[Paper Review] Quantum search algorithm by adiabatic evolution under a priori probability

Zhaohui Wei, Mingsheng Ying|ArXiv.org|Dec 15, 2004

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper proposes an adiabatic quantum search algorithm that leverages prior knowledge of solution probabilities to reduce runtime, showing that only the amplitude of the solution state in the initial superposition affects evolution time. It further demonstrates that scaling the initial and final Hamiltonians by a factor can reduce the algorithm's runtime to a constant independent of database size, a feature absent in standard Grover’s algorithm.

ABSTRACT

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.

Motivation & Objective

• To investigate how prior knowledge of solution probability can be used to accelerate adiabatic quantum search.
• To analyze whether the runtime of adiabatic quantum search can be arbitrarily controlled via Hamiltonian scaling.
• To compare the performance of adiabatic search with and without prior probability information.
• To demonstrate that adiabatic evolution enables runtime control not possible in standard gate-model quantum algorithms.

Proposed method

• The algorithm uses adiabatic evolution with a time-dependent Hamiltonian H(t) = (1-s)H₀ + sHₘ, where s(t) increases from 0 to 1.
• The initial Hamiltonian H₀ = I - |ψ₀⟩⟨ψ₀| prepares a uniform superposition, while Hₘ = I - |m⟩⟨m| marks the solution state.
• The runtime T is estimated using the adiabatic condition involving the minimum energy gap g_min and the maximum rate of change D_max.
• Theoretical analysis shows that T ≈ (1/aₘ) × (π/(2ε)) when the initial state has amplitude aₘ for the solution, proving only the solution amplitude affects runtime.
• For prior probability p_M in subset A_M containing the marked item, the runtime is T ≈ √(n_M / p_M) × (π/(2ε)), where n_M is the subset size.
• By scaling H₀ and Hₘ by a factor c > 0, the runtime scales as T₂ = T₁ / c, enabling constant-time search when c = √N.

Experimental results

Research questions

• RQ1 Can prior knowledge of solution probability be used to reduce the runtime of adiabatic quantum search?
• RQ2 Does the amplitude of the solution state in the initial superposition uniquely determine the adiabatic evolution time?
• RQ3 Can the runtime of adiabatic quantum search be arbitrarily reduced by scaling the Hamiltonians?
• RQ4 Is the runtime scaling property of adiabatic algorithms absent in standard gate-model quantum algorithms like Grover’s?

Key findings

• When prior probability is known, the adiabatic search runtime becomes T ≈ √(n_M / p_M) × (π/(2ε)), which is faster than the standard O(√N) runtime when p_M is large.
• The mean runtime over all possible solution locations is bounded by T_mean ≤ (π/(2ε)) × √N, with equality only when all subsets have proportional prior probabilities.
• Scaling the initial and final Hamiltonians by a factor c reduces the runtime by the same factor, so T₂ = T₁ / c.
• By setting the Hamiltonians to scale with √N, the runtime can be reduced to a constant T ≈ π/(2ε), independent of database size N.
• The adiabatic algorithm’s ability to adjust runtime via Hamiltonian scaling is a feature not present in standard Grover’s algorithm.
• The analysis shows that adiabatic evolution allows efficient handling of complex initial states with prior probability, unlike Grover’s algorithm where such analysis is difficult.