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Speedup in Quantum Adiabatic Evolution Algorithm

Joonwoo Bae, Younghun Kwon

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving instances of NP-complete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one through the evolution. In this letter, we show a way of speedup in quantum adiabatic evolution algorithm,using the extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides the new point of view to reduce the running time. It is generally expected that quantum computation may solve the problems which are hard in classical context. Shor’s quantum factorization algorithm[1] and Grover’s quantum search algorithm[2] are the preludes to such expectation. Factorization problem, which is to factorize a large number, needs exponential time resource when a known classical algorithm is applied. However, Shor’s algorithm solves the problem with exponential speedup. Grover algorithm finds a target with quadratic speedup, compared to any known classical search algorithms. Quantum adiabatic evolution algorithm was proposed by Farhi et al. and applied to solve instances of NPcomplete

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Speedup in Quantum Adiabatic Evolution Algorithm

Joonwoo Bae, Younghun Kwon

The quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving NPcomplete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one in the evolution. In this paper we show a way of speedup in the quantum adiabatic evolution algorithm, using an extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides a new point of view to reduce the running time.

arXiv:quant-ph/0205048•Zeitschrift für Naturforschung A•Apr 1, 2003•0 citations

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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 $\sqrt{N},$ 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.

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Continuous-time Quantum Algorithms: Searching and Adiabatic Computation

Lawrence M. Ioannou

One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. \nSoon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi et. al. [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. \nI begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. \nUPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.

•Jan 1, 2002•4 citations
Quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving instances of NP-complete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one through the evolution. In this letter, we show a way of speedup in quantum adiabatic evolution algorithm,using the extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides the new point of view to reduce the running time. It is generally expected that quantum computation may solve the problems which are hard in classical context. Shor’s quantum factorization algorithm[1] and Grover’s quantum search algorithm[2] are the preludes to such expectation. Factorization problem, which is to factorize a large number, needs exponential time resource when a known classical algorithm is applied. However, Shor’s algorithm solves the problem with exponential speedup. Grover algorithm finds a target with quadratic speedup, compared to any known classical search algorithms. Quantum adiabatic evolution algorithm was proposed by Farhi et al. and applied to solve instances of NPcomplete
The quantum adiabatic evolution algorithm suggested by Farhi et al. was effective in solving NPcomplete problems. The algorithm is governed by the adiabatic theorem. Therefore, in order to reduce the running time, it is essential to examine the minimum energy gap between the ground level and the next one in the evolution. In this paper we show a way of speedup in the quantum adiabatic evolution algorithm, using an extended Hamiltonian. We present the exact relation between the energy gap and the elements of the extended Hamiltonian, which provides a new point of view to reduce the running time.
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.
The authors analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'. Adiabatic quantum computation is a novel paradigm for the design of quantum algorithms; it is truly quantum in the sense that it can be used to speed up searching by a quadratic factor over any classical algorithm. On the question of whether this new paradigm may be used to efficiently solve NP-complete problems on a quantum computer, we show that the usual query complexity arguments cannot be used to rule out a polynomial time solution. On the other hand, we argue that the adiabatic approach may be thought of as a kind of 'quantum local search'. We design a family of minimization problems that is hard for such local search heuristics, and establish an exponential lower bound for the adiabatic algorithm for these problems. This provides insights into the limitations of this approach. It remains an open question whether adiabatic quantum computation can establish an exponential speed-up over traditional computing or if there exists a classical algorithm that can simulate the quantum adiabatic process efficiently.
The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speedup over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.
Exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we argue that second-order transitions -- typically associated with broken or restored symmetries -- should be advantageous in comparison to first-order transitions. Guided by simple examples we construct an alternative adiabatic algorithm for the NP-complete problem {\em Exact Cover 3}. We show numerically that its average performance (for the considered cases up to O{20}{\cal O}\{20\}O{20} qubits) is better than that of the conventional scheme. The run-time of adiabatic algorithms is not just determined by the minimum value of the fundamental energy gap (between the ground state and the exited states), but also by its curvature at the critical point. The proposed symmetry-restoring adiabatic quantum algorithm only contains contributions linear and quadratic in the Pauli matrices and can be generalized to other problem Hamiltonians which are decomposed of terms involving one and two qubits. We show how the factoring problem can be cast into such a quadratic form. These findings suggest that adiabatic quantum algorithms can solve a large class of NP problems much faster than the Grover search routine (which corresponds to a first-order transition and yields a quadratic enhancement only).
One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. \nSoon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi et. al. [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. \nI begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. \nUPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.