Particle-preserving fermionic shadows with mode-independent sample complexity

Abstract: We consider the problem of learning expectation values of particle-preserving operators with respect to an unknown $η$-particle $n$-mode fermionic state via classical shadows. Our main application is to estimating overlaps with arbitrary Slater determinant states: While it is known that such overlaps can, in the average case, be learnt to a fixed additive precision with a constant number of samples, the best-known worst case bound is $\mathcal{O}(\sqrt n \log n)$; here we improve this to $\mathcal{O}(η\logη)$, achieving a mode-independent sample cost. Our procedure is also computationally efficient, requiring only classical post-processing which for a generic dense orbital runs in time $\mathcal{O}(nη^2)$. For the task of estimating the expectation value of a general particle-preserving quadratic fermionic observable $h$, we prove a sample complexity bound of $\mathcal{O}(η|h_0|_2^2)$, where $h_0$ is the traceless component of $h$; the associated classical post-processing scales as $\mathcal{O}(n^2η)$. Finally, we discuss implementation of the required randomization: in a first-quantized encoding, approximate unitary designs give circuit depths polylogarithmic in the number of modes, contrasting with linear-depth requirements for nearest-neighbor second-quantized matchgate implementations. On the technical side, our proof reduces the extremal shadow variance to harmonic analysis on the AIII symmetric space $U(n)/(U(η)\times U(n-η))$ and evaluates the resulting integral using techniques from the theories of Jacobi ensembles and orthogonal polynomials, in a calculation which may be of independent interest.

• The paper derives optimal, mode-independent sample complexity bounds for estimating Slater determinant overlaps and particle-preserving quadratic observables.
• It employs harmonic analysis on symmetric spaces and orthogonal polynomial techniques to obtain explicit variance expressions.
• The work demonstrates efficient classical and circuit post-processing methods, significantly reducing quantum resource requirements for fermionic state estimation.

Succinct Summary and Central Contributions

The paper "Particle-preserving fermionic shadows with mode-independent sample complexity" (2606.27254) investigates classical shadow protocols for learning expectation values of particle-preserving operators on $\eta$-particle $n$-mode fermionic states. Its primary technical advance is the derivation of tight, mode-independent worst-case bounds for the sample complexity needed to estimate overlaps with arbitrary Slater determinant states and particle-preserving quadratic fermionic observables. The results leverage harmonic analysis on the AIII symmetric space $U(n)/(U(\eta)\times U(n-\eta))$, with integrals evaluated through Jacobi ensemble and orthogonal polynomial techniques. The analysis yields bounds optimal up to constant factors and demonstrates computational efficiency for classical post-processing.

Classical Shadows for Fermionic Observables

Classical shadow algorithms estimate expectation values from randomized measurements on copies of an unknown quantum state. For fermionic systems, shadow protocols typically exploit random unitary circuits or matchgates, with particular attention to particle-preserving ensembles. The authors focus on learning two critical families of observables:

• Slater determinant overlaps: Estimating $\Tr[\rho \Pi_\phi]$, where $\rho$ is an $\eta$-particle fermionic state and $\Pi_\phi$ is a Slater determinant.
• Particle-preserving quadratic observables: Estimating $\Tr[\rho h]$ for quadratic $h$ preserving fermion number.

Prior worst-case bounds for these tasks scale with $O(\sqrt{n}\log n)$ for Slater overlaps or $n$0 for quadratic observables [wan2023matchgate]. This work reduces those scalings to $n$1 and $n$2 respectively, demonstrating mode-independence in the sample complexity.

Analytical Techniques and Sample Complexity

The variance analysis for shadow estimators is grounded in harmonic analysis on the symmetric space $n$3. The proofs exploit zonal spherical function theory and Jacobi polynomial ensembles, yielding explicit expressions for the estimator variance:

• For overlap estimation with any Slater determinant $n$4, the sample complexity is $n$5, tight for Slater input states of fixed filling ratios.
• For particle-preserving quadratic observables $n$6, the variance scales as $n$7, where $n$8 is traceless.

These results contrast sharply with previous bounds scaling in $n$9; the improvement is substantial when $U(n)/(U(\eta)\times U(n-\eta))$0. The analysis exposes the representation-theoretic interplay between operators and the shadow measurement channel, allowing the use of explicit spectral decompositions and orthogonal polynomial summations.

Computational and Circuit Considerations

The classical post-processing cost for extracting Slater overlaps from shadow data scales as $U(n)/(U(\eta)\times U(n-\eta))$1, and for quadratic observables as $U(n)/(U(\eta)\times U(n-\eta))$2. These costs are optimal given the input representations. The authors also discuss randomized unitary implementations for shadow protocols:

• In first quantized encodings, circuit depths for approximate unitary designs are polylogarithmic in the number of modes ($U(n)/(U(\eta)\times U(n-\eta))$3).
• In second quantized encodings (matchgate circuits), circuit depths required for particle-preserving 3-designs scale linearly with $U(n)/(U(\eta)\times U(n-\eta))$4, with formal lower bounds against sublinear-depth local ensembles.

Thus, the improved sample complexity is compatible with practical circuit and classical computation demands.

Implications and Future Directions

The results present both theoretical and practical advancements. Tight sample complexity bounds are crucial for reducing the quantum resources required for learning fermionic observables, especially in resource-constrained scenarios such as near-term quantum devices. The work clarifies the distinction between average-case and worst-case efficiency, providing guarantees for adversarial input cases. The use of symmetric space analysis, combined with explicit random matrix and polynomial calculations, supplies a robust template for future classical shadow work and design of measurement protocols.

Practically, these protocols are immediately applicable for quantum chemistry and condensed matter applications requiring efficient estimation of RDMs and overlaps. They also inform the design of measurement and tomographic routines for fermionic states with fixed particle number.

Theoretically, the connection to symmetric space harmonic analysis points to further intersections between quantum information and random matrix theory, algebraic combinatorics, and representation theory. Potential future topics include the extension of these bounds to more general operator families, composite particle sectors, or to shadow learning for other symmetry-restricted quantum systems.

Conclusion

This paper establishes mode-independent, worst-case optimal sample complexity bounds for particle-preserving fermionic shadow protocols. The analytical framework leverages symmetric space techniques and explicit variance calculations, leading to significant improvements in quantum resource efficiency. The classical and circuit complexities are well-controlled, and the results lay the foundation for efficient quantum state estimation in many-body fermionic systems. The theoretical development enriches the landscape of classical shadow tomography and suggests new avenues for algorithmic and analytical advances in quantum measurement protocols.