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arXiv:2402.19230 · npj Quantum Information

A simple and efficient joint measurement strategy for estimating fermionic observables and Hamiltonians

Joanna Majsak, Daniel McNulty, Michał Oszmaniec

Published Apr 7, 2025• 4 citations
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Abstract

We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an N mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to ϵ precision using $${\mathcal{O}}(N\log (N)/{\epsilon }^{2})$$ and $${\mathcal{O}}({N}^{2}\log (N)/{\epsilon }^{2})$$ measurement rounds respectively, matching the performance offered by fermionic classical shadows1,2. In certain settings, such as a rectangular lattice of qubits which encode an N mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth $${\mathcal{O}}({N}^{1/2})$$ with $${\mathcal{O}}({N}^{3/2})$$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth $${\mathcal{O}}(N)$$ and $${\mathcal{O}}({N}^{2})$$ two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.

DOI
10.1038/s41534-025-00957-7

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We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an N mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to ϵ precision using O(Nlog⁡(N)/ϵ2){\mathcal{O}}(N\log (N)/{\epsilon }^{2})O(Nlog(N)/ϵ2) and O(N2log⁡(N)/ϵ2){\mathcal{O}}({N}^{2}\log (N)/{\epsilon }^{2})O(N2log(N)/ϵ2) measurement rounds respectively, matching the performance offered by fermionic classical shadows1,2. In certain settings, such as a rectangular lattice of qubits which encode an N mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth O(N1/2){\mathcal{O}}({N}^{1/2})O(N1/2) with O(N3/2){\mathcal{O}}({N}^{3/2})O(N3/2) two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth O(N){\mathcal{O}}(N)O(N) and O(N2){\mathcal{O}}({N}^{2})O(N2) two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.
An important class of fermionic observables, relevant in tasks such as fermionic partial tomography and estimating energy levels of chemical Hamiltonians, are the binary measurements obtained from the product of anti-commuting Majorana operators. In this work, we investigate efficient estimation strategies of these observables based on a joint measurement which, after classical post-processing, yields all sufficiently unsharp (noisy) Majorana observables of even-degree. By exploiting the symmetry properties of the Majorana observables, as described by the braid group, we show that the incompatibility robustness, i.e., the minimal classical noise necessary for joint measurability, relates to the spectral properties of the Sachdev-Ye-Kitaev (SYK) model. In particular, we show that for an n mode fermionic system, the incompatibility robustness of all degree- 2 k Majorana observables satisfies Θ ( n − k / 2 ) for k ≤ 5 . Furthermore, we present a joint measurement scheme achieving the asymptotically optimal noise, implemented by a small number of fermionic Gaussian unitaries and sampling from the set of all Majorana monomials. Our joint measurement, which can be performed via a randomization over projective measurements, provides rigorous performance guarantees for estimating fermionic observables comparable with fermionic classical shadows.
Estimation of expectation values of incompatible observables is an essential practical task in quantum computing, especially for approximating energies of chemical and other many-body quantum systems. In this Letter, we introduce a method for this purpose based on performing a single joint measurement that can be implemented locally and whose marginals yield noisy (unsharp) versions of the target set of noncommuting Pauli observables. We derive bounds on the number of experimental repetitions required to estimate energies up to a certain precision. We compare this strategy to the classical shadow formalism and show that our method yields the same performance as the locally biased classical shadow protocol. We also highlight some general connections between the two approaches by showing that classical shadows can be used to construct joint measurements and vice versa. Finally, we adapt the joint measurement strategy to minimise the sample complexity when the implementation of measurements is assumed noisy. This can provide significant efficiency improvements compared to known generalizations of classical shadows to noisy scenarios.
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.
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.
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%.