Adaptive Syndrome Extraction

TL;DR

This work introduces adaptive syndrome extraction as a scheme to improve code performance and reduce the quantum error correction cycle time by measuring only the stabilizer generators that are likely to provide useful syndrome information.

Abstract

Device error rates on current quantum computers have improved enough to where demonstrations of error correction below break-even are now possible. Still, the circuits required for quantum error correction introduce significant overhead and sometimes inject more errors than they correct. In this work, we introduce adaptive syndrome extraction as a scheme to improve code performance and reduce the quantum error correction cycle time by measuring only the stabilizer generators that are likely to provide useful syndrome information. We provide a concrete example of the scheme through the [[4,2,2]] code concatenated with a hypergraph product code and a syndrome extraction cycle that uses quantum error detection to modify the syndrome extraction circuits in real time. Compared to non-concatenated codes and non-adaptive syndrome extraction, we find that the adaptive scheme achieves over an order of magnitude lower logical error rates while requiring fewer CNOT gates and physical qubits. Furthermore, we show how to achieve fault-tolerant universal logical computation with [[4,2,2]]-concatenated hypergraph product codes.

Figures (15)

• Figure 1: A QED+QEC concatenated code. Physical qubits are first encoded in a small error detecting code, such as the $[[4,2,2]]$ Iceberg code. The logical qubits of many Iceberg code blocks then act as the physical qubits of some high-rate qLDPC code.
• Figure 2: Partitioning the qubits of a $[[52,4,4]]$ HGP code formed from two copies of a $[6,2,4]$ classical code. (a) $Z$-type generator $S_Z(1,3)$ (blue) and $X$-type generator $S_X(4,4)$ (red) obtained by taking rows and columns of the (transpose) parity check matrix of the classical code. (b) Canonical logical operators $\overline{Z}_2$ and $\overline{X}_2$. The qubit highlighted both blue and red is the pivot qubit for that logical pair. (c) Diagonal qubits (black) and twin qubits (yellow).
• Figure 3: Assignment of physical HGP qubits to logical $[[4,2,2]]$ qubits. Physical HGP qubits are numbered starting from the top left qubit of the $L$ sector, going row-by-row, and finishing on the bottom right qubit of the $R$ sector. Consecutive diagonal qubits, e.g. qubits $\{1,8\}$ and $\{47,52\}$ are assigned to the same Iceberg code block. Twin qubits, e.g. qubits $\{5,23\}$ and $\{44,50\}$ are assigned to the same Iceberg block.
• Figure 4: Circuit to measure an $X$-type concatenated HGP generator. A hook error that occurs in the middle of the circuit will propagate to a large number of detectable errors.
• Figure 5: Logical error rate per round, $\epsilon_L$ as a function of the physical error rate, $p$. (a) $[[4,2,2]]$-concatenated HGP codes where the syndrome extraction circuit is adaptive. (b) Non-concatenated HPG codes using normal syndrome extraction. (c) $[[4,2,2]]$-concatenated HGP codes where the syndrome extraction circuit is not adaptive; that is, every generator is measured in every round. Error bars indicate the standard deviation on $\epsilon_L$, Eq. \ref{['eq:ler_per_round_error']}. The dashed lines and right y-axis show the number of CNOT gates in the syndrome extraction circuit averaged over the $r = 100$ rounds.

Theorems & Definitions (17)

• Theorem 2
• proof
• Definition 3
• Theorem 4: Clifford Gates for Concatenated HGP Code
• Definition 5: campbell2019theory
• Definition 6: campbell2019theory
• Lemma 7: Soundness of "Transposed" Concatenated Code
• proof
• Lemma 8: Good Soundness of Thickened Concatenated Code
• proof