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

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 shadows^1,2. 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.