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International Journal of Quantum Information

SIMPLE SETS OF MEASUREMENTS FOR UNIVERSAL QUANTUM COMPUTATION AND GRAPH STATE PREPARATION

Yasuhiro Takahashi

Published Sep 1, 2010• 2 citations
Publisher

Abstract

We consider the problem of minimizing resources required for universal quantum computation using only projective measurements. The resources we focus on are observables, which describe projective measurements, and ancillary qubits. We show that the set of observables {Z ⊗ X, ( cos θ)X + ( sin θ)Y all θ ∈ [0, 2π)} with one ancillary qubit is universal for quantum computation. The set is simpler than a previous one in the sense that one-qubit projective measurements described by the observables in the set are ones only in the (X, Y) plane of the Bloch sphere. The proof of the universality immediately implies a simple set of observables that is approximately universal for quantum computation. Moreover, the proof implies a simple set of observables for preparing graph states efficiently.

DOI
10.1142/s0219749910006642

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We consider the problem of minimizing resources required for universal quantum computation using only projective measurements. The resources we focus on are observables, which describe projective measurements, and ancillary qubits. We show that the set of observables {Z ⊗ X, ( cos θ)X + ( sin θ)Y all θ ∈ [0, 2π)} with one ancillary qubit is universal for quantum computation. The set is simpler than a previous one in the sense that one-qubit projective measurements described by the observables in the set are ones only in the (X, Y) plane of the Bloch sphere. The proof of the universality immediately implies a simple set of observables that is approximately universal for quantum computation. Moreover, the proof implies a simple set of observables for preparing graph states efficiently.
We consider the problem of minimizing the resources required for approximate universality in measurement-only quantum computation. This problem is important not only for realizing a quantum computer, but also for understanding the computational power of quantum computation. The resources we focus on are observables, which describe projective measurements, and ancillary qubits. We show that, if we are allowed to use two ancillary qubits, the set of observables { cos (π/8)X - sin (π/8)Y ,Z ⊗ X} is approximately universal for quantum computation. This is the first construction of an approximately universal set consisting only of one one-qubit observable and one two-qubit observable. Using the proof of the approximate universality, we also show that, if we are allowed to use two initialized ancillary qubits, one two-qubit observable is sufficient for graph state preparation. The use of only one two-qubit observable is optimal in terms of the number of observables available and the number of qubits to be measured jointly.
Since quantum measurement is universal for quantum computation (Nielsen [3]), the minimization of the resources required for measurement‐based quantum computation is a crucial point ([3, 1, 4, 5, 6]). We have shown in [6] that a set of observables composed of one two‐qubit measurement and three one‐qubit measurements form a universal set for quantum computation if one ancillary qubit is available as an additional resource. This set is minimal in terms of two‐qubit measurements, because at least one multi‐qubit operation is needed to create entanglement, and is also minimal in terms of ancillary qubits, because at least one ancillary qubit is required to simulate reversible transformations with projective measurements only. We discuss in this paper the required number of one‐qubit measurements for a set of observables to be universal. A set composed of one two‐qubit measurement and only two one‐qubit measurements is proved universal, at the cost of a second ancillary qubit. Thus an additional ancillary qubit seems to be required to obtain a smaller universal set of measurements. One may conclude to a trade off between ancillary qubits and observables in measurement‐based quantum computation.
We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction [quant-ph/0108020]. Then, we prove the universality of simple discrete sets of 2-qubit measurements, again using a scheme simplifying the initial construction [quant-ph/0111122]. Finally, we show how to use a single 4-qubit measurement to achieve universal quantum computation, and outline a proof for the universality of almost all maximally entangling 4-qubit measurements.
We present a finite set of projective measurements that, together with quantum memory and preparation of the |0> state, suffice for universal quantum computation. This extends work of Nielsen [quant-ph/0108020], who proposed a scheme in which an arbitrary unitary operation on n qubits can be simulated using only projective measurements on at most 2n qubits. All measurements in our set involve two qubits, except two measurements which involve three qubits. Thus we improve by one the upper bound, implied by Nielsen's results, on the maximum number of qubits needed to participate in any single measurement to achieve universal quantum computation. Each of our measurements is two-valued, and each can be expressed mathematically as a Boolean combination of single-qubit measurements.
We improve the upper bound on the minimal resources required for measurement-only quantum computation (M A Nielsen 2003 Phys. Rev. A 308 96–100; D W Leung 2004 Int. J. Quantum Inform. 2 33; S Perdrix 2005 Int. J. Quantum Inform. 3 219–23). Minimizing the resources required for this model is a key issue for experimental realization of a quantum computer based on projective measurements. This new upper bound also allows one to reply in the negative to the open question presented by Perdrix (2004 Proc. Quantum Communication Measurement and Computing ) about the existence of a trade-off between observable and ancillary qubits in measurement-only QC.
We define the class of register-logic graphs and prove that any uniformly deterministic measurement-based quantum computation (MBQC) where the inputs coincide with the outputs must be driven on such graphs by measurements in the YZYZYZ plane of the Bloch sphere. This observation is revisited in the context that goes beyond uniform determinism, where we present a universal YZYZYZ-plane-only measurement pattern and establish a connection between YZYZYZ-plane-only and XZXZXZ-plane-only patterns. These results conclude the line of research on universal patterns with measurements restricted to one of the principal planes of the Bloch sphere. We further demonstrate, within the framework of the Parity Architecture, that YZYZYZ-plane patterns with the register-logic graph can be embedded into another graph with purely local interactions, and we extend this case to the scenario of universal quantum computation.