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.
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.
Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only O(3klogk-1N) unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only O(N2) unique measurement circuits, each of which requires only a local O(N)-depth measurement circuit. We prove a lower bound of ω(ϵ-2Nk) on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic k-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of ω anticommuting two-body fermionic operators with only O(ω) gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in O(N4/ω) time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.
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.
Many applications of quantum simulation require to prepare and then characterize quantum states by performing an efficient partial tomography to estimate observables corresponding to k-body reduced density matrices (k-RDMs). For instance, variational algorithms for the quantum simulation of chemistry usually require that one measure the fermionic 2-RDM. While such marginals provide a tractable description of quantum states from which many important properties can be computed, their determination often requires a prohibitively large number of circuit repetitions. Here we describe a method by which all elements of k-body qubit RDMs acting on N qubits can be directly measured with a number of circuits scaling as O(3klogk−1N), an exponential improvement in N over prior art. Next, we show that if one is able to implement a linear depth circuit on a linear array prior to measurement, then one can directly measure all elements of the fermionic 2-RDM using only O(N2) circuits. We prove that this result is asymptotically optimal, thus establishing an exponential separation between the number of circuits required to directly measure all elements of qubit versus fermion RDMs. We further demonstrate a technique to estimate the expectation value of any linear combination of fermionic 2-RDM elements using O(N4/ω) circuits, each with only O(ω) gates on a linear array where ω≤N is a free parameter. We expect these results will improve the viability of many proposals for near-term quantum simulation.
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 introduce a direct estimation framework for reconstructing multiple density matrix elements of an unknown quantum state using classical shadow tomography. Traditional direct measurement protocols (DMPs), while effective for individual elements, suffer from poor scalability due to post-selection losses and the need for element-specific measurement configurations. In contrast, our method, DMP-ST, leverages random Clifford or biased mutually unbiased basis measurements to enable global estimation: a single dataset suffices to estimate arbitrary off-diagonal entries with high accuracy. We prove that estimating K off-diagonal matrix elements up to additive error ε requires only O(logK/ε2) samples, achieving exponential improvement over conventional DMPs. The number of required measurement configurations can also be exponentially reduced for large K. When extended to full state tomography, DMP-ST attains trace distance error ≤ε with sample complexity O(d3logd/ε2), which is closed to the optimal scaling for single-copy measurements. Moreover, biased MUB measurements reduce sample complexity by a constant factor than random Clifford measurements. This work provides both theoretical guarantees and explicit protocols for efficient, entrywise quantum state reconstruction. It significantly advances the practicality of direct tomography, especially for high-dimensional systems and near-term quantum platforms.
We consider classical shadows of fermion wavefunctions with η particles occupying n modes. We prove that all k-Reduced Density Matrices (RDMs) may be simultaneously estimated to an average variance of ε2 using at most (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 k-RDM with O(k2η) classical complexity. Our sample complexity is a super-exponential improvement over the O((kn)ε2k) scaling of prior approaches as n can be arbitrarily larger than η, which is common in natural problems. Our method, in the worst-case of half-filling, still provides a factor of 4k advantage in sample complexity, and also estimates all η-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most O(ε21) samples, which is additionally independent of η.