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arXiv:0709.0128 · Bulletin of the American Physical Society

A Universal Operator Theoretic Framework for Quantum Fault Tolerance

Gerald N. Gilbert, Michael Hamrick, Yaakov S. Weinstein, Vaneet Aggarwal, Calderbank, A. Robert

Published Sep 2, 2007• 0 citations
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Abstract

In this paper we introduce a universal operator theoretic framework for quantum fault tolerance. This incorporates a top-down approach that implements a system-level criterion based on specification of the full system dynamics, applied at every level of error correction concatenation. This leads to more accurate determinations of error thresholds than could previously be obtained. This is demonstrated both formally and with an explicit numerical example. The basis for our approach is the Quantum Computer Condition (QCC), an inequality governing the evolution of a quantum computer. We show that all known coding schemes are actually special cases of the QCC. We demonstrate this by introducing a new, operator theoretic form of entanglement assisted quantum error correction, which incorporates as special cases all known error correcting protocols, and is itself a special case of the QCC.

DOI
10.48550/arxiv.0709.0128

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In this paper we introduce a universal operator theoretic framework for quantum fault tolerance. This incorporates a top-down approach that implements a system-level criterion based on specification of the full system dynamics, applied at every level of error correction concatenation. This leads to more accurate determinations of error thresholds than could previously be obtained. This is demonstrated both formally and with an explicit numerical example. The basis for our approach is the Quantum Computer Condition (QCC), an inequality governing the evolution of a quantum computer. We show that all known coding schemes are actually special cases of the QCC. We demonstrate this by introducing a new, operator theoretic form of entanglement assisted quantum error correction, which incorporates as special cases all known error correcting protocols, and is itself a special case of the QCC.
In this paper we present a new unified theoretical framework that describes the full dynamics of quantum computation. Our formulation allows any questions pertaining to the physical behavior of a quantum computer to be framed, and in principle, answered. We refer to the central organizing principle developed in this paper, on which our theoretical structure is based, as the *Quantum Computer Condition* (QCC), a rigorous mathematical statement that connects the irreversible dynamics of the quantum computing machine, with the reversible operations that comprise the quantum computation intended to be carried out by the quantum computing machine. Armed with the QCC, we derive a powerful result that we call the *Encoding No-Go Theorem*. This theorem gives a precise mathematical statement of the conditions under which fault-tolerant quantum computation becomes impossible in the presence of dissipation and/or decoherence. In connection with this theorem, we explicitly calculate a universal critical damping value for fault-tolerant quantum computation. In addition we show that the recently-discovered approach to quantum error correction known as "operator quantum error-correction" (OQEC) is a special case of our more general formulation. Our approach furnishes what we will refer to as "operator quantum fault-tolerance" (OQFT). In particular, we show how the QCC allows one to derive error thresholds for fault tolerance in a completely general context. We prove the existence of solutions to a class of time-dependent generalizations of the Lindblad equation. Using the QCC, we also show that the seemingly different circuit, graph- (including cluster-) state, and adiabatic paradigms for quantum computing are in fact all manifestations of a single, universal paradigm for all physical quantum computation.
The standard approach to quantum fault tolerance is to calculate error thresholds on basic gates in the limit of arbitrarily many concatenation levels. In contrast this paper takes the number of qubits and the target implementation accuracy as given, and provides a framework for engineering the constrained quantum system to the required tolerance. The approach requires solving the full dynamics of the quantum system for an arbitrary admixture (biased or unbiased) of Pauli errors. The inaccuracy between ideal and implemented quantum systems is captured by the supremum of the Schatten-k norm of the difference between the ideal and implemented density matrices taken over all density matrices. This is a more complete analysis than the standard approach, where an intricate combination of worst case assumptions and combinatorial analysis is used to analyze the special case of equiprobable errors. Conditions for fault tolerance are now expressed in terms of error regions rather than a single number (the standard error threshold). In the important special case of a stochastic noise model and a single logical qubit, an optimization over all 2×2 density matrices is required to obtain the full dynamics. The complexity of this calculation is greatly simplified through reduction to an optimization over only three projectors. Error regions are calculated for the standard 5- and 7-qubit codes. Knowledge of the full dynamics makes it possible to design sophisticated concatenation strategies that go beyond repeatedly using the same code, and these strategies can achieve target fault tolerance thresholds with fewer qubits.
The standard approach to quantum fault tolerance is to calculate error thresholds on basic gates in the limit of arbitrarily many concatenation levels. In contrast this paper takes the number of qubits and the target implementation accuracy as given, and provides a framework for engineering the constrained quantum system to the required tolerance. The approach requires solving the full dynamics of the quantum system for an arbitrary admixture (biased or unbiased) of Pauli errors. The inaccuracy between ideal and implemented quantum systems is captured by the supremum of the Schatten k-norm of the difference between the ideal and implemented density matrices taken over all density matrices. This is a more complete analysis than the standard approach, where an intricate combination of worst case assumptions and combinatorial analysis is used to analyze the special case of equiprobable errors. Conditions for fault tolerance are now expressed in terms of error regions rather than a single number (the standard error threshold). In the important special case of a stochastic noise model and a single logical qubit, an optimization over all 2×2 density matrices is required to obtain the full dynamics. The complexity of this calculation is greatly simplified through reduction to an optimization over only three projectors. Error regions are calculated for the standard 5- and 7-qubit codes. Knowledge of the full dynamics makes it possible to design sophisticated concatenation strategies that go beyond repeatedly using the same code, and these strategies can achieve target fault tolerance thresholds with fewer qubits.
We extend standard fault tolerance theory and Kitaev’s model for quantum computation so as to enable quantitative determination of design parameters for quantum computers that ensure that the overall computation yields a correct final result with some prescribed probability, as opposed to merely ensuring that the desired final quantum state is obtained. Our extension allows us to explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result with some prescribed success probability and demonstrate a clear connection between error correction and fault tolerance.
Fault tolerance is an indispensable prerequisite for constructing large-scale universal quantum computers. Drawing philosophies from classical computer architecture, this paper presents a hardware-agnostic three-layer high-level architectural framework for generic fault-tolerant quantum computation. Guided by the real execution workflows of fault-tolerant quantum algorithms, the proposed model is decoupled from specific physical qubit hardware platforms and quantum error correction codes, serving as a universal abstract standard rather than a platform-specific implementation scheme. Special attention is devoted to the intermediate Fault-Tolerance Layer, which serves as the architectural bridge between application-level logical programs and hardware-level execution. We systematically characterize its five internal components, the interfaces and data exchanged among them, and the execution, correction, and adaptation paths that together enable logical synthesis, fault-tolerant resources management, decoding, and runtime fault-tolerant control. An end-to-end example is further provided to illustrate the full-stack operating pipeline of fault-tolerant quantum algorithms under this framework. Given the increasing emphasis on modular, heterogeneous, and cross-layer fault-tolerant quantum systems, our architecture provides a unified foundational model for organizing such designs.