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arXiv:0805.3202

Topological cluster state quantum computing

Austin G. Fowler, Kovid Goyal

Published May 21, 2008• 0 citations
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Abstract

The quantum computing scheme described in Phys. Rev. Lett. 98, 190504 (2007), when viewed as a cluster state computation, features a 3-D cluster state, novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only, a threshold error rate approaching 1% and low overhead arbitrarily long-range logical gates. In this work, we review the scheme in detail framing discussion solely in terms of the required 3-D cluster state and its stabilizers.

DOI
10.48550/arxiv.0805.3202

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The quantum computing scheme described in Phys. Rev. Lett. 98, 190504 (2007), when viewed as a cluster state computation, features a 3-D cluster state, novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only, a threshold error rate approaching 1% and low overhead arbitrarily long-range logical gates. In this work, we review the scheme in detail framing discussion solely in terms of the required 3-D cluster state and its stabilizers.
Introduction Cluster states were introduced in Chapter 18, along with some of the approaches to achieving fault tolerance in measurement-based quantum computing. In this chapter, we describe an extremely promising fault-tolerant cluster state quantum computing scheme [RH07, RHG07] with a threshold error rate of 7.46 × 10 −3 , low overhead arbitrarily long-range logical gates and novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only. Detailed proposed implementations of this scheme exist for ion traps [SJ09] and single photons with cavity mediated interactions [DGI+07]. The discussion is organized as follows. In Section 20.2, we describe the topological cluster state, which is a specific three-dimensional (3D) cluster state, and give a brief overview of what topological cluster state quantum computing involves. Section 20.3 describes logical qubits in more detail and how to initialize them to ∣0 L 〉 and ∣+ L 〉 and measure them in the Z L and X L bases. State injection, the non-fault-tolerant construction of arbitrary logical states, is covered in Section 20.4. Logical gates, namely the logical identity gate and the logical CNOT gate, are discussed in Section 20.5 along with their byproduct operators. Section 20.6 describes the errorcorrection procedure. In Section 20.7, we calculate an estimate for the threshold of this scheme. Section 20.8 presents an analysis of the overhead as a function of both the circuit size and the error rate.
Topological error correction--a novel method to actively correct errors based on cluster states with topological properties--has the highest order of tolerable error rates known to date (10^{-2}). Moreover, the scheme requires only nearest-neighbour interaction, particularly suitable for most physical systems. Here we report the first experimental demonstration of topological error correction with an 8-qubit optical cluster state. In the experiment, it is shown that a correlation can be protected against a single error on any single qubit. In addition, when all qubits are simultaneously subjected to errors with equal probability, the effective error rate is significantly reduced, clearly verifying the advantage of topological error correction. The quantum gate with the error rate below the threshold is within the current experimental technology. We believe topological error correction should be a critical ingredient for the future large-scale quantum computation.
Topological error correction provides an effective method to correct errors in quantum computation. It allows quantum computation to be implemented with a higher error threshold and high tolerating loss rates. We present a topological error correction scheme with continuous variables based on an eight-partite Gaussian cluster state. We show that topological quantum correlation between two modes can be protected against a single quadrature phase displacement error occurring on any mode and some of two errors occurring on two modes. More interestingly, some cases of errors occurring on three modes can also be recognized and corrected, which is different from the topological error correction with discrete variables. We show that the final error rate after correction can be further reduced if the modes are subjected to identical errors occurring on all modes with equal probability. The presented results provide a feasible scheme for topological error correction with continuous variables and it can be experimentally demonstrated with a Gaussian cluster state.