Iterative Phase Estimation
Estimating an eigenphase one bit at a time using a single ancilla qubit and classical feedback.
Trading qubits for repetitions
Textbook quantum phase estimation uses a register of t ancilla qubits and an inverse quantum Fourier transform. Iterative phase estimation (IPE) achieves the same precision with a single ancilla qubit reused across t rounds, plus classical feedback. This suits hardware short on qubits, at the cost of more sequential circuit executions.
Reading bits from least significant
IPE determines the binary digits of phi from the least significant bit upward. In each round it applies a controlled U raised to a power that isolates one bit, applies a classically computed feedback phase to cancel the contributions of bits already known, then measures the ancilla in the X basis to read the current bit.
One round
- Prepare the ancilla with a Hadamard.
- Apply controlled-U^{2^{t-1-k}} for round k.
- Apply an RZ feedback rotation by an angle built from the bits measured so far.
- Apply a Hadamard and measure; the outcome is the next bit of phi.
The feedback phase
After measuring the lower bits b_0, b_1, ..., the feedback angle for the next round is -2 pi times the binary fraction formed by those bits, shifted appropriately. This cancels their phase contribution so the current measurement cleanly reveals the next bit. The classical controller updates this angle between rounds, which is why IPE needs mid-circuit measurement and feedforward.
Trade-offs
IPE uses one ancilla instead of t, a major saving on constrained devices, and replaces the QFT with simple single-qubit rotations. Its costs are the requirement for real-time classical feedback and mid-circuit measurement, and sensitivity to errors that can corrupt a bit and propagate through the feedback. Statistical robustness is improved by repeating rounds and majority voting. A related single-ancilla scheme without adaptive feedback is Kitaev phase estimation.