Bayesian phase difference estimation based on single-photon projective measurement

The estimation of quantum phase differences plays an important role in quantum simulation and quantum computation, yet existing quantum phase estimation algorithms face critical limitations in noisy intermediate-scale quantum (NISQ) devices due to their excessive depth and circuit complexity. We demonstrate a high-precision phase difference estimation protocol based on the Bayesian phase difference estimation algorithm and single-photon projective measurement. The iterative framework of the algorithm, combined with the independence from controlled unitary operations, inherently mitigates circuit depth and complexity limitations. Through an experimental realization on the photonic system, we demonstrate high-precision estimation of diverse phase differences, showing root-mean-square errors (RMSE) below the standard quantum limit $\mathcal{O}(1/\sqrt{N})$ and reaching the Heisenberg scaling $\mathcal{O}(1/N)$ after a certain number of iterations. Our scheme provides a critical advantage in quantum resource-constrained scenarios, and advances practical implementations of quantum information tasks under realistic hardware constraints.