Quantum Interference-Induced Bhattacharyya Distance

Abstract

We propose a quantum distance measure between probability distributions encoded in quantum states based on the fragility of quantum interference under entangling evolution. The Quantum Interference-Induced Bhattacharyya Distance (QIBD) is defined through a single-ancilla interferometric circuit in which an interaction Hamiltonian generates correlation-dependent phases that modulate interference visibility. When the interaction vanishes, QIBD reduces to the classical Bhattacharyya distance; however, for entangling interactions, it cannot be expressed as a function of fidelity alone. Numerical simulations demonstrate that QIBD responds to correlation structure in ways that overlap-based measures do not, suggesting potential utility in contexts where interaction-aligned correlations are physically relevant.

Figures (2)

• Figure 1: Interferometric circuit for measuring QIBD. An ancilla qubit controls the preparation of a superposition between amplitude-encoded states $\ket{\psi_p}$ and $\ket{\psi_q}$ through a sequence consisting of a unitary $U_p$, followed by controlled-$U_p^\dagger$ and controlled-$U_q$. Note that, $\ket{\psi_p}=U_p\ket{0}^{\otimes n}$, and $\ket{\psi_q}=U_q\ket{0}^{\otimes n}$. The interaction unitary $U_E$ is applied asymmetrically (conditioned on the ancilla state) to induce configuration-dependent phases. The ancilla is then measured to extract interference visibility, from which QIBC is obtained.
• Figure 2: Numerical evaluation of QIBD for $n=5$ qubits (32-dimensional probability space). (a) Dependence on interaction strength $\alpha$ for fixed Gaussian distributions with $(\mu_p,\sigma_p)=(5.0,1.5)$ and $(\mu_q,\sigma_q)=(9.0,2.0)$. Classical Bhattacharyya distance (blue dashed line, $D_{\mathrm{classical}}=1.321$) remains invariant under changes in $\alpha$, while QIBD (black circles) increases monotonically from the classical value at $\alpha=0$ to $D_{\mathrm{QIBD}}=2.994$ at $\alpha=0.8$. The smooth growth reflects progressive accumulation of interaction-induced phase dispersion. (b) Response to correlation parameter $\theta$ at fixed interaction strength $\alpha=1$. The distribution $q_\theta(x)$ is constructed to have correlations aligned with $H_E$, controlled by $\theta$. QIBD (blue circles) exhibits strong sensitivity, increasing from approximately $4.1$ at $\theta=0$ to $4.3$ at $\theta=0.8$, while classical distance (black squares) shows weak variation, growing from $1.45$ to $2.17$ over the same range. The differential response demonstrates the sensitivity of QIBD to correlation structure aligned with the probing Hamiltonian.