Figure 1

Quantum circuit for measurement-based phase estimation algorithm (MPEA) using quantum state tomograph (QST) approach. (a) The circuit for the MPEA using QST. From top to bottom, an index register, a target register, and an interacting register are prepared in the states $|0\rangle$, ${\rho }_B$, and $|{\phi }_A\rangle$, respectively. The index register is a single qubit used as control qubit; the target register is used to represent the state of subsystem $B$; and the interacting register is used to represent the state of subsystem A, which interacts with $B$. (b) The circuit for performing $m$ projective measurements with period $\tau$. In the circuit, the unitary transformation $U=\exp (-\mathit{iHt})$, and we set $t$ such that $m$ projective measurements are performed successfully on subsystem $A$ separated by a time interval $\tau$, while the unitary transformation of the whole system evolves for time $t$.Reuse & Permissions

Figure 2

Quantum circuit for a measurement-based phase estimation algorithm (MPEA) using measured quantum Fourier transformation. (a) The circuit for the MPEA. From top to bottom, an index register, a target register, and an interacting register are prepared in the states $(|0\rangle +|1\rangle )/\sqrt{2}$, ${\rho }_B$, and $|{\phi }_A\rangle$, respectively. The index register is a single qubit used as control qubit and to perform mQFT; the target register is used to represent the state of subsystem $B$; the interacting register is used to represent the state of subsystem A, which interacts with $B$. The circuit in the dotted square in (a) is used for obtaining the $n$th binary digit of the real part of the phase factor. (b) A sequence of circuits inside the dashed squares is used to replace the circuit in the dashed square in (a), in order to resolve from the ($n-1$)th to the first binary digits of the real part of the phase factor. Here, $H$ is the Hadamard gate, $Q_k$ is a single-qubit rotation as defined in Eq. (27), and $R_k$ is a single-qubit rotation in quantum Fourier transform. (c) The circuit for the $k$th partial measurement $W(k)$ with period $\tau$. In the circuit, for the unitary transformation $U=\exp (-\mathit{iHt})$, we set $t$ such that all projective measurements are performed successfully on subsystem $A$ while the unitary transformation of the whole system evolves for time $t$.Reuse & Permissions

Figure 3

Survival probability $P(m)$ ($▴$) and fidelity $F(m)$ ($•$) for $|{\phi }_A\rangle =|1\rangle$ versus $m$, the number of successful measurement, for the Jaynes-Cummings model.Reuse & Permissions

Figure 4

Survival probability $P(m)$ for $|{\phi }_A\rangle =|1\rangle$ versus $m$, the number of successful measurements, by using the ${\sigma }_z$ eigenstate as the measurement basis for the axial symmetry model.Reuse & Permissions