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arXiv:1805.02645 · Physical Review A

Implementing quantum algorithms on temporal photonic cluster states

Daiqin Su, Krishna Kumar Sabapathy, Casey R. Myers, Haoyu Qi, Christian Weedbrook, Kamil Brádler

Published Sep 14, 2018• 19 citations
arXivHTMLPDF

Abstract

Implementing quantum algorithms is essential for quantum computation. We study the implementation of three quantum algorithms on a two-dimensional temporal continuous-variable cluster state. We first review the generation of temporal cluster states and the implementation of gates using the measurement-based model. Alongside this we discuss methods to introduce non-Gaussianity into the cluster states. The first algorithm we consider is Gaussian boson sampling in which only Gaussian unitaries need to be implemented. Taking into account the fact that input states are also Gaussian, the errors due to the effect of finite squeezing can be partially corrected and a high fidelity can be achieved, provided a moderate amount of online squeezing is available. This helps to construct a large Gaussian boson sampling machine. The second algorithm is the continuous-variable instantaneous quantum polynomial circuit in which one needs to implement non-Gaussian gates, such as the cubic phase gate. We discuss several methods of implementing the cubic phase gate and fit them into the temporal cluster state architecture. The third algorithm is the continuous-variable version of Grover's search algorithm, the main challenge of which is the implementation of the inversion operator. We propose a method to implement the inversion operator by injecting a resource state into a teleportation circuit. The resource state is simulated using the Strawberry Fields quantum software package.

DOI
10.1103/physreva.98.032316

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Implementing quantum algorithms is essential for quantum computation. We study the implementation of three quantum algorithms on a two-dimensional temporal continuous-variable cluster state. We first review the generation of temporal cluster states and the implementation of gates using the measurement-based model. Alongside this we discuss methods to introduce non-Gaussianity into the cluster states. The first algorithm we consider is Gaussian boson sampling in which only Gaussian unitaries need to be implemented. Taking into account the fact that input states are also Gaussian, the errors due to the effect of finite squeezing can be partially corrected and a high fidelity can be achieved, provided a moderate amount of online squeezing is available. This helps to construct a large Gaussian boson sampling machine. The second algorithm is the continuous-variable instantaneous quantum polynomial circuit in which one needs to implement non-Gaussian gates, such as the cubic phase gate. We discuss several methods of implementing the cubic phase gate and fit them into the temporal cluster state architecture. The third algorithm is the continuous-variable version of Grover's search algorithm, the main challenge of which is the implementation of the inversion operator. We propose a method to implement the inversion operator by injecting a resource state into a teleportation circuit. The resource state is simulated using the Strawberry Fields quantum software package.
We propose an experimental design for universal continuous-variable quantum computation that incorporates recent innovations in linear-optics-based continuous-variable cluster state generation and cubic-phase gate teleportation. The first ingredient is a protocol for generating the bilayer-square-lattice cluster state (a universal resource state) with temporal modes of light. With this state, measurement-based implementation of Gaussian unitary gates requires only homodyne detection. Second, we describe a measurement device that implements an adaptive cubic-phase gate, up to a random phase-space displacement. It requires a two-step sequence of homodyne measurements and consumes a (non-Gaussian) cubic-phase state.
Quantum computers offer impressive computational speed-ups over their present-day (classical) counterparts. In the measurement-based model, quantum computation is driven by single-site measurements on a large entangled quantum state known as a cluster state. This thesis explores extensions of the measurement-based model for quantum computation in qubit and continuous-variable systems. Within the qubit setting, we consider the task of characterizing how well a small-scale measurement-based quantum device can perform logic gates. We adapt a pre-existing scheme known as randomized benchmarking into the setting of measurement-based quantum computation on a one-dimensional cluster state. A key feature of randomized benchmarking is that it uses random sequences of gates. We show how the intrinsic randomness of measurement-based quantum computation can be harnessed when implementing them. Within the continuous-variable setting, we consider optical cluster states that can be generated with current technology. We propose a compact method for generating universal cluster states based on optical-parametric-oscillator technology. We consider how finite squeezing effects manifest in computation and show that pre-existing measurement-based protocols are suboptimal. We propose new measurement-based protocols that have better noise properties, compactness, and circuit flexibility. As an application, we introduce a measurement-based method for implementing interferometry. In this model, the finite squeezing noise can be dealt with as a photon-loss process. Building further on this work, we investigate the resource requirements of a measurement-based boson-sampling device, proving simultaneous efficiency in time, space, and squeezing (energy) resources. These results offer new insights into how to build, use, and characterize a measurement-based quantum computer.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.
Continuous-variable cluster states offer a potentially promising method of implementing a quantum computer. This paper extends and further refines theoretical foundations and protocols for experimental implementation. We give a cluster-state implementation of the cubic phase gate through photon detection, which, together with homodyne detection, facilitates universal quantum computation. In addition, we characterize the offline squeezed resources required to generate an arbitrary graph state through passive linear optics. Most significantly, we prove that there are universal states for which the offline squeezing per mode does not increase with the size of the cluster. Simple representations of continuous-variable graph states are introduced to analyze graph state transformations under measurement and the existence of universal continuous-variable resource states.
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.
Measurement-based quantum computation with continuous variables in an optical setup shows great promise toward implementation of large-scale quantum computation, where the time-domain multiplexing approach enables us to generate the large-scale cluster state used to perform measurement-based quantum computation. To make effective use of the advantage of the time-domain multiplexing approach, in this paper, we propose the method to generate the large-scale three-dimensional cluster state which is a platform for topologically protected measurement-based quantum computation. Our method combines a time-domain multiplexing approach with a divide-and-conquer approach and has two advantages for implementing large-scale quantum computation. First, the squeezing level for verification of the entanglement of the three-dimensional cluster states is experimentally feasible. The second advantage is the robustness against analog errors derived from the finite squeezing of continuous variables during topologically protected measurement-based quantum computation. Therefore, our method is a promising approach to implement large-scale quantum computation with continuous variables.