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arXiv:quant-ph/0511065

Fault-Tolerant Quantum Computation for Local Leakage Faults

Panos Aliferis, Barbara M. Terhal

Published Nov 7, 2005• 1 citation
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Abstract

We provide a rigorous analysis of fault-tolerant quantum computation in the presence of local leakage faults. We show that one can systematically deal with leakage by using appropriate leakage-reduction units such as quantum teleportation. The leakage noise is described by a Hamiltonian and the noise is treated coherently, similar to general non-Markovian noise analyzed in Refs. quant-ph/0402104 and quant-ph/0504218. We describe ways to limit the use of leakage-reduction units while keeping the quantum circuits fault-tolerant and we also discuss how leakage reduction by teleportation is naturally achieved in measurement-based computation.

DOI
10.48550/arxiv.quant-ph/0511065

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We provide a rigorous analysis of fault-tolerant quantum computation in the presence of local leakage faults. We show that one can systematically deal with leakage by using appropriate leakage-reduction units such as quantum teleportation. The leakage noise is described by a Hamiltonian and the noise is treated coherently, similar to general non-Markovian noise analyzed in Refs. quant-ph/0402104 and quant-ph/0504218. We describe ways to limit the use of leakage-reduction units while keeping the quantum circuits fault-tolerant and we also discuss how leakage reduction by teleportation is naturally achieved in measurement-based computation.
We provide a rigorous analysis of fault-tolerant quantum computation in the presence of local leakage faults. We show that one can systematically deal with leakage by using appropriate leakage-reduction units such as quantum teleportation. The leakage noise is described microscopically, by Hamiltonian couplings, and the noise is treated coherently, similar to general non-Markovian noise analyzed in Refs. \cite{Terhal04} and \cite{Aliferis05b}. We describe ways to limit the use of leakage-reduction units while keeping the quantum circuits fault-tolerant and we also discuss how leakage reduction by teleportation is naturally achieved in measurement-based computation.
Physical implementations of cryptographic algorithms leak information, which makes them vulnerable to so-called side-channel attacks.Cryptography is now used in an ever-increasing variety of scenarios, and the assumption that the execution of cryptosystems is physically insulated is often not realistic.The field of leakage resilience proposes to mitigate side-channel attacks by designing protocols that are secure even if information leaks during execution.In this work, we study leakage-resilient computation, which concerns the problem of performing secure universal computation in the presence of leakage.Fault-tolerant quantum computation is concerned with the problem of noise in quantum computers.Since it is very hard to insulate quantum systems from noise, fault tolerance proposes schemes for performing computations correctly even if some noise is present.It turns out that there exists a connection between leakage resilience and fault tolerance.In this work, we show that leakage in a classical circuit is a form of noise, when the circuit is interpreted as quantum.We then prove that for an arbitrary leakage model, there exists a corresponding noise model in which a circuit that is fault-tolerant against the noise model is also resilient against the given leakage model.We also show how to use constructions for fault tolerance to implement classical circuits that are secure in specific leakage models.This is done by establishing criteria in which quantum circuits can be converted into classical circuits in such a way that the leakage resilience property is preserved.Using these criteria, we convert an implementation of universal fault-tolerant quantum computation into a classical leakageresilient compiler, i.e., a scheme that compiles an arbitrary circuit into a circuit of the same functionality that is leakage-resilient.vi 4.9 (a): Circuito de codificação para |0 no código de Steane.O circuito que codifica |+ é o mesmo, com a diferença de que alvos e controles são revertidos, e estados |0 são revertidos em |+ e vice-versa.(b):
We derive a threshold result for fault-tolerant quantum computation for local non-Markovian noise models. The role of error amplitude in our analysis is played by the product of the elementary gate time t0{t}_{0}t0​ and the spectral width of the interaction Hamiltonian between system and bath. We discuss extensions of our model and the applicability of our analysis.
Fault-tolerant quantum computation traditionally incurs substantial resource overhead, with both qubit and time overheads scaling polylogarithmically with the size of the computation. While prior work by Gottesman showed that constant qubit overhead is achievable under stochastic noise using quantum low-density parity-check (QLDPC) codes, it has remained an open question whether similar guarantees hold under more general, non-stochastic noise models. In this work, we address this question by considering a general circuit-level noise model defined via the diamond norm, which captures both stochastic and non-stochastic noise, including coherent and amplitude damping noise. We prove that constant qubit overhead fault-tolerant quantum computation is achievable in this general setting, using QLDPC codes with constant rate and linear minimum distance. To establish our result, we develop a fault-tolerant error correction scheme and a method for implementing logic gates under general circuit noise. These results extend the theoretical foundations of fault-tolerant quantum computation and offer new directions for fault-tolerant architectures under realistic noise models.
We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on Nˉ\bar{N}Nˉ logical qudits of depth Tˉ\bar{T}Tˉ, we construct a fault-tolerant circuit on N=poly(Nˉ)N=\text{poly}(\bar{N})N=poly(Nˉ) physical qudits of depth Tˉ⋅Nˉo(1)\bar{T}\cdot\bar{N}^{o(1)}Tˉ⋅Nˉo(1), which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number N1−o(1)N^{1-o(1)}N1−o(1) of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits. Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance. Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.