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Quantum Information and Computation

Constructing arbitrary Steane code single logical qubit fault-tolerant gates

Austin G. Fowler

Published Sep 1, 2011• 71 citations
Publisher

Abstract

We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.

DOI
10.26421/qic11.9-10-10

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We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.
We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.
We present a method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbtrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation, however, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the 7-qubit Steane code, it is shown that the longest practical gates sequences can be found. We also analyse the practicality of the fault-tolerant approximations of the phase rotation gates used in Shor’s algorithm and find that simple non-fault-tolerant phase rotations are more robust for realistic error rates. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined. In large-scale quantum computation, every qubit of data is encoded across multiple physical qubits to form a logical qubit permitting quantum error correction
Using transversal gates is a straightforward and efficient technique for fault-tolerant quantum computing. Since transversal gates alone cannot be computationally universal, they must be combined with other approaches such as magic state distillation, code switching, or code concatenation to achieve universality. In this paper we propose an alternative approach for universal fault-tolerant quantum computing, mainly based on the code concatenation approach proposed in [T. Jochym-O'Connor and R. Laflamme, Phys. Rev. Lett. 112, 010505 (2014)], but in a nonuniform fashion. The proposed approach is described based on nonuniform concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, as well as the 5-qubit code with the 15-qubit Reed-Muller code, which lead to two 49-qubit and 47-qubit codes, respectively. These codes can correct any arbitrary single physical error with the ability to perform a universal set of fault-tolerant gates, without using magic state distillation.
It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal.This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation.Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes.Here, we demonstrate precisely the existence of such gates.In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement.Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes.We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach.One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates.Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.
Fault-tolerant logical operations for qubits encoded by Calderbank-Shor-Steane codes are discussed, with emphasis on methods that apply to codes of high rate, encoding kkk qubits per block with k>1k>1k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into XXX and ZZZ parts. Optimized methods to move logical qubits around and to achieve controlled-NOT and Toffoli gates are discussed. It is found that the number of time steps required to complete a fault-tolerant quantum computation is the same when k>1k>1k>1 as when k=1k=1k=1.