Square Root Fidelity Estimation

• Square Root Fidelity Estimation is defined by functionals quantifying the overlap of quantum states through square root operations on density operators and trace functions.
• Quantum algorithms utilize amplitude estimation, phase estimation, and block-encoding techniques to achieve optimal resource scaling for fidelity evaluation.
• Robust protocols combine classical sampling and quantum methods to manage statistical risk and noise in state verification, enhancing overall estimator performance.

Square root fidelity estimation comprises a suite of methodologies and theoretical advances for the accurate, resource-efficient quantification of the similarity between quantum states, and the robust combination of adaptive estimates in classical importance sampling. In quantum information theory, square root fidelity typically refers to functionals such as $$F(\rho, \sigma) = \operatorname{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})$$ or simplified variants specialized to pure or low-rank cases. The notion of “square root” in this context also links to efficient design of estimators whose guarantees and error propagation are inherently related to the behavior of the square root function, appearing both in fidelity formulas and statistical risk bounds.

1. Mathematical Formulations for Square Root Fidelity

In quantum information, fidelity quantifies the overlap between two quantum states, commonly expressed for general mixed states as

$$F(\rho, \sigma) = \operatorname{Tr}\left(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right).$$

For specific cases:

• Between a pure state $|\psi\rangle$ and a mixed state $\rho$, the fidelity reduces to $$F(\rho, |\psi\rangle) = \sqrt{\langle\psi|\rho|\psi\rangle}$$ (Fang et al., 30 Jun 2025).
• Uhlmann-Jozsa fidelity can be written as $$F(\rho, \sigma) = \left(\operatorname{Tr}\sqrt{\rho\,\sigma}\right)^2$$ (Starke et al., 2023), with the square root acting on the product of density operators and the trace taken of the result.
• For certain protocols and risk measures, classical fidelity (e.g., Bhattacharyya coefficient) employs the sum $\left(\sum_k \sqrt{p_k^\mu p_k^\nu}\right)^2$ over observed measurement statistics (Seshadri et al., 2021).

The square root operation here is both a function of positive semidefinite matrices and a critical operation in estimator construction.

2. Quantum Algorithms for Square Root Fidelity Estimation

Significant advances have been made in quantum algorithms that directly estimate the square root fidelity, achieving optimal or near-optimal resource scaling:

• Query-Optimal Approach for Pure vs. Mixed States: Utilizing purified quantum query access and circuit constructions that encode the desired fidelity into an amplitude, square root amplitude estimation yields additive-error $\varepsilon$ approximation to $F(\rho,|\psi\rangle)$ with $\Theta(1/\varepsilon)$ queries (Fang et al., 30 Jun 2025). The circuit is engineered so that a measurement on designated registers reveals $F^2(\rho,|\psi\rangle)$, from which the square root can be extracted efficiently by embedding the amplitude (Wang, 29 Aug 2024).
• Phase Estimation Protocols for Pure States: Defining Householder reflections $R_\phi$ and $R_\psi$ about pure states, the product $R_\phi R_\psi$ features eigenvalues $-e^{\mp i 2\gamma}$, with $\gamma = \arcsin(|\langle\phi|\psi\rangle|)$ (Wang et al., 28 Oct 2024). Quantum phase estimation on $R_\phi R_\psi$ with input $|\phi\rangle$ yields a direct estimator of $|\langle\phi|\psi\rangle|$, achieving optimal sample complexity $\Theta(1/\varepsilon^2)$ when converting query algorithms to sample-based algorithms via samplizers.
• Block-Encoding and Polynomial Eigenvalue Techniques: For general mixed states, block-encoding enables polynomial transformation (QSVT) of the singular values to extract roots, allowing quantum singular value transformation to obtain block-encodings of $\sqrt{\rho}$ or $\sqrt{\sigma}$ (Gilyén et al., 2022, Wang et al., 2021). A composite operator forms $F(\rho, \sigma)$, and amplitude estimation computes the trace efficiently, with performance depending on the effective rank $r$ and error parameter $\varepsilon$.
• Quantum Amplitude Estimation Enhanced Protocols: Direct fidelity estimation for generic states combines classical shadow tomography and quantum amplitude estimation. Utilizing Clifford sampling and QAE, the protocol estimates $$F(\psi, \rho) = (\operatorname{E}_{C,b} |\langle b|C|\psi\rangle|^2) - 1$$ with resource cost $\mathcal{O}(\sqrt{d})$ in the Hilbert space dimension $d = 2^n$ (Vairogs et al., 10 Dec 2024).

Algorithmic guarantees are rigorously established. Lower bounds show that additive accuracy estimation for fidelity (or its square root) must scale polynomially in the rank or dimension, confirming the optimality of the presented methods where possible (Gilyén et al., 2022, Wang et al., 28 Oct 2024).

