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arXiv:2610.24686

Efficient Quantum Algorithm for Mixed-State Quantum Geometry

Chenyang Wang, Qi Zhao, Xiao Yuan, Xiaoxia Cai

quant-phSubmitted Oct 17, 2026• Announced Oct 20, 2026• 0 citations

Abstract

Mixed-state quantum geometry plays a fundamental role in quantum metrology, multiparameter estimation, and the characterization of quantum many-body systems. Yet, unlike its pure-state counterpart, accessing mixed-state geometry requires solving an ill-conditioned Lyapunov equation for the symmetric logarithmic derivative (SLD), posing a major challenge for resource-efficient quantum computation. Here we address the challenge and develop a novel quantum algorithmic framework for efficiently accessing the quantum Fisher information matrix (QFIM) and quantum geometric tensor (QGT) of mixed states. Our key insight is to uncover a previously unexplored connection between the positive semidefinite structure of density matrices and exponentially convergent Gauss-Laguerre quadrature, enabling an efficient integral-based construction of the SLD. A rigorous complexity analysis further establishes the resource efficiency of our algorithm, demonstrating logarithmic precision dependence and double-logarithmic ancillary-qubit overhead at fixed conditioning. Building on this construction, we achieve near-Heisenberg precision scaling for diagonal QFI estimation without quantum phase estimation and construct unified block encodings of the full QFIM and QGT under a bounded spectral-gap assumption. A complementary regularized algorithm extends efficient diagonal estimation to exponentially small spectral gaps under a bounded-response condition. Our results establish a resource-efficient route to coherent mixed-state quantum geometry, enabling its direct integration into quantum linear-algebra algorithms without classical reconstruction and opening new possibilities for quantum-enhanced multiparameter metrology.

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Mixed-state quantum geometry plays a fundamental role in quantum metrology, multiparameter estimation, and the characterization of quantum many-body systems. Yet, unlike its pure-state counterpart, accessing mixed-state geometry requires solving an ill-conditioned Lyapunov equation for the symmetric logarithmic derivative (SLD), posing a major challenge for resource-efficient quantum computation. Here we address the challenge and develop a novel quantum algorithmic framework for efficiently accessing the quantum Fisher information matrix (QFIM) and quantum geometric tensor (QGT) of mixed states. Our key insight is to uncover a previously unexplored connection between the positive semidefinite structure of density matrices and exponentially convergent Gauss-Laguerre quadrature, enabling an efficient integral-based construction of the SLD. A rigorous complexity analysis further establishes the resource efficiency of our algorithm, demonstrating logarithmic precision dependence and double-logarithmic ancillary-qubit overhead at fixed conditioning. Building on this construction, we achieve near-Heisenberg precision scaling for diagonal QFI estimation without quantum phase estimation and construct unified block encodings of the full QFIM and QGT under a bounded spectral-gap assumption. A complementary regularized algorithm extends efficient diagonal estimation to exponentially small spectral gaps under a bounded-response condition. Our results establish a resource-efficient route to coherent mixed-state quantum geometry, enabling its direct integration into quantum linear-algebra algorithms without classical reconstruction and opening new possibilities for quantum-enhanced multiparameter metrology.

Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.

For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.

The fidelity estimation between two quantum states is crucial for quantum computation and information science. However, an efficacious method for this, especially for mixed states and higher-dimensional density matrices, remains elusive. While there are many existing algorithms on computing the fidelity between two pure states, there is not much work on how to obtain the fidelity between two mixed states. Here, an efficient quantum algorithm for the fidelity estimation is proposed, based primarily on the density matrix exponentiation and interferometeric scheme for mixed states, with a time complexity of

O(κ^2N^2/ε^7)

, where

N

is the system size,

κ

is the larger of the condition number of the density matrices and

ε

is a precision error. This algorithm may serve as a resource-efficient technique to deduce fidelity of any two (pure or mixed) unknown or known quantum states, when the density matrices of the quantum states commute with each other.

Abstract We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.

In this work, we introduce an experimentally viable scheme to enhance the simultaneous estimation precision of the couplings

G_{mc}

and

G_{mb}

, with a particular focus on the performance of heterodyne detection. By comparing simultaneous and individual estimation strategies, we demonstrate that the simultaneous approach offers a notable advantage in our system. To support this, we compute the quantum Fisher information matrices (QFIMs) based on the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD). Our results show that the quantum Cramér Rao bound (QCRB) associated with the RLD is consistently lower than that of the SLD, indicating superior estimation precision. From a physical standpoint, this improvement reflects the system's enhanced capacity to encode, transfer, and extract quantum information while allowing optimal control of fundamental interactions. We show that increasing the Rabi frequency, cavity loss rate, and the average number of photons and phonons, combined with reduced mechanical damping and temperature, enhances the system's sensitivity to the coupling parameters. These mechanisms act on the available quantum resources, such as entanglement, squeezing, and state purity, leading to more precise estimations. Furthermore, our analysis reveals that under certain conditions, heterodyne detection can closely approach the ultimate precision set by the QFIM. This suggests that a measurement strategy based on heterodyne detection can offer an efficient and practical route for estimating the couplings

G_{mc}

and

G_{mb}

, paving the way for high precision hybrid quantum sensors.

Abstract The Quantum Fisher Information Matrix (QFIM) quantifies how sensitive parameterized quantum states are to changes in their parameters. Recently, it has been used to improve variational quantum algorithm optimization through geometry-aware techniques. However, estimating the QFIM—particularly its off-block-diagonal elements—requires substantial resources. To address this, we introduce a novel protocol that efficiently computes these elements for commuting-block variational circuits. Our approach reduces the number of quantum state preparations from

O(m^2)

O ( m 2 ) to

O(L^2)

O ( L 2 ) , where m is the number of parameters and L the number of circuit layers. This also lowers classical measurement and post-processing requirements, improving computational efficiency.