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arXiv:quant-ph/0702241 · International Journal of Quantum Information

LIMITATIONS OF SOME SIMPLE ADIABATIC QUANTUM ALGORITHMS

Lawrence M. Ioannou, Michele Mosca

Published Jun 1, 2008• 11 citations
arXivHTMLPDF

Abstract

Let H(t) = (1 - t/T)H0 + (t/T)H1, t ∈ [0,T], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.

DOI
10.1142/s0219749908003724

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Let H(t) = (1 - t/T)H0 + (t/T)H1, t ∈ [0,T], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.
Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.
We examine the driving Hamiltonian in the analog analogue of Grover's algorithm by Farhi and Gutmann. For a quantum system with a given Hamiltonian E∣w><w∣E|w> < w|E∣w><w∣, we explicitly show that while the driving Hamiltonian E∣s><s∣E|s> < s|E∣s><s∣ optimally produces the state ∣w>|w>∣w> from an initial state ∣s>|s>∣s>, the driving Hamiltonian E′∣s><s∣(E′≠E)E^{\prime}|s> < s|(E^{\prime} \ne E)E′∣s><s∣(E′=E) does not provide any speedup compared even with a classical computation.
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
Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization prob-lem is encoded in the ground state of a "problem Hamiltonian " Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian i where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We
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.
We describe a class of the Hamiltonians, for which the local adiabaticity condition is satisfied automatically. We consider the controllable adiabatic interpolations between the initial and final Hamiltonians. We show that the quantum search by the adiabatic quantum computation can be mapped into this class. Assuming that the interpolation is infinitely differentiable and the target adiabatic ground state is nondegenerate and separated by a gap from the rest of the spectrum, it is shown that one can obtain an exponentially small error in the algorithm run time T betweenthe finale adiabatic eigenstate and the actual state of the system. The optimal for quantum search scaling of T as square root of the system size is also satisfied.