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arXiv:1207.0253 · Physical Review A

Generating and verifying graph states for fault-tolerant topological measurement-based quantum computing in two-dimensional optical lattices

Jaewoo Joo, Emilio Alba, Juan José García‐Ripoll, Timothy P. Spiller

Published Jul 26, 2013• 4 citations
arXivHTMLPDF

Abstract

We propose two schemes for implementing graph states useful for fault-tolerant topological measurement-based quantum computation in two-dimensional (2D) optical lattices. We show that bilayer cluster and surface-code states can be created by global single-row and controlled-$Z$ operations. The schemes benefit from the accessibility of atom addressing on 2D optical lattices and the existence of an efficient verification protocol which allows us to ensure the experimental feasibility of measuring the fidelity of the system against the ideal graph state. The simulation results show potential for a physical realization toward fault-tolerant measurement-based quantum computation against dephasing and unitary phase errors in optical lattices.

DOI
10.1103/physreva.88.012328

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We propose two schemes for implementing graph states useful for fault-tolerant topological measurement-based quantum computation in two-dimensional (2D) optical lattices. We show that bilayer cluster and surface-code states can be created by global single-row and controlled-ZZZ operations. The schemes benefit from the accessibility of atom addressing on 2D optical lattices and the existence of an efficient verification protocol which allows us to ensure the experimental feasibility of measuring the fidelity of the system against the ideal graph state. The simulation results show potential for a physical realization toward fault-tolerant measurement-based quantum computation against dephasing and unitary phase errors in optical lattices.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2)({Z}^{1/2})(Z1/2), and TTT (Z1/4)({Z}^{1/4})(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the TTT gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-ZZZ errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.
Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation—the construction of fault-tolerant graph states—does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error-correction protocols particularly well suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex yields improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.
In this thesis, we consider steps towards building a useful quantum computer in the presence of realistic errors. We start by presenting a novel scheme for fault-tolerant quantum computation with Rydberg atoms. The scheme uses topological error correction codes to provide fault-tolerance and utilises electromagnetically induced transparency to perform entangling gates between qubits. We examine the benefits of this scheme compared to alternative approaches, and we investigate the errors that are likely to occur and how these can be managed. We then consider how to perform quantum computation with a surface code based quantum computer that has permanently faulty components. We study both faulty links between qubits, which prevent qubits from interacting, and faulty qubits themselves; such faults could affect a wide variety of physical implementations, including schemes that use superconducting qubits and those that use trapped ions. The approach we propose for dealing with these faulty components requires minimal modification to the quantum operations required by the surface code and should therefore be implementable by almost all current and future proposals for surface code based quantum devices. Finally, we consider how to perform fault-tolerant quantum computation with components that fail probabilistically. This section is largely motivated by quantum computation with linear optics and therefore focuses on quantum computation with cluster states. We show that these probabilistic failures can be tolerated providing the failure rate is suitably low, and we present methods for increasing error thresholds in the presence of such errors.
We describe an efficient, fully fault-tolerant implementation of Measurement-Based Quantum Computation (MBQC) in the 3D cluster state. The two key novelties are (i) the introduction of a lattice defect in the underlying cluster state and (ii) the use of the Rudolph-Grover rebit encoding. Concretely, (i) allows for a topological implementation of the Hadamard gate, while (ii) does the same for the phase gate. Furthermore, we develop general ideas towards circuit compaction and algorithmic circuit verification, which we implement for the Reed-Muller code used for magic state distillation. Our performance analysis highlights the overall improvements provided by the new methods.
We present a scheme to improve the noise threshold for fault-tolerant topological one-way computation with constant overhead. Certain cluster states of finite size, say star clusters, are constructed with logical qubits through an efficient verification process to achieve high fidelity. Then, the star clusters are connected near-deterministically with verification to form a three-dimensional cluster state to implement topological one-way computation. The necessary postselection for verification is localized within the star clusters, ensuring the scalability of computation. By using the Steane seven-qubit code for the logical qubits, this scheme works with a high error rate of 22%2 and reasonable resources comparable to or less than those for the other fault-tolerant schemes. A higher noise threshold would be achieved by adopting a larger code.
Fault-tolerant measurement-based quantum computation (MBQC) with recent progress on quantum technologies leads to a promising scalable platform for realizing quantum computation, conducted by preparing a large-scale graph state over many qubits and performing single-qubit measurements on the state. With fault-tolerant MBQC, even if the graph-state preparation suffers from errors occurring at an unknown physical error rate, we can suppress the effect of the errors. Verifying graph states is vital to test whether we can conduct MBQC as desired even with such errors. However, problematically, existing state-of-the-art protocols for graph-state verification by fidelity estimation have required measurements on many copies of the entire graph state and hence have been prohibitively costly in terms of the number of qubits and the runtime. We here construct an efficient alternative framework for testing graph states for fault-tolerant MBQC based on the theory of property testing. Our test protocol accepts with high probability when the physical error rate is small enough to make fault-tolerant MBQC feasible and rejects when the rate is above the threshold of fault-tolerant MBQC. The novelty of our protocol is that we use only a single copy of the NNN-qubit graph state and single-qubit Pauli measurements only on a constant-sized subset of the qubits; thus, the protocol has a constant runtime independently of NNN. Furthermore, we can immediately use the rest of the graph state for fault-tolerant MBQC if the protocol accepts. These results achieve a significant advantage over prior art for graph-state verification in the number of qubits and the total runtime. Consequently, our work offers a new route to a fast and practical framework for benchmarking large-scale quantum state preparation.