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arXiv:1206.0800 · Physical Review Letters

Proof of Finite Surface Code Threshold for Matching

Austin G. Fowler

Published Nov 2, 2012• 55 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.109.180502

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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.
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.
Quantum error correction will be a necessary component towards realizing scalable quantum computers with physical qubits. Theoretically, it is possible to perform arbitrarily long computations so long as the error rate is below a threshold value. The two-dimensional surface code permits relatively high fault-tolerant thresholds at the ∼ 1% level, and only requires a latticed network of qubits with nearest-neighbor interactions. I will discuss our implementation of a sub-section of the larger fabric using three transmon qubits and two linking microwave resonators. We demonstrate high-fidelity control over the sub-section surface code strand, verified via quantum prcoess tomography and randomized benchmarking experiments. Our fixed-frequency qubit approach relies on the two-qubit cross-resonance microwave driving interaction, which is now one of many microwave-based entangling gate protocols. I will also discuss the prospects to scale to surface code plaquette level experiments.
Quantum error correction will be a necessary component towards realizing scalable quantum computers with physical qubits. Theoretically, it is possible to perform arbitrarily long computations if the error rate is below a threshold value. The two-dimensional surface code permits relatively high fault-tolerant thresholds at the ~1% level, and only requires a latticed network of qubits with nearest-neighbor interactions. Superconducting qubits have continued to steadily improve in coherence, gate, and readout fidelities, to become a leading candidate for implementation into larger quantum networks. Here we describe characterization experiments and calibration of a system of four superconducting qubits arranged in a planar lattice, amenable to the surface code. Insights into the particular qubit design and comparison between simulated parameters and experimentally determined parameters are given. Single- and two-qubit gate tune-up procedures are described and results for simultaneously benchmarking pairs of two-qubit gates are given. All controls are eventually used for an arbitrary error detection protocol described in separate work [Corcoles et al., Nature Communications, 6, 2015].
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.
We present a comprehensive and self-contained simplified review of the quantum computing scheme of Raussendorf et al. [Phys. Rev. Lett. 98, 190504 (2007); N. J. Phys. 9, 199 (2007)], which features a two-dimensional nearest-neighbor coupled lattice of qubits, a threshold error rate approaching 1%, natural asymmetric and adjustable strength error correction, and low overhead arbitrarily long-range logical gates. These features make it one of the best and most practical quantum computing schemes devised to date. We restrict the discussion to direct manipulation of the surface code using the stabilizer formalism, both of which we also briefly review, to make the scheme accessible to a broad audience.
The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studies for the threshold with correlated error, they are only the lower bound rather than the exact value, which offers potential for higher error thresholds. Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the maximum-likelihood threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We present analytical constraints for the error correction threshold of surface codes, and by applying these constraints, we numerically obtain the error threshold higher than the existing values. Our error threshold is both an upper bound and achievable.