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 $O$ of Majorana degree $2k$, we improve the previously known sample-complexity bound of $\mathcal{O}(n^{2k}\|O\|_\infty^2)$ to $\mathcal{O}(n^{k}\|O\|_\infty^2)$, 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 $t=1$, $V=4$, and for a target additive precision of 0.1, this reduces the number of required shots from $\sim 10^9$ to $\sim 10^5$. That is, the new bound reduces the required number of shots by approximately $99.98\%$.
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 O of Majorana degree 2k, we improve the previously known sample-complexity bound of O(n2k∥O∥∞2) to 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=1, V=4, and for a target additive precision of 0.1, this reduces the number of required shots from ∼109 to ∼105. That is, the new bound reduces the required number of shots by approximately 99.98%.
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 O(nlogn); here we improve this to 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). For the task of estimating the expectation value of a general particle-preserving quadratic fermionic observable h, we prove a sample complexity bound of O(η∥h0∥22), where h0 is the traceless component of h; the associated classical post-processing scales as 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)) 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 study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown d-dimensional quantum state ρ and a known set of observables {Oi}i=1m, the goal is to estimate expectation values {tr(Oiρ)}i=1m to accuracy ε in Lp-norm, using possibly adaptive measurements that act on O(polylog(d)) number of copies of ρ at a time. We focus on the regime where ε is below an instance-dependent threshold.
Our main contribution is an instance-optimal characterization of the sample complexity as Θ~(Γp/ε2), where Γp is a function of {Oi}i=1m defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with L∞-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form ∑i=1mαiOi with ∥α∥q=1 (where q is dual to p) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of Θ(Γpob/ε2). We then show Θ~(Γp/ε2) is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, Γ∞ob=Γ∞. In both cases, allowing c-copy measurements improves the sample complexity by at most Ω(1/c).
Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.
We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a protocol that, given any m∈N and ε≤O(d−12), measures O(log(m)/ε2) copies of an unknown mixed state ρ∈Cd×d and outputs a classical description of ρ which can then be used to estimate any collection of m observables to within additive accuracy ε. Previously, even for the simpler task of shadow tomography -- where the m observables are known in advance -- the best known rates either scaled benignly but suboptimally in all of m,d,ε, or scaled optimally in ε,m but had additional polynomial factors in d for general observables. Intriguingly, we also show via dimensionality reduction, that we can rescale ε and d to reduce to the regime where ε≤O(d−1/2). Our algorithm draws upon representation-theoretic tools recently developed in the context of full state tomography.
Shadow tomography has appeared as a powerful tool for estimating observables on quantum computers from a small number of samples. We show that shadow-tomography-inspired ideas can offer similarly improved sample scaling for estimating observables on tensor network states on classical computers after proper adaptation. We develop strategies for both spin (bosonic) and fermionic systems, tailored to the contraction requirements of tensor networks, and generate scaling improvements of factors of O(N) to O(N3) (where N is system size), depending on the specific task and system type. For the important and difficult task of evaluating the expectation value of long-range interacting Hamiltonians, we achieve the optimal O(1) overall scaling (up to logarithmic factors) for an arbitrarily fixed relative Monte Carlo error in both spin and fermionic systems. Additionally, we show that shadow estimators offer more stable gradients of observables in variational optimization tasks than standard Monte Carlo estimators. We demonstrate practical advantage by simulating systems with long-range interactions, including the 2D long-range Heisenberg model and an ab-initio quantum chemistry Hamiltonian.
Abstract Accurate estimation of fermionic observables is essential for advancing quantum physics and chemistry. The fermionic classical shadow (FCS) method offers an efficient framework for estimating these observables without requiring a transformation into a Pauli basis for k-local Fermionic observables. However, the random matchgate circuits in FCS require linear-depth circuits with a brickwork structure, which presents significant challenges for near-term quantum devices with limited computational resources. To address this limitation, we introduce an adaptive-depth fermionic classical shadow (ADFCS) protocol designed to reduce the circuit depth while maintaining the sample complexity. Through theoretical analysis and numerical fitting, we establish that the required depth for approximating a fermionic observable H is upper bounded by O ( max { d int 2 ( H ) / log n , d int ( H ) } ) when the locality k is a constant, where d int is the interaction distance of H and n is the number of qubits. We demonstrate the effectiveness of the ADFCS protocol through numerical experiments, which show similar accuracy to the traditional FCS method while requiring significantly fewer quantum resources. Additionally, we apply ADFCS to compute the expectation value of the Kitaev chain Hamiltonian, further validating its performance in practical scenarios. Our findings suggest that ADFCS enables more efficient quantum simulations when the locality k is constant, reducing circuit depth while preserving the sample complexity and offering a viable solution for near-term quantum devices.
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 k, we can simultaneously estimate all k-body fermionic correlations of an N-mode η-particle state with a given variance ε2 using only Ok(ηk/ε2) samples, which are independent of the system size N. We further establish a matching information-theoretic lower bound Ωk(ηk/ε2) for any adaptive protocol based on single-copy measurements, showing that the (ηk,ε)-dependence is optimal up to constants depending only on k. Furthermore, numerical studies show a 20-fold query reduction for one-body correlation estimation at N=200, η=20, and 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=288 are reduced by factors of 148 for energy density and 161 for spin structure factor estimation. For molecular energy derivatives with respect to atomic coordinates, our method reduces sample requirements by factors of 53 for Cr2 and 18.5 for [Fe2S2]2− at N=128. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.