Projection Coefficients Estimation in Continuous-Variable Quantum Circuits

TL;DR

The paper addresses the problem of extracting the Maclaurin coefficients of holomorphic functions in the Segal–Bargmann space by translating them into quantum amplitudes of a single-mode CV system. It introduces a Projection Coefficients Algorithm that computes $c_n$ from inner products in $\mathcal{H}_{SB}$ and analyzes truncation error via $\|E_N\|^2 = \pi \sum_{n>N}|c_n|^2 n!$. It then presents a CV quantum circuit implementation that prepares the state $|f\rangle$ corresponding to $f(z)$ using a state-preparation oracle $\hat{U}_f$, and retrieves $c_n$ through photon-number-resolved detection for magnitudes and an interferometric scheme for phases. The approach provides a scalable, measurement-based alternative to classical numerical integration, with practical utility for visualizing coefficient sequences, assessing convergence, and characterizing non-Gaussian states in quantum optics. This work bridges complex analysis, approximation theory, and quantum information processing by exploiting the Bargmann isomorphism between holomorphic functions and Fock states, and it discusses implications for CV probability theory and numerical analysis.

Abstract

In this work, we propose a continuous-variable quantum algorithm to compute the projection coefficients of a holomorphic function in the Segal--Bargmann space by leveraging its isometric correspondence with single-mode quantum states. Using CV quantum circuits, we prepare the state $\ket{f}$ associated with $f(z)$ and extract the coefficients $c_n = \braket{n}{f}$ via photon-number-resolved detection, enhanced by interferometric phase referencing to recover full complex amplitudes. This enables direct quantum estimation and visualization of the coefficient sequence -- offering a scalable, measurement-based alternative to classical numerical integration for functional analysis and non-Gaussian state characterization.

Figures (7)

• Figure 1: The projection coefficients $c_{n}$ with $n\in \{0,10\}$ for the function $f(z)=e^{z}$.
• Figure 2: Comparison of Maclaurin Coefficients and Projection Coefficients for $e^{z}$ and $n\in \{0,10\}$.
• Figure 3: The imaginary and real parts of projection coefficients for $f(z)=e^{iz}$.
• Figure 4: The magnitudes of the projection coefficients are shown for (a) $\sin(z)$ and (b) $\cos(z)$. For $\sin(z)$, the leading terms occur at $n=1$ and $n=3$. For $\cos(z)$, the leading terms appear at $n=0$ and $n=2$.
• Figure 5: The decay of $|c_n|^2 n!$ and cumulative truncation error for the function $e^{z}$ with $n\in \{0,20\}$.