Quantum advantage in temporally flat measurement-based quantum computation

Michael de Oliveira1,2,3, Luís S. Barbosa1,2,3, and Ernesto F. Galvão3,4

1University of Minho, Department of Computer Science, Braga, Portugal
2INESC TEC, Braga, Portugal
3International Iberian Nanotechnology Laboratory (INL) Av. Mestre Jose Veiga, 4715-330, Braga, Portugal
4Instituto de Física, Universidade Federal Fluminense Av. Gal. Milton Tavares de Souza s/n, Niterói, RJ, 24210-340, Brazil

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Several classes of quantum circuits have been shown to provide a quantum computational advantage under certain assumptions. The study of ever more restricted classes of quantum circuits capable of quantum advantage is motivated by possible simplifications in experimental demonstrations. In this paper we study the efficiency of measurement-based quantum computation with a completely flat temporal ordering of measurements. We propose new constructions for the deterministic computation of arbitrary Boolean functions, drawing on correlations present in multi-qubit Greenberger, Horne, and Zeilinger (GHZ) states. We characterize the necessary measurement complexity using the Clifford hierarchy, and also generally decrease the number of qubits needed with respect to previous constructions. In particular, we identify a family of Boolean functions for which deterministic evaluation using non-adaptive MBQC is possible, featuring quantum advantage in width and number of gates with respect to classical circuits.

Quantum computing promises to deliver computational advantage with respect to the best classical algorithms for many tasks. Rigorous results quantifying this advantage are rare, and help focus the research on the crucial quantum resources that deliver better-than-classical performance. This quantum advantage can happen with respect to different resources: the total number of gates required, the depth of the resulting circuits, or the size of the memory used (known as circuit width).

In this work we analyse the evaluation of Boolean functions, something that quantum computers can do using the correlated outcomes of measurements on entangled Greenberger-Horne-Zeilinger (GHZ) states of many qubits. This variant of measurement-based quantum computation requires no adaptivity, so all qubits can be measured simultaneously. This flat temporal structure of the computational process results, in some cases, in very economical quantum circuits. We identify the characteristics of a Boolean function that determine how many qubits are needed, and the required measurement precision. For a particular family of Boolean functions we show there is a rigorous advantage in width and number of gates with respect to the corresponding family of classical circuits. In the future, our techniques may help devise better ways of using quantum resources also for adaptive circuits displaying more computational expressivity.

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