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arXiv:1005.2456 · Physical Review Letters

Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors

S. D. Barrett, Thomas M. Stace

Published Nov 9, 2010• 164 citations
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Abstract

Many proposals for fault tolerant quantum computation (FTQC) suffer detectable loss processes. 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.

DOI
10.1103/physrevlett.105.200502

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Many proposals for fault tolerant quantum computation (FTQC) suffer detectable loss processes. 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.
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.
Many proposals for quantum information processing are subject to detectable loss errors. In this Letter, we show that topological error correcting codes, which protect against computational errors, are also extremely robust against losses. We present analytical results showing that the maximum tolerable loss rate is 50%, which is determined by the square-lattice bond percolation threshold. This saturates the bound set by the no-cloning theorem. Our numerical results support this and show a graceful trade-off between tolerable thresholds for computational and loss errors.
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.
The threshold theorem indicates that if errors are all local and their rates are below a certain threshold, it is possible to implement large scale quantum computation with arbitrarily small error based on active quantum error correction (QEC). However, most fault-tolerant quantum computation (FTQC) schemes require enormous overhead to achieve this rate by increasing either the concatenation levels for concatenated codes or the distance (hence the size) for topological codes. As a result, a logical qubit is usually encoded in more than thousands of physical qubits, even when the error rate is below the threshold. In this thesis, we try to reduce the resource overhead of FTQC. We first study a quantum computation scheme using clouds of atoms trapped on atomic chips. It suggests that this scheme is robust to noise to some extent and has potential to be scalable. Secondly, we propose a FTQC scheme of encoding many logical qubits into large code blocks. The error performance of such scheme is numerically studied. Preparation of various ancilla states necessary for computation is also explored. Thirdly, we explore the idea of introducing energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation (HQC) can be implemented in stabilizer codes with the existence of such energy protection. Especially, we studied fault-tolerant HQC in surface codes in detail. This scheme opens the possibility for self-correcting quantum computation in 2D lattice.
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.
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.