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Resources for measurement−based quantum computation: A unifying view

Simon Perdrix

Published Jan 1, 2005• 1 citation
OpenAlex

Abstract

Alternative quantum computing models to the standard unitary-based model have been independently introduced and developed. One of them, introduced by Nielsen [11] is based on projective measurements only. Another, the one-way quantum computation introduced by Briegel and Raussendorf [13] consists in applying local measurements on an offline-prepared entangled state, called cluster state. The model via measurements only has been successively improved [12], decreasing the resources (in terms of number of different projective measurements) used for a universal quantum computation. The last improvement is based on state transfer [15]. The resources used by the one-way quantum computation are not limited to local measurements, since the preparation of the initial cluster state has to be taken into account. It turns out that the resources used by the one-way quantum computation if the cluster state is prepared by means of projective measurements [9] are similar to the resources used by the quantum computation via measurements only based on state transfer. Moreover, the measurement-based preparation of the initial cluster state leads to a natural unification of these models. If, instead of a measurement-based preparation, a unitary-based preparation of the cluster state is considered, a natural unification between the one-way quantum computation and an hybrid model allowing both unitary transformation and projective measurements is obtained [3].

Related papers

Unifying Quantum Computation with Projective Measurements only and One-Way Quantum Computation

Philippe Jorrand, Perdrix, S

Abstract. Quantum measurement is universal for quantum computation (Nielsen [4], Raussendorf [7,8]). Two models for performing measurement-based quantum computation exist: the one-way quantum computer was introduced by Briegel and Raussendorf [7], and quantum computation via projective measurements only by Nielsen [4]. The more recent development of this second model is based on state transfers [6] instead of teleportation. From this development, a finite but approximate quantum universal family of observables is exhibited, which includes only one two-qubit observable, while others are one-qubit observables [6]. In this article, an infinite but exact quantum universal family of observables is proposed, including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation, i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison, which was initiated by Cirac and Verstraete [9], closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation. 1

•Jan 1, 2004•29 citations

Unifying quantum computation with projective measurements only and one-way quantum computation

Philippe Jorrand, Simon Perdrix

Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.

arXiv:quant-ph/0404125•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jun 1, 2005•12 citations

Focus on Measurement-Based Quantum Information Processing

Terry Rudolph, Jian-Wei Pan

The last two decades have witnessed the rapid growth of quantum information processing as an active field of multidisciplinary science and technology. Tremendous theoretical and experimental progress has been made in quantum communication and quantum computation, and it seems clear that some of this progress will soon result in practical applications outside the laboratory. Quantum computation aims to build a quantum computer and to explore the origins and limits of quantum computing power. The original circuit model of a quantum computer, due to Deutsch, has a strong formal similarity with the circuit model of a classical computer. Both models depict a computation as processing via the wiring of a number of fiducial gates, each of which dynamically couples a small number of input (qu)bits in a controlled manner. In other words, unitary evolution is the basic mechanism of information processing in the circuit model of quantum computers. Starting with a given fixed state of many qubits, measurement based quantum computation (MBQC) processes information by applying a sequence of measurements to designated qubits in designated bases. This is remarkable, given the probabilistic nature of quantum measurement and the murky physical status of the dynamics of 'wavefunction collapse'. The final result of the computation is determined from the classical data of all the measurement outcomes. There exist two principle models: the teleportation based model and the 'one-way quantum computer' of 'cluster state computation'. The formalism of cluster states has proven to be a powerful way of describing the essential entanglement resources needed to perform quantum information processing. Unlike the circuit model, the MBQC models have no natural classical counterpart. There are a number of physical architectures for which measurements on the relevant systems are more easily performed than controlled unitary evolution. Moreover, it appears the MBQC models can offer novel computational benefits, especially regarding issues of fault-tolerance and parallelizability of algorithms. As a result, the theory of MBQC is now shaping the latest experimental proposals across the full spectrum of quantum information processing technologies. Naturally, understanding the scope of measurement based quantum computation is also yielding many insights into what is necessary (as opposed to merely sufficient) for us to harness the power of quantum computers. Quantum communication aims to offer efficient and secure ways of information exchange, directly based on quantum mechanics. Photons serve as natural information carriers, owing to their robustness against decoherence. A major problem for photonic communication systems is due to photon loss. It turns out that this difficulty can be overcome by the quantum repeater, a measurement-based technique. The basic idea is to divide the transmission channel into many segments and perform some quantum measurement on each segment. Additional measurement-based protocols and technologies have also been demonstrated to improve the quality of quantum communication, such as entanglement swapping and purification. The articles in this invited Focus Issue cover topics both in measurement-based quantum communication and computation, but the emphasis is given to quantum computation. A significant number of articles deal with the preparation of cluster states, the experimental realization of a one-way quantum computer, and the associated efficiency and benefits. We hope that you will find that some of these new results are of interest to you, and further hope that the issue will increase your interest in this interdisciplinary research field if you are a non-expert. 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New Journal of Physics•Jun 29, 2007•2 citations

Measurement-based quantum computation on cluster states

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We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.

