[Paper Review] Spacetime codes of Clifford circuits

Nicolas Delfosse, Adam Paetznick|arXiv (Cornell University)|Apr 12, 2023

Quantum Computing Algorithms and ArchitectureComputer Science11 citations

TL;DR

The paper introduces the outcome code and its associated spacetime stabilizer code to detect and correct faults in any Clifford circuit. It provides polynomial-time constructions, links to LDPC decoding, and a circuit-centered fault correction workflow.

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.

Motivation & Objective

• Motivate fault tolerance for Clifford circuits in noisy quantum hardware.
• Show that the set of all possible measurement outcomes of a Clifford circuit forms a linear code (the outcome code).
• Construct a stabilizer code (the spacetime code) from the outcome code to enable fault decoding via stabilizer decoding.
• Provide efficient algorithms to compute outcome checks and spacetime-code generators.
• Demonstrate how a most likely error decoder for the spacetime code yields a most likely fault decoder for the circuit.
• Explore conditions under which the spacetime code is LDPC and develop sparsification techniques to produce low-weight checks.

Proposed method

• Define the outcome code as the linear code of possible outcome bit-strings from a Clifford circuit (Theorem 1, Corollary 2).
• Develop Algorithm 1 to compute a complete set of checks for the outcome code.
• Construct the spacetime code by backward accumulation of outcome checks through the circuit (Theorem 2).
• Show how to convert a most likely error decoder for the spacetime code into a circuit decoder that outputs a most likely fault configuration (Theorem 3).
• Introduce Algorithm 3 to generate low-weight stabilizer generators for the spacetime code, enabling LDPC decoding.
• Explain the backward/forward propagation (cumulant/back-cumulant) formalism and prove their adjoint relationship (Proposition 3).
• Provide a framework to use any LDPC decoder (e.g., Union-Find, Renormalization Group, Belief Propagation) on the resulting spacetime code.

Figure 1: Construction of codes from a Clifford circuit. Given a Clifford circuit as input, a modified stabilizer simulation, Algorithm 1 , produces the outcome code. The corresponding spacetime code can then be constructed by accumulating measurement observables from the outcome code backward throu

Experimental results

Research questions

• RQ1 Can the set of all possible outcome bit-strings of a Clifford circuit be characterized as a linear code (outcome code)?
• RQ2 How can one systematically construct a stabilizer spacetime code from a given Clifford circuit?
• RQ3 How can a decoder for the spacetime code be transformed into a practical circuit fault decoder?
• RQ4 Under what conditions can the spacetime code be made LDPC, and how can one generate low-weight stabilizers for efficient decoding?
• RQ5 Is there a circuit-centric, automated method to correct faults in Clifford circuits using only the circuit as input?

Key findings

• The outcome string set of any Clifford circuit forms a linear code (the outcome code).
• A stabilizer spacetime code can be constructed from the outcome code, allowing fault correction to be cast as stabilizer decoding.
• A most likely fault decoder for the spacetime code yields a most likely circuit fault decoder (Theorem 3).
• Algorithms exist to construct outcome checks (Algorithm 1) and low-weight spacetime-code generators (Algorithm 3) in polynomial time.
• The spacetime code can be LDPC under certain conditions, enabling efficient LDPC decoders for fault correction.
• The framework applies to a broad class of Clifford circuits, including those with intermediate and multi-qubit measurements, and can be integrated with various LDPC decoding strategies.

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 th