Adaptive-depth randomized measurement for fermionic observables

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 when the locality k is a constant, where dint 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.