Particle-preserving fermionic shadows with mode-independent sample complexity

We consider the problem of learning expectation values of particle-preserving operators with respect to an unknown $\eta$-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 $\mathcal{O}(\eta\log\eta)$, 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 $\mathcal{O}(n\eta^2)$. For the task of estimating the expectation value of a general particle-preserving quadratic fermionic observable $h$, we prove a sample complexity bound of $\mathcal{O}(\eta \|h_0\|_2^2)$, where $h_0$ is the traceless component of $h$; the associated classical post-processing scales as $\mathcal{O}(n^2\eta )$. 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 $U(n)/(U(\eta)\times U(n-\eta))$ 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.