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arXiv:quant-ph/0209092 · Journal of the Physical Society of Japan

Hamiltonian and Measuring Time for Analog Quantum Search

Jin-Yuan Hsieh, Che-Ming Li, Der-San Chuu

Published Nov 15, 2005• 1 citation
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Abstract

We derive in this study a Hamiltonian to solve with certainty the analog quantum search problem analogue to the Grover algorithm. The general form of the initial state is considered. Since the evaluation of the measuring time for finding the marked state by probability of unity is crucially important in the problem, especially when the Bohr frequency is high, we then give the exact formula as a function of all given parameters for the measuring time.

DOI
10.1143/jpsj.74.2945

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We derive in this study a Hamiltonian to solve with certainty the analog quantum search problem analogue to the Grover algorithm. The general form of the initial state is considered. Since the evaluation of the measuring time for finding the marked state by probability of unity is crucially important in the problem, especially when the Bohr frequency is high, we then give the exact formula as a function of all given parameters for the measuring time.
The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state. The probability to measure a marked state as a function of time is calculated, and found to depend strongly on the specific initial state. The form of the function, though, remains as it is in the case of initial pure state. We study the role of the von Neumann entropy of the initial state, and show that the entropy cannot be a measure for the usefulness of the algorithm. We give few examples and show that for some extremely mixed initial states (carrying high entropy), the generalized Grover algorithm is considerably faster than any classical algorithm.
Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First-order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments. Our results imply that Grover's algorithm is robust against modest noise in the amplitude initialization procedure.
We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system with a Hamiltonian of the form E∣w〉〈w∣E|w〉〈w|E∣w〉〈w∣ where ∣w〉|w〉∣w〉 is an unknown (normalized) state. The problem is to produce ∣w〉|w〉∣w〉 by adding a Hamiltonian (independent of ∣w〉)|w〉)∣w〉) and evolving the system. If ∣w〉|w〉∣w〉 is chosen uniformly at random we can (with high probability) produce ∣w〉|w〉∣w〉 in a time proportional to N1/2/E{N}^{1/2}/EN1/2/E. If ∣w〉|w〉∣w〉 is instead chosen from a fixed, known orthonormal basis we can also produce ∣w〉|w〉∣w〉 in a time proportional to N1/2/E{N}^{1/2}/EN1/2/E and we show that this time is optimally short. This restricted problem is an analog analogue to Grover's algorithm, a computation on a conventional (!) quantum computer that locates a marked item from an unsorted list of NNN items in a number of steps proportional to N1/2{N}^{1/2}N1/2.
Grover's algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T \sim O(\sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.
We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system described by an N dimensional Hilbert space with a Hamiltonian of the form E∣w><w∣E |w >< w|E∣w><w∣ where ∣w>| w>∣w> is an unknown (normalized) state. We show how to discover ∣w>| w >∣w> by adding a Hamiltonian (independent of ∣w>| w >∣w>) and evolving for a time proportional to N1/2/EN^{1/2}/EN1/2/E. We show that this time is optimally short. This process is an analog analogue to Grover's algorithm, a computation on a conventional (!) quantum computer which locates a marked item from an unsorted list of N items in a number of steps proportional to N1/2N^{1/2}N1/2.
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel are also studied.