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AIP conference proceedings

Qubit vs Observable Resource Trade-Offs in Measurement-Based Quantum Computation

Simon Perdrix

Published Jan 1, 2004• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.1834406

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•Jan 1, 2004•29 citations
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 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.
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.
Quantum measurement is universal for quantum computation. The model of quantum computation introduced by Nielsen and further developed by Leung relies on a generalized form of teleportation. In order to simulate any n-qubit unitary transformation with this model, four auxiliary qubits are required. Moreover Leung exhibited a universal family of observables composed of one one-qubit measurement and four two-qubit measurements. We introduce a model of quantum computation via measurements only, relying on state transfer: state transfer only retains the part of teleportation which is necessary for computation. In order to simulate any n-qubit unitary transformation with this new model, only one auxiliary qubit is required. Moreover we exhibit a universal family of observables composed of three one-qubit measurements and only one two-qubit measurement. This model improves those of Nielsen and Leung in terms of both the number of auxiliary qubits and the number of two-qubit measurements required for quantum universality. In both cases, the minimal amounts of necessary resources are now reached: one auxiliary qubit (because measurement is destructive) and one two-qubit measurement (for creating entanglement).
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.
In measurement-based quantum computation (MBQC), a special highly entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and postmeasurement corrections. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realized in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns which have quantum circuit representation as formalized by the notion of flow, with nnn inputs, nnn outputs, and mmm total qubits, we show that only a minimum of n+1n+1n+1 and mmm qubits are required, while the number of required qubits can be as high as $m\ensuremath{-}2$ for measurement patterns which implement a unitary but do not have a quantum circuit representation, as formalized by the notion of generalized flow (gflow). We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realize the measurement pattern.
Abstract. Quantum measurement is universal for quantum computation (Nielsen [4], Raussendorf [7,8]). Two models for performing measurement-based quantum computation exist: the one-way quantum computer was introduced by Briegel and Raussendorf [7], and quantum computation via projective measurements only by Nielsen [4]. The more recent development of this second model is based on state transfers [6] instead of teleportation. From this development, a finite but approximate quantum universal family of observables is exhibited, which includes only one two-qubit observable, while others are one-qubit observables [6]. In this article, an infinite but exact quantum universal family of observables is proposed, including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation, i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison, which was initiated by Cirac and Verstraete [9], closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation. 1