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arXiv:quant-ph/0602004 · Physical Review A

Error models for mode mismatch in linear optics quantum computing

Peter P. Rohde, Timothy C. Ralph

Published Jun 9, 2006• 33 citations
arXivHTMLPDF

Abstract

One of the most significant challenges facing the development of linear optics quantum computing (LOQC) is mode mismatch, whereby photon distinguishability is introduced within circuits, undermining quantum interference effects. We examine the effects of mode mismatch on the parity (or fusion) gate, the fundamental building block in several recent LOQC schemes. We derive simple error models for the effects of mode mismatch on its operation, and relate these error models to current fault-tolerant-threshold estimates.

DOI
10.1103/physreva.73.062312

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One of the most significant challenges facing the development of linear optics quantum computing (LOQC) is mode mismatch, whereby photon distinguishability is introduced within circuits, undermining quantum interference effects. We examine the effects of mode mismatch on the parity (or fusion) gate, the fundamental building block in several recent LOQC schemes. We derive simple error models for the effects of mode mismatch on its operation, and relate these error models to current fault-tolerant-threshold estimates.
Proposals for linear optical quantum computation (LOQC) rely on quantum interference between single photons which, when coupled with photon detection, provides a mechanism to generate photonic entanglement. However, ideal quantum interference between photons only occurs if the photons are completely indistinguishable in all degrees of freedom except the one being interfered. In practice, no photons will be perfectly indistinguishable and therefore it is important to understand how using partially-distinguishable photons will affect the quantum information within the linear optical quantum computer. Here we describe how states of partially-distinguishable photons are transformed by linear-optical networks and measured by photon-counting detectors. In addition we show how to calculate the logical content of a given photonic state containing partially-distinguishable photons. Using a realistic model of single photon sources, we then apply these tools to calculate the resulting logical error rates on recent proposals for universal LOQC. Understanding the connection between physical imperfections and logical errors is crucial in assessing the fault-tolerance prospects of a quantum computation platform. Our results will help inform error correction strategies and allow one to specify the necessary fabrication and operational tolerances required for scalability.
Quantum computing is an exciting field that promises great improvements to our ability to solve certain intractable classes of problem. Implementing a quantum computer will require a physical system in which quantum effects are strongly evident. Many different physical systems have been proposed for this task, including ion traps, quantum dots, neutral atoms and photons. This thesis focuses on optical implementations of a quantum computer, and specifically on single-photon designs, as opposed to those utilising coherent states. Photonic systems have shown a great deal of promise for quantum computing tasks since it was shown by Knill, Laflamme and Milburn (KLM) that scalable efficient quantum computing could be performed using only linear optical elements and measurement. This was made viable through the use of error-correcting codes. The work in this thesis covers the use of error-correcting codes in single-photon linear optics quantum computing (LOQC). The central feature of these schemes is the parity code introduced by KLM and used to overcome the gate errors that are inherent in this type of LOQC. First is a discussion of an improved scheme that incorporates the fusion gates used in Browne and Rudolph's cluster state proposal in order to simplify the encoding and gate procedures. This results in high-probability gates with a resource usage comparable to the cluster state schemes. The remainder of the thesis examines the application of further error correcting codes to handle other errors that may arise in any practical implementation of an optical quantum computer. The most significant type of error in any optical system is photon loss, and hence an extension of the parity code is presented that allows loss errors to be detected and corrected. Procedures for performing universal quantum computation whilst employing this encoding are also described. These enable the establishment of loss thresholds for parity-state based LOQC, revealing this code is capable of handling up to 10\% loss in all elements during general computation, or 17\% if it is used solely as an active memory. Although loss is the most common type of optical error, other types of error will also become an issue when attempting to produce large quantum computing systems. An analysis of the error thresholds and resource usage which can be expected when applying a general error correcting code such as the Steane code to this type of LOQC system is also presented. The error thresholds and resource requirements established are greater than those found for cat state optical quantum computing, but fall below those calculated for cluster states. However, the resource requirements of the scheme are significantly smaller than the cluster state requirements for an equivalent circuit. Finally, a new experiment is proposed, achievable with today's technology, for testing the loss-tolerant encoding described earlier. Expansions of the experiment that would demonstrate tolerance of actual experimental error are also discussed.
The realisation of a fault-tolerant universal quantum computer requires the ability to implement and control large scale quantum systems while maintaining physical noise levels sufficiently low to avoid errors in the computation. Fusion-Based Quantum Computation (FBQC) – a framework in which entangling fusion measurements are performed between constant-size entangled resource states – has emerged as a promising approach to achieve this, especially in the context of Linear Optical Quantum Computing (LOQC). Most FBQC schemes proposed so far use physical fusions between two qubits. In this thesis, we study the impact of using multi-way fusions, which act on more physical qubits simultaneously, on the error tolerance and resource overhead of linear optical FBQC architectures. The main challenge of LOQC is that entangling operations are inherently non-deterministic, leading to large resource overheads. The success probability is known to decrease exponentially with the number of input qubits. We introduce novel fusion circuits which succeed at higher rates and/or entangle subsets of inputs upon failure. We present a framework to analyse the impact of noise on these circuits and describe the resulting errors as effective Pauli error channels on the input qubits. We also examine the application of multi-way fusion circuits within a linear optical FBQC architecture. We find that they can be used to generate resource states with significantly reduced Pauli error rates and footprint compared to two-way fusions alone, in turn resulting in linear optical FBQC architectures with higher error tolerance and lower resource overheads. We also investigate their use as primitives of the computation, applied between different resource states, and find evidence that two-way fusions remain preferable in this context due to their lower erasure rate.
A heavy focus for optical quantum computing is the introduction of error correction, and the minimization of resource requirements. We detail a complete encoding and manipulation scheme designed for circuit-model linear optics quantum computing, incorporating scalable operations and loss-tolerant architecture. We also calculate loss thresholds for the gate systems described, with an efficiency of 8383%83 required for the single-qubit operations, and 9090%90 for the two-qubit controlled-NOT gate.
Typically linear optical quantum computing (LOQC) models assume that all input photons are completely indistinguishable. In practice there will inevitably be nonidealities associated with the photons and the experimental setup which will introduce a degree of distinguishability between photons. We consider a nondeterministic optical controlled-NOT gate, a fundamental LOQC gate, and examine the effect of temporal and spectral distinguishability on its operation. We also consider the effect of utilizing nonideal photon counters, which have finite bandwidth and time response.
One intriguing but difficult route to scalable quantum information processing (QIP) is linear optical quantum computing (LOQC). The systematic growth of LOQC is examined in this research. It focuses on significant experimental developments, fault-tolerant methods, and the enduring difficulties influencing the field area. In addition, it highlights the incorporation of Gottesman-Kitaev-Preskill (GKP) encoding which is an essential step towards fault-tolerant quantum computation. We analyze the importance of the Knill-Laflamme-Milburn (KLM) protocol in addressing the probabilistic limits of two-qubit gates. Strong quantum structures are made possible by this synergy, which increases resistance to phase errors and photon loss. Notwithstanding these developments, problems with state preparation, measurement accuracy, and error correction still exist, requiring multidisciplinary efforts to improve methods and investigate fresh ideas.Beyond its scope, the knowledge gathered from LOQC may have an impact on secure quantum communication networks and hybrid quantum systems. Enhancing error correction codes, creating new quantum optical technologies, and encouraging cooperation within quantum computing paradigms should be the main goals of future research. The present work intends to stimulate more developments and push QIP towards scalable and useful quantum computation by reviewing the state of LOQC today and defining strategic directions.