Optimum phase estimation with two control qubits

Phase estimation is used in many quantum algorithms, particularly in order to estimate energy eigenvalues for quantum systems. When using a single qubit as the probe (used to control the unitary we wish to estimate the eigenvalue of), it is not possible to measure the phase with a minimum mean-square error. In standard methods, there would be a logarithmic (in error) number of control qubits needed in order to achieve this minimum error. Here, we show how to perform this measurement using only two control qubits, thereby reducing the qubit requirements of the quantum algorithm. To achieve this task, we prepare the optimal control state one qubit at a time, at the same time as applying the controlled unitaries and inverse quantum Fourier transform. As each control qubit is measured, it is reset to | 0 ⟩ then entangled with the other control qubit, so only two control qubits are needed.