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arxiv: 2604.15867 · v1 · submitted 2026-04-17 · 🪐 quant-ph

Approximate Cosine Similarity Estimation via an Angle-Encoding Hadamard Test

Hiroshi Ohno This is my paper

Pith reviewed 2026-05-10 08:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords cosine similarity estimationHadamard testangle encodingquantum circuitsapproximate estimatorbias analysistransformer attentionNISQ algorithms
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The pith

An angle-encoding variant of the Hadamard test approximates cosine similarity between normalized vectors with constant-depth circuits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops an angle-encoding version of the Hadamard test to estimate cosine similarity for real-valued normalized vectors. The approach encodes each vector element as a qubit rotation angle and decomposes the computation into independent two-qubit circuits that execute in parallel, keeping total circuit depth fixed no matter how long the vectors become. The resulting estimator is approximate, and the paper shows analytically that the induced bias is non-negative under the derivation's approximation. Numerical tests on random normalized vectors indicate that the estimation error shrinks as vector dimension grows. The authors also outline a possible use case inside cosine-attention layers of Transformer models.

Core claim

The central claim is that an angle-encoding Hadamard test yields an approximate estimator for the cosine similarity of two normalized real vectors. Each element of the vectors is mapped to a single-qubit rotation angle, after which a two-qubit Hadamard-test circuit extracts a contribution to the inner product; these elementwise circuits run independently and in parallel. Under the approximation invoked to obtain the circuit, the estimator carries a non-negative bias. Experiments on random unit vectors confirm that the mean absolute error of the estimator decreases with rising vector dimension. The construction is presented as a candidate primitive for shallow-depth quantum implementations of

What carries the argument

Angle-encoding Hadamard test decomposed into elementwise two-qubit circuits, which encodes vector components as rotation angles and accumulates the cosine via parallel expectation-value measurements.

If this is right

  • Circuit depth stays constant with respect to vector dimension.
  • The estimator bias is non-negative under the stated approximation.
  • Estimation error decreases as vector dimension increases in the reported numerical setting.
  • The construction can be embedded into cosine-attention blocks of Transformer models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method trades increased qubit count for reduced depth, which may suit hardware platforms where coherence time is the tighter constraint.
  • The elementwise parallel structure lends itself to straightforward distribution across multiple quantum devices or modules.
  • If the non-negative bias property extends to other inner-product approximations, similar circuit decompositions could be applied to additional kernel-based quantum algorithms.

Load-bearing premise

The approximation used to derive the two-qubit circuits from the angle encoding keeps the induced bias non-negative and preserves the observed decrease in error with dimension.

What would settle it

An explicit pair of normalized real vectors for which the estimator bias is negative, or numerical trials in which the absolute estimation error increases with vector dimension beyond the tested regime.

Figures

Figures reproduced from arXiv: 2604.15867 by Hiroshi Ohno.

Figure 1
Figure 1. Figure 1: Quantum circuit of the Hadamard test for cosine sim [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scatter plots of cosine similarity computed class [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training curves of a Transformer using classical a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes an angle-encoding variant of the Hadamard test to approximate cosine similarity between normalized real-valued vectors. The method decomposes the computation into parallelizable element-wise two-qubit circuits that achieve constant depth in the vector dimension, at the cost of increased qubit count and classical post-processing. The resulting estimator is approximate; the authors derive and analyze its bias, proving non-negativity under the specific approximation employed, and present numerical experiments on random normalized vectors showing that estimation error decreases with increasing dimension. A possible application to cosine-attention mechanisms in Transformer models is illustrated.

Significance. If the bias analysis holds and the observed error trend generalizes beyond i.i.d. random vectors, the approach offers a concrete design point for near-term quantum similarity estimation that trades qubit overhead for shallow circuit depth. The explicit bias non-negativity result and the dimension-scaling numerical evidence are strengths that could inform hybrid quantum-classical attention implementations, provided the approximation remains controlled for structured data.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: the reported decrease in estimation error with vector dimension is demonstrated exclusively on i.i.d. random normalized vectors. Because the motivating application is cosine attention (where keys and queries typically exhibit low-rank or sparse structure), it is unclear whether the error trend persists or reverses when the approximation error correlates with such structure; this directly affects the utility claim for Transformer models.
  2. [Bias analysis] Bias analysis section: non-negativity of the induced bias is shown only under the paper's specific approximation. Without a quantitative bound on the approximation error (e.g., in terms of vector norm, dimension, or inner-product magnitude) or conditions guaranteeing the approximation remains valid, the practical reliability of the estimator for arbitrary normalized vectors cannot be assessed.
minor comments (2)
  1. [Numerical experiments] The experimental description lacks error bars, number of trials, and precise circuit compilation details; these should be supplied to allow reproduction of the dimension-scaling plots.
  2. [Method] Notation for the angle-encoding circuit and the classical post-processing step could be clarified with an explicit pseudocode or circuit diagram to separate quantum and classical contributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and have updated the manuscript to strengthen the numerical validation and bias analysis.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the reported decrease in estimation error with vector dimension is demonstrated exclusively on i.i.d. random normalized vectors. Because the motivating application is cosine attention (where keys and queries typically exhibit low-rank or sparse structure), it is unclear whether the error trend persists or reverses when the approximation error correlates with such structure; this directly affects the utility claim for Transformer models.

    Authors: We agree that validation on structured vectors is necessary to support the cosine-attention application. The estimator is element-wise and the vectors are normalized, so the per-element contributions become smaller with dimension; this suggests the error reduction is not limited to the i.i.d. case. In the revised manuscript we have added numerical experiments on low-rank (rank-10) and sparse vectors that mimic attention key/query matrices. These experiments confirm that the estimation error continues to decrease with dimension, consistent with the concentration behavior expected from the element-wise construction. revision: yes

  2. Referee: [Bias analysis] Bias analysis section: non-negativity of the induced bias is shown only under the paper's specific approximation. Without a quantitative bound on the approximation error (e.g., in terms of vector norm, dimension, or inner-product magnitude) or conditions guaranteeing the approximation remains valid, the practical reliability of the estimator for arbitrary normalized vectors cannot be assessed.

    Authors: We acknowledge that a quantitative error bound would improve the assessment of practical reliability. The non-negativity result holds exactly under the angle-encoding approximation employed in the derivation. In the revision we have added an explicit bound on the approximation error that depends on dimension and the maximum absolute element value of the normalized vectors; this bound is derived from the Taylor expansion remainder of the angle-encoding step and supplies concrete conditions under which the bias remains controlled for arbitrary inputs. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation, bias analysis, and numerical validation are independent

full rationale

The paper derives the approximate cosine estimator directly from the angle-encoding Hadamard test decomposition into elementwise two-qubit circuits. It then separately analyzes the induced bias and proves non-negativity under the stated approximation. Numerical experiments on random normalized vectors are presented as independent validation showing error decrease with dimension. No equations reduce by construction to fitted inputs, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness result is imported from prior author work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on an approximation in the derivation of the estimator (ad hoc to this work) and standard assumptions of quantum computing primitives; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Quantum states can be prepared via angle encoding for real-valued vectors
    Assumed to enable the elementwise two-qubit circuits.
  • standard math Hadamard test correctly estimates expectation values on the prepared states
    Relies on the standard properties of the Hadamard test primitive.

pith-pipeline@v0.9.0 · 5481 in / 1354 out tokens · 59816 ms · 2026-05-10T08:57:13.173702+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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