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arXiv:1209.0510

A bridge to lower overhead quantum computation

Austin G. Fowler, Simon J. Devitt

Published Sep 4, 2012• 58 citations
arXivHTMLPDF

Abstract

Two primary challenges stand in the way of practical large-scale quantum computation, namely achieving sufficiently low error rate quantum gates and implementing interesting quantum algorithms with a physically reasonable number of qubits. In this work we address the second challenge, presenting a new technique, bridge compression, which enables remarkably low volume structures to be found that implement complex computations in the surface code. The surface code has a number of highly desirable properties, including the ability to achieve arbitrarily reliable computation given sufficient qubits and quantum gate error rates below approximately 1%, and the use of only a 2-D array of qubits with nearest neighbor interactions. As such, our compression technique is of great practical relevance.

DOI
10.48550/arxiv.1209.0510

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Two primary challenges stand in the way of practical large-scale quantum computation, namely achieving sufficiently low error rate quantum gates and implementing interesting quantum algorithms with a physically reasonable number of qubits. In this work we address the second challenge, presenting a new technique, bridge compression, which enables remarkably low volume structures to be found that implement complex computations in the surface code. The surface code has a number of highly desirable properties, including the ability to achieve arbitrarily reliable computation given sufficient qubits and quantum gate error rates below approximately 1%, and the use of only a 2-D array of qubits with nearest neighbor interactions. As such, our compression technique is of great practical relevance.
The topological quantum error correction (TQEC) scheme is promising for scalable and reliable quantum computing. A TQEC circuit can be modeled by a three-dimensional diagram, and the implementation resource of a TQEC circuit is abstracted to its space-time volume. Implementing a quantum algorithm with a reasonable physical qubit number and reasonable computation time is challenging for large-scale practical problems. Therefore, minimizing the space-time volume of a TQEC circuit becomes a crucial issue. Previous work shows that bridge compression can greatly compress TQEC circuits, but it was performed only manually. It is desirable to develop automated compression techniques for TQEC circuits to achieve low-overhead, large-scale quantum computations. In this paper, we present the first work that can automatically perform bridge compression on TQEC circuits. Compared with the state-of-the-art method, experimental results show that our proposed algorithm can averagely reduce space-time volumes by 83%.
Large-scale quantum computation will only be achieved if experimentally implementable quantum error correction procedures are devised that can tolerate experimentally achievable error rates. We describe an improved decoding algorithm for the Kitaev surface code, which requires only a two-dimensional square lattice of qubits that can interact with their nearest neighbors, that raises the tolerable quantum gate error rate to over 1%. The precise maximum tolerable error rate depends on the error model, and we calculate values in the range 1.1--1.4% for various physically reasonable models. These values represent a very high threshold error rate calculated in a constrained setting.
The field of quantum computation currently lacks a formal proof of experimental feasibility. Qubits are fragile and sophisticated quantum error correction is required to achieve reliable quantum computation. The surface code is a promising quantum error correction code, requiring only a physically reasonable 2D lattice of qubits with nearest neighbor interactions. However, existing proofs that reliable quantum computation is possible using this code assume the ability to measure four-body operators and, despite making this difficult to realize assumption, require that the error rate of these operator measurements is less than 10(-9), an unphysically low target. High error rates have been proved tolerable only when assuming tunable interactions of strength and error rate independent of distance, which is also unphysical. In this work, given a 2D lattice of qubits with only nearest neighbor two-qubit gates, and single-qubit measurement, initialization, and unitary gates, all of which have error rate p, we prove that arbitrarily reliable quantum computation is possible provided p < 7.4 × 10(-4), a target that many experiments have already achieved. This closes a long-standing open problem, formally proving the experimental feasibility of quantum computation under physically reasonable assumptions.
Computer architectures which exploit quantum mechanical effects can solve computing tasks that are otherwise impossible to perform. A quantum computer operates on a number of small quantum mechanical systems, known as quantum bits, or qubits. Since these systems are realized on the scale of atoms, they are very prone to errors. Errors occur when the environment interacts with the qubits, a process called decoherence. It is widely accepted that it will not be possible to shield qubits completely from the outside world. If one were to perform a quantum computation on the qubits directly, then after a short period of time the information present in the qubits would be lost. To counter decoherence the state of a qubit can be encoded into multiple physical ones. This is called a quantum error correcting code. Performing quantum error correction allows one to extend the life time of the encoded qubit arbitrarily, assuming that the rate of errors remains below a certain threshold value. The use of quantum codes creates an overhead in resources, as for every logical qubit many more physical qubits are needed. The resource overhead for fault-tolerance is problematic, since realizing qubits will be costly, and in the early stages of building quantum computers the number of physical qubits will be limited. The currently favored coding architecture is the toric code and its variant the surface code in which the physical qubits are put on a square grid in which interactions are only between nearest neighbors. In this thesis we will explore quantum codes in which qubits interact as if they were nearest neighbors in more exotic spaces. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnitude. We also show how to read and write the encoded qubits while keeping their connectivity low. In the second part we consider codes in which qubits are layed-out according to a four- dimensional geometry. Such codes allow for much simpler decoding schemes compared to codes which are two-dimensional. In particular, measurements do not necessarily have to be repeated to obtain reliable information about the error and the classical hardware performing the error correction is greatly simplified. We perform numerical simulations to analyze the performance of these codes using decoders based on local updates. We also introduce a novel decoder based on techniques from machine learning and image recognition to decode four-dimensional codes.
In quantum computer architectures, low connectivity poses a major challenge in implementing efficient error correction codes such as surface codes that require high connectivity degrade when direct interaction between qubits is limited, making accurate error correction and reliable computation difficult. To implement surface codes with distances of 2 and 3 in quantum computer structures with limited connectivity, we analyze resources by the number of qubits and by the number of CNOT gates, suggesting a way to expand connectivity using routing qubits and SWAP gates.
The fragile nature of quantum information limits our ability to construct large quantities of quantum bits suitable for quantum computing. An important goal, therefore, is to minimize the amount of resources required to implement quantum algorithms, many of which are serial in nature and leave large numbers of qubits idle much of the time unless compression techniques are used. Furthermore, quantum error-correcting codes, which are required to reduce the effects of noise, introduce additional resource overhead. We consider a strategy for quantum circuit optimization based on topological deformation in the surface code, one of the best performing and most practical quantum error-correcting codes. Specifically, we examine the problem of minimizing computation time on a two-dimensional qubit lattice of arbitrary, but fixed dimension, and propose two algorithms for doing so.