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STATE TRANSFER INSTEAD OF TELEPORTATION IN MEASUREMENT-BASED QUANTUM COMPUTATION

Simon Perdrix

Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, four auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of one one-qubit measurement and four two-qubit measurements. We introduce a model of quantum computation via measurements only, relying on state transfer: state transfer only retains the part of teleportation which is necessary for computation. In order to simulate any n-qubit unitary transformation with this new model, only one auxiliary qubit is required. Moreover we exhibit a universal family of observables composed of three one-qubit measurements and only one two-qubit measurement. This model improves those of Nielsen and Leung in terms of both the number of auxiliary qubits and the number of two-qubit measurements required for quantum universality. In both cases, the minimal amounts of necessary resources are now reached: one auxiliary qubit (because measurement is destructive) and one two-qubit measurement (for creating entanglement).

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D. Gross, Jens Eisert

We establish a framework which allows one to construct novel schemes for measurement-based quantum computation. The technique develops tools from many-body physics-based on finitely correlated or projected entangled pair states-to go beyond the cluster-state based one-way computer. We identify resource states radically different from the cluster state, in that they exhibit nonvanishing correlations, can be prepared using nonmaximally entangling gates, or have very different local entanglement properties. In the computational models, randomness is compensated in a different manner. It is shown that there exist resource states which are locally arbitrarily close to a pure state. We comment on the possibility of tailoring computational models to specific physical systems.

