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arXiv:quant-ph/0006031 · Physical Review A

Arbitrary phases in quantum amplitude amplification

Peter Høyer

Published Oct 11, 2000• 160 citations
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Abstract

We consider the use of arbitrary phases in quantum amplitude amplification, which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by $\mathrm{tan}(\ensuremath{\varphi}/2)=\mathrm{tan}(\ensuremath{\varphi}/2)(1\ensuremath{-}2a),$ where \ensuremath{\varphi} and \ensuremath{\varphi} are the phases used and where a is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase errors of order up to $1/\sqrt{a}$ can be tolerated in amplitude amplification.

DOI
10.1103/physreva.62.052304

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We consider the use of arbitrary phases in quantum amplitude amplification, which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by tan(\ensuremathφ/2)=tan(\ensuremathφ/2)(1\ensuremath−2a),\mathrm{tan}(\ensuremath{\varphi}/2)=\mathrm{tan}(\ensuremath{\varphi}/2)(1\ensuremath{-}2a),tan(\ensuremathφ/2)=tan(\ensuremathφ/2)(1\ensuremath−2a), where \ensuremath{\varphi} and \ensuremath{\varphi} are the phases used and where a is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase errors of order up to 1/a1/\sqrt{a}1/a​ can be tolerated in amplitude amplification.
The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).
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.
Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.
We present a general phase matching condition for the quantum search algorithm with arbitrary unitary transformation and arbitrary phase rotations. We show by an explicit expression that the phase matching condition depends both on the unitary transformation U and the initial state. Assuming that the initial amplitude distribution is an arbitrary superposition sinθ_0 |1> + cosθ_0 e^{iδ} |2> with |1> = {1 / sinβ} \sum_k |τ_k> and |2> = {1 / cosβ} \sum_{i \ne τ}|i> , where |τ_k> is a marked state and \sinβ= \sqrt{\sum_k|U_{τ_k 0}|^2} is determined by the matrix elements of unitary transformation U between |τ_k> and the |0> state, then the general phase matching condition is tan{θ/ 2} [cos 2β+ tanθ_0 cosδsin 2β]= tan{ϕ/ 2} [1-tanθ_0 sinδsin 2βtan{θ/ 2}], where θand ϕare the phase rotation angles for |0> and |τ_k>, respectively. This generalizes previous conclusions in which the dependence of phase matching condition on UUU and the initial state has been disguised. We show that several phase conditions previously discussed in the literature are special cases of this general one, which clarifies the question of which condition should be regarded as exact.
Amplitude amplification techniques are the basic building blocks in many quantum algorithms. Amplifying the probability of success to the highest possible probability to find solutions in the search space is an essential aim. To simplify the construction of the hardware, the search operators with fixed phases must be used to handle the whole search space. This paper presents more reliable fixed-phase quantum search algorithm, where the phase shifts that used in the oracle and the diffusion operator are not matched, in contrary to the wide belief in literature that the matching condition between the phase shifts is the best choice. This paper shows that there are many phase shifts which can be used for the search operators to give better results when the number of solutions is known in advance. The minimum probability of success of the proposed algorithm is 99.61% in quadratic speed-up which is better than any other known fixed operator quantum search algorithms.
Quantum search/amplitude amplification algorithms are designed to be able to amplify the amplitude in the target state linearly with the number of operations. Since the probability is the square of the amplitude, this results in the success probability rising quadratically with the number of operations. This paper presents a new kind of quantum search algorithm in which the amplitude of the target state, itself increases quadratically with the number of operations. However, the domain of applications of this is much more limited than standard amplitude amplification.