Particle-preserving fermionic shadows with mode-independent sample complexity

• URL: http://arxiv.org/abs/2606.27254v1
• Date: Thu, 25 Jun 2026 16:37:29 GMT
• Title: Particle-preserving fermionic shadows with mode-independent sample complexity
• Abstract summary: We consider the problem of learning expectation values of particle-preserving operators with respect to an unknown $$-particle $n$-mode fermionic state.<n>Our main application is to estimating overlaps with arbitrary Slater determinant states.<n>In a first-quantized encoding, approximate unitary designs give circuit depths polylogarithmic in the number of modes.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Dequantization and Hardness of Spectral Sum Estimation [1.0323063834827415]
  We give new dequantization and hardness results for estimating spectral sums of matrices, such as the logdeterminant.<n>We complement our classical upper bound with $mathsfDQC1$-completeness results for estimating specific spectral sums.
  arXiv  Detail & Related papers  (2025-09-24T14:44:53Z)
• Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond [40.79840141270367]
  Given an unnormalized probability density $piproptomathrme-V$, estimating its normalizing constant $Z=int_mathbbRdmathrme-V(x)mathrmdx$ or free energy $F=-log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning.<n>We propose a new algorithm based on reverse diffusion samplers, establish a framework for analyzing its complexity, and empirically demonstrate its efficiency in tackling multimodality.
  arXiv  Detail & Related papers  (2025-02-07T00:05:28Z)
• Obtaining Lower Query Complexities through Lightweight Zeroth-Order Proximal Gradient Algorithms [65.42376001308064]
  We propose two variance reduced ZO estimators for complex gradient problems. We improve the state-of-the-art function complexities from $mathcalOleft(minfracdn1/2epsilon2, fracdepsilon3right)$ to $tildecalOleft(fracdepsilon2right)$.
  arXiv  Detail & Related papers  (2024-10-03T15:04:01Z)
• Measuring quantum relative entropy with finite-size effect [53.64687146666141]
  We study the estimation of relative entropy $D(rho|sigma)$ when $sigma$ is known.<n>Our estimator attains the Cram'er-Rao type bound when the dimension $d$ is fixed.
  arXiv  Detail & Related papers  (2024-06-25T06:07:20Z)
• Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs [56.237917407785545]
  We consider the problem of learning an $varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.
  arXiv  Detail & Related papers  (2024-05-10T09:58:47Z)
• Classical shadows of fermions with particle number symmetry [0.0]
  We provide an estimator for any $k$-RDM with $mathcalO(k2eta)$ classical complexity. Our method, in the worst-case of half-filling, still provides a factor of $4k$ advantage in sample complexity.
  arXiv  Detail & Related papers  (2022-08-18T17:11:12Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Complexity of zigzag sampling algorithm for strongly log-concave distributions [6.336005544376984]
  We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. We prove that the zigzag sampling algorithm achieves $varepsilon$ error in chi-square divergence with a computational cost equivalent to $Obigl.
  arXiv  Detail & Related papers  (2020-12-21T03:10:21Z)
• Fermionic partial tomography via classical shadows [0.0]
  We propose a tomographic protocol for estimating any $ k $-body reduced density matrix ($ k $-RDM) of an $ n $-mode fermionic state. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting.
  arXiv  Detail & Related papers  (2020-10-30T06:28:26Z)
• Hybrid Stochastic-Deterministic Minibatch Proximal Gradient: Less-Than-Single-Pass Optimization with Nearly Optimal Generalization [83.80460802169999]
  We show that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model. For loss factors, we prove that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model.
  arXiv  Detail & Related papers  (2020-09-18T02:18:44Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.