Quantum Algorithms for Fidelity Susceptibility: From Quantum Criticality to Metrology

The fidelity susceptibility serves as a universal probe for quantum phase transitions, offering an order-parameter-free metric that captures ground-state sensitivity to Hamiltonian perturbations and exhibits critical scaling. Classical computation of this quantity, however, is limited by exponential Hilbert space growth and correlation divergence near criticality, restricting analyses to small or specialized systems. Here, we present a quantum algorithm that achieves efficient and Heisenberg-limited estimation of fidelity susceptibility through a novel resolvent reformulation and pseudoinverse block encoding leveraging quantum singular value transformation and amplitude estimation for norm evaluation. This constitutes the first quantum algorithm for fidelity susceptibility with optimal precision scaling. Moreover, for frustration-free Hamiltonians, we show that the resolvent can be approximated with a further quadratic speedup. Beyond fidelity susceptibility, our resolvent-based framework applies broadly to observables expressible through spectral sums or linear response functions, suggesting a systematic approach to quantum algorithm design for many-body physics. Our Letter bridges quantum many-body physics and algorithmic design, enabling scalable exploration of quantum criticality with applications in materials simulation and metrology on fault-tolerant quantum platforms.