Scalable Postselection of Quantum Resources

TL;DR

This work presents an approach to lower the overhead of quantum computing using scalable postselection, based on directly postselecting sub-circuits with a size extensive in the code distance using decoder soft information, and introduces a metric, the partial gap, that estimates what the logical gap of a resource state will be after it is consumed.

Abstract

The large overhead imposed by quantum error correction is a critical challenge to the realization of quantum computers, and motivates searching for alternative error correcting codes and fault-tolerant circuit constructions. Postselection is a powerful tool that builds large programs out of probabilistically generated sub-circuits, and has been shown to increase the threshold of quantum error correction based on fusing fixed-size resource states or concatenated codes. In this work, we present an approach to lower the overhead of quantum computing using scalable postselection, based on directly postselecting sub-circuits with a size extensive in the code distance using decoder soft information. We introduce a metric, the partial gap, that estimates what the logical gap of a resource state will be after it is consumed, and show that postselection based on the partial gap leads to scalable improvements in the logical error rate. In the specific context of implementing logical gates via teleportation through a cluster state, we demonstrate that scalable postselection provides a $4\times$ reduction in the overhead per logical gate, at the same logical error probability.

Figures (4)

• Figure 1: a) Fusion-based computation achieves higher thresholds by constructing circuits by joining fixed-size resource states that are postselected on absence of detected errors. b) Our approach to scalable postselection joins resource states with a size extensive in the code distance that are scalably postselected based on a soft decoder output called the partial gap.
• Figure 2: Partial gap for the repetition code. (a) Logical error rate after $d+1$ rounds of syndrome extraction on a distance $d$ repetition code, binned by $G_P$ (circles) or $\tilde{G}_P$ (stars) predicted from the first $d$ layers of syndromes. The dashed lines show logistic models \ref{['eq:logistic']} with scaling factors $\alpha \in \{0.9, 1\}$. (b) Scatter plot showing the correlation between $G_P$ and $\tilde{G}_P$ for the repetition code at many distances ($p=10^{-3}$). The solid line shows $\tilde{G}_P=G_P$. (c) Logical error rate $\bar{p}$ as a function of physical error rate with postselection using $G_P$ at rejection rate $r=1/2$. The error rate with no postselection is shown for reference (stars). Three regimes are indicated schematically denoting above-threshold (I), bulk-error-limited (II) and boundary-error-limited (III). The results are virtually indistinguishable if the approximate partial gap $\tilde{G}_P$ is used instead for postselection (not shown).
• Figure 3: Partial gap for the surface code. (a) A schematic matching graph with visible syndrome $\sigma_v$ (red diamonds), and the most likely hidden syndrome $\sigma_h$ (green diamonds). The most likely corrections for each logical outcome are shown as colored strings (blue, purple). The critical string is the logical operator at the top of the cube which is mostly blue and partially purple. (b) Making an adjustment to $\sigma_h$, shown as a green diamond at the top of the cube, only slightly alters the logical operator. However, its partition into blue and purple sub-strings is greatly affected. (c) Logical error rate against the approximate partial gap $\hat{G}_P$ ($p=2 \times 10^{-3}$). The grey curves show logistic behavior (\ref{['eq:logistic']}) for $\alpha = 0.9$ and $\alpha = 1.0$. (d) Logical error rate after $d+2$ rounds of surface code simulation, after postselecting on $\hat{G}_P$ using the inner $d$ layers with $r=1/2$ (circles) and without postselection (stars). The dashed lines show exponential fits to the behavior far below threshold, extrapolating to an effective threshold of $2.01\%$ for boundary errors. (e) A regression of $\bar{p}$ against $d$ for each value $p$ gives a distance scaling coefficient $\Lambda$. Without postselection, we expect constant $\Lambda \cdot p$. With postselection, we see enhanced $p$-dependence, which we quantify with a power-law fit with slope $m \approx -0.505$ (green line). This implies effective distance augmented by a factor of $1.505$.
• Figure 4: a) An example compute cycle leveraging postselected resource states. The cycle begins with a state $\Psi$ in memory. We then prepare a resource state implementing a single logical gate $\ket{U}$ and $d$ layers of parity checks. The state is rejected with probability $r$, incurring an overhead factor of $(1-r)^{-1}$. After accepting a state, we teleport $\Psi$ through $\ket{U}$ and return it to memory before the next operation. b) Logical error rate per logical gate, as a function of overhead $\varkappa$ for various strategies ($p = 2 \times 10^{-3}$). Colored curves correspond to a given distance, varying the reject rate (dots denote $r \in \{0, 0.01, 0.1, 0.5\}$). Squares show larger distances (without postselection) for reference, along with a spline connecting the $r=0$ points to guide the eye.