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Quantum Information and Computation

Threshold error rates for the toric and planar codes

D.S. Wang, Austin G. Fowler, Ashley M. Stephens, Lloyd C. L. Hollenberg

Published May 1, 2010• 90 citations
Publisher

Abstract

The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1\%. This result was obtained for the toric code, from which the planar code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and planar codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By applying one and two qubit depolarizing errors of equal probability $p$, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of $p_{\textrm{th}} = 15.5\%$ under ideal syndrome extraction, and $p_{\textrm{th}} = 7.8 \times 10^{-3}$ for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.

DOI
10.26421/qic10.5-6-6

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Building a quantum computer is a formidable challenge. In this thesis, we focus on two projects, which tackle very different aspects of quantum computation, and yet still share a common goal in hopefully getting us closer to implementing a quantum computer on a large scale. The first project involves a numerical error threshold calculation of a quantum error correcting code called a surface code. These are local check codes, which means that only nearest neighbour interaction is required to determine where errors occurred. This is an important advantage over other approaches, as in many physical systems, doing operations on arbitrarily spaced qubits is often very difficult. An error threshold is a measure of how well a given error correcting scheme performs. It gives the experimentalists an idea of which approaches to error correction hold greater promise. We simulate both toric and planar variations of a surface code, and numerically calculate a threshold value of approximately $6.0 \\times 10^{-3}$, which is comparable to similar calculations done by others \\cite{Raussendorf2006,Raussendorf2007,Wang2009}. The second project deals with coupling superconducting flux qubits together. It expands the scheme presented in \\cite{Plourde2004} to a three qubit, two coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the coupling strength changes in terms of the dimensions of the couplers. We explore two cases, the first where the interaction energy between two nearest neighbour qubits is high, while the coupling to the third qubit is as negligible as possible, as well as a case where all the coupling energies are as small as possible. Although only an initial step, a similar scheme can in principle be extended further to implement a lattice required for computation on a surface code.

•Oct 2, 2009•0 citations
The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1%. This result was obtained for the toric code, from which the planar code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and planar codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By applying one and two qubit depolarizing errors of equal probability ppp, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of pth=15.5%p_{\textrm{th}} = 15.5\%pth​=15.5% under ideal syndrome extraction, and pth=7.8×10−3p_{\textrm{th}} = 7.8 \times 10^{-3}pth​=7.8×10−3 for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.
The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By direct simulation of this error correction scheme, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of p = 15.5% under ideal syndrome extraction, and p = 7.8 10^-3 for the non-ideal case, in agreement with prior work. Simulations of the surface code indicate that the threshold is close to that of the toric code.
Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1% (NJoP 9:199, 2007), (arXiv:0905.0531). However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The colour code of (PRL 97:180501, 2006) is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the colour code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.1%.
Quantum computing platforms based on arrays of tweezer-confined neutral atoms have recently emerged as a competitive modality thanks to a direct path toward high qubit count, rapidly advancing operation fidelities, and their ability to execute circuits with arbitrary qubit connectivity. These features will enable the use of efficient error correction schemes with high encoding-rates, time-efficient decoding, and resource-efficient architectures based on transversal gates. With these goals in mind, recent state of the art neutral atom demonstrations focus on the transition from the use of physical qubits to error-corrected logical qubits, but to date there has been no demonstration of repeated error correction scalable to arbitrary depth. Here, we demonstrate many cycles of syndrome extraction in a toric quantum error correcting code, using mid-circuit measurement and replacement of lost qubits, including reloading of a qubit reservoir for indefinite coherent operation. We characterize the logical error rate after up to 90 cycles, showing that logical information can be preserved through multiple rounds of qubit reloading. Comparing two distances of the code up to 8 rounds of syndrome extraction shows a lower absolute logical error rate for the larger distance code.
Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1\%. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The color code of is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the color code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25\% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3\%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.10 \pm 0.01\%.
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.
Building a quantum computer is a formidable challenge. In this thesis, we focus on two projects, which tackle very different aspects of quantum computation, and yet still share a common goal in hopefully getting us closer to implementing a quantum computer on a large scale. The first project involves a numerical error threshold calculation of a quantum error correcting code called a surface code. These are local check codes, which means that only nearest neighbour interaction is required to determine where errors occurred. This is an important advantage over other approaches, as in many physical systems, doing operations on arbitrarily spaced qubits is often very difficult. An error threshold is a measure of how well a given error correcting scheme performs. It gives the experimentalists an idea of which approaches to error correction hold greater promise. We simulate both toric and planar variations of a surface code, and numerically calculate a threshold value of approximately 6.0times10−36.0 \\times 10^{-3}6.0times10−3, which is comparable to similar calculations done by others \\cite{Raussendorf2006,Raussendorf2007,Wang2009}. The second project deals with coupling superconducting flux qubits together. It expands the scheme presented in \\cite{Plourde2004} to a three qubit, two coupler scenario. We study L-shaped and line-shaped coupler geometries, and show how the coupling strength changes in terms of the dimensions of the couplers. We explore two cases, the first where the interaction energy between two nearest neighbour qubits is high, while the coupling to the third qubit is as negligible as possible, as well as a case where all the coupling energies are as small as possible. Although only an initial step, a similar scheme can in principle be extended further to implement a lattice required for computation on a surface code.