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arXiv:1001.4860 · Physical Review Letters

Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables

Ryuji Ukai, Noriaki Iwata, Yuji Shimokawa, Seiji Armstrong, Alberto Politi, Jun–ichi Yoshikawa, Peter van Loock, Akira Furusawa

Published Jun 17, 2011• 144 citations
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Abstract

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.

DOI
10.1103/physrevlett.106.240504

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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.
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.
We experimentally demonstrate a controlled-phase gate for continuous variables using a cluster-state resource of four optical modes. The two independent input states of the gate are coupled with the cluster in a teleportation-based fashion. As a result, one of the entanglement links present in the initial cluster state appears in the two unmeasured output modes as the corresponding entangling gate acting on the input states. The genuine quantum character of this gate becomes manifest and is verified through the presence of entanglement at the output for a product two-mode coherent input state. By combining our gate with the recently reported module for single-mode Gaussian operations [R. Ukai et al., Phys. Rev. Lett. 106, 240504 (2011)], it is possible to implement any multimode Gaussian operation as a fully measurement-based one-way quantum computation.
In the previous chapter we have discussed some important quantum information processing techniques based on continuous variables, or ‘qunats’. In this chapter, we describe methods for extending these techniques to quantum computation with optical qunats. We will first show in Section 9.1 how the optical qunat states are initialized using single-mode squeezing, and we give optical implementations of some of the most important single-qunat gates, such as the Heisenberg-Weyl operators and the Fourier transform. In Section 9.2, we study the theory of two-mode Gaussian operations, and we construct a general optical circuit that can implement any two-mode Gaussian operation. In Section 9.3, we introduce the stabilizer formalism for qunats, and use it to derive a Gottesman-Knill theorem. This leads to the general requirement of nonlinear gates for qunats, and their optical implementation will be discussed in Section 9.4. Having thus established the optical implementation of universal quantum computing with continuous variables, we extend this to the one-way model in Section 9.5, where we introduce cluster states for continuous variables. Finally, in Section 9.6, we discuss various quantum error-correction techniques for qunat quantum computing. Single-mode optical qunat gates We have already seen in Chapter 2 that the single qunat gates are fundamentally different from qubit gates. In the context of optical implementations, this difference arises due to the complementary nature of qubits and qunats: qubits that are represented by single photons can be considered particles, whereas qunats are encoded in the wave properties of the optical field.
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.
Measurement-based one-way quantum computation using cluster states as resources provides an efficient model to perform computation and information processing of quantum codes. Arbitrary Gaussian quantum computation can be implemented sufficiently by long single-mode and two-mode gate sequences. However, continuous variable gate sequences have not been realized so far due to an absence of cluster states larger than four submodes. Here we present the first continuous variable gate sequence consisting of a single-mode squeezing gate and a two-mode controlled-phase gate based on a six-mode cluster state. The quantum property of this gate sequence is confirmed by the fidelities and the quantum entanglement of two output modes, which depend on both the squeezing and controlled-phase gates. The experiment demonstrates the feasibility of implementing Gaussian quantum computation by means of accessible gate sequences. Measurement-based one-way quantum computation with cluster states is an efficient route to processing quantum information, yet gate sequences for large states remain elusive. Su et al. present a continuous variable squeezed gate and controlled-phase gate sequence using a six-mode cluster state.