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arXiv:quant-ph/0510231 · Physical Review Letters

Fault-Tolerant Quantum Computation with Long-Range Correlated Noise

Dorit Aharonov, Alexei Kitaev, John P. Preskill

Published Feb 7, 2006• 243 citations
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Abstract

We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r(D).

DOI
10.1103/physrevlett.96.050504

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We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r(D).
I study the effectiveness of fault-tolerant quantum computation against correlated Hamiltonian noise, and derive a sufficient condition for scalability. Arbitrarily long quantum computations can be executed reliably provided that noise terms acting collectively on kkk system qubits are sufficiently weak, and decay sufficiently rapidly with increasing kkk and with increasing spatial separation of the qubits.
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.
We investigate the efficacy of topological quantum error-correction in correlated noise model which permits collective coupling of all the codeword qubits to the same non-Markovian environment. In this noise model, the probability distribution over set of phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial likelihood of the occurring large numbers of qubits errors. We find that in the presence of noise correlation, one cannot guarantee arbitrary high computational accuracy simply by incrementing the codeword size while retaining constant noise level per qubit operation. However, if instead, per-operation qubit error probability in an n-qubits long codeword is reduced O(\sqrt{n}) times below the accuracy threshold, arbitrarily accurate quantum computation becomes feasible with acceptable scaling of the codeword size. Our results suggest that progressively reducing noise level in qubits and gates is as important as continuously integrating more qubits to realize scalable and reliable quantum computer.
We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.
We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated error-detecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated error-correcting codes) not to strongly-correlated adversarial noise. Our rigorously established lower bound on the accuracy threshold, 1.04×10−31.04\times 10^{-3}1.04×10−3, is well below Knill's numerical estimates.