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.

© 2011 Optical Society of America

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