Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2508.00052

Learning shadows to predict quantum ground state correlations

Pierre-Gabriel Rozon, Kartiek Agarwal

Published Jul 31, 2025• 0 citations
arXivHTMLPDF

Abstract

We introduce a variational scheme inspired by classical shadow tomography to compute ground state correlations of quantum spin Hamiltonians. Shadow tomography allows for efficient reconstruction of expectation values of arbitrary observables from a bag of repeated, randomized measurements, called snapshots, on copies of the state $ρ$. The prescription allows one to infer expectation values of $M$ $k-$local observables to accuracy $ε$ using just $N \sim 3^k \text{log}M /ε^2$ snapshots when measurements are performed in locally random bases. Turning this around, a bag of snapshots can be considered an efficient representation of the state $ρ$, particularly for estimating low-weight observables, such as terms in a local Hamiltonian needed to estimate the energy. Inspired by this, we consider a variational scheme wherein a bag of $N$ parametrized snapshots is used to represent the putative ground state of a desired local spin Hamiltonian and optimized to lower the energy with respect to it. Additional constraints in the form of positivity of reduced density matrices, motivated by work in quantum chemistry, are employed to ensure compatibility of the predicted correlations with the underlying Hilbert space. Despite the limited set of imposed constraints, we show that the optimized ensemble of snapshots yields fairly accurate estimates of the ground state energy and other correlations, including those that do not appear in the set of constraints used in optimization.

DOI
10.48550/arxiv.2508.00052

Related papers

Classical shadow tomography with locally scrambled quantum dynamics

Hong-Ye Hu, Soonwon Choi, Yi‐Zhuang You

We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local-basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.

arXiv:2107.04817•Physical Review Research•Apr 14, 2023•89 citations

Constrained Shadow Tomography for Molecular Simulation on Quantum Devices

Irma Avdic, Yuchen Wang, Michael Rose +4 more

Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating $N$-representability constraints and nuclear-norm regularization into the optimization, the method builds an $N$-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.

quant-pharXiv:2511.09717•Chemical Science•Nov 14, 2025•0 citations

Shadow tomography for classical tensor network simulations

Jiace Sun, Garnet Kin-Lic Chan

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 $O(N^{3})$ (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.

quant-pharXiv:2606.29133•Jun 30, 2026•0 citations

Triply Efficient Shadow Tomography

Robbie King, David Gosset, Robin Kothari +1 more

We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k -body fermionic or k -local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k -local Pauli operators with constant k . However, we prove that for k -body fermionic observables and the full set of n -qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.

arXiv:2404.19211•PRX Quantum•Feb 27, 2025•23 citations

Shadow tomography based on informationally complete positive operator-valued measure

Atithi Acharya, Siddhartha Saha, Anirvan M. Sengupta

Recently introduced shadow tomography protocols use ``classical shadows'' of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang et al. [Huang, Kueng, and Preskill, Nat. Phys. 16, 1050 (2020)], requires additional circuits for performing certain random unitary transformations. In this paper, we avoid these transformations but employ an arbitrary informationally complete positive operator-valued measure and show that such a procedure can compute $k$-bit correlation functions for quantum states reliably. We also show that, for this application, we do not need the median of means procedure of Huang et al. Finally, we discuss the contrast between the computation of correlation functions and fidelity of reconstruction of low rank density matrices.

Physical Review A•Nov 16, 2021•38 citations

Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry

Yuki Koizumi, Kaito Wada, Toshinori P. Takama +1 more

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 $\varepsilon^2$ using only $O_k(η^k/\varepsilon^2)$ samples, which are independent of the system size $N$. We further establish a matching information-theoretic lower bound $Ω_k(η^k/\varepsilon^2)$ for any adaptive protocol based on single-copy measurements, showing that the $(η^k,\varepsilon)$-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 $\varepsilon=10^{-2}$, 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 $\mathrm{Cr}_2$ and $18.5$ for $[\mathrm{Fe}_2\mathrm{S}_2]^{2-}$ at $N=128$. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.

quant-pharXiv:2606.30601•Jun 30, 2026•0 citations
We introduce a variational scheme inspired by classical shadow tomography to compute ground state correlations of quantum spin Hamiltonians. Shadow tomography allows for efficient reconstruction of expectation values of arbitrary observables from a bag of repeated, randomized measurements, called snapshots, on copies of the state ρρρ. The prescription allows one to infer expectation values of MMM k−k-k−local observables to accuracy εεε using just N∼3klogM/ε2N \sim 3^k \text{log}M /ε^2N∼3klogM/ε2 snapshots when measurements are performed in locally random bases. Turning this around, a bag of snapshots can be considered an efficient representation of the state ρρρ, particularly for estimating low-weight observables, such as terms in a local Hamiltonian needed to estimate the energy. Inspired by this, we consider a variational scheme wherein a bag of NNN parametrized snapshots is used to represent the putative ground state of a desired local spin Hamiltonian and optimized to lower the energy with respect to it. Additional constraints in the form of positivity of reduced density matrices, motivated by work in quantum chemistry, are employed to ensure compatibility of the predicted correlations with the underlying Hilbert space. Despite the limited set of imposed constraints, we show that the optimized ensemble of snapshots yields fairly accurate estimates of the ground state energy and other correlations, including those that do not appear in the set of constraints used in optimization.
We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local-basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.
Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating NNN-representability constraints and nuclear-norm regularization into the optimization, the method builds an NNN-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.
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)O(N)O(N) to O(N3)O(N^{3})O(N3) (where NNN 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)O(1)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.
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k -body fermionic or k -local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k -local Pauli operators with constant k . However, we prove that for k -body fermionic observables and the full set of n -qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.
Recently introduced shadow tomography protocols use ``classical shadows'' of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang et al. [Huang, Kueng, and Preskill, Nat. Phys. 16, 1050 (2020)], requires additional circuits for performing certain random unitary transformations. In this paper, we avoid these transformations but employ an arbitrary informationally complete positive operator-valued measure and show that such a procedure can compute kkk-bit correlation functions for quantum states reliably. We also show that, for this application, we do not need the median of means procedure of Huang et al. Finally, we discuss the contrast between the computation of correlation functions and fidelity of reconstruction of low rank density matrices.
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.