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arXiv:1111.1774 · Physical Review A

Layer-by-layer generation of cluster states

Katherine L. Brown, Dominic C. Horsman, Viv M. Kendon, William J. Munro

Published May 7, 2012• 14 citations
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Abstract

Cluster states can be used to perform measurement-based quantum computation. The cluster state is a useful resource, because once it has been generated only local operations and measurements are needed to perform universal quantum computation. In this paper, we explore techniques for quickly and deterministically building a cluster state. In particular, we consider generating cluster states on a qubus quantum computer, a computational architecture which uses a continuous variable ancilla to generate interactions between qubits. We explore several techniques for building the cluster, with the number of operations required depending on whether we allow the ability to destroy previously created controlled-phase links between qubits. In the case where we cannot destroy these links, we show how to create an $n\ifmmode\times\else\texttimes\fi{}m$ cluster using just $3nm\ensuremath{-}2n\ensuremath{-}\ensuremath{\lceil}3m/2\ensuremath{\rceil}+3$ operations. This gives more than a factor of 2 savings over a naive method. Further savings can be obtained if we include the ability to destroy links, in which case we only need $\ensuremath{\lceil}\frac{1}{3}(8nm\ensuremath{-}4n\ensuremath{-}4m\ensuremath{-}8)\ensuremath{\rceil}$ operations. Unfortunately, the latter scheme is more complicated so choosing the correct order to interact the qubits is considerably more difficult. A halfway scheme, that keeps a modular generation but saves additional operations over never destroying links requires only $3nm\ensuremath{-}2n\ensuremath{-}2m+4$ operations. The first scheme and the last scheme are the most practical for building a cluster state because they split up the generation into the repetition of simple sections.

