[Paper Review] Generalized Grover Search Algorithm for Arbitrary Initial Amplitude Distribution

David Biron, Ofer Biham+3 more|arXiv (Cornell University)|Jan 28, 1998

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper generalizes Grover's quantum search algorithm to work with arbitrary initial amplitude distributions, deriving exact solutions for amplitude evolution via linear difference equations. The key contribution is an optimal measurement time T ∼ O(√(N/r)) and a probability bound dependent only on the standard deviation of initial amplitudes, extending Grover's method beyond uniform initial states.

ABSTRACT

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.

Motivation & Objective

• To extend Grover's quantum search algorithm beyond uniform initial amplitude distributions.
• To model the time evolution of amplitudes in marked and unmarked states under arbitrary initial distributions.
• To derive an optimal measurement time T that depends only on the average amplitudes of marked and unmarked states.
• To establish a bound on the success probability of measuring a marked state based on the standard deviation of initial amplitude distributions.
• To provide exact analytical solutions for amplitude evolution using first-order linear difference equations.

Proposed method

• Formulate first-order linear difference equations describing the time evolution of amplitudes in marked and unmarked states.
• Solve these difference equations exactly to track amplitude dynamics over time.
• Express the optimal measurement time T as T ∼ O(√(N/r)), where N is total states and r is marked states.
• Derive a bound on the success probability of measuring a marked state using the standard deviation of initial amplitude distributions.
• Analyze the dependence of optimal time and success probability on initial average amplitudes and variance.
• Validate the results by showing that the generalized algorithm reduces to standard Grover's algorithm under uniform initial amplitudes.

Experimental results

Research questions

• RQ1 How does the optimal measurement time in Grover's algorithm depend on non-uniform initial amplitude distributions?
• RQ2 Can the success probability of measuring a marked state be bounded using only the standard deviation of initial amplitudes?
• RQ3 What is the exact time evolution of amplitudes in marked and unmarked states under arbitrary initial distributions?
• RQ4 Does the generalized algorithm retain the √(N/r) scaling when initial amplitudes are non-uniform?
• RQ5 Under what conditions does the generalized algorithm achieve maximum success probability?

Key findings

• The optimal measurement time is T ∼ O(√(N/r)), which depends only on the initial average amplitudes of marked and unmarked states.
• The probability of measuring a marked state is bounded by a function that depends solely on the standard deviation of the initial amplitude distributions.
• Exact analytical solutions for amplitude evolution are derived using first-order linear difference equations.
• The generalized algorithm reduces to standard Grover's algorithm when initial amplitudes are uniform.
• The success probability bound is tighter when the initial amplitude distribution has lower variance.
• The results confirm that the √(N/r) scaling remains valid even for non-uniform initial amplitude distributions.