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arXiv:2606.27254

Particle-preserving fermionic shadows with mode-independent sample complexity

Maxwell West, M. Cerezo, Martin Larocca

quant-phSubmitted Jun 25, 2026• Announced Jun 26, 2026• 0 citations
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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.

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We consider the problem of learning expectation values of particle-preserving operators with respect to an unknown ηηη-particle nnn-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 O(nlog⁡n)\mathcal{O}(\sqrt n \log n)O(n​logn); here we improve this to O(ηlog⁡η)\mathcal{O}(η\logη)O(ηlogη), 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 O(nη2)\mathcal{O}(nη^2)O(nη2). For the task of estimating the expectation value of a general particle-preserving quadratic fermionic observable hhh, we prove a sample complexity bound of O(η∥h0∥22)\mathcal{O}(η\|h_0\|_2^2)O(η∥h0​∥22​), where h0h_0h0​ is the traceless component of hhh; the associated classical post-processing scales as O(n2η)\mathcal{O}(n^2η)O(n2η). 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)×U(n−η))U(n)/(U(η)\times U(n-η))U(n)/(U(η)×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.
We consider classical shadows of fermion wavefunctions with ηηη particles occupying nnn modes. We prove that all kkk-Reduced Density Matrices (RDMs) may be simultaneously estimated to an average variance of ε2ε^{2}ε2 using at most (ηk)(1−η−kn)k1+n1+n−k/ε2\binomη{k}\big(1-\frac{η-k}{n}\big)^{k}\frac{1+n}{1+n-k}/ε^{2}(kη​)(1−nη−k​)k1+n−k1+n​/ε2 measurements in random single-particle bases that conserve particle number, and provide an estimator for any kkk-RDM with O(k2η)\mathcal{O}(k^2η)O(k2η) classical complexity. Our sample complexity is a super-exponential improvement over the O((nk)kε2)\mathcal{O}(\binom{n}{k}\frac{\sqrt{k}}{ε^{2}})O((kn​)ε2k​​) scaling of prior approaches as nnn can be arbitrarily larger than ηηη, which is common in natural problems. Our method, in the worst-case of half-filling, still provides a factor of 4k4^{k}4k advantage in sample complexity, and also estimates all ηηη-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most O(1ε2)\mathcal{O}(\frac{1}{ε^{2}})O(ε21​) samples, which is additionally independent of ηηη.
Classical shadow tomography is widely touted as supplying a family of methods for extracting information from quantum systems with polynomially scaling sample-complexities. In the current era of quantum computers possessing on the order of hundreds of qubits, however, polynomial scaling can nonetheless be prohibitive. Thus, there is a strong practical need for obtaining sample-complexity bounds which are as tight as possible. Here we address this in the case of fermionic (matchgate) shadows. For an arbitrary observable OOO of Majorana degree 2k2k2k, we improve the previously known sample-complexity bound of O(n2k∥O∥∞2)\mathcal{O}(n^{2k}\|O\|_\infty^2)O(n2k∥O∥∞2​) to O(nk∥O∥∞2)\mathcal{O}(n^{k}\|O\|_\infty^2)O(nk∥O∥∞2​), which is asymptotically tight. For example, for the estimation of the energy per mode of an open fermionic 50-site Hubbard chain with hopping and on site strenghts respectively given by t=1t=1t=1, V=4V=4V=4, and for a target additive precision of 0.1, this reduces the number of required shots from ∼109\sim 10^9∼109 to ∼105\sim 10^5∼105. That is, the new bound reduces the required number of shots by approximately 99.98%99.98\%99.98%.
Predicting local fermionic correlations is a central task in quantum many-body physics, as these correlations encode many physically relevant local observables. The ubiquitous particle-number symmetry imposes strong structural constraints on quantum states, suggesting that local correlations should be learned with fewer samples than by symmetry-agnostic approaches. However, it has remained unclear whether such a provable advantage exists in collective learning of local correlations. Here, we develop a framework of number-conserving fermionic-shadow tomography based on random orbital rotations. We prove that, for every given order kkk, we can simultaneously estimate all kkk-body fermionic correlations of an NNN-mode ηηη-particle state with a given variance ε2\varepsilon^2ε2 using only Ok(ηk/ε2)O_k(η^k/\varepsilon^2)Ok​(ηk/ε2) samples, which are independent of the system size NNN. We further establish a matching information-theoretic lower bound Ωk(ηk/ε2)Ω_k(η^k/\varepsilon^2)Ωk​(ηk/ε2) for any adaptive protocol based on single-copy measurements, showing that the (ηk,ε)(η^k,\varepsilon)(ηk,ε)-dependence is optimal up to constants depending only on kkk. Furthermore, numerical studies show a 20-fold query reduction for one-body correlation estimation at N=200N=200N=200, η=20η=20η=20, and ε=10−2\varepsilon=10^{-2}ε=10−2, compared with the best alternative including Heisenberg-limited estimation methods. Relative to fermionic Gaussian-unitary shadows, sample costs for the square-lattice Fermi-Hubbard model at N=288N=288N=288 are reduced by factors of 148148148 for energy density and 161161161 for spin structure factor estimation. For molecular energy derivatives with respect to atomic coordinates, our method reduces sample requirements by factors of 535353 for Cr2\mathrm{Cr}_2Cr2​ and 18.518.518.5 for [Fe2S2]2−[\mathrm{Fe}_2\mathrm{S}_2]^{2-}[Fe2​S2​]2− at N=128N=128N=128. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.
We propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of an n-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all k-RDM elements to additive precision ϵ requires on the order of (n/k)k^{3/2}log(n)/ϵ^{2} repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for k≥2, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2-5 times more repetitions.
Quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC) is a promising route to using quantum computers for strongly correlated electronic structure problems, but its scalability is limited by a classical post-processing bottleneck: the measured quantum trial state must be used to evaluate overlaps, force biases, and local energies for many walkers throughout the AFQMC propagation. Existing matchgate-shadow approaches avoid the exponential cost of generic classical shadows, but their post-processing remains expensive for large systems. In this work, we show that particle-number-preserving fermionic shadows, implemented as orbital-rotation shadows, are able to drastically reduce QC-AFQMC post-processing costs. We derive efficient estimators for overlaps and their derivatives, giving O(N3)O(N^3)O(N3) scaling for overlaps and force biases and O(N4)O(N^4)O(N4) scaling for local energies per shadow and walker and with O(N)O(N)O(N) Cholesky vector. For fixed normalized target determinants, we also prove that the overlap estimator is unbiased with variance upper bounded by O(η)O(η)O(η), where ηηη is the particle number, yielding a sufficient budget of O(η/ε2)O(η/ε^2)O(η/ε2) ideal independent measurements for additive overlap error εεε. We combine these estimators with a novel virtual-correlation-energy treatment and a GPU implementation and demonstrate QC-AFQMC using orbital-rotation shadows. Performance projections for πππ-conjugated molecules up to pentacene with a (22e, 22o) active space and 356 total orbitals show that the full classical post-processing workflow can be completed in slightly more than two days on 8×\times×A100 graphics processing units (GPUs). These results significantly reduce classical post-processing costs for QC-AFQMC and motivate further improvements in preparation and measurement of the trial state on quantum hardware.
Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes. As part of our technical analysis we obtain ε\varepsilonε-approximate relative-error kkk-designs over the nnn-qubit symplectic group in depth O(nk+log⁡(1/ε))\mathcal{O}(nk+\log(1/\varepsilon))O(nk+log(1/ε)), which is optimal with respect to nnn, and may be of some independent interest.