Z$ errors. Several nontrivial lower and upper bounds for the decodable region are constructed analytically by analyzing the properties of the homological difference, equal minus the logarithm of the maximum-likelihood decoding probability for a given syndrome. Numerical results include an upper bound for the decodable region from specific heat calculations in associated Ising models and a minimum-weight decoding threshold of approximately $7\%$.

Figure 1

The $(p,T)$ phase diagram of the two random-bond Ising models associated with QHP codes from a (3,4) Gallager ensemble, labeled “(3,4) QHP” and “dual (3,4) QHP,” where the latter model includes a summation over additional spin variables corresponding to the codewords, see Eq. (5). Here $p$ is the error probability and $T$ the dimensionless temperature. Positions of the specific heat maxima at different $p$, extrapolated to infinite system size, are shown with solid blue circles and solid red boxes with ad hoc linear fits [red open boxes connected with a dotted line correspond to parabolic extrapolation in Fig. 5]. The two transition lines intersect the $p=0$ axis in approximately mutually dual temperatures and are located, respectively, above and below the critical temperature for the square-lattice random-bond Ising model (dashed-dotted green line; data from Ref. [33]). The ML decoding problem corresponds to the points on the Nishimori line shown with a solid black line; a temperature not on the Nishimori line corresponds to a suboptimal decoder which assumes an incorrect error probability. The magenta-shaded region shows the lower bound for the decodable region from Theorem t2; the rightmost point of this region coincides with the lower bound obtained by analyzing minimum-energy decoder in Ref. [16] (dashed magenta downward arrow). The forest-green solid vertical arrow shows our numerical estimate for the minimum-weight decoding threshold, see Sec. 4b. The temperatures $T_{\max }$ and its dual, $T_{\max }^{*}$ from Theorem t3, are marked with a pair of horizontal arrows separated by a gray bar on the vertical axis; the lower arrow corresponds to the analytical upper temperature bound of the decodable region. A more accurate upper bound from the present data is given by the “dual (3,4) QHP” line.

Figure 2

Decoder failure probability as a function of bit-flip error probability for three (3,4) QHP codes (left) and the rotated toric codes (right). The decoding was performed over 1024 error realizations for (3,4) QHP codes and over 4096 error realizations for toric codes. The corresponding decoding pseudothreshold is close to $p=7.0\%$ for (3,4) QHP codes and to $10.4\%$ for toric codes.

Figure 3

Specific heat $C$ vs dimensionless temperature $T$ for (3,4) QHP models with distances (a) $d=4$, (b) $d=6$, and (c) $d=8$, at $p$ values as indicated on panel (a). Each curve contains data points from the feedback-optimized parallel tempering simulation where ordered and disordered configurations are used as initial states. The peak positions are extrapolated to infinite distance to obtain the transition temperatures (see Fig. 5).

Figure 4

Specific heat $C$ divided by the number of bonds $n$ vs dimensionless temperature $T$ for (3,4) QHP models at (a) $p=0$, (b) $p=2\%$, and (c) $p=10\%$, with code distances as indicated in the captions. Each curve contains data points from the feedback-optimized parallel tempering simulation. The annealing plot contains data points from upward and downward temperature sweeps. Relatively weak variation of the peak height with system size is indicative of a discontinuous transition. Open symbols in plot (c) show the data obtained with annealing, which agrees with the parallel tempering data (filled symbols).

Figure 5

(a) Finite-size scaling of the dimensionless temperatures $T_{\max }$ where specific heat reaches the maximum for (3,4) QHP models vs the inverse square of the code distance, with the fraction of flipped bonds $p$ as indicated. For the point $p=0$ and $d=10$ where we could not eliminate the hysteresis, the average temperature was used. (b) The same for the dual (3,4) QHP models. To accommodate the increased curvature, for $p=0.08$ and 0.10 we also used parabolic fits, $T_{\max }=T_c+A/d^2+B/d^4$, which results in a significantly reduced extrapolated value of $T_c$ for $p=0.10$. The $T_c$ values from parabolic fits are shown in Fig. 1 with open red symbols.