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Noise Thresholds for Optical Cluster-State Quantum Computation

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Noise Thresholds for Optical Cluster-State Quantum Computation PHYSICAL REVIEW A 73, 052306 ͑2006͒ Noise thresholds for optical cluster-state quantum computation Christopher M. Dawson,1 Henry L. Haselgrove,1,2,* and Michael A. Nielsen1 1School of Physical Sciences, The University of Queensland, Queensland 4072, Australia 2Information Sciences Laboratory, Defence Science and Technology Organisation, Edinburgh 5111, Australia ͑Received 23 January 2006; published 9 May 2006͒ 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 3ϫ10−3 and the depolarization probability is less than 10−4. Our fault-tolerant protocol involves a number of innovations, including a method for syndrome extraction known as telecorrection, whereby repeated syndrome measure- ments are guaranteed to agree. This paper is an extended version of Dawson et al. ͓Phys. Rev. Lett. 96, 020501 ͑2006͔͒. DOI: 10.1103/PhysRevA.73.052306 PACS number͑s͒: 03.67.Lx I. INTRODUCTION further simplified in ͓14͔, where it is estimated that only tens of optical elements will be required to implement a single Optical systems have many significant advantages for logical gate. The resulting proposal for optical cluster-state quantum computation, such as the ease of performing single- quantum computing thus appears to offer an extremely prom- qubit manipulations, long decoherence times, and efficient ising approach to quantum computation. Recent experiments readout. Unfortunately, standard linear-optical elements ͓15͔ have demonstrated the construction of simple optical alone are unsuitable for quantum computation, as they do not cluster states. A recent review of work on optical quantum enable photons to interact. This difficulty can, in principle, computation, including cluster-based approaches, is ͓16͔. be resolved by making use of nonlinear-optical elements While the optical cluster-state proposals ͓11,14͔ present ͓1,2͔, at the price of requiring large nonlinearities that are encouraging progress, for them to be considered credible ap- currently difficult to achieve. proaches to fully scalable quantum computation, it is neces- An alternate approach was developed by Knill, Laflamme, sary to consider the effects of noise. In particular, it is nec- and Milburn ͑KLM͓͒3͔, who proposed using measurement essary to establish a noise threshold theorem for the optical to effect entangling interactions between optical qubits. Us- cluster-state proposals. A noise threshold theorem proves the ing this idea, KLM developed a scheme for scalable quantum existence of a constant noise threshold value, such that pro- computation based on linear-optical elements, together with vided the amount of noise per elementary operation is below high-efficiency photodetection, feedforward of measurement this level, it is possible to efficiently simulate a quantum computation of arbitrary length, to arbitrary accuracy, using results, and single-photon generation. KLM thus showed that appropriate error-correction techniques. In the standard scalable optical quantum computation is in principle pos- ͓ ͔ quantum circuit model of computation such a threshold has sible. Experimental demonstrations 4–8 of several of the been known to exist since the mid-1990s ͑see Chapter 10 of basic elements of KLM’s scheme have been achieved. ͓17͔ for an introduction and references͒. However, the opti- Despite these successes, the obstacles to fully scalable cal cluster-state proposals are not based on the circuit model, optical quantum computation with the KLM approach re- but rather on the cluster-state model of computation, and thus main formidable. The biggest challenge is to perform a two- a priori it is not obvious that a similar noise threshold need qubit entangling gate in the near-deterministic fashion re- hold. quired for scalable quantum computation. KLM propose an Fortunately, recent work ͓18–20͔ has shown that the fault- ingenious scheme showing that this is possible in principle, tolerance techniques developed for the circuit model can be but with a considerable overhead: doing a single entangling adapted for use in the cluster-state model, and used to prove gate with high probability of success requires tens of thou- ͑ ͓ ͔͒ the existence of a noise threshold for noisy cluster-state com- sands of optical elements. Several proposals e.g., 9,10 puting. The earliest work ͓18,19͔ established the existence of have been made to reduce this overhead, but it still remains a threshold for clusters, without obtaining a value. Reference formidable even in these improved schemes. ͓ ͔ ͓ ͔͑ ͓ ͔͒ 20 argued that in a specific noise model, the cluster thresh- A recent proposal 11 cf. 12 combines the basic ele- old is no more than an order of