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arXiv:1106.3377 · Physical Review A

Computational power and correlation in a quantum computational tensor network

Keisuke Fujii, Tomoyuki Morimae

Published Mar 30, 2012• 10 citations
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Abstract

We investigate relationships between computational power and correlation in resource states for quantum computational tensor network, which is a general framework for measurement-based quantum computation. We find that if the size of resource states is finite, not all resource states allow correct projective measurements in the correlation space, which is related to nonvanishing two-point correlations in the resource states. On the other hand, for infinite-size resource states, we can always implement correct projective measurements if the resource state can simulate arbitrary single-qubit rotations, since such a resource state exhibits exponentially decaying two-point correlations. This implies that a many-body state whose two-point correlation cannot be upper bounded by an exponentially decaying function cannot simulate arbitrary single-qubit rotations.

DOI
10.1103/physreva.85.032338

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We investigate relationships between computational power and correlation in resource states for quantum computational tensor network, which is a general framework for measurement-based quantum computation. We find that if the size of resource states is finite, not all resource states allow correct projective measurements in the correlation space, which is related to nonvanishing two-point correlations in the resource states. On the other hand, for infinite-size resource states, we can always implement correct projective measurements if the resource state can simulate arbitrary single-qubit rotations, since such a resource state exhibits exponentially decaying two-point correlations. This implies that a many-body state whose two-point correlation cannot be upper bounded by an exponentially decaying function cannot simulate arbitrary single-qubit rotations.
In the framework of quantum computational tensor network [D. Gross and J. Eisert, Phys. Rev. Lett. {\bf98}, 220503 (2007)], which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form, and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straightforward for general resource states.
In the framework of quantum computational tensor network, which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form (or a matrix-product form), and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straightforward for general resource states.
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 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 discuss the potential and limitations of Gaussian cluster states for measurement-based quantum computing. Using a framework of Gaussian-projected entangled pair states, we show that no matter what Gaussian local measurements are performed on systems distributed on a general graph, transport and processing of quantum information are not possible beyond a certain influence region, except for exponentially suppressed corrections. We also demonstrate that even under arbitrary non-Gaussian local measurements, slabs of Gaussian cluster states of a finite width cannot carry logical quantum information, even if sophisticated encodings of qubits in continuous-variable systems are allowed for. This is proven by suitably contracting tensor networks representing infinite-dimensional quantum systems. The result can be seen as sharpening the requirements for quantum error correction and fault tolerance for Gaussian cluster states and points toward the necessity of non-Gaussian resource states for measurement-based quantum computing. The results can equally be viewed as referring to Gaussian quantum repeater networks.
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.