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arXiv:quant-ph/9702029 · Physical Review A

Theory of fault-tolerant quantum computation

Daniel Gottesman

Published Jan 1, 1998• 949 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.57.127

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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 states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.
Performing quantum computing that is robust against noise will require that all operations are fault-tolerant, meaning that they succeed with high probability even if a limited number of errors occur. We address the problem of fault-tolerantly implementing logical operators on quantum error-correcting codes – operators that apply logic gates to information protected by such codes. Specifically, we investigate what classes of logical operators are possible by particular approaches in important types of codes, especially topological stabiliser codes. We also analyse what fundamental limitations constrain the goal of realising fault-tolerant quantum computing by such implementations and how these limitations can be overcome. \n \nWe begin by presenting necessary background theory on quantum computing, quantum error-correcting codes and fault tolerance. We then specifically consider the approach to fault tolerance of locality-preserving logical operators in topological stabiliser codes. We present a method for determining the set of such operators admitted by a wide range of such codes and apply this method to important examples such as surface codes and colour codes. Next, we consider the alternative approach of implementing logical operators in topological stabiliser codes with defects, especially by the technique of braiding. We show that such approaches are fundamentally limited, but that effective schemes can nonetheless be constructed, both within these limitations and by circumventing them. We then consider fault tolerance in a more general context. We prove a highly general no-go theorem in this context, applicable to a wide range of stabiliser codes. We also show that this proof illuminates how it can be circumvented and provides perspective on a range of fault-tolerant schemes. Finally, we conclude by reviewing how these results collectively address our research questions and suggesting future work.
The first theme of this book-quantum error correction-has now been pre­ sented. We carry on in this chapter with a discussion of fault-tolerant quantum computing (FTQC), which is our second theme. The discussion will take place in the context of quantum stabilizer codes and is based on Refs. [1-6]. See also Refs. [7,8]. Section 5.1 introduces the idea of fault-tolerance and makes ba­ sic definitions, while Section 5.2 presents a procedure for doing fault-tolerant quantum error correction. The theory of FTQC is then taken up in the fol­ lowing five sections. Section 5.3 examines the action of unitary operators on a quantum stabilizer code. It introduces two important groups: (i) the normalizer of the stabilizer S whose elements are encoded unitary operations; and (ii) the Clifford group which is generated by the Hadamard, phase, and CNOT gates. The discussion focuses on unitary operations that belong to the inter­ section of these two groups. By combining measurements on ancilla qubits (Section 5.4) with a four-qubit unitary operation introduced in Section 5.5, it will be possible to fault-tolerantly implement these gates on encoded data for stabilizer codes that encode a single qubit. Section 5.6 presents an extension of this approach that works for stabilizer codes encoding multiple qubits, and Section 5.7 completes the theoretical development by showing how a faulttolerant encoded Toffoli gate can be constructed for any quantum stabilizer code. By this point a theoretical framework is in place that can construct a set of fault-tolerant encoded quantum gates that is capable of universal quan­ tum computation using any quantum stabilizer code. The final two sections illustrate the theory by applying it to the [5,1,3] and [4,2,2] stabilizer codes. A brief summary of universal sets of quantum gates is given in Appendix C.
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.