magnitude lower than the ments of the KLM approach with the cluster-state model of ͓ ͔ ͓ ͔ threshold for circuits. The most recent work 21 combines quantum computation 13 to achieve a reduction in com- ideas from cluster-state computing with topological error plexity of many orders of magnitude. This scheme has been correction to obtain a cluster threshold. However, neither ͓20͔ nor ͓21͔ is of direct relevance to the optical cluster-state proposal, since they make use of deterministic entangling *Electronic address: [email protected] gates, which are not available in linear optics, and the noise 1050-2947/2006/73͑5͒/052306͑26͒ 052306-1 ©2006 The American Physical Society DAWSON, HASELGROVE, AND NIELSEN PHYSICAL REVIEW A 73, 052306 ͑2006͒ model does not include any process analogous to photon sible. The main outcome of our paper is a series of threshold loss. regions, with the different regions corresponding to varying The present paper studies in detail the value of the noise assumptions regarding the relative noise strength of quantum threshold for optical cluster-state computing. We use numeri- memory, and the use of different quantum error-correcting cal simulations to estimate the threshold for a particular codes. Qualitatively, we find that our fault-tolerant protocols fault-tolerant protocol, for two different quantum codes. The are substantially more resistant to photon loss noise than they ͓ ͔ paper is an extended version of an earlier work 22 , which are to depolarizing noise, with threshold values of approxi- ͑ ͒ provided an overview but few details of the protocol and mately 6ϫ10−3 for photon loss noise ͑in the limit of zero simulation techniques used, and only a brief summary of depolarization noise͒, and 3ϫ10−4 for depolarizing noise ͑in results. the limit of no photon loss͒. When both types of noise are Our threshold analysis is tailored to the dominant sources present in the system, a typical value in the threshold region of noise in optical implementations of quantum computing. has a strength of 3ϫ10−3 for photon loss noise, and a depo- In particular, our simulations involve three different sources larization probability of 10−4. of noise: ͑a͒ the inherent nondeterminism of the entangling ͑ ͒ ͑ ͒ Our fault-tolerant protocol involves a number of innova- gates used to build up the cluster; b photon loss; and c ͑ ͒ depolarizing noise. The strength of the noise source ͑a͒ is tions in addition to those already described, including 1 the regarded as essentially fixed, while the strengths of ͑b͒ and development of special techniques to deal with the inherent ͑ ͒ ͑c͒ are regarded as variables that can be changed by im- nondeterminism of the entangling optical gates; 2 heavy proved engineering. Note that most existing work on thresh- use of the ability to parallelize cluster-state computations ͓ ͔ olds ͑e.g., ͓20,21,23,24͔͒ in either clusters or circuits focuses 27 , and the ability to do as much of the computation offline ͑ ͒ on abstract noise models based on depolarizing noise, and as possible; and 3 as a special case of the previous point, neglects sources ͑a͒ and ͑b͒. we develop a method for doing fault-tolerant syndrome mea- Noise sources ͑a͒ and ͑b͒ likely dominate actual experi- surement which we call telecorrection. This has the striking ments, and our protocol attempts to cope with these very property that repeated measurements of the syndrome are ͑ ͒ efficiently. The protocols for decoding and correction can be guaranteed to agree which helps increase the threshold , un- made to take advantage of the knowledge the experimenter like in standard protocols, where measurements only some- has of the locations of error types ͑a͒ and ͑b͒. For example, times agree. the well-known Steane seven-qubit code is usually used to The structure of the paper is as follows. In Sec. II we correct a depolarization error on a single qubit. A more effi- briefly overview the required background on cluster-state cient use of the code is possible, in which it is used to correct computation and the optical cluster-state proposal. Section photon loss or nondeterministic gate failure errors on as III describes our assumptions about the physical setting: many as two qubits. what physical resources we are allowed, what quantum gates Although noise sources ͑a͒ and ͑b͒ will dominate, sources we can perform, and what noise is present in the system. of noise other than ͑a͒ and ͑b͒ will also be present in experi- Section IV describes briefly how we simulate noisy cluster- ments, and so it is important that our fault-tolerant protocol state computations. This is a surprisingly subtle topic, due to and analysis also deals with those. This is why we include the multiple noise sources in our model, which is why it noise source ͑c͒, as a proxy for all additional noise effects.
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