Single-scan quantum process tomography

The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\varepsilon$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\chi$ matrix characterizing the process $\varepsilon$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.

Figure 1

Illustration of (a) single-qubit QPT requiring a total of eight NMR scans, (b) AAPT requiring two NMR scans, and (c) SSPT requiring a single-scan NMR experiment. In each case, dotted lines are used to indicate the single-quantum elements of the density matrix that are directly observable. Other elements are observed by converting them to observable single-quantum coherences by using certain unitary operations in a subsequent scan (or scans).

Figure 2

Quantum circuit for SSPT.

Figure 3

(a) Molecular structure of iodotrifluoroethylene, (b) table of Hamiltonian and relaxation parameters, and (c) NMR pulse sequence to demonstrate SSPT. The pulse sequence for preparing the $|00\rangle$ pseudopure state is shown in the inset of (c).

Figure 4

Bar plots showing experimental $\chi$ matrices for various quantum processes obtained using SSPT. In each case, the left and right bar plots correspond to the real and imaginary parts, respectively, and the fidelities are indicated in parentheses.

Figure 5

Simulated fidelity of various processes as a function of noise $\eta$.

Figure 6

(a) Experimental values of $|{\chi }_{\mathrm{EE}}|$ ($|{\chi }_{\mathrm{ZZ}}|$) shown by closed squares (closed circles). The solid lines illustrate theoretical values of ${\chi }_{\mathrm{EE}}$ and ${\chi }_{\mathrm{ZZ}}$. The crosses indicate the gate fidelities $F({\chi }_{\mathrm{expt}},{\chi }_{\mathrm{theor}})$. The bar plots correspond to experimental $\left|\chi \right|$ matrices at (b) $\Phi =0$, (c) $\Phi =0.64\pi$, (d) $\Phi =\pi$, and (e) $\Phi =3.43\pi$.