Approximate Cosine Similarity Estimation via an Angle-Encoding Hadamard Test

TL;DR

This paper introduces an angle-encoding variant of the Hadamard test for estimating cosine similarity between vectors, emphasizing shallow circuit depth and analyzing its bias and accuracy.

Contribution

It proposes a parallelizable, shallow-depth quantum method for cosine similarity estimation using angle encoding, with bias analysis and practical insights.

Findings

Estimation error decreases with increasing vector dimension.

The bias of the estimator is non-negative under the approximation.

Potential application to cosine-attention in Transformer models.

Abstract

The Hadamard test is a standard quantum primitive for estimating inner products and expectation values, but in data-processing settings its practical utility is often limited by the cost of preparing amplitude-encoded quantum states. In this study, we investigate an angle-encoding variant of the Hadamard test for estimating cosine similarity between normalized real-valued vectors. The proposed method decomposes the similarity computation into elementwise two-qubit Hadamard-test circuits that can, in principle, be executed in parallel, resulting in constant circuit depth with respect to the vector dimension at the expense of a larger qubit footprint and classical post-processing. Because the resulting estimator is approximate, we analyze the induced bias and show that it is non-negative under the approximation used in our derivation. Numerical experiments on random normalized vectors show that, in the tested setting, the estimation error decreases as the vector dimension increases. We further illustrate a possible application to cosine-attention-based Transformer models. These results suggest that the angle-encoding Hadamard test may provide a useful design point for near-term similarity estimation when shallow circuit depth is preferred over compact qubit usage.