Efficient decoding of random errors for quantum expander codes

Fawzi, Omar; Grospellier, Antoine; Leverrier, Anthony

doi:10.1145/3188745.3188886

Cited by 35 publications

(56 citation statements)

References 26 publications

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“…Another limitation of our work is the very small threshold value that it yields. While the threshold is usually expected to lie between 10 −3 and 10 −2 for the best constructions based on code concatenation, we expect our value to be several orders of magnitude smaller, as this was already the case in Gottesman's paper [14] and in our previous work with perfect syndrome measurement [8]. Part of the explanation is due to the very crude bounds that we obtain via percolation theory arguments.…”

Section: Introductionmentioning

confidence: 57%

“…In order to analyze random errors of linear weight, we show using percolation theory that, with high probability, the error forms clusters in the sense of connected α-subsets (Definition 20 and Lemma 27). This is similar to the analysis in [17,8], except that we use the syndrome adjacency graph of the code (as in [14]) to establish the "locality" of the decoding algorithm, implying that each cluster of the error is corrected independently of the other ones (Lemma 19). Using the fact that clusters are of size bounded by the minimum distance of the code, the result on low weight errors shows that the size of E ⊕Ê is controlled by the syndrome error size.…”

Section: Main Results and Proof Techniquesmentioning

confidence: 79%

“…In a recent work, the analysis was extended to the case of random errors (either independent and identically distributed, or local stochastic) provided that the syndrome extraction is performed perfectly and under a stricter condition on the expansion of the graph [8].…”

Section: Quantum Expander Codesmentioning

confidence: 99%

“…Theorem 8 (Fawzi, Grospellier, Leverrier [8]). Let G = (A ∪ B, E) be a (d A , d B )-biregular (γ, δ)-expanding graph with δ < 1/8.…”

Section: Quantum Expander Codesmentioning

confidence: 99%

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Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

Fawzi

Grospellier

Leverrier

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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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 of 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. on k qubits with |C| locations 1 , O(log log(|C|/ε)) levels of encoding are needed, which translates into a polylog(|C|/ε) space overhead. While this might seem like a reasonably small overhead, this remains rather prohibitive in practice, and more importantly, it raises the question of whether this value is optimal. In this paper, we consider a realistic model for quantum computing where the quantum gates are noisy, but all classical computation is assumed to be fast and error-free. Note that if classical gates are also noisy, then it is known that classical fault-tolerance cannot be obtained with constant overhead [20,10].In a breakthrough paper, Gottesman has shown that the polylogarithmic overhead was maybe not necessary after all, and that polynomial-time computations could be performed with a noisy circuit with only a constant overhead [14]. In fact, the constant can even be taken arbitrarily close to 1 provided that the physical error is sufficiently rate small. In order to overcome the polylogarithmic barrier, Gottesman suggested to use quantum error correcting codes with constant rate. More precisely, the idea is to encode the logical qubits in large blocks, but still of size sublinear in k. The encoding can still be made fault-tolerant thanks to concatenation, but this only yields an overhead polylogarithmic in the block size, and choosing a sufficiently small block size to a sub-linear overhead for encoding. Gates are then performed with Knill's technique by teleporting the appropriate encoded states. Overall, apart from the initial preparation and final measurement, the encoded circuit alternates between steps applying a gate of the original circuit on the encoded state with Knill's technique, and steps of error correction for the quantum code consisting of a measurement of the syndrome, running a (sufficiently fast) classical decoding algorithm and applying the necessary correc...

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Section: Introductionmentioning

confidence: 57%

Section: Main Results and Proof Techniquesmentioning

confidence: 79%

Section: Quantum Expander Codesmentioning

confidence: 99%

“…Theorem 8 (Fawzi, Grospellier, Leverrier [8]). Let G = (A ∪ B, E) be a (d A , d B )-biregular (γ, δ)-expanding graph with δ < 1/8.…”

Section: Quantum Expander Codesmentioning

confidence: 99%

See 2 more Smart Citations

Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

Fawzi

Grospellier

Leverrier

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

show abstract

“…But near optimal (or very fast suboptimal) decoding algorithms are already proposed for these codes [28][29][30][31], which exploit their regular lattice structure. In contrast, for quantum LDPC codes, which are defined on random graphs, only recently has a decoding algorithm been found for the special family of expander codes [7,32,33] and the general case remains open. Our main motivation to study this problem is that quantum LDPC codes have the potential of greatly reducing the overhead required to realize robust quantum processors [34,35].…”

mentioning

confidence: 99%

Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes

Liu

Poulin

2019

Phys. Rev. Lett.

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Belief-propagation (BP) decoders play a vital role in modern coding theory, but they are not suitable to decode quantum error-correcting codes because of a unique quantum feature called error degeneracy. Inspired by an exact mapping between BP and deep neural networks, we train neural BP decoders for quantum low-density parity-check (LDPC) codes with a loss function tailored to error degeneracy. Training substantially improves the performance of BP decoders for all families of codes we tested and may solve the degeneracy problem which plagues the decoding of quantum LDPC codes.

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Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation

Yamasaki,

Koashi

2024

Nat. Phys.

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Scaling up quantum computers to attain substantial speedups over classical computing requires fault tolerance. Conventionally, protocols for fault-tolerant quantum computation demand excessive space overheads by using many physical qubits for each logical qubit. A more recent protocol using quantum analogues of low-density parity-check codes needs only a constant space overhead that does not grow with the number of logical qubits. However, the overhead in the processing time required to implement this protocol grows polynomially with the number of computational steps. To address these problems, here we introduce an alternative approach to constant-space-overhead fault-tolerant quantum computing using a concatenation of multiple small-size quantum codes rather than a single large-size quantum low-density parity-check code. We develop techniques for concatenating different quantum Hamming codes with growing size. As a result, we construct a low-overhead protocol to achieve constant space overhead and only quasi-polylogarithmic time overhead simultaneously. Our protocol is fault tolerant even if a decoder has a non-constant runtime, unlike the existing constant-space-overhead protocol. This code concatenation approach will make possible a large class of quantum speedups with feasibly bounded space overhead yet negligibly short time overhead.

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Efficient decoding of random errors for quantum expander codes

Cited by 35 publications

References 26 publications

Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes

Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation

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