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arXiv:0706.3401 · Physical Review A

Measurement-based quantum computation beyond the one-way model

D. Gross, Jens Eisert, Norbert Schuch, David Pérez-Garcı́a

Published Nov 19, 2007• 236 citations
arXivHTMLPDF

Abstract

We introduce schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in Gross and Eisert [Phys. Rev. Lett. 98, 220503 (2007); quant-ph/0609149]. Our method makes use of tools from many-body physics---matrix product states, finitely correlated states, or projected entangled pairs states---to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem---how to realize quantum computation---was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting and present a large number of examples. We find computational schemes, which differ from the original one-way computer, for example, in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may, for example, exhibit nonvanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.

DOI
10.1103/physreva.76.052315

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We introduce schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in Gross and Eisert [Phys. Rev. Lett. 98, 220503 (2007); quant-ph/0609149]. Our method makes use of tools from many-body physics---matrix product states, finitely correlated states, or projected entangled pairs states---to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem---how to realize quantum computation---was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting and present a large number of examples. We find computational schemes, which differ from the original one-way computer, for example, in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may, for example, exhibit nonvanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.
We establish a framework which allows one to systematically construct novel schemes for measurement-based quantum computation. The technique utilizes 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 also identify resource states with radically different entanglement properties than the cluster state. It is shown that there exist universal resource states which are locally arbitrarily close to a pure state. We find that non-vanishing two-point correlation functions are no obstacle to universality. An explicit example for a resource state is presented, which can partly be prepared by gates with non-maximal entangling power. Finally, we comment on the possibility of tailoring computational models to specific physical systems as, e.g. in linear optical experiments.
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.
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.
Abstract: Quantum computation is a novel way of information processing which 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 phys-ical 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. 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 en-tanglement and quantum correlations. We also give examples for ground states of simple Hamiltonians which enable universal quantum computation by local measurements. 1 ar
Fermions, as a major class of quantum particles, provide platforms for quantum information processing beyond the possibilities of spins or bosons, which have been studied more extensively. One particularly interesting model to study, in view of recent progress in manipulating ultracold fermion gases, is the fermionic version of measurement-based quantum computation (MBQC), which implements full quantum computation with only single-site measurements on a proper fermionic many-body resource state. However, it is not known which fermionic states can be used as the resource states for MBQC and how to find them. In this paper, we generalize the framework of spin MBQC to fermions. In particular, we provide a general formalism to construct many-body entangled fermion resource states for MBQC based on the fermionic projected entangled pair state representation. We give a specific fermionic state which enables universal MBQC and demonstrate that the nonlocality inherent in fermion systems can be properly taken care of with suitable measurement schemes. Such a framework opens up possibilities of finding MBQC resource states which can be more readily realized in the laboratory.
We present strictly efficient schemes for scalable measurement-based quantum computing using continuous-variable systems: These schemes are based on suitable non-Gaussian resource states, ones that can be prepared using interactions of light with matter systems or even purely optically. Merely Gaussian measurements such as optical homodyning as well as photon counting measurements are required, on individual sites. These schemes overcome limitations posed by Gaussian cluster states, which are known not to be universal for quantum computations of unbounded length, unless one is willing to scale the degree of squeezing with the total system size. We establish a framework derived from tensor networks and matrix product states with infinite physical dimension and finite auxiliary dimension general enough to provide a framework for such schemes. Since in the discussed schemes the logical encoding is finite dimensional, tools of error correction are applicable. We also identify some further limitations for any continuous-variable computing scheme from which one can argue that no substantially easier ways of continuous-variable measurement-based computing than the presented one can exist.