Symmetry-Based Perspectives on Hamiltonian Quantum Search Algorithms and Schrodinger's Dynamics between Orthogonal States

TL;DR

This paper investigates the limitations of Hamiltonian quantum search algorithms, especially when transitioning between orthogonal states, highlighting the role of symmetry and the necessity of time-dependent Hamiltonians or higher-dimensional spaces for optimal evolution.

Contribution

It provides a detailed analysis of the constraints on time-optimal quantum evolutions between orthogonal states using constant Hamiltonians, emphasizing the importance of system symmetry.

Findings

Constant Hamiltonians cannot achieve time-optimal evolution between orthogonal states in two-dimensional subspaces.

Time-dependent Hamiltonians or higher-dimensional evolutions are required for non-trivial state transitions.

Symmetry within the system fundamentally limits the effectiveness of analog quantum search methods.

Abstract

It is known that the continuous-time variant of Grover's search algorithm is characterized by quantum search frameworks that are governed by stationary Hamiltonians, which result in search trajectories confined to the two-dimensional subspace of the complete Hilbert space formed by the source and target states. Specifically, the search approach is ineffective when the source and target states are orthogonal. In this paper, we employ normalization, orthogonality, and energy limitations to demonstrate that it is unfeasible to breach time-optimality between orthogonal states with constant Hamiltonians when the evolution is limited to the two-dimensional space spanned by the initial and final states. Deviations from time-optimality for unitary evolutions between orthogonal states can only occur with time-dependent Hamiltonian evolutions or, alternatively, with constant Hamiltonian evolutions in higher-dimensional subspaces of the entire Hilbert space. Ultimately, we employ our quantitative analysis to provide meaningful insights regarding the relationship between time-optimal evolutions and analog quantum search methods. We determine that the challenge of transitioning between orthogonal states with a constant Hamiltonian in a sub-optimal time is closely linked to the shortcomings of analog quantum search when the source and target states are orthogonal and not interconnected by the search Hamiltonian. In both scenarios, the fundamental cause of the failure lies in the existence of an inherent symmetry within the system.