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arXiv:1708.05627 · Physical Review A

Fault-tolerant quantum computation with nondeterministic entangling gates

James M. Auger, Hussain Anwar, Mercedes Gimeno-Segovia, Thomas M. Stace, Dan E. Browne

Published Mar 16, 2018• 40 citations
arXivHTMLPDF

Abstract

Performing entangling gates between physical qubits is necessary for building a large-scale universal quantum computer, but in some physical implementations---for example, those that are based on linear optics or networks of ion traps---entangling gates can only be implemented probabilistically. In this work, we study the fault-tolerant performance of a topological cluster state scheme with local nondeterministic entanglement generation, where failed entangling gates (which correspond to bonds on the lattice representation of the cluster state) lead to a defective three-dimensional lattice with missing bonds. We present two approaches for dealing with missing bonds; the first is a nonadaptive scheme that requires no additional quantum processing, and the second is an adaptive scheme in which qubits can be measured in an alternative basis to effectively remove them from the lattice, hence eliminating their damaging effect and leading to better threshold performance. We find that a fault-tolerance threshold can still be observed with a bond-loss rate of 6.5% for the nonadaptive scheme, and a bond-loss rate as high as 14.5% for the adaptive scheme.

DOI
10.1103/physreva.97.030301

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Performing entangling gates between physical qubits is necessary for building a large-scale universal quantum computer, but in some physical implementations---for example, those that are based on linear optics or networks of ion traps---entangling gates can only be implemented probabilistically. In this work, we study the fault-tolerant performance of a topological cluster state scheme with local nondeterministic entanglement generation, where failed entangling gates (which correspond to bonds on the lattice representation of the cluster state) lead to a defective three-dimensional lattice with missing bonds. We present two approaches for dealing with missing bonds; the first is a nonadaptive scheme that requires no additional quantum processing, and the second is an adaptive scheme in which qubits can be measured in an alternative basis to effectively remove them from the lattice, hence eliminating their damaging effect and leading to better threshold performance. We find that a fault-tolerance threshold can still be observed with a bond-loss rate of 6.5% for the nonadaptive scheme, and a bond-loss rate as high as 14.5% for the adaptive scheme.
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.
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.
The possibility of practical large-scale quantum computation is largely dictated by the error threshold and qubit overhead of the error correcting scheme. To achieve this goal, architectures that perform fault tolerance quantum computation (FTQC) with 3D cluster states have been designed; however, the current benchmarks for such constructions are still below general experimental requirements. In light of this, we propose a new 3D FTQC scheme by exploiting geometric tolerance to erasure errors using cluster states with novel lattice structures, specifically, by replacing the physical qubits with small-sized error detection codes that allow simple applications of logical CZ gates. We are able to locate Pauli errors locally and effectively convert them into loss errors. With a combination of code concatenation and multi-staged decoding, the scheme is able to achieve higher performance than traditional cubic lattices in protecting quantum information. We have surveyed a wide range of possible concatenation schemes and Monte-Carlo simulations have been performed on the decoders to study their respective scaling behavior. The scheme has demonstrated an overall decrement in active qubit overhead and an improvement of the error threshold, which varies from 10 % to 160% depending on the noise model, indicating a relatively higher rate of fault tolerance compared with previous cluster states-based codes.
Many proposals for fault tolerant quantum computation (FTQC) suffer detectable loss processes. This includes schemes based on optics, trapped atoms or ions and atomic ensembles [1]. Here we show that topological FTQC schemes, which are known to have high error thresholds, are also extremely robust against losses. We demonstrate that these schemes tolerate loss rates up to 24.9%, determined by bond percolation on a cubic lattice. Our numerical results show that these schemes retain good performance when loss and computational errors are simultaneously present.
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.