This thesis considers various paradigms of quantum computation in an attempt to understand the nature of the underlying physics. A standard approach is to consider unitary computation on pure input states, such that the outcome of the computation is determined by single computational basis measurement on the output state. It has been shown that there exists equivalent models of computation, such as measurement based quantum computing (MBQC), which provide insight into the role of entanglement and measurement. Furthermore, constraining or relaxing available resources can directly impacts the power of the computation, allowing one to gauge their role in the process. Here, we first extend known constructions such as Matrix Product States, MBQC and the oneclean qubit model to a mixed state formalism, in an attempt to develop computational models where noise acting on the physical resources, as might be experienced in laboratory settings, may be mapped to logical noise on the computation. Next, we introduce Measurement-Based Classical Computing, an essentially classical model of computation, wherein the complexity hard wired into probability distributions generated via quantum means yields surprising non classical results. Finally, we consider postselection the ability to discard displeasing measurement outcomes and argue that it may be used in a tame way, which does not provide a dramatic increase in computational power. From here, we develop a new Hamiltonian, based on a circuit to Hamiltonian construction, presenting evidence of QMA-hardness.
This thesis considers various paradigms of quantum computation in an attempt to understand the nature of the underlying physics. A standard approach is to consider unitary computation on pure input states, such that the outcome of the computation is determined by single computational basis measurement on the output state. It has been shown that there exists equivalent models of computation, such as measurement based quantum computing (MBQC), which provide insight into the role of entanglement and measurement. Furthermore, constraining or relaxing available resources can directly impacts the power of the computation, allowing one to gauge their role in the process. Here, we first extend known constructions such as Matrix Product States, MBQC and the oneclean qubit model to a mixed state formalism, in an attempt to develop computational models where noise acting on the physical resources, as might be experienced in laboratory settings, may be mapped to logical noise on the computation. Next, we introduce Measurement-Based Classical Computing, an essentially classical model of computation, wherein the complexity hard wired into probability distributions generated via quantum means yields surprising non classical results. Finally, we consider postselection the ability to discard displeasing measurement outcomes and argue that it may be used in a tame way, which does not provide a dramatic increase in computational power. From here, we develop a new Hamiltonian, based on a circuit to Hamiltonian construction, presenting evidence of QMA-hardness.
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
Measurement-based quantum computation (MBQC) is a model of quantum computation, in which computation proceeds via adaptive single qubit measurements on a multiqubit quantum state. It is computationally equivalent to the circuit model. Unlike the circuit model, however, its classical analog is little studied. Here we present a classical analog of MBQC whose computational complexity presents a rich structure. To do so, we identify uniform families of quantum computations [refining the circuits introduced by Bremner Proc. R. Soc. A 467, 459 (2010)] whose output is likely hard to exactly simulate (sample) classically. We demonstrate that these circuit families can be efficiently implemented in the MBQC model without adaptive measurement and, thus, can be achieved in a classical analog of MBQC whose resource state is a probability distribution which has been created quantum mechanically. Such states (by definition) violate no Bell inequality, but, if widely held beliefs about computational complexity are true, they, nevertheless, exhibit nonclassicality when used as a computational resource—an imprint of their quantum origin.
Measurement-Based Quantum Computing (MBQC) is an alternative to the quantum circuit model, whereby the computation proceeds via measurements on an entangled resource state. Noise processes are a major experimental challenge to the construction of a quantum computer. Here, we investigate how noise processes affecting physical states affect the performed computation by considering MBQC on a one-dimensional cluster state. This allows us to break down the computation in a sequence of building blocks and map physical errors to logical errors. Next, we extend the Matrix Product State construction to mixed states (which is known as Matrix Product Operators) and once again map the effect of physical noise to logical noise acting within the correlation space. This approach allows us to consider more general errors than the conventional Pauli errors, and could be used in order to simulate noisy quantum computation.
Measurement-based quantum computation (MBQC) has emerged as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that takes unitary operations as fundamental. Among measurement-based quantum computation methods the recently introduced one-way quantum computer stands out as basic and fundamental. The key idea is to start from an entangled state and then use measurements and one-qubit unitaries, which may be dependent on the outcomes of measurements, to guide the computation. The main point is that one never has to perform unitaries on more than one qubit at a time after the initial preparation of an entangled state. The “programs” that one writes in this model are traditionally called “patterns.” In this chapter, we develop a rigorous mathematical model underlying measurement-based quantum computation. We give syntax, operational semantics, denotational semantics, and an algebra of programs derived from the denotational semantics. We also present a rewrite theory and prove a general standardization theorem that allows all programs to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements, and expressiveness theorems. We use our general measurement calculus not just to formalize the one-way model but also several other measurement-based models, e.g., Teleportation, Phase, and Pauli models, and present compositional embeddings of them into and from the one-way model.
Measurement-based quantum computation (or the one-way quantum computation) is a model in which a series of adaptive single qubit measurements upon an entangled resource state can produce classical output data equivalent to an arbitrary quantum logic circuit. Loosely speaking, the correlations in these measurements can be considered the resource which allows the computation to be performed. Bell inequalities demonstrate that quantum mechanics permits the outcomes of sets of measurements to be correlated in ways incompatible with the predictions of any local hidden variable theory, including classical physics. In the Bell inequality, and multi-party generalisations, a set of single-qubit measurements are performed, on spatially separated systems. Correlations in these measurements can act as a signature that no local hidden variable description of the experiment is possible. At first sight, we see superficial similarities between these settings. In both cases, non-classical correlations play a central role. However, there are also some key differences. Most importantly, measurement-based quantum computing requires adaptive measurements, at odds with the spatial-separated nature of Bell inequality experiments. Nevertheless some striking connections have been reported [1], [2] particularly between GHZ-type paradoxes and deterministic MBQC computations, and between CHSH-type Bell inequalities and non-deterministic MBQC correlations [1], [3]. In my talk I will survey these works and some recent results [3], [4] from my group at UCL, in which a number of connections between multi-party Bell inequalities and MBQC are explored. [1] J. Anders and D.E. Browne, "The Computational Power of Correlations", Physical Review Letters 102, 050502 (2009) [2] R. Raussendorf, "Quantum computation, discreteness, and contextuality", arxiv:0907.5449 [3] M. J. Hoban, E.T. Campbell, K. Loukopolous, D.E. Browne, "Non-adaptive measurement-based quantum computation and multi-party Bell inequalties", in preparation. [4] M. J. Hoban and D.E. Browne, in preparation.
Quantum computation is a novel way of information processing that allows, for certain classes of problems, exponential speedups over classical computation. Various models of quantum computation exist, such as the adiabatic, circuit, and measurement-based models. They have been proven equivalent in their computational power, but operate very differently. As such, they may be suitable for realization in different physical systems, and also offer different perspectives on open questions such as the precise origin of the quantum speedup. Here, we give an introduction to the one-way quantum computer, a scheme of measurement-based quantum computation (MBQC). In this model, the computation is driven by local measurements on a carefully chosen, highly entangled state. We discuss various aspects of this computational scheme, such as the role of entanglement and quantum correlations. We also give examples for ground states of simple Hamiltonians that enable universal quantum computation by local measurements.