Matchgate Shadows for Fermionic Quantum Simulation

Abstract

Classical shadows" are estimators of an unknown quantum state, constructed from suitably distributed random measurements on copies of that state [1]. Here, we analyze classical shadows obtained using random matchgate circuits, which correspond to fermionic Gaussian unitaries. We prove that the first three moments of the Haar distribution over the continuous group of matchgate circuits are equal to those of the discrete uniform distribution over only the matchgate circuits that are also Clifford unitaries; thus, the latter forms a "matchgate 3-design." This implies that the classical shadows resulting from the two ensembles are functionally equivalent. We show how one can use these matchgate shadows to efficiently estimate inner products between an arbitrary quantum state and fermionic Gaussian states, as well as the expectation values of local fermionic operators and various other quantities, thus surpassing the capabilities of prior work. As a concrete application, this enables us to apply wavefunction constraints that control the fermion sign problem in the quantum-classical auxiliary-field quantum Monte Carlo algorithm (QC-AFQMC) [2], without the exponential post-processing cost incurred by the original approach.

Improved machine learning algorithm for predicting ground state properties

“…The output 𝑦 ℓ in the training data can be obtained by measuring Tr(𝑂𝜌(𝑥 ℓ )) for the same observable 𝑂 multiple times and averaging the outcomes. Alternatively, we can use the classical shadow formalism [41][42][43][44][45]53] that performs randomized Pauli measurements on 𝜌(𝑥 ℓ ) to predict Tr(𝑂𝜌(𝑥 ℓ )) for a wide range of observables 𝑂. Theorem 1 and the classical shadow formalism together yield the following corollary for predicting ground state representations. We present the proof of Corollary 1 in Appendix C 2.…”

“…However, we may be interested in training an ML model that can predict Tr(𝑂𝜌(𝑥)) for a wide range of observables 𝑂. In this setting, one could consider a classical dataset {𝑥 ℓ , 𝜎 𝑇 (𝜌(𝑥 ℓ ))} 𝑁 ℓ=1 generated by performing classical shadow tomography [41][42][43][44][45] on the ground state 𝜌(𝑥 ℓ ) for each 𝑥 ℓ in ℓ = 1, . .…”

“…In contrast, the improved ML algorithm has a sample complexity of 𝑁 = log(𝑛)2 polylog(1/𝜖) , which is quasi-polynomial in 1/𝜖. In combination with the classical shadow formalism [41][42][43][44][45], the proposed ML algorithm also yields the same reduction in sample and time complexity compared to [29] for predicting ground state representations.…”

Classical Shadows With Noise

“…A number of works based on classical shadows have appeared since its introduction in [11]. These include a generalization of classical shadows to the fermionic setting [47][48][49] and the use of classical shadows to estimate expectation values of molecular Hamiltonians [50] and to detect bipartite entanglement in a many-body mixed state by estimating moments of the partially transposed density matrix [51]. In addition, the first experimental implementation of classical shadows was carried out by Struchalin et al in a quantum optical experiment with high-dimensional spatial states of photons [52].…”

“…Subsequent to version 1 [53] of our manuscript, several new extensions and applications of classical shadows have been developed [48,49,. Amongst these are extensions of the classical shadows framework to quantum channels [71,72] and to more general ensembles, like locally scrambled unitary ensembles [69] and Pauli-invariant unitary ensembles [77].…”