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arXiv:quant-ph/0204044 · Journal of Physics A Mathematical and General

Energy and efficiency of adiabatic quantum search algorithms

Saurya Das, Randy Kobes, G. Kunstatter

Published Mar 6, 2003• 34 citations
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Abstract

We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a database of N items. In particular, we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time-dependent Hamiltonian temporarily to a maximum of ∝√ N , it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the √ N bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time-dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.

DOI
10.1088/0305-4470/36/11/313

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We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a database of N items. In particular, we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time-dependent Hamiltonian temporarily to a maximum of ∝√ N , it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the √ N bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time-dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N,\sqrt{N},N​, where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.
We present an extension of adiabatic quantum computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.
We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size NNN can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.
We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least ppp in time of order N(1+p/ε)\sqrt{N}(1+p/\varepsilon)N​(1+p/ε), and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.