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arXiv:1207.6131 · Quantum Information and Computation

Sufficient condition on noise correlations for scalable quantum computing

John P. Preskill

Published Mar 1, 2013• 60 citations
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Abstract

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 $k$ system qubits are sufficiently weak, and decay sufficiently rapidly with increasing $k$ and with increasing spatial separation of the qubits.

DOI
10.26421/qic13.3-4-1

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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 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).
Over three decades after the formulation of early quantum algorithms, the realization of large-scale, fault-tolerant quantum computers remains constrained within the Noisy Intermediate-Scale Quantum (NISQ) regime. This paper critically re-evaluates the foundational assumptions underlying the Quantum Threshold Theorem and scalable Quantum Error Correction (QEC). We contrast classical error-suppression paradigms—specifically von Neumann’s majority-logic architecture for unreliable components—with the physical realities of continuous SU(2) symmetries in Hilbert space. We argue that the physical implementation of fault-tolerant quantum computing relies on a highly idealized model of uncorrelated, local Markovian noise, whose physical realizability at macroscopic scales remains contested. By examining interaction Hamiltonians in highly entangled macroscopic states, we demonstrate that collective decoherence channels and spatial noise correlations scale nonlinearly with register size n. This correlated phase drift acts as a directional force field—analogous to a long-wavelength hydrodynamic gradient across a field of coupled oscillators—rendering spatial averaging and independent syndrome measurements mathematically incapable of distinguishing systemic logic from collective environmental shifts. Consequently, we propose that universal quantum computers should not be conceptualized as scalable digital Turing machines, but rather as highly specialized, analog physical emulators bound by fundamental physical limits of continuous symmetry preservation.
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.
This dissertation addresses the issue of noise in quantum information processing devices. It is common knowledge that quantum states are particularly fragile to the effects of noise. In order to perform scalable quantum computation, it is necessary to suppress effective noise to levels which depend on the size of the computation. Various theoretical proposals have discussed how this can be achieved, under various assumptions about properties of the noise and the availability of qubits. We discuss new approaches to the suppression of noise, and propose experimental protocols characterizing the noise. \n \nIn the first part of the dissertation, we discuss a number of applications of teleportation to fault-tolerant quantum computation. We demonstrate how measurement-based quantum computation can be made inherently fault-tolerant by exploiting its relationship to teleportation. We also demonstrate how continuous variable quantum systems can be used as ancillas for computation with qubits, and how information can be reliably teleported between these different systems. Building on these ideas, we discuss how the necessary resource states for teleportation can be prepared by allowing quantum particles to be scattered by qubits, and investigate the feasibility of an implementation using superconducting circuits. \n \nIn the second part of the dissertation, we propose scalable experimental protocols for extracting information about the noise. We concentrate on information which has direct practical relevance to methods of noise suppression. In particular, we demonstrate how standard assumptions about properties of the noise can be tested in a scalable manner. The experimental protocols we propose rely on symmetrizing the noise by random application of unitary operations. Depending on the symmetry group use, different information about the noise can be extracted. We demonstrate, in particular, how to estimate the probability of a small number of qubits being corrupted, as well as how to test for a necessary condition for noise correlations. We conclude by demonstrating how, without relying on assumptions about the noise, the information obtained by symmetrization can also be used to construct protective encodings for quantum states.