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Optimal phase change for a generalized Grover's algorithm

Christopher Cardullo, Min Kang

TL;DR

The paper addresses how to optimize the phase change $\phi$ at each iteration of a generalized Grover's algorithm when the initial amplitude vector is arbitrary. It derives a closed-form one-step probability $P(|\tau\rangle)$ as a function of $(\phi,\alpha,\theta,N)$ and shows that the optimum $\phi$ can be found from a simple optimization; for Hadamard initialization, this reduces to $\phi=\pi$ for most of the search. It identifies a threshold probability $P_r(N)$ and angle boundary $\alpha_0$ where the optimal phase departs from $\pi$, with $P_r(N)=\frac{1}{2}\left(1+\frac{N-6}{\sqrt{N^2-8N+32}}\right)$ and $\alpha_0=\frac{1}{2}\cot^{-1}\left(\frac{-N+6}{2\sqrt{N-1}}\right)$, noting that the threshold tends to 1 as $N$ grows. For complex initial amplitudes, the optimal phase depends non-trivially on the amplitude's structure and is obtained via the presented optimization formula, indicating potential speedups in certain regimes.

Abstract

We study the generalized Grover's algorithm with an arbitrary amplitude vector to find the optimal phase change for maximizing the gain in probability for the target of each iteration. In the classic setting of Grover's algorithm with a real initial amplitude vector, we find that a phase change of $π$ stays optimal until the probability of observing the target is quite close to 1. We provide a formula for identifying this cut-off point based on the size of the data set. When the amplitude is truly complex, we find that the optimal phase change depends non-trivially on the complexity of the amplitude vector. We provide an optimization formula to identify the required optimal phase change.

Optimal phase change for a generalized Grover's algorithm

TL;DR

The paper addresses how to optimize the phase change at each iteration of a generalized Grover's algorithm when the initial amplitude vector is arbitrary. It derives a closed-form one-step probability as a function of and shows that the optimum can be found from a simple optimization; for Hadamard initialization, this reduces to for most of the search. It identifies a threshold probability and angle boundary where the optimal phase departs from , with and , noting that the threshold tends to 1 as grows. For complex initial amplitudes, the optimal phase depends non-trivially on the amplitude's structure and is obtained via the presented optimization formula, indicating potential speedups in certain regimes.

Abstract

We study the generalized Grover's algorithm with an arbitrary amplitude vector to find the optimal phase change for maximizing the gain in probability for the target of each iteration. In the classic setting of Grover's algorithm with a real initial amplitude vector, we find that a phase change of stays optimal until the probability of observing the target is quite close to 1. We provide a formula for identifying this cut-off point based on the size of the data set. When the amplitude is truly complex, we find that the optimal phase change depends non-trivially on the complexity of the amplitude vector. We provide an optimization formula to identify the required optimal phase change.

Paper Structure

This paper contains 5 sections, 4 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimizing the probability of a generic step
  4. Optimizing when initializing with the Hadamard gate
  5. Conclusion & Further Research

Key Result

Theorem 1

Suppose we carry out the generalized Grover's iteration defined in eq:IterationMatrix2 with the general initial vector given by $\mathbf{v}_m^\top = [\sin(\alpha)e^{i\theta},\ \cos(\alpha)]^\top$ from eq:GenAmp, then $P(\ket{\tau})$, the probability of the target vector $\ket{\tau}$, is given by the

Figures (3)

  • Figure 1: Comparison of optimal phase change found by local search (left) and global search (right) for $N = 2^{10}$
  • Figure 2: Difference between phase change determined by \ref{['eq:SimplePhi']} and optimal phase change found by global search for $N=2^{10}$
  • Figure 3: Difference in probability improvement from one iteration between the optimal phase change, and $\pi$ for $N=2^{10}$

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • ...and 1 more