A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians

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 $\epsilon$ 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 shadows. 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.

11 + 10 pages, 7 figures. v3: accepted in npj Quantum Information

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Cited by in corpus (2)

• Fermionic Magic Resources of Quantum Many-Body Systems
• Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables