1. Constant overhead quantum fault tolerance with quantum expander codes
- Author
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Anthony Leverrier, Antoine Grospellier, Omar Fawzi, Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Cryptologie symétrique, cryptologie fondée sur les codes et information quantique (COSMIQ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Omar Fawzi acknowledges support from the ANR through the project ACOM. Antoine Grospellier and Anthony Leverrier acknowledge support from the ANR through the QuantERA project QCDA., ANR-18-CE47-0011,ACOM,Une Théorie Algorithmique de la Communcation(2018), ANR-18-QUAN-0004,QCDA,Quantum Code Design and Architectures(2018), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-École normale supérieure - Lyon (ENS Lyon)
- Subjects
0303 health sciences ,General Computer Science ,Computer science ,Concatenated error correction code ,Computation ,020206 networking & telecommunications ,02 engineering and technology ,Topology ,Expander code ,03 medical and health sciences ,[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph] ,Qubit ,0202 electrical engineering, electronic engineering, information engineering ,Overhead (computing) ,Quantum ,Decoding methods ,030304 developmental biology ,Quantum computer - Abstract
The threshold theorem is a seminal result in the field of quantum computing asserting that arbitrarily long quantum computations can be performed on a faulty quantum computer provided that the noise level is below some constant threshold. This remarkable result comes at the price of increasing the number of qubits (quantum bits) by a large factor that scales polylogarithmically with the size of the quantum computation we wish to realize. Minimizing the space overhead for fault-tolerant quantum computation is a pressing challenge that is crucial to benefit from the computational potential of quantum devices. In this paper, we study the asymptotic scaling of the space overhead needed for fault-tolerant quantum computation. We show that the polylogarithmic factor in the standard threshold theorem is in fact not needed and that there is a fault-tolerant construction that uses a number of qubits that is only a constant factor more than the number of qubits of the ideal computation. This result was conjectured by Gottesman who suggested to replace the concatenated codes from the standard threshold theorem by quantum error-correcting codes with a constant encoding rate. The main challenge was then to find an appropriate family of quantum codes together with an efficient classical decoding algorithm working even with a noisy syndrome. The efficiency constraint is crucial here: bear in mind that qubits are inherently noisy and that faults keep accumulating during the decoding process. The role of the decoder is therefore to keep the number of errors under control during the whole computation. On a technical level, our main contribution is the analysis of the SMALL-SET-FLIP decoding algorithm applied to the family of quantum expander codes . We show that it can be parallelized to run in constant time while correcting sufficiently many errors on both the qubits and the syndrome to keep the error under control. These tools can be seen as a quantum generalization of the BIT-FLIP algorithm applied to the (classical) expander codes of Sipser and Spielman.
- Published
- 2021