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arXiv:quant-ph/0507192 · Journal of Optics B Quantum and Semiclassical Optics

Adaptive phase measurements in linear optical quantum computation

Timothy C. Ralph, Austin P. Lund, Howard M. Wiseman

Published Sep 14, 2005• 30 citations
arXivHTMLPDF

Abstract

Photon counting induces an effective non-linear optical phase shift in certain states derived by linear optics from single photons. Although this non-linearity is non-deterministic, it is sufficient in principle to allow scalable linear optics quantum computation (LOQC). The most obvious way to encode a qubit optically is as a superposition of the vacuum and a single photon in one mode—so-called ‘single-rail’ logic. Until now this approach was thought to be prohibitively expensive (in resources) compared to ‘dual-rail’ logic where a qubit is stored by a photon across two modes. Here we attack this problem with real-time feedback control, which can realize a quantum-limited phase measurement on a single mode, as has been recently demonstrated experimentally. We show that with this added measurement resource, the resource requirements for single-rail LOQC are not substantially different from those of dual-rail LOQC. In particular, with adaptive phase measurements an arbitrary qubit state can be prepared deterministically .

DOI
10.1088/1464-4266/7/10/007

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The main topic of this thesis is linear optics and the implementation of quantum operations (measurements, quantum channels, and unitary rotations) on optical systems. \nIn the opening chapter the basic notions needed to understand the rest of the thesis will be explained. \nThese notions include defining a quantum state, measurement, quantum channel and the linear optics tool set. \n \nThe work in this thesis takes both fundamental and practical approaches to studying linear optical networks. \nFor instance in the first chapter a proof is provided that shows that any unitary on a single mode Fock state can be realised with linear optics. \nThe proof is constructive, however the approach to realising the unitary is not suitable for experimental implementation \nbecause it requires complicated ancilla states. \nAs in the KLM proposal the procedure works only stochastically however by allowing the size of the ancilla to grow \nthe probability of failure can be made arbitrarly small. \n \nFurthermore we investigate the realisation of arbitrary channels in a specific encoding that we call a $d$-rail encoding. \nThe only ancilla state that we allow is a vacuum ancilliary state and further restrictions were considered (e.g. photon counting). \nA proof is provided that using these resources only random unitaries can be applied deterministically using linear optics. An expression for the optimal probability of success for realising \nmore general channels with these resources is also discussed. \n \nAs a final topic we also investigate the realisation of a quantum non-demolition measurement onto the dual rail qubit space. \nThe investigation is a blend of both fundamental and practical approaches. \nTo begin we employ a modified KLM-like procedure and show that the scheme can be realised perfectly but stochastically. \nThe probability that the proper measurement is made can be made arbitrarly close to one using a suitably large ancilla state. \nIn addition we consider an existing scheme \\cite{gisin10a} which uses practical sources (two single photon sources) to perform the measurement. \nThe scheme does not realise the true measurement but instead has a free parameter in it which is the transmittiviy of a beamsplitter. The measurement will project onto a space that has a vacuum component. By adjusting the transmittivity of this beamsplitter the vacuum component can be made arbitrarly small but only at the expense of the probability of success of the procedure. \nIn this thesis a modification that can be made to eliminate the vacuum component without changing the sources is introduced. The modification is surprisingly simple and only involves the addition of a single beamsplitter. In the proposal for the original amplifier it was used in simulations for DIQKD that included device imperfections. \nTo show the improvement of our modification these DIQKD simulations are reproduced using the modified amplifier and its results are compared to the results of the original amplifier.

