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arXiv:quant-ph/0610082 · Physical Review Letters

Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions

Robert Raussendorf, Jim Harrington

Published May 11, 2007• 812 citations
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Abstract

We present a scheme of fault-tolerant quantum computation for a local architecture in two spatial dimensions. The error threshold is 0.75% for each source in an error model with preparation, gate, storage, and measurement errors.
DOI
10.1103/physrevlett.98.190504

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The performance of fault-tolerant quantum computation with concatenated codes using local gates in small numbers of spatial dimensions is discussed. It is shown that a threshold result still exists in three, two, or one spatial dimensions when next-to-nearest-neighbour gates are available, and explicit constructions are presented. In two or three dimensions, it is also shown how nearest-neighbour gates can give a threshold result. In all cases, it is simply demonstrated that a threshold exists, and no attempt to optimize the error correction circuit or to determine the exact value of the threshold is made. The additional overhead due to the fault-tolerance in both space and time is polylogarithmic in the error rate per logical gate.
We describe a fault-tolerant version of the one-way quantum computer using a cluster state in three spatial dimensions. Topologically protected quantum gates are realized by choosing appropriate boundary conditions on the cluster. We provide equivalence transformations for these boundary conditions that can be used to simplify fault-tolerant circuits and to derive circuit identities in a topological manner. The spatial dimensionality of the scheme can be reduced to two by converting one spatial axis of the cluster into time. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors. The operational overhead is poly-logarithmic in the circuit size.
A scheme for fault-tolerant quantum computation (FTQC) with probabilistic two-qubit gates is proposed. The error thresholds for FTQC are estimated by numerical simulations, where the conditional error probability (the error probability under the condition of ``success'') is taken into account in addition to the failure probability. As a result, it turns out that FTQC is possible even if the failure probability is as high as 10% (5%) under the condition that the conditional error probability is lower than about 0.03% (0.3%). These results mean that the present scheme has a great advantage in using probabilistic two-qubit gates.
A major challenge in practical quantum computation is the ineludible errors caused by the interaction of quantum systems with their environment. Fault-tolerant schemes, in which logical qubits are encoded by several physical qubits, enable to the output of a higher probability of correct logical qubits under the presence of errors. However, strict requirements to encode qubits and operators render the implementation of a full fault-tolerant computation challenging even for the achievable noisy intermediate-scale quantum technology. Especially the threshold for fault-tolerant computation still lacks experimental verification. Here, based on an all-optical setup, we experimentally demonstrate the existence of the threshold for the fault-tolerant protocol. Four physical qubits are represented as the spatial modes of two entangled photons, which are used to encode two logical qubits. The experimental results clearly show that when the error rate is below the threshold, the probability of correct output in the circuit, formed with fault-tolerant gates, is higher than that in the corresponding non-encoded circuit. In contrast, when the error rate is above the threshold, no advantage is observed in the fault-tolerant implementation. The developed high-accuracy optical system may provide a reliable platform to investigate error propagation in more complex circuits with fault-tolerant gates.