3. Minimax, Robustness, and Statistical Aspects

Classical and quantum fidelity estimation protocols incorporate statistical minimax and robustness analyses, with risk bounds intimately related to the square root structure:

• Minimax Estimation and Confidence Intervals: The affine estimator framework yields confidence intervals for fidelity that are nearly minimax optimal for arbitrary measurement settings (Seshadri et al., 2021). The risk is governed by the classical fidelity between measurement outcome distributions:

$$\mathcal{R}_* = \frac{1}{2} \sqrt{1 - (\delta/2)^{2/R}}$$

with sample complexity $R = \mathcal{O}(\log(1/\delta)/\varepsilon^2)$. This “root-fidelity” behavior is rooted in the Hellinger affinity.

• Handling Arbitrary Noise and Measurement Collapse: In entanglement distribution, fidelity estimation protocols for Bell and GHZ states employ random sampling, local Pauli measurements, and preprocessing operations such as probabilistic rotations that diagonalize the density matrix without altering fidelity (Ruan, 2022, Ruan, 18 Aug 2024). These protocols achieve minimum mean squared error (MSE) by maximizing Fisher information over separable measurements, and are proven to be robust to heterogeneous and correlated noise.
• Efficient Two-Stage Estimation in Classical Diffusion Models: In square-root diffusion (CIR), high-frequency sampling enables construction of practical two-stage Gaussian quasi-likelihood estimators whose asymptotic covariance structures are explicitly tractable (Cheng et al., 2021). This ensures high-fidelity parameter recovery in classical ergodic processes relevant to finance and statistical physics.

4. Simplified Expressions and Algebraic Perspectives

The algebraic structure of square root fidelity expressions can be streamlined using cyclic properties and expansion techniques:

• Alternative Derivations for Uhlmann–Jozsa Fidelity: The equivalence $$F(\rho, \sigma) = (\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2 = (\operatorname{Tr}\sqrt{\rho\sigma})^2$$ is established leveraging operator power series expansion and the cyclic property of the trace (Starke et al., 2023). This eliminates the need for similarity analysis and enables direct computation with lower numerical overhead.
• Generalization to Other Functionals: The same reasoning extends to more complex quantities such as $\alpha$-$z$ relative Rényi divergences:

$$D_{(\alpha, z)}(\rho||\sigma) = \frac{1}{\alpha - 1} \log\{\operatorname{Tr}[ (\sigma^{(1-\alpha)/2z}\rho^{\alpha/z}\sigma^{(1-\alpha)/2z})^z ]\}$$

and for $(\alpha, z) = (1/2, 1/2)$ one recovers $-2\log[F(\rho, \sigma)]$, tightly linking fidelity estimation and quantum divergence measures.

5. Practical Implementation, Resource Scaling, and Applications

The reviewed protocols span a range of implementation requirements and applications:

Protocol Class	Resource Scaling	Notable Features
Query-optimal pure/mixed estimation	$\Theta(1/\varepsilon)$ queries	Direct amplitude embedding, quadratic speedup (Fang et al., 30 Jun 2025, Wang, 29 Aug 2024)
Sample-optimal pure state estimation	$\Theta(1/\varepsilon^2)$ samples	Samplized phase estimation, optimal samplizer construction (Wang et al., 28 Oct 2024)
Block-encoding/mixed state estimation	$\mathcal{O}(\operatorname{poly}(r,1/\varepsilon))$	QSVT, spectral sampling, hardness for QSZK (Wang et al., 2021, Gilyén et al., 2022)
Classical/quantum combined estimation	$\mathcal{O}(\sqrt{d})$ quantum, finite measurement	Classical shadows + QAE, quadratic speedup over prior art (Vairogs et al., 10 Dec 2024)
Adaptive importance sampler (classical)	$K$ samples, $x=1/2$ exponent	Simplicity, minimax variance inflation bound $\leq 9/8$ (Owen et al., 2019)

Applications span state verification and certification, entanglement quality control in quantum networks, efficient quantum property testing, tomography, and algorithmic benchmarking. Randomized local measurement protocols have particular relevance in distributed quantum tasks and networked entanglement distribution, where they enable implementation-friendly minimum-error estimation under arbitrary noise (Ruan, 18 Aug 2024).

6. Future Directions and Remaining Challenges

In quantum settings, open problems include generalizing query/sampling optimal schemes to arbitrary mixed states and measurement models, and developing resource-aware trade-offs for hybrid estimation protocols, particularly when explicit access to purification or circuit descriptions may be unavailable.

In classical Monte Carlo settings, the square root rule (Owen et al., 2019) points to a broader set of “midpoint” strategies for robust, near-optimal pooling across adaptive stages, and is being generalized to handle more complex variance decay patterns and model mis-specification.

Overall, square root fidelity estimation defines a critical intersection of matrix function analysis, quantum algorithm engineering, and statistical estimation theory, with recent results pushing the limits of resource efficiency, robustness, and generality for state similarity and quality assessment.