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arXiv:quant-ph/0311142

Non-Probabilistic Termination of Measurement-based Quantum Computation

Philippe Jorrand, Simon Perdrix

Published Nov 20, 2003• 2 citations
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Abstract

Nielsen [quant-ph/0108020] introduced a model of quantum computation by measurement-based simulation of unitary computations. In this model, a consequence of the non-determinism of quantum measurement is the probabilistic termination of simulations. This means that the time when simulation terminates for a given computation is probabilistic, and this simulation may even never end. We introduce (section 3) a measurement-based model with non probabilistic termination, which permits, unlike existing models, to predict the time of termination. This new scheme is a modification of Nielsen's. After an introduction to Nielsen's scheme (section 1), an analysis of different temporal organisations of elementary simulations within Nielsen's scheme (section 2) leads to the non probabilistic model.

DOI
10.48550/arxiv.quant-ph/0311142

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Nielsen [quant-ph/0108020] introduced a model of quantum computation by measurement-based simulation of unitary computations. In this model, a consequence of the non-determinism of quantum measurement is the probabilistic termination of simulations. This means that the time when simulation terminates for a given computation is probabilistic, and this simulation may even never end. We introduce (section 3) a measurement-based model with non probabilistic termination, which permits, unlike existing models, to predict the time of termination. This new scheme is a modification of Nielsen's. After an introduction to Nielsen's scheme (section 1), an analysis of different temporal organisations of elementary simulations within Nielsen's scheme (section 2) leads to the non probabilistic model.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
We develop a formal model for distributed measurement-based quantum computations, adopting an agent-based view, such that computations are described locally where possible. Because the network quantum state is in general entangled, we need to model it as a global structure, reminiscent of global memory in classical agent systems. Local quantum computations are described as measurement patterns. Since measurement-based quantum computation is inherently distributed, this allows us to extend naturally several concepts of the measurement calculus [V. Danos, E. Kashefi and P. Panangaden, The measurement calculus (2004), arXiv:quant-ph/0412135], a formal model for such computations. Our goal is to define an assembly language, i.e. we assume that computations are well-defined and we do not concern ourselves with verification techniques. The operational semantics for systems of agents is given by a probabilistic transition system, and we define operational equivalence in a way that it corresponds to the notion of bisimilarity. With this in place, we prove that teleportation is bisimilar to a direct quantum channel, and this also within the context of larger networks.
We present unified, systematic derivations of schemes in the two known measurement-based models of quantum computation. The first model (introduced by Raussendorf and Briegel, [Phys. Rev. Lett. 86, 5188 (2001)]) uses a fixed entangled state, adaptive measurements on single qubits, and feedforward of the measurement results. The second model (proposed by Nielsen, [Phys. Lett. A 308, 96 (2003)] and further simplified by Leung, [Int. J. Quant. Inf. 2, 33 (2004)]) uses adaptive two-qubit measurements that can be applied to arbitrary pairs of qubits, and feedforward of the measurement results. The underlying principle of our derivations is a variant of teleportation introduced by Zhou, Leung, and Chuang, [Phys. Rev. A 62, 052316 (2000)]. Our derivations unify these two measurement-based models of quantum computation and provide significantly simpler schemes.
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).
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
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.