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arXiv:2304.05943

Spacetime codes of Clifford circuits

Nicolas Delfosse, Adam Paetznick

Published Apr 12, 2023• 8 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2304.05943

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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.
Quantum error-correcting codes provide a powerful framework for analyzing and designing fault-tolerant Clifford circuits. In this work, we construct the subsystem spacetime code and use it to characterize the distances of Clifford circuits. We show that its gauge group can be explicitly constructed and classified using the masking properties of the instantaneous stabilizer groups and consists precisely of faults that are harmless to the circuit. We use this framework to identify where a dynamical code loses distance and to analyze how the fault distance changes when circuits are composed from smaller components. Our results provide a versatile tool for analyzing and designing fault-tolerant circuits, with potential applications to the construction of new dynamical codes.
Recent advances in quantum error-correction (QEC) have shown that it is often beneficial to understand fault-tolerance as a dynamical process, a circuit with redundant measurements that help correct errors, rather than as a static code equipped with a syndrome extraction circuit. Spacetime codes have emerged as a natural framework to understand error correction at the circuit level while leveraging the traditional QEC toolbox. Here, we introduce a framework based on chain complexes and chain maps to model spacetime codes and transformations between them. We show that stabilizer codes, quantum circuits, and decoding problems can all be described using chain complexes, and that the equivalence of two spacetime codes can be characterized by specific maps between chain complexes, the fault-tolerant maps, that preserve the number of encoded qubits, fault distance, and minimum-weight decoding problem. As an application of this framework, we extend the foliated cluster state construction from stabilizer codes to any spacetime code, showing that any Clifford circuit can be transformed into a measurement-based protocol with the same fault-tolerant properties. To this protocol, we associate a chain complex which encodes the underlying decoding problem, generalizing previous cluster state complex constructions. Our method enables the construction of cluster states from non-CSS, subsystem, and Floquet codes, as well as from logical Clifford operations on a given code.
Error probability distribution associated with a given Clifford measurement circuit is described exactly in terms of the circuit error-equivalence group, or the circuit subsystem code previously introduced by Bacon, Flammia, Harrow, and Shi. This gives a prescription for maximum-likelihood decoding with a given measurement circuit. Marginal distributions for subsets of circuit errors are also analyzed; these generate a family of related asymmetric LDPC codes of varying degeneracy. More generally, such a family is associated with any quantum code. Implications for decoding highly-degenerate quantum codes are discussed.
We introduce a low-overhead approach for detecting errors in arbitrary Clifford circuits on arbitrary qubit connectivities. Our method is based on the framework of spacetime codes, and is particularly suited to near-term hardware since it has a much milder overhead in qubits and gates compared to error correction, while achieving a better sampling overhead than existing error mitigation methods. We present efficient algorithms for finding valid checks that are simultaneously low weight, satisfy connectivity constraints, and cover large detecting regions within the circuit. Using this approach, we experimentally demonstrate error detection on circuits of up to 50 logical qubits containing 2450 CZ gates, and show physical to logical fidelity gains of up to 236×236\times236×. Furthermore, we show our algorithm can efficiently find checks in universal circuits, but the space of valid checks diminishes exponentially with the non-Cliffordness of the circuit. These theoretical and experimental results suggest that Clifford-dominated circuits are promising candidates for near-term quantum advantage.
A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.
Quantum error correction is the art of protecting fragile quantum information through suitable encoding and active interventions. After encoding kkk logical qubits into n>kn>kn>k physical qubits using a stabilizer code, this amounts to measuring stabilizers, decoding syndromes, and applying an appropriate correction. Although quantum information can be protected in this way, it is notoriously difficult to manipulate encoded quantum data without introducing uncorrectable errors. Here, we introduce a mathematical framework for constructing hardware-tailored quantum circuits that implement any desired Clifford unitary on the logical level of any given stabilizer code. Our main contribution is the formulation of this task as a discrete optimization problem. We can explicitly integrate arbitrary hardware connectivity constraints. As a key feature, our framework naturally incorporates an optimization over all Clifford gauges (differing only in their action outside the code space) of a desired logical circuit. In this way, we find, for example, fault-tolerant and teleportation-free logical Hadamard circuits for the [[8,3,2]][[8,3,2]][[8,3,2]] code. From a broader perspective, we turn away from the standard generator decomposition approach and instead focus on the holistic compilation of entire logical circuits, leading to significant savings in practice. Our work introduces both the necessary mathematics and open-source software to compile hardware-tailored logical Clifford circuits for stabilizer codes.