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arXiv:2208.08964

Classical shadows of fermions with particle number symmetry

Guang Hao Low

Published Aug 18, 2022• 13 citations
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Abstract

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 $ε^{2}$ using at most $\binomη{k}\big(1-\frac{η-k}{n}\big)^{k}\frac{1+n}{1+n-k}/ε^{2}$ measurements in random single-particle bases that conserve particle number, and provide an estimator for any $k$-RDM with $\mathcal{O}(k^2η)$ classical complexity. Our sample complexity is a super-exponential improvement over the $\mathcal{O}(\binom{n}{k}\frac{\sqrt{k}}{ε^{2}})$ 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 $4^{k}$ advantage in sample complexity, and also estimates all $η$-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most $\mathcal{O}(\frac{1}{ε^{2}})$ samples, which is additionally independent of $η$.

DOI
10.48550/arxiv.2208.08964

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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.
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.
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.
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%.
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor (3/2)k(3/2)^k(3/2)k for kkk-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within 2/(d+2)2/(d+2)2/(d+2) of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains (3/2)k(3/2)^k(3/2)k. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.
Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We present a simplified, combinatorial analysis of a recently proposed instantiation of this approach based on the ensemble of unitaries that are both fermionic Gaussian and Clifford. Using this analysis, we derive a corrected expression for the variance of the estimator. We then show how this leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state (provably) and for an XXX-like operator of the form (∣0⟩⟨ψ∣|\mathbf 0\rangle\langleψ|∣0⟩⟨ψ∣ + h.c.) (via numerical evidence). We also construct much smaller ensembles of measurement bases that yield the exact same quantum channel, which may help with compilation. We use these tools to show that an nnn-electron, mmm-mode Slater determinant can be learned to within εεε fidelity given O(n2m7log⁡(m/δ)/ε2)O(n^2 m^7 \log(m / δ) / ε^2)O(n2m7log(m/δ)/ε2) samples of the Slater determinant.