Ancilla-assisted quantum state tomography in multiqubit registers

The standard method of quantum state tomography (QST) relies on the measurement of a set of noncommuting observables, realized in a series of independent experiments. Ancilla-assisted QST (AAQST) proposed by Nieuwenhuizen and co-workers [Phys. Rev. Lett. 92, 120402 (2004)] greatly reduces the number of independent measurements by exploiting an ancilla register in a known initial state. In suitable conditions AAQST allows mapping out density matrix of an input register in a single experiment. Here we describe methods for explicit construction of AAQST experiments in multiqubit registers. We also report nuclear magnetic resonance studies on AAQST of (i) a two-qubit input register using a one-qubit ancilla in an isotropic liquid-state system and (ii) a three-qubit input register using a two-qubit ancilla register in a partially oriented system. The experimental results confirm the effectiveness of AAQST in such multiqubit registers.

Figure 1

Minimum number of independent experiments required for QST (with zero ancilla) and AAQST.

Figure 2

Molecular structure of iodotrifluoroethylene and the table of Hamiltonian parameters and relaxation constants. Chemical shifts (diagonal elements) and $J$-coupling constants (off-diagonal elements) are shown in Hz. The longitudinal relaxation time constants ($T_1$) and effective transverse relaxation time constants ($T_2^{*}$) for each of the three spins are shown in seconds.

Figure 3

AAQST results for thermal equilibrium state ${\rho }_1$ (left column), and that of state ${\rho }_2$ (right column), described in the text. The reference spectra is in the top trace. The spectra corresponding to the real part ($R_{j\nu ,j{\nu }^{\prime }}^{\left(1\right)}$, middle trace) and the imaginary part ($S_{j\nu ,j{\nu }^{\prime }}^{\left(1\right)}$, bottom trace) of the ${}^{19}$F signal are obtained in a single shot AAQST experiment. The bar plots correspond to theoretically expected states (top row) and those obtained from AAQST experiments (bottom row). Fidelities of the states are 0.997 and 0.99, respectively, for the two density matrices.

Figure 4

Average fidelity versus noise level.

Figure 5

Molecular structure of 1-bromo-2,4,5-trifluorobenzene, and the table of Hamiltonian parameters in Hz: chemical shifts (diagonal elements) and effective coupling constants ($J+2D$) (off-diagonal elements). The longitudinal relaxation time constant ($T_1$) and effective transverse relaxation time constant ($T_2^{*}$) for each spin are also shown.

Figure 6

AAQST results for thermal equilibrium state, i.e., $({\sigma }_z^1+{\sigma }_z^2+{\sigma }_z^3)/2$. The reference spectrum is in the top trace. The spectra corresponding to the real part ($R_{j\nu ,j{\nu }^{\prime }}^{\left(1\right)}$, middle trace) and the imaginary part ($R_{j\nu ,j{\nu }^{\prime }}^{\left(1\right)}$, bottom trace) of the ${}^{19}$F signal are obtained in a single shot AAQST experiment. The bar plots correspond to theoretically expected states (top row) and those obtained from AAQST experiments (bottom row). Fidelity of the AAQST state is 0.98.

Figure 7

AAQST results for state ${\rho }_2$ described in the text. The reference spectrum is in the top trace. The real (middle trace) and the imaginary spectra (bottom trace) are obtained in a single shot AAQST experiment. The bar plots correspond to theoretically expected states (top row) and those obtained from AAQST experiments (bottom row). Fidelity of the AAQST state is 0.95.