•Oct 4, 2011•0 citations
Photon counting induces an effective non-linear optical phase shift in certain states derived by linear optics from single photons. Although this non-linearity is non-deterministic, it is sufficient in principle to allow scalable linear optics quantum computation (LOQC). The most obvious way to encode a qubit optically is as a superposition of the vacuum and a single photon in one mode—so-called ‘single-rail’ logic. Until now this approach was thought to be prohibitively expensive (in resources) compared to ‘dual-rail’ logic where a qubit is stored by a photon across two modes. Here we attack this problem with real-time feedback control, which can realize a quantum-limited phase measurement on a single mode, as has been recently demonstrated experimentally. We show that with this added measurement resource, the resource requirements for single-rail LOQC are not substantially different from those of dual-rail LOQC. In particular, with adaptive phase measurements an arbitrary qubit state can be prepared deterministically .
Photonic quantum systems are among the most promising architectures for quantum computers. It is well known that for dual-rail photons effective non-linearities and near-deterministic non-trivial two-qubit gates can be achieved via the measurement process and by introducing ancillary photons. While in principle this opens a legitimate path to scalable linear optical quantum computing, the technical requirements are still very challenging and thus other optical encodings are being actively investigated. One of the alternatives is to use single-rail encoded photons, where entangled states can be deterministically generated. Here we prove that even for such systems universal optical quantum computing using only passive optical elements such as beam splitters and phase shifters is not possible. This no-go theorem proves that photon bunching cannot be passively suppressed even when extra ancilla modes and arbitrary number of photons are used. Our result provides useful guidance for the design of optical quantum computers.
Photonic quantum systems are among the most promising architectures for quantum computers. It is well known that for dual-rail photons effective non-linearities and near-deterministic non-trivial two-qubit gates can be achieved via the measurement process and by introducing ancillary photons. While in principle this opens a legitimate path to scalable linear optical quantum computing, the technical requirements are still very challenging and thus other optical encodings are being actively investigated. One of the alternatives is to use single-rail encoded photons, where entangled states can be deterministically generated. Here we prove that even for such systems universal optical quantum computing using only passive optical elements such as beam splitters and phase shifters is not possible. This no-go theorem proves that photon bunching cannot be passively suppressed even when extra ancilla modes and arbitrary number of photons are used. Our result provides useful guidance for the design of optical quantum computers.
As primitives for entanglement generation, controlled phase gates have a central role in quantum computing. Especially in ideas realizing instances of quantum computation in linear optical gate arrays, a closer look can be rewarding. In such architectures, all effective nonlinearities are induced by measurements. Hence the probability of success is a crucial parameter of such quantum gates. In this paper, we discuss this question for controlled phase gates that implement an arbitrary phase with one and two control qubits. Within the class of post-selected gates in dual-rail encoding with vacuum ancillas, we identify the optimal success probabilities. We construct networks that allow for implementation using current experimental capabilities in detail. The methods employed here appear specifically useful with the advent of integrated linear optical circuits, providing stable interferometers on monolithic structures.
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.
Any quantum state of the radiation field, sliced in small non-overlapping space-time bins is a collection of single-rail qubits, each spanning the vacuum and single-photon Fock state of a mode. Quantum logic on these qubits would enable arbitrary measurements on information-bearing light, but is hard due to the lack of strong nonlinearities. With unentangled ancilla single-rail qubits, an 888-port interferometer and photon detection, we show any single-rail qubit measurement in the XYXYXY Bloch plane is realizable with success probability 147/256147/256147/256, which beats the prior-known 1/21/21/2 limit.
The main topic of this thesis is linear optics and the implementation of quantum operations (measurements, quantum channels, and unitary rotations) on optical systems. \nIn the opening chapter the basic notions needed to understand the rest of the thesis will be explained. \nThese notions include defining a quantum state, measurement, quantum channel and the linear optics tool set. \n \nThe work in this thesis takes both fundamental and practical approaches to studying linear optical networks. \nFor instance in the first chapter a proof is provided that shows that any unitary on a single mode Fock state can be realised with linear optics. \nThe proof is constructive, however the approach to realising the unitary is not suitable for experimental implementation \nbecause it requires complicated ancilla states. \nAs in the KLM proposal the procedure works only stochastically however by allowing the size of the ancilla to grow \nthe probability of failure can be made arbitrarly small. \n \nFurthermore we investigate the realisation of arbitrary channels in a specific encoding that we call a ddd-rail encoding. \nThe only ancilla state that we allow is a vacuum ancilliary state and further restrictions were considered (e.g. photon counting). \nA proof is provided that using these resources only random unitaries can be applied deterministically using linear optics. An expression for the optimal probability of success for realising \nmore general channels with these resources is also discussed. \n \nAs a final topic we also investigate the realisation of a quantum non-demolition measurement onto the dual rail qubit space. \nThe investigation is a blend of both fundamental and practical approaches. \nTo begin we employ a modified KLM-like procedure and show that the scheme can be realised perfectly but stochastically. \nThe probability that the proper measurement is made can be made arbitrarly close to one using a suitably large ancilla state. \nIn addition we consider an existing scheme \\cite{gisin10a} which uses practical sources (two single photon sources) to perform the measurement. \nThe scheme does not realise the true measurement but instead has a free parameter in it which is the transmittiviy of a beamsplitter. The measurement will project onto a space that has a vacuum component. By adjusting the transmittivity of this beamsplitter the vacuum component can be made arbitrarly small but only at the expense of the probability of success of the procedure. \nIn this thesis a modification that can be made to eliminate the vacuum component without changing the sources is introduced. The modification is surprisingly simple and only involves the addition of a single beamsplitter. In the proposal for the original amplifier it was used in simulations for DIQKD that included device imperfections. \nTo show the improvement of our modification these DIQKD simulations are reproduced using the modified amplifier and its results are compared to the results of the original amplifier.