Spacetime codes of Clifford circuits

TL;DR

We address fault tolerance for quantum computation by developing a circuit-centric framework that turns circuit faults in Clifford circuits into stabilizer-code errors. The core idea is to extract an outcome code from circuit measurements and then construct a spacetime stabilizer code whose decoders implement a most-likely fault correction, enabling an efficient, automated path from circuit input to fault mitigation. The approach extends prior circuit-to-code mappings to include intermediate and multi-qubit measurements and provides algorithms to generate low-weight checks, with conditions under which the spacetime code is LDPC. The framework offers a flexible, automated route to syndrome extraction, decoding, and fault correction that can adapt to various architectures and supports LDPC-based decoders for scalable operation.

Abstract

We propose a scheme for detecting and correcting faults in any Clifford circuit. The scheme is based on the observation that the set of all possible outcome bit-strings of a Clifford circuit is a linear code, which we call the outcome code. From the outcome code we construct a corresponding stabilizer code, the spacetime code. Our construction extends the circuit-to-code construction of Bacon, Flammia, Harrow and Shi [2], revisited recently by Gottesman [16], to include intermediate and multi-qubit measurements. With this correspondence, we reduce the problem of correcting faults in a circuit to the well-studied problem of correcting errors in a stabilizer code. More precisely, a most likely error decoder for the spacetime code can be transformed into a most likely fault decoder for the circuit. We give efficient algorithms to construct the outcome and spacetime codes. We also identify conditions under which these codes are LDPC, and give an algorithm to generate low-weight checks, which can then be combined with effcient LDPC code decoders.

Figures (2)

• Figure 1: Construction of codes from a Clifford circuit. Given a Clifford circuit as input, a modified stabilizer simulation, Algorithm \ref{['algo:circuit_checks']}, produces the outcome code. The corresponding spacetime code can then be constructed by accumulating measurement observables from the outcome code backward through the circuit. See Section \ref{['subsec:check_operators']}. Low-weight generators of the spacetime code are obtained by sparsification with Algorithm \ref{['algo:spacetime_code_local_genertors']}. The dashed arrow indicates that sparsification is not possible for all circuits. All constructions run in polynomial time. The LDPC spacetime code can be decoded using any quantum LDPC code decoder, thereby providing an automated and efficient means of correcting faults given only the circuit as an input.
• Figure 2: A depth-three circuit with Pauli measurements and unitary Clifford gates. Pauli faults are supported on the white circles. We show a fault operator in (a) and its back-cumulant in (b) obtained by propagation of faults backward. In this circuit, the third measurement is redundant and the three measurement outcomes $o_1, o_2, o_3$ satisfy $o_1 + o_2 + o_3 = 0 \pmod 2$. This defines a check of the outcome code. The stabilizer generator of the spacetime code corresponding to this check is obtained by the backward accumulation of the three measurements through the circuit. This results in the weight-four Pauli operator shown in (b) supported on the circles.

Theorems & Definitions (57)

• Proposition 1: Effect of faults
• proof
• Proposition 2: Explicit cumulant and back-cumulant
• proof
• Corollary 1
• proof
• Proposition 3: Adjoint of the accumulator
• proof
• Theorem 1: Outcome code of a Clifford circuit
• Lemma 1: Stabilizer update rules