Iterative quantum phase estimation with cQED encoding

Quantum phase estimation is a cornerstone algorithm for determining eigenvalues of unitary operators or Hamiltonians with Heisenberg-limited precision. Conventional implementations rely on deep controlled-unitary operations together with an inverse quantum Fourier transform, resulting in substantial circuit depth and hardware overhead. Here, we propose a conceptually simple and experimentally feasible alternative that exploits the toolbox of circuit quantum electrodynamics. The protocol extracts the phase through a sequence of binary threshold tests, eliminating the need for an inverse quantum Fourier transform. A bosonic mode serves as an efficient quantum memory in which the binary digits of the phase are encoded into the direction of phase-space rotations. These digits are then read out sequentially via high-fidelity homodyne measurements. We show that the protocol achieves Heisenberg scaling in estimation precision while simultaneously providing exponentially suppressed failure probability. By replacing the ancillary circuit with a bosonic degree of freedom, the scheme significantly reduces hardware complexity and offers a practical route toward implementing high-precision quantum phase estimation on circuit quantum electrodynamics platforms.