Quantum circuit for measuring an operator's generalized expectation values and its applications to non-Hermitian winding numbers

We propose a general quantum circuit based on the swap test for measuring the quantity $\langle \psi_1 | A | \psi_2 \rangle$ of an arbitrary operator $A$ with respect to two quantum states $|\psi_{1,2}\rangle$. This quantity is frequently encountered in many fields of physics, and we dub it the generalized expectation as a two-state generalization of the conventional expectation. We apply the circuit, in the field of non-Hermitian physics, to the measurement of generalized expectations with respect to left and right eigenstates of a given non-Hermitian Hamiltonian. To efficiently prepare the left and right eigenstates as the input to the general circuit, we also develop a quantum circuit via effectively rotating the Hamiltonian pair $(H,-H^\dagger)$ in the complex plane. As applications, we demonstrate the validity of these circuits in the prototypical Su-Schrieffer-Heeger model with nonreciprocal hopping by measuring the Bloch and non-Bloch spin textures and the corresponding winding numbers under periodic and open boundary conditions (PBCs and OBCs), respectively. The numerical simulation shows that non-Hermitian spin textures building up these winding numbers can be well captured with high fidelity, and the distinct topological phase transitions between PBCs and OBCs are clearly characterized. We may expect that other non-Hermitian topological invariants composed of non-Hermitian spin textures, such as non-Hermitian Chern numbers, and even significant generalized expectations in other branches of physics would also be measured by our general circuit, providing a different perspective to study novel properties in non-Hermitian as well as other physics realized in qubit systems.