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arXiv:quant-ph/0601066 · Physical Review A

Noise thresholds for optical cluster-state quantum computation

Christopher M. Dawson, Henry L. Haselgrove, Michael A. Nielsen

Published May 9, 2006• 79 citations
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Abstract

In this paper we do a detailed numerical investigation of the fault-tolerant threshold for optical cluster-state quantum computation. Our noise model allows both photon loss and depolarizing noise, as a general proxy for all types of local noise other than photon loss noise. We obtain a threshold region of allowed pairs of values for the two types of noise. Roughly speaking, our results show that scalable optical quantum computing is possible in the combined presence of both noise types, provided that the loss probability is less than $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and the depolarization probability is less than ${10}^{\ensuremath{-}4}$. Our fault-tolerant protocol involves a number of innovations, including a method for syndrome extraction known as telecorrection, whereby repeated syndrome measurements are guaranteed to agree. This paper is an extended version of Dawson et al. [Phys. Rev. Lett. 96, 020501 (2006)].

DOI
10.1103/physreva.73.052306

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In this paper we do a detailed numerical investigation of the fault-tolerant threshold for optical cluster-state quantum computation. Our noise model allows both photon loss and depolarizing noise, as a general proxy for all types of local noise other than photon loss noise. We obtain a threshold region of allowed pairs of values for the two types of noise. Roughly speaking, our results show that scalable optical quantum computing is possible in the combined presence of both noise types, provided that the loss probability is less than 3\ifmmode×\else\texttimes\fi10\ensuremath−33\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}3\ifmmode×\else\texttimes\fi10\ensuremath−3 and the depolarization probability is less than 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4. Our fault-tolerant protocol involves a number of innovations, including a method for syndrome extraction known as telecorrection, whereby repeated syndrome measurements are guaranteed to agree. This paper is an extended version of Dawson et al. [Phys. Rev. Lett. 96, 020501 (2006)].
The one-way quantum computing model introduced by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)] shows that it is possible to quantum compute using only a fixed entangled resource known as a cluster state, and adaptive single-qubit measurements. This model is the basis for several practical proposals for quantum computation, including a promising proposal for optical quantum computation based on cluster states [M. A. Nielsen, Phys. Rev. Lett. (to be published), quant-ph/0402005]. A significant open question is whether such proposals are scalable in the presence of physically realistic noise. In this paper we prove two threshold theorems which show that scalable fault-tolerant quantum computation may be achieved in implementations based on cluster states, provided the noise in the implementations is below some constant threshold value. Our first threshold theorem applies to a class of implementations in which entangling gates are applied deterministically, but with a small amount of noise. We expect this threshold to be applicable in a wide variety of physical systems. Our second threshold theorem is specifically adapted to proposals such as the optical cluster-state proposal, in which nondeterministic entangling gates are used. A critical technical component of our proofs is two powerful theorems which relate the properties of noisy unitary operations restricted to act on a subspace of state space to extensions of those operations acting on the entire state space. We expect these theorems to have a variety of applications in other areas of quantum-information science.
Quantum information processing is the coherent manipulation of a quantum state for the purpose of performing an information processing task. It is known that certain tasks can in principle be performed much more efficiently using quantum information processing rather than ordinary classical processing. Examples include factoring, secure key distribution, and the simulation of quantum systems. However, significant theoretical and technological obstacles stand in the road to achieving these promised benefits. It is exceedingly difficult to construct devices that are able to achieve an extensive level of control of a quantum state while keeping that state well isolated from the effects of noise. Although there is considerable experimental effort underway to build devices that are less noisy and more controllable, there is also a significant theoretical program aimed at finding schemes that make such physical limitations have less effect on the overall reliability of a device. The crowning achievement is the theory of fault-tolerant quantum error-correction, which shows that a noisy quantum device can be efficiently made to behave as though it were noise free, so long as the amount of noise present is below the “noise threshold”. A central result of this thesis is the calculation of the noise threshold for optical cluster-state quantum computing. Optical cluster-state quantum computing is one of the most promising proposals for the physical implementation of a quantum computer (i.e., a generic quantum information processing device). Previous studies of the value of the threshold, that considered other physical implementations, do not apply to optical quantum computing due to the unusual features of the optical proposal such as nondeterministic gates and photon loss. We present the first detailed analysis of the value of the noise threshold for this proposal. Our analysis involves a number of innovations, including a method for error-correction known as telecorrection, whereby repeated error-syndrome measurements are guaranteed to agree due to the use of teleportation during the correction process.  This thesis also considers how to overcome limits to the amount of physical control able to be applied to a quantum device. We ask, if a quantum device can only control limited parts of its quantum state, can that device still be used to achieve a useful information processing task? We consider simple networks of interacting quantum spins where only a few of the spins are controllable, and consider the problem of using this system for high-fidelity quantum communication (i.e., using the system as a simple “quantum wire”). Without control, such systems generally yield a very poor communication fidelity. We show that a very simple scheme that involves controlling a small number of the spins can be used to greatly increase the fidelity across the entire network. The scheme is designed using techniques of state encoding. The thesis also considers the problem of engineering the interactions in a quantum device so that the ground state is a quantum error-correcting code. Such a problem is an important part of the proposal for naturally fault-tolerant systems, which have the ability to resist noise whilst using little or no external control. Our results prove that a certain important class of quantum error-correcting codes, the so called nondegenerate codes, cannot be the eigenstate of any physically-plausible quantum system. This result places significant restrictions on the design of naturally fault-tolerant devices, and sheds light on why current proposals for natural fault tolerance use codes that are degenerate.
We investigate a scheme for topological quantum computing using optical hybrid qubits and make an extensive comparison with previous all-optical schemes. We show that the photon loss threshold reported by Omkar et al. [Phys. Rev. Lett. 125, 060501 (2020)] can be improved further by employing postselection and multi-Bell-state-measurement-based entangling operations to create a special cluster state, known as Raussendorf lattice for topological quantum computation. In particular, the photon loss threshold is enhanced up to $5.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which is the highest reported value given a reasonable error model. This improvement is obtained at the price of consuming more resources by an order of magnitude compared with the scheme in the aforementioned reference. Nevertheless, this scheme remains resource-efficient compared with other known optical schemes for fault-tolerant quantum computation.
A scheme for linear optical implementation of fault-tolerant quantum computation is proposed, which is based on an error-detecting code. Each computational step is mediated by transfer of quantum information into an ancilla system embedding error-detection capability. Photons are assumed to be subjected to both photon loss and depolarization, and the threshold region of their strengths for scalable quantum computation is obtained, together with the amount of physical resources consumed. Compared to currently known results, the present scheme reduces the resource requirement, while yielding a comparable threshold region.
Abstract Fault-tolerant cluster states form the basis for scalable measurement-based quantum computation. Recently, new stabilizer codes for scalable circuit-based quantum computation have been introduced that have very high thresholds under biased noise where the qubit predominantly suffers from one type of error, e.g. dephasing. However, extending these advances in stabilizer codes to generate high-threshold cluster states for biased noise has been a challenge, as the standard method for foliating stabilizer codes to generate fault-tolerant cluster states does not preserve the noise bias. In this work, we overcome this barrier by introducing a generalization of the cluster state that allows us to foliate stabilizer codes in a bias-preserving way. As an example of our approach, we construct a foliated version of the XZZX code which we call the XZZX cluster state. We demonstrate that under a circuit-level-noise model, our XZZX cluster state has a threshold more than double the usual cluster state when dephasing errors are more likely than errors that cause bit flips by a factor of order ~100 or more.