Sequential measurements as a resource in ancilla-assisted quantum metrology

We present a protocol for simultaneous estimation of both quadratures of a displacement channel on a quantum harmonic oscillator using a single ancilla qubit, without complex state preparation or adaptive control. The qubit is coupled to the quantum harmonic oscillator via controlled displacements of strength A and undergoes N rounds of measurement and reset, while the oscillator coherently accumulates the displacement signal. The non-commuting displacements imprint an order-dependent geometric phase at each round, giving a quantum Fisher information that scales as O(A² N³). Although the measurement record space grows as 2N, we show that exact likelihoods for any individual measurement record can be computed in O(N²) time. With this, we show that measurement in the computational basis approximately saturates the quantum bound and maintains the O(A² N³) scaling. Furthermore, the periodic information extraction makes the protocol robust to decoherence. Our results provide a framework for the design and analysis of sequential measurement in ancilla-assisted protocols.