DOI
10.1103/physreva.85.052305

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Efficient generation of cluster states is crucial for engineering large-scale measurement-based quantum computers. Hybrid matter-optical systems offer a robust, scalable path to this goal. Such systems have an ancilla which acts as a bus connecting the qubits. Ancilla-driven schemes are important for chip-based quantum computing architectures, where the flying ancilla mediates between static qubits. Hybrid architectures form a natural substrate for measurement-based quantum computing (MBQC) [1], one type of which (the topological model) has the best error threshold for quantum computing [2]. In this type of processing a highly-entangled cluster state is generated, and then computation performed by sequential qubit measurements. The quantum processing task is to generate the cluster state, after which it becomes a matter of measurement and classical communication. In physically-realizable implementations the cluster is prepared dynamically, a few layers at a time. As the cluster state is the fundamental quantum resource of a measurement-based computation, it becomes extremely important to make it as error-free as possible. Errors in constructing the cluster can propagate rapidly through a computation because of the highly-entangled nature of the state, leading to failure of the computation. Hybrid systems are susceptible to specific error types that other systems are not, because of the use of the mediating ancilla. In cases where the ancilla is not destroyed after each gate there is a nonzero probability of errors propagating through ancilla reuse. FIG. 1: A cluster of 4 bricks of length b = 5 qubits, each made using a different bus. We present the optimal scheme for dynamic fault-tolerant 2-D cluster state generation in hybrid systems where the mediating system can be used for more than a single gate operation without being reset. We divide the cluster state into “Lego bricks”, each of which uses a single bus. We give the optimal method for constructing the bricks, reducing the number of system-bus entanglements. We then show how to determine the block size based on the error threshold of the system being used. We find that, even when the probability of error in the system is high, this scheme can still deliver significant efficiency savings through bus reuse, enabling a larger cluster to be generated. By reducing the time required to prepare sections of the cluster, bus reuse more than doubles the size of the computational workspace that can be used before decoherence effects dominate [3]. A simple example of the bricks is shown in figure 1. They are always the same height (m direction), but have length b (n direction) determined by [3] 1 ¡1 − exp[−(6b + 4)γτ − 16bηβ2]¢ ≤ ε. (1) 2 For a given set of experimental parameters γ, τ, η and β, and desired dephasing limit to ε, this determines b. If we use one bus per CPhase gate to generate a brick, we have 1 ¡1 − exp[−16bγτ − 4ηβ2]¢ ≤ ε. (2) 2 Comparing equations (1) and (2), we find our Lego scheme produces less qubit dephasing than using one bus per CPhase gate provided ηβ2 < ∼ γτ/2. Compared with 8mn − 4(m + n) bus operations with no bus reuse, for large clusters, the Lego scheme uses less than half for b > 2. And even for b = 1, the reduction is O(5mn) compared to O(8mn), equivalent to the method of [4] for five qubits per bus. This will thus be the method of choice for any ancilla-based cluster generation that allows bus reuse. Our results are directly applicable to bus-based experimental production of cluster states, enabling the same FIG. 2: Dynamic generation using multiple ancillas. resources to produce dynamically-generated cluster states of twice the size compared to single-gate bus use, see fig. 2. For multi-bus dynamic schemes, fully scalable operation can be achieved with half the coherence time compared to single-gate buses, needing as few as 20 CPhase gates per bus. We thank Aram Harrow, Ashley Stephens and Simon Devitt for useful discussions. CH was supported by EU project QAP and the Bristol Centre for Nanoscience and Quantum Information. KLB is supported by a UK Engineering and Physical Sciences Research Council industrial CASE studentship from Hewlett-Packard, VMK is supported by a UK Royal Society University Research Fellowship and WJM acknowledges partial support from the EU project HIP and MEXT in Japan.
In this thesis I explore using the qubus for quantum computing. The qubus is an architecture of quantum computing, where a continuous variable ancilla is used to generate operations between matter qubits. I concentrate on using the qubus for two purposes - quantum simulation, and generating cluster states. Quantum simulation is the idea of using a quantum computer to simulate a quantum system. I focus on conducting a simulation of the BCS Hamiltonian. I demonstrate how to perform the necessary two qubit operations in a controlled fashion using the qubus. In particular I demonstrate an O(N3) saving over an implementation on an NMR computer, and a factor of 2 saving over a naıve technique. I also discuss how to perform the quantum Fourier transform on the qubus quantum computer. I show that it is possible to perform the quantum Fourier transform using just, 24⌊N/2⌋ + 7N − 6, this is an O(N) saving over a naıve method. In the second part of the thesis, I move on, and consider generating cluster states using the qubus. A cluster state, is a universal resource for one-way or measurement-based computation. In one-way computation, the pre-generated, entangled resource is used to perform calculations, which only require local corrections and measurement. I demonstrate that the qubus can generate cluster states deterministically, and in a relatively short time. I discuss several techniques of cluster state generation, one of which is optimal, given the physical architecture we are using. This can generate an n × m cluster in only 3nm − 2n − 2m + 4 operations. The alternative techniques look at generating a cluster using layers or columns, allowing it to be built dynamically, while the cluster is used to perform calculations. I then move on, and discuss problems with error accumulation in the generation process.
Measurement based quantum computation requires the generation of a cluster state (quantum resource) prior to starting a computation. Generation of this entangled state can be difficult with many schemes already proposed. We present an abstract scheme which can create 2D cluster states as a universal resource for quantum computing. We find a linear scaling of grid size with cluster depth. The scheme is also capable of creating more exotic topologies including 3D structures and the unit cell for topological error correction. We note its relevance to the cavity QED scheme in [30] although it could be applied to various architectures.
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.
Efficient generation of cluster states is crucial for engineering large-scale measurement-based quantum computers. Hybrid matter-optical systems offer a robust, scalable path to this goal. Such systems have an ancilla which acts as a bus connecting the qubits. We show that by generating the cluster in smaller sections of interlocking bricks, reusing one ancilla per brick, the cluster can be produced with maximal efficiency, requiring fewer than half the operations compared with no bus reuse. By reducing the time required to prepare sections of the cluster, bus reuse more than doubles the size of the computational workspace that can be used before decoherence effects dominate. A row of buses in parallel provides fully scalable cluster-state generation requiring only 20 controlled-phase gates per bus use.