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Anthony Leverrier

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5 papers

1 author row

SODA Conference 2023 Conference Paper

Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes

Anthony Leverrier
Gilles Zémor

We introduce and analyse an efficient decoder for quantum Tanner codes that can correct adversarial errors of linear weight. Previous decoders for quantum low-density parity-check codes could only handle adversarial errors of weight. We also work on the link between quantum Tanner codes and the Lifted Product codes of Panteleev and Kalachev, and show that our decoder can be adapted to the latter. The decoding algorithm alternates between sequential and parallel procedures and converges in linear time.

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FOCS Conference 2022 Conference Paper

Quantum Tanner codes

Anthony Leverrier
Gilles Zémor

Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c 3 locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.

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FOCS Conference 2018 Conference Paper

Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

Omar Fawzi
Antoine Grospellier
Anthony Leverrier

We prove that quantum expander codes can be combined with quantum fault-tolerance techniques to achieve constant overhead: the ratio between the total number of physical qubits required for a quantum computation with faulty hardware and the number of logical qubits involved in the ideal computation is asymptotically constant, and can even be taken arbitrarily close to 1 in the limit o small physical error rate. This improves on the polylogarithmic overhead promised by the standard threshold theorem. To achieve this, we exploit a framework introduced by Gottesman together with a family of constant rate quantum codes, quantum expander codes. Our main technical contribution is to analyze an efficient decoding algorithm for these codes and prove that it remains robust in the presence of noisy syndrome measurements, a property which is crucial for fault-tolerant circuits. We also establish two additional features of the decoding algorithm that make it attractive for quantum computation: it can be parallelized to run in logarithmic depth, and is single-shot, meaning that it only requires a single round of noisy syndrome measurement.

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STOC Conference 2018 Conference Paper

Efficient decoding of random errors for quantum expander codes

Omar Fawzi
Antoine Grospellier
Anthony Leverrier

We show that quantum expander codes, a constant-rate family of quantum low-density parity check (LDPC) codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Zémor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman’s construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of α-percolation: for a random subset E of vertices of a given graph, we consider the size of the largest connected α-subset of E , where X is an α-subset of E if | X ∩ E | ≥ α | X |.

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FOCS Conference 2015 Conference Paper

Quantum Expander Codes

Anthony Leverrier
Jean-Pierre Tillich
Gilles Zémor

We present an efficient decoding algorithm for constant rate quantum hyper graph-product LDPC codes which provably corrects adversarial errors of weight proportional to the code minimum distance, or equivalently to the square-root of the block length. The algorithm runs in time linear in the number of qubits, which makes its performance the strongest to date for linear-time decoding of quantum codes. The algorithm relies on expanding properties, not of the quantum code's factor graph directly, but of the factor graph of the original classical code it is constructed from.

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