Accuracy versus run time in an adiabatic quantum search

Rezakhani, A. T.; Pimachev, Artem; Lidar, Daniel A.

doi:10.1103/physreva.82.052305

Cited by 71 publications

(116 citation statements)

References 71 publications

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“…Similar arguments go through for SA and all the other algorithms that we consider. We note that PHWO problems are strictly toy problems since these problems are typically represented by highly nonlocal Hamiltonians (see Appendix B) and thus are not physically implementable, in the same sense that the adiabatic Grover search problem is unphysical [21,22]. Nevertheless, these problems provide us with important insights into the mechanisms behind a quantum speedup (or lack thereof), and illuminate the relative advantages of one computational model over another.…”

Section: -3mentioning

confidence: 99%

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

2016

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Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semiclassical potential arising from the spin-coherent path-integral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoided-level crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spin-vector dynamics, is at least as efficient as diabatic QA. Finally, in a different example with a convex cost function, the diabatic transitions result in a speedup relative to both adiabatic QA with tunneling and classical spin-vector dynamics.

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Section: -3mentioning

confidence: 99%

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

2016

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“…More recently with the developement of the quantum annealing [28] and adiabatic quantum computation [4], a renewed interest has been devoted to the subject in condensed matter and quantum information [29][30][31][32]. Here we investigate for the first time the QSL of the dynamics of a first order QPT in the adiabatic version of Grover's search algo-rithm (GSA) [33,34] and of a second order QPT [35] in the Lipkin-Meshkov-Glick (LMG) model. Specifically we consider the problem of converting the ground state on one side of the critical point into the ground state on the opposite side in the fastest and most accurate way by selecting an optimal time-dependence of the control field.…”

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confidence: 99%

Speeding up critical system dynamics through optimized evolution

Caneva¹,

Calarco²,

Fazio³

et al. 2011

Phys. Rev. A

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The number of defects which are generated on crossing a quantum phase transition can be minimized by choosing properly designed time-dependent pulses. In this work we determine what are the ultimate limits of this optimization. We discuss under which conditions the production of defects across the phase transition is vanishing small. Furthermore we show that the minimum time required to enter this regime is T ∼ π/∆, where ∆ is the minimum spectral gap, unveiling an intimate connection between an optimized unitary dynamics and the intrinsic measure of the Hilbert space for pure states. Surprisingly, the dynamics is non-adiabatic, this result can be understood by assuming a simple two-level dynamics for the many-body system. Finally we classify the possible dynamical regimes in terms of the action s = T ∆. PACS numbers:The rapid progress in the experimental realization and manipulation of quantum systems [1] is opening the rich and intriguing perspective of the exploitation of quantum physics to realize quantum technologies like quantum simulators [2] and quantum computers [3,4]. These achievements pave the way to the simulation of condensed matter systems giving the possibility of studying different states of matter in controlled experiments [5]. Despite the impressive results obtained so far, this is a formidable technological and theoretical challenge due to the complexity of the systems in analysis and the experimental requirements. Indeed, the level of control needed on the quantum system is unprecedented: one should be able to prepare a system in a desired initial state, perform the desired evolution and finally measure the state in a very precise way. Moreover, the whole experiment should be performed faster than the system decoherence time that eventually will destroy any quantum information capability. Quantum optimal control (OC) theory, the study of optimization strategies to improve the outcome of a quantum process, can be an extremely powerful tool to cope with these issues [6-10]. It allows not only to optimize the desired experiment outcome but also to speed up the process itself. Traditionally employed in atomic and molecular physics [11,12], OC has been recently applied with success to the optimization of the dynamics of many-body systems [13][14][15], allowing to achieve the ultimate bound imposed by quantum mechanics, the so called quantum speed limit (QSL) [16]. Indeed as intuitively suggested by the time-energy uncertainty principle, the time required by a state to reach another distinguishable state has to be longer than the inverse of its energy fluctuations [17]. This implies that a quantum system cannot evolve at an arbitrary speed in its Hilbert space, but a minimum time is required to perform a transformation between orthogonal states [18][19][20][21][22]. For time-independent Hamiltonians this bound has been exactly determined [16]; the QSL has been formally generalized also to time-dependent Hamiltonians, but so far has been computed only in a few simple cases [13,[23][24][25][26]. A s...

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“…This fact was also pointed out in Ref. [11,13]. In this work we choose six typical adiabatic paths s(t) to systematically demonstrate how to optimally control these time derivatives to reduce the computational error.…”

Section: Introductionmentioning

confidence: 99%

“…The most popular choice so * Electronic address: wubiao@pku.edu.cn far is the linear path, s(t) = t/T . Other choices were proposed in literature [11]. People has also tried to optimize the adiabatic path s(t) using geometrization [4].…”

Section: Introductionmentioning

confidence: 99%

Optimizing the quantum adiabatic algorithm

2016

Phys. Rev. A

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In quantum adiabatic algorithm, as the adiabatic parameter s(t) changes slowly from zero to one with finite rate, a transition to excited states inevitably occurs and this induces an intrinsic computational error. We show that this computational error depends not only on the total computation time T but also on the time derivatives of the adiabatic parameter s(t) at the beginning and the end of evolution. Previous work (Phys. Rev. A 82, 052305) also suggested this result. With six typical paths, we systematically demonstrate how to optimally design an adiabatic path to reduce the computational errors. Our method has a clear physical picture and also explains the pattern of computational error. In this paper we focus on quantum adiabatic search algorithm although our results are general.

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Accuracy versus run time in an adiabatic quantum search

Cited by 71 publications

References 71 publications

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Speeding up critical system dynamics through optimized evolution

Optimizing the quantum adiabatic algorithm

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