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arXiv:quant-ph/9912040

Robust quantum computation by simulation

Seth Lloyd, Benjamin Rahn, Charlene Ahn

Published Dec 8, 1999• 4 citations
arXivHTMLPDF

Abstract

Simulation of quantum systems that provide intrinsically fault-tolerant quantum computation is shown to preserve fault tolerance. Errors committed in the course of simulation are eliminated by the natural error-correcting features of the systems simulated. Two examples are explored, toric codes and non-abelian anyons. The latter is shown to provide universal robust quantum computation via simulation.

DOI
10.48550/arxiv.quant-ph/9912040

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Simulation of quantum systems that provide intrinsically fault-tolerant quantum computation is shown to preserve fault tolerance. Errors committed in the course of simulation are eliminated by the natural error-correcting features of the systems simulated. Two examples are explored, toric codes and non-abelian anyons. The latter is shown to provide universal robust quantum computation via simulation.
This thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code. Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead. In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment. Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.
Classically simulating the dynamics of anyonic excitations in two-dimensional quantum systems is likely intractable in general because such dynamics are sufficient to implement universal quantum computation. However, processes of interest for the study of quantum error correction in anyon systems are typically drawn from a restricted class that displays significant structure over a wide range of system parameters. We exploit this structure to classically simulate, and thereby demonstrate the success of, an error-correction protocol for a quantum memory based on the universal Fibonacci anyon model. We numerically simulate a phenomenological model of the system and noise processes on lattice sizes of up to $128\ifmmode\times\else\texttimes\fi{}128$ sites, and find a lower bound on the error-correction threshold of approximately 0.125 errors per edge, which is comparable to those previously known for Abelian and (nonuniversal) non-Abelian anyon models.
In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a theory of fault-tolerant operations on stabilizer codes based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-quantum-bit code.
Quantum error correction and fault-tolerant quantum computation are two fundamental concepts which make quantum computing feasible. While providing a theoretical means with which to ensure the arbitrary accuracy of any quantum circuit, fault-tolerant error correction is predicated upon the robust preparation of logical states. An optimal direct circuit and a more complex fault-tolerant circuit for the preparation of the [[7,1,3]] Steane logical-zero are simulated in the presence of discrete quantum errors to quantify the regime within which fault-tolerant preparation of logical states is preferred.
This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, ηηη, is smaller than a constant threshold, ηcη_cηc​. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.
One reason why building large-scale quantum computers will be particularly challenging is that quantum computers are far more susceptible to making errors than conventional digital computers. I will explain the principles of quantum error correction and fault-tolerant quantum computation, which can enable a properly designed quantum computer with imperfect components to achieve good reliability. Some recent developments in the theory of quantum fault tolerance will be reviewed, including quantum error-correcting codes for systems described by continuous quantum variables, robust storage of quantum information based on topological principles, and applications of quantum error-correcting codes to quantum cryptography.