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arXiv:0910.0660 · Physical Review A

Universal linear Bogoliubov transformations through one-way quantum computation

Ryuji Ukai, Jun–ichi Yoshikawa, Noriaki Iwata, Peter van Loock, Akira Furusawa

Published Mar 16, 2010• 63 citations
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Abstract

We show explicitly how to realize an arbitrary linear unitary Bogoliubov (LUBO) transformation on a multimode quantum state through homodyne-based one-way quantum computation. Any LUBO transformation can be approximated by means of a fixed, finite-sized, sufficiently squeezed Gaussian cluster state that allows for the implementation of beam splitters (in form of three-mode connection gates) and general one-mode LUBO transformations. In particular, we demonstrate that a linear four-mode cluster state is a sufficient resource for an arbitrary one-mode LUBO transformation. Arbitrary-input quantum states including non-Gaussian states could be efficiently attached to the cluster through quantum teleportation.

DOI
10.1103/physreva.81.032315

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We show explicitly how to realize an arbitrary linear unitary Bogoliubov (LUBO) transformation on a multimode quantum state through homodyne-based one-way quantum computation. Any LUBO transformation can be approximated by means of a fixed, finite-sized, sufficiently squeezed Gaussian cluster state that allows for the implementation of beam splitters (in form of three-mode connection gates) and general one-mode LUBO transformations. In particular, we demonstrate that a linear four-mode cluster state is a sufficient resource for an arbitrary one-mode LUBO transformation. Arbitrary-input quantum states including non-Gaussian states could be efficiently attached to the cluster through quantum teleportation.
We describe a generalization of the cluster-state model of quantum computation to continuous-variable systems, along with a proposal for an optical implementation using squeezed-light sources, linear optics, and homodyne detection. For universal quantum computation, a nonlinear element is required. This can be satisfied by adding to the toolbox any single-mode non-Gaussian measurement, while the initial cluster state itself remains Gaussian. Homodyne detection alone suffices to perform an arbitrary multimode Gaussian transformation via the cluster state. We also propose an experiment to demonstrate cluster-based error reduction when implementing Gaussian operations.
One-way quantum computation is a very promising candidate to fulfill the capabilities of quantum information processing. Here we demonstrate an important set of unitary operations for continuous variables using a linear cluster state of four entangled optical modes. These operations are performed in a fully measurement-controlled and completely unconditional fashion. We implement three different levels of squeezing operations and a Fourier transformation, all of which are accessible by selecting the correct quadrature measurement angles of the homodyne detections. Though not sufficient, these linear transformations are necessary for universal quantum computation.
The linear optical creation of Gaussian cluster states, a potential resource for universal quantum computation, is investigated. First, using Bloch-Messiah reduction, we show how to achieve canonical cluster-state generation, otherwise based on pairwise acting quantum nondemolition gates, by off-line squeezers and beam splitters. Moreover, we find that, in terms of squeezing resources, the canonical states are rather wasteful. Hence we propose a systematic way to create a whole family of cluster-type states, including potentially cheaper states. Any given cluster (or graph) state can be realized this way. As an example, we consider a protocol in which a single-mode quantum state propagates through a multiple-rail cluster. Such a redundant encoding may reduce errors due to finite squeezing.
This paper theoretically demonstrates a compact method for the production of cluster states, based on recent experiments with multimode cavities, by using a so-called multi-pixel homodyne detection (MPHD) system and classical post-processing. This effectively allows to incorporate the linear optic network in the stage of the measurement. The physical transformations which the modes undergo due to the measurement (i.e., propagation from the cavity to the multipixel homodyne detector and multipixel homodyne detection) can be modeled by symplectic matrices, acting on the vector which regroups the annihilation operators associated to the various input squeezed mode. The method proposed can mimic the action of a desired matrix Uthif there exists a suitable combination of the experimental parameters such that UMPHD= Uth. It is therefore important to characterize from a theoretical point of view the class of unitary operations which are experimentally implementable by the use of this method. A condition which allows to test the implementability of a given unitary matrix Uth, and to find the experimental parameters to use in order to emulate it is determined. Some examples of cluster state generation by this method are presented. Finally, how this procedure can be generalized to perform Gaussian quantum computation is further analyzed. In particular, the possibility of implementing the elementary operations which compose a Gaussian universal set, including the Fourier transform of a single-mode input state and a two-mode CZinteraction, is studied.
In this paper we propose two strategies for decreasing the error of arbitrary single-mode Gaussian transformations implemented using one-way quantum computation on a four-node linear cluster state. We show that it is possible to minimize the error of the arbitrary single-mode Gaussian transformation by a proper choice of the weight coefficients of the cluster state. We modify the computation scheme by adding a non-Gaussian state obtained using a cubic phase gate as one of the nodes of the cluster. This further decreases the computation error. We evaluate the efficiencies of the proposed optimization schemes comparing the probabilities of the error correction of the quantum computations with and without optimizations. We show that for some transformations, the error probability can be reduced by up to 900 times.
We demonstrate a quadratic phase gate for one-way quantum computation in the continuous-variable regime. This canonical gate, together with phase-space displacements and Fourier rotations, completes the set of universal gates for realizing any single-mode Gaussian transformation such as arbitrary squeezing. As opposed to previous implementations of measurement-based squeezers, the current gate is fully controlled by the local oscillator phase of the homodyne detector. Verifying this controllability, we give an experimental demonstration of the principles of one-way quantum computation over continuous variables. Moreover, we can observe sub-shot-noise quadrature variances in the output states, confirming that nonclassical states are created through cluster computation.