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arXiv:quant-ph/0303008 · Physical Review Letters

Deterministic Linear Optics Quantum Computation with Single Photon Qubits

Nadav Yoran, Benni Reznik

Published Jul 17, 2003• 138 citations
arXivHTMLPDF

Abstract

We suggest an efficient scheme for quantum computation with linear optical elements, where the qubits are encoded in single photon states. The scheme reduces the resources required per logical gate by several orders of magnitude, compared to an earlier proposal of Knill, Laflamme, and Milburn, while the resource overhead per gate is independent of the length of the computation. A central feature of the scheme, enabling these improvements, is the prior construction of a "linked" photon state designed according to the particular quantum circuit one wishes to process. Once this state has been successfully prepared, the computation is pursued deterministically by a sequence of teleportation steps.

DOI
10.1103/physrevlett.91.037903

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We suggest an efficient scheme for quantum computation with linear optical elements, where the qubits are encoded in single photon states. The scheme reduces the resources required per logical gate by several orders of magnitude, compared to an earlier proposal of Knill, Laflamme, and Milburn, while the resource overhead per gate is independent of the length of the computation. A central feature of the scheme, enabling these improvements, is the prior construction of a "linked" photon state designed according to the particular quantum circuit one wishes to process. Once this state has been successfully prepared, the computation is pursued deterministically by a sequence of teleportation steps.
In the previous chapter we developed some of the basic aspects of quantum information processing with single photons as qubits. Apart from noting the obvious benefit of using light for quantum communication, we identified some difficulties in manipulating quantum information that is encoded in photons. In particular, it is difficult to construct two-qubit gates for photonic qubits. In this chapter we will have to face this difficulty head-on in our discussion of quantum computation with single photons and linear-optical elements. The possibility of a quantum computer based on single photons, linear optics, and photon counting was demonstrated in a landmark paper by Knill, Laflamme, and Milburn in 2001. Their protocol is commonly referred to as the ‘ klm protocol’. In subsequent years, the klm protocol has been dramatically slimmed down in terms of complexity and the necessary resources to create the universal set of quantum gates. For pedagogical reasons we give the streamlined version here, and we briefly describe the original klm protocol in Appendix 2. We will make extensive use of the results in Section 1.4 about mode transformations, and Section 2.3 about cluster-state quantum computing. We start this chapter, however, with a description of linear-optical networks, and how they fail as deterministic quantum computers. In Section 6.2 we discuss the principle of post-selection, and how this can be used to create probabilistic gates with deterministic feed-forward control.
One of the earliest proposals for implementing quantum computation was based on encoding each qubit in two optical modes, each containing exactly one photon. However, it is extremely difficult to unitarily couple two optical modes containing few photons. In 2001, Knill, Laflamme, and Milburn (KLM) found a way to circumvent this restriction and implement efficient quantum computation using only passive linear optics, photodetectors, and single photon sources. This efficient linear optical quantum computing (LOQC) is distinct from other linear optical schemes that are not efficiently scalable. The technological requirements for even the simplest gates in the KLM proposal initially seemed out of reach. However, Ralph, White, Munro and Milburn soon showed that, in principle, demonstrations of key gates and techniques could be achieved with current technology. The objective of the research undertaken in this project was to produce, in three years, a prototype two-qubit gate for photons using the KLM linear optics quantum computation approach, and to develop a blue-print for a multiple qubit device that might be implemented over a longer time scale. A non-deterministic two-qubit gate was demonstrated that operates with high fidelity. A number of key tasks were demonstrated using this gate, and several theoretical investigations of short, medium, and longer term issues for the LOQC approach were successfully completed.
Optical quantum computation was limited to a few photon applications such as quantum cryptography and partial teleportation until 2001. Then the possibility of performing arbitrary logic conditional on certain detection outcomes was introduced by Knill, Laflamme and Milburn ( Nature 2001 409 46). Simple circuits consisting of single photon sources, entangled pair sources, linear optical elements and high efficiency photon counting detectors can be used to develop probabilistic two-photon logic gates. Initially such gates show a low probability of success but can approach the success rate of deterministic (non-linear) logic at the cost of added complexity. These developments have led to the emergence of a novel field of research aimed at improving the efficiency of linear logic and developing the efficient sources and detectors for its implementation. This topical issue of Journal of Optics B: Quantum and Semiclassical Optics is devoted to recent advances in this field. Key to all linear optics gates is the interference effect between separate photons that have no previous history. In this issue we include the first experiment to show such an effect (Rarity et al ), on which many later experiments are based. These interference effects can then be used to entangle separate pairs of particles into higher order entangled states (Zou et al 2005 J. Opt. B: Quantum Semiclass. Opt. 7 119–121) and also to establish entanglement between matter qubits (Kok et al ). Quantum dot single photon sources suitable for linear optics quantum logic are discussed in the paper by Unitt et al while a waveguide source of pair photons is presented by Ducci et al . The difficulties in scaling linear optics quantum computing arise from the low success probabilities of the gates. Introducing a small amount of non-linearity could allow unit efficiency gates to be designed (Munro et al ) while large non-linearities generated by the collective effects in EIT could be used to implement a wide variety of quantum logic functions (Petrosyan). The development of algorithms suitable for optical implementation is addressed by Ellinas et al . Key contributors to this issue are collaborating in the project IST:38864 RAMBOQ which has a primary goal of developing linear optics computation tools and techniques.
Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn [2001, Nature (London) 409, 46] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. The original theory and its improvements are reviewed, and a few examples of experimental two-qubit gates are given. The use of realistic components, the errors they induce in the computation, and how these errors can be corrected is discussed.
We introduce a scheme for linear optics quantum computation, that makes no use of teleported gates, and requires stable interferometry over only the coherence length of the photons. We achieve a much greater degree of efficiency and a simpler implementation than previous proposals. We follow the "cluster state" measurement based quantum computational approach, and show how cluster states may be efficiently generated from pairs of maximally polarization entangled photons using linear optical elements. We demonstrate the universality and usefulness of generic parity measurements, as well as introducing the use of redundant encoding of qubits to enable utilization of destructive measurements--both features of use in a more general context.