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arXiv:2107.01218

Behavior of Analog Quantum Algorithms.

Lucas T. Brady, Lucas Kocia, Przemysław Bienias, Aniruddha Bapat, Alexey Vyacheslavovich Gorshkov

Published Mar 1, 2021• 14 citations
arXivHTMLPDF

Abstract

Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA).These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters.In this work we connect all these algorithms to the optimal analog procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state.Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern.Furthermore, the ratios of the QAOA bangs is determined by the smooth, non-oscillatory part of the optimal procedure.We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure.All of this shows that these analog algorithms are all different limits and approximations of the same optimal quantum protocol.In the appendix, we present a new algorithm for better approximating the optimal protocol using the insights from the main body.

DOI
10.2172/1856736

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Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA).These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters.In this work we connect all these algorithms to the optimal analog procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state.Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern.Furthermore, the ratios of the QAOA bangs is determined by the smooth, non-oscillatory part of the optimal procedure.We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure.All of this shows that these analog algorithms are all different limits and approximations of the same optimal quantum protocol.In the appendix, we present a new algorithm for better approximating the optimal protocol using the insights from the main body.
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The quantum approximate optimization algorithm (QAOA) is a near-term quantum algorithm aimed at solving combinatorial optimization problems. Since its introduction, various generalizations have emerged, spanning modifications to the initial state, phase unitaries, and mixer unitaries. In this work, we present an analytical study of broad families of QAOA variants. We begin by examining a family of QAOA with product mixers, which includes single-body mixers parametrized by multiple variational angles, and derive exact analytical expressions for the cost expectation on weighted problem graphs in the single-layer ansatz setting. We then analyze a family of QAOA that employs many-body Grover-type mixers, deriving analogous analytical expressions for weighted problem hypergraphs in the setting of arbitrarily many circuit ansatz layers. For both families, we allow individual phase angles for each node and edge (hyperedge) in the problem graph (hypergraph). Our results reveal that, in contrast to product mixers, the Grover mixer is sensitive to contributions from cycles of all lengths in the problem graph, exhibiting a form of non-locality. Our study advances the understanding of QAOA's behavior in general scenarios, providing a foundation for further theoretical exploration.
The Quantum Approximate Optimization Algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a Quantum Annealing (QA) protocol. This work formalizes this connection formally and empirically, showing the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource -- i.e. integrated angles or total time -- and which in QAOA also contain a higher temperature arising from the Trotterization. This also means QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, T∼1/pT\sim 1/pT∼1/p, and to the integrated angles -- or integrated interactions in QA.