ASSESSMENT OF CURRENT RESULTS AND OUTLOOK ON FUTURE EFFORTS

QUANTUM COMPUTING

LINEAR OPTICS QUANTUM COMPUTATION

A. Physical approach and perspective

Optical quantum computing (OQC) exploits measurement-based quantum computing schemes with photons as physical qubits. The interaction between separate photonic qubits is induced by measurement, as opposed to a direct interaction via nonlinear media. The two main physical architectures for OQC are based on proposals by Knill, Laflamme and Milburn [1], the KLM architecture, and by Raussendorf and Briegel [2], the one-way quantum computer with cluster states:

• KLM allows universal and scalable OQC using only single photons, linear optics and photon counting. The by-now seminal KLM work was based on the important findings of Gottesman, Chuang and Nielsen concerning the role of teleportation for universal quantum computing. The physical resources for universal (optical) quantum computation in the KLM scheme are multi-particle entangled states and (entangling) multi-particle projective measurements. A recent review of the scheme along with suggested improvements is presented in [3].
• Cluster state quantum computing has become an exciting alternative to existing proposals for quantum computing, and linear optics is one possible implementation. It consists of a highly entangled multi-particle state, called a cluster state, combined with single-qubit measurements and feed-forward, which are sufficient to implement scalable, universal quantum computation. Different algorithms only require a different "pattern" of single qubit operations on a sufficiently large cluster state. Since only single-particle projections together with the ability to construct the initial highly entangled cluster-state are needed to operate such a one-way quantum computer, the cluster state approach opens up the possibility of universal scalable quantum computation for systems where deterministic two-qubit entangling gates are infeasible or difficult. It only suffices to have the ability to probabilistically entangle pairs of qubits in order to efficiently construct arbitrarily large cluster states, such as post-selection of a beam-splitter based Bell-state analyser.

B. State of the art (November 2004)

Important key elements for linear optics quantum computation, namely the generation of entangled states, quantum state teleportation and entanglement swapping have already been realized early in the field (e.g. teleportation in 1997 and entanglement swapping in 1998). The latest developments include realization of entanglement purification, freely propagating teleported qubits and feed-forward technology.

Several practical designs implementing the KLM scheme have subsequently been developed. Experimental methods for ultra-precise photonic quantum state creation, which serve as ancillas in the measurement-based schemes, now achieve typical fidelities above 99%. Using coincident detection there have been a range of demonstrations of nondeterministic two-qubit gates: a fully-characterized two-photon gate operating with >90% fidelity, four-photon [=CNOT=] gates both with entangled ancilla and with teleportation, a KLM nonlinear sign shift gate and a three-photon simulation of the entangled-ancilla gate. These gates can be made scalable with additional resources. Several of these gates have been used in simple applications such as demonstrations of quantum error encoding and generalized non-destructive quantum measurement circuits of two classical logic gates and Bell measurement for teleportation.

Proposals for the optical implementation of cluster state quantum computing have been put forward recently and are promising significant reductions in physical resources by two orders of magnitude as compared to the original KLM scheme. Separately, a variety of modifications to KLM has been suggested to also reduce resource requirements.

Enabling technologies for OQC are:

• Characterization of photonic quantum states and processes. A complete, tomographic, characterization of individual devices is indispensable for error-correction and has quite progressed within the last years. Quantum process tomography, invented by Chuang and Nielsen, can be used to fully characterize a quantum gate, probing either with a range of input states or a single multi-qubit state. Based on the information gained from a complete process characterization it was recently shown how to estimate (and bound) the error probability per gate.
• Development of single photons and/or entangled photon sources is required for OQC. Currently, no source of timed single photons or entanglement is available. In the meantime, bright, albeit non-deterministic sources of correlated photons or entangled-photon pairs are critical to allow on-going evaluation of circuit technology as it is developed. Ultra bright and compact sources, some fiber-coupled to improve mode quality, have been developed. Relatively brighter sources (though not yet in absolute terms) have been demonstrated using periodically-poled nonlinear waveguides.
• High-fidelity multi-qubit measurements (in the KLM scheme) and reliable preparation of multi-qubit states (in both the KLM and the cluster state scheme).

C. Strengths and weaknesses

Current drawbacks of the OQC approach are low photon creation rates, low photon detection efficiencies, and the difficulties with intermediate storage of photons in a quantum memory (see also section 4.1.3). Advantages are obviously low decoherence (due to the photon's weak coupling to the environment), ultrafast processing, compatibility to fiber optics and integrated optics technologies and, in principle, straightforward scalability of resources. However, the low efficiencies quoted above are presently an important practical limitation to scalability, in the sense that they damp exponentially the success probability of most quantum operations.

D. Challenges

The main challenges for OQC can be summarized as follows:

• To achieve fault-tolerant quantum computing. The basic elements of fault-tolerance for OQC are becoming well understood. It has recently been shown that also optical cluster state QC may be performed in a fault-tolerant manner. Error models for KLM-style OQC have found that error thresholds for gates are above 1.78%.
• To reduce the resources required for OQC further and to find the limiting bounds on the required resources.
• To achieve massive parallelism of qubit processing by investing in source and detector technologies. Specifically, the development of high-flux sources of single photons and of entangled photons as well as photon-number resolving detectors will be of great benefit to achieve this goal.
• To generate high-fidelity, large multi-photon (or, more generally, many-particle) entangled states. This will be of crucial importance for cluster state quantum computing.
• To implement OQC architectures on smaller, integrated circuits. All of the current technologies involve either free space optics or combinations of free-space optics and optical fibers. To achieve long term scale-up, it will be essential to move to waveguide and integrated optics.

E. Key references

[1] E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).

[2] R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).

[3] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135–174, (2007); arXiv:quant-ph/0512071