arXiv:quant-ph/0609149•Physical Review Letters•May 31, 2007•253 citations
Alternative quantum computing models to the standard unitary-based model have been independently introduced and developed. One of them, introduced by Nielsen [11] is based on projective measurements only. Another, the one-way quantum computation introduced by Briegel and Raussendorf [13] consists in applying local measurements on an offline-prepared entangled state, called cluster state. The model via measurements only has been successively improved [12], decreasing the resources (in terms of number of different projective measurements) used for a universal quantum computation. The last improvement is based on state transfer [15]. The resources used by the one-way quantum computation are not limited to local measurements, since the preparation of the initial cluster state has to be taken into account. It turns out that the resources used by the one-way quantum computation if the cluster state is prepared by means of projective measurements [9] are similar to the resources used by the quantum computation via measurements only based on state transfer. Moreover, the measurement-based preparation of the initial cluster state leads to a natural unification of these models. If, instead of a measurement-based preparation, a unitary-based preparation of the cluster state is considered, a natural unification between the one-way quantum computation and an hybrid model allowing both unitary transformation and projective measurements is obtained [3].
Abstract. Quantum measurement is universal for quantum computation (Nielsen [4], Raussendorf [7,8]). Two models for performing measurement-based quantum computation exist: the one-way quantum computer was introduced by Briegel and Raussendorf [7], and quantum computation via projective measurements only by Nielsen [4]. The more recent development of this second model is based on state transfers [6] instead of teleportation. From this development, a finite but approximate quantum universal family of observables is exhibited, which includes only one two-qubit observable, while others are one-qubit observables [6]. In this article, an infinite but exact quantum universal family of observables is proposed, including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation, i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison, which was initiated by Cirac and Verstraete [9], closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation. 1
Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.
The last two decades have witnessed the rapid growth of quantum information processing as an active field of multidisciplinary science and technology. Tremendous theoretical and experimental progress has been made in quantum communication and quantum computation, and it seems clear that some of this progress will soon result in practical applications outside the laboratory. Quantum computation aims to build a quantum computer and to explore the origins and limits of quantum computing power. The original circuit model of a quantum computer, due to Deutsch, has a strong formal similarity with the circuit model of a classical computer. Both models depict a computation as processing via the wiring of a number of fiducial gates, each of which dynamically couples a small number of input (qu)bits in a controlled manner. In other words, unitary evolution is the basic mechanism of information processing in the circuit model of quantum computers. Starting with a given fixed state of many qubits, measurement based quantum computation (MBQC) processes information by applying a sequence of measurements to designated qubits in designated bases. This is remarkable, given the probabilistic nature of quantum measurement and the murky physical status of the dynamics of 'wavefunction collapse'. The final result of the computation is determined from the classical data of all the measurement outcomes. There exist two principle models: the teleportation based model and the 'one-way quantum computer' of 'cluster state computation'. The formalism of cluster states has proven to be a powerful way of describing the essential entanglement resources needed to perform quantum information processing. Unlike the circuit model, the MBQC models have no natural classical counterpart. There are a number of physical architectures for which measurements on the relevant systems are more easily performed than controlled unitary evolution. Moreover, it appears the MBQC models can offer novel computational benefits, especially regarding issues of fault-tolerance and parallelizability of algorithms. As a result, the theory of MBQC is now shaping the latest experimental proposals across the full spectrum of quantum information processing technologies. Naturally, understanding the scope of measurement based quantum computation is also yielding many insights into what is necessary (as opposed to merely sufficient) for us to harness the power of quantum computers. Quantum communication aims to offer efficient and secure ways of information exchange, directly based on quantum mechanics. Photons serve as natural information carriers, owing to their robustness against decoherence. A major problem for photonic communication systems is due to photon loss. It turns out that this difficulty can be overcome by the quantum repeater, a measurement-based technique. The basic idea is to divide the transmission channel into many segments and perform some quantum measurement on each segment. Additional measurement-based protocols and technologies have also been demonstrated to improve the quality of quantum communication, such as entanglement swapping and purification. The articles in this invited Focus Issue cover topics both in measurement-based quantum communication and computation, but the emphasis is given to quantum computation. A significant number of articles deal with the preparation of cluster states, the experimental realization of a one-way quantum computer, and the associated efficiency and benefits. We hope that you will find that some of these new results are of interest to you, and further hope that the issue will increase your interest in this interdisciplinary research field if you are a non-expert. Focus on Measurement-Based Quantum Information Processing Contents A proof-of-principle experiment of eliminating photon-loss errors in cluster states Wei-Bo Gao, Xiao-Qi Zhou, Jin Zhang, Tao Yang and Jian-Wei Pan Continuous variable multipartite entanglement and optical implementations of quantum communication networks Yimin Lian, Changde Xie and Kunchi Peng Generalized flow and determinism in measurement-based quantum computation Daniel E Browne, Elham Kashefi, Mehdi Mhalla and Simon Perdrix The entanglement of the four-photon cluster state C Schmid, N Kiesel, W Wieczorek and H Weinfurter Optical zeno gate: bounds for fault tolerant operation Patrick M Leung and Timothy C Ralph Towards experimental entanglement connection with atomic ensembles in the single excitation regime Julien Laurat, Chin-wen Chou, Hui Deng, Kyung Soo Choi, Daniel Felinto, Hugues de Riedmatten and H J Kimble Towards minimal resources of measurement-based quantum computation Simon Perdrix Experimental realization of a quantum game on a one-way quantum computer Robert Prevedel, Andre Stefanov, Philip Walther and Anton Zeilinger Fundamentals of universality in one-way quantum computation M Van den Nest, W Dur, A Miyake and H J Briegel Loss tolerant linear optical quantum memory by measurement-based quantum computing Michael Varnava, Daniel E Browne and Terry Rudolph Graph state generation with noisy mirror-inverting spin chains Stephen R Clark, Alexander Klein, Martin Bruderer and Dieter Jaksch One-way quantum computing in adecoherence-free subspace M S Tame, M Paternostro and M S Kim Cluster state preparation using gates operating at arbitrary success probabilities K Kieling, D Gross and J Eisert Topological fault-tolerance in cluster state quantum computation R Raussendorf, J Harrington and K Goyal Strategies for the preparation of large cluster states using non-deterministic gates Peter P Rohde and Sean D Barrett A repeat-until-success quantum computing scheme A Beige, Y L Lim and L C Kwek Efficient growth of complex graph states via imperfect path erasure Earl T Campbell, Joseph Fitzsimons, Simon C Benjamin and Pieter Kok Investigation of a single-photon source based on quantum interference T B Pittman, J D Franson and B C Jacobs Entanglement and local information access for graph states Damian Markham, Akimasa Miyake and Shashank Virmani The efficiencies of generating cluster states with weak nonlinearities Sebastien G R Louis, Kae Nemoto, W J Munro and T P Spiller A direct approach to fault-tolerance in measurement-based quantum computation via teleportation Marcus Silva, Vincent Danos, Elham Kashefi and Harold Ollivier Experimental ancilla-assisted qubit transmission against correlated noise T Yamamoto, R Nagase, J Shimamura, S K Ozdemir, M Koashi and N Imoto Terry Rudolph, Imperial College, London, UK Jian-Wei Pan, University of Science and Technology of China, Hefei, People's Republic of China and Universitat Heidelberg, Germany
We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.
Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, four auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of one one-qubit measurement and four two-qubit measurements. We introduce a model of quantum computation via measurements only, relying on state transfer: state transfer only retains the part of teleportation which is necessary for computation. In order to simulate any n-qubit unitary transformation with this new model, only one auxiliary qubit is required. Moreover we exhibit a universal family of observables composed of three one-qubit measurements and only one two-qubit measurement. This model improves those of Nielsen and Leung in terms of both the number of auxiliary qubits and the number of two-qubit measurements required for quantum universality. In both cases, the minimal amounts of necessary resources are now reached: one auxiliary qubit (because measurement is destructive) and one two-qubit measurement (for creating entanglement).
We establish a framework which allows one to construct novel schemes for measurement-based quantum computation. The technique develops tools from many-body physics-based on finitely correlated or projected entangled pair states-to go beyond the cluster-state based one-way computer. We identify resource states radically different from the cluster state, in that they exhibit nonvanishing correlations, can be prepared using nonmaximally entangling gates, or have very different local entanglement properties. In the computational models, randomness is compensated in a different manner. It is shown that there exist resource states which are locally arbitrarily close to a pure state. We comment on the possibility of tailoring computational models to specific physical systems.