Interference and Measurement: Changing amplitude phase information to amplitude magnitude information

TL;DR

This paper investigates how interference and measurement can convert phase-encoded information in quantum amplitudes into amplified magnitude information, enabling selective enhancement of favorable solutions without relying on a full phase-estimation protocol. The authors develop a phase-amplification protocol using a two-register setup with $U_{ab}$ and $U_a$ that produces evolved states $|\psi_m\rangle$ with amplitudes $(e^{i\phi_x}-1)^m$, and they show how iterative successful measurements amplify high-phase components toward $\pi$. They apply this technique to MaxCut within a QAOA-like framework and to minimum vertex cover by tailoring the Hamiltonians, demonstrating that while no quantum advantage arises, the approach can reduce the number of classical checks and provide richer information about the phase distribution and subspaces. Through analytical and numerical examples, they show how measurement sequences reveal $g(\theta)$, the distribution of phases, and how probabilities bound by simple formulas can inform about the landscape of solutions. Overall, the work offers a practical method to probe and manipulate phase landscapes in combinatorial optimization problems using interference and postselection.

Abstract

There are quantum procedures that encode the solutions to a problem in the phases of quantum amplitudes. This happens in some quantum optimization algorithms in which the value of a function to be maximized or minimized is represented by a phase. An example of this is the QAOA algorithm for the MaxCut problem in which one encodes the number of edges connecting the sets resulting from a partition of the vertices of a graph into phases of amplitudes of a quantum state. Another is the minimum vertex cover problem in which the number of edges included in the cover is encoded in phases. Here we want to see what can be done if we only use simple aspects of quantum mechanics, interference and measurement, to manipulate the magnitudes of the amplitudes whose phases encode the relevant information. The idea is to use constructive interference to enhance the amplitudes that contain useful information and destructive interference to suppress those that do not. We examine examples, both analytically and numerically. We also show how the results of sequences of measurements can be used to gain information about the landscape of solutions.

Figures (3)

• Figure 1: Analysis of MaxCut solution probabilities: (a) Shows how total probability of optimal solutions varies for different line graph sizes. (b) Success probabilities for both individual and sequential measurements across graph types. (c) Comparison between grid and star-ring behavior.
• Figure 2: Phase distribution of states for a few well-known graphs.
• Figure 3: 10th iteration of the amplification procedure on the 4x4 grid graph. $g(\theta)$ here is $2^{16}$ times the probability distribution and is normalized this way for comparison to Fig. 2b. Note the peak has shifted from around 1.5 radians to more than 2 radians, making it likelier to sample a better solution.