Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor

We experimentally explore the state space of three qubits on a nuclear magnetic resonance (NMR) quantum-information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a nontrivial combination of Greenberger-Horne-Zeilinger (GHZ) and $W$-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and $W$ states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and $W$ states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the $W$ state, we demonstrate this method of reconstruction by comparing it to the directly tomographed three-qubit state.

Figure 1

Molecular structure, NMR parameters, and ${}^{19}\mathrm{F}$ thermal equilibrium spectrum of trifluoroiodoethylene. The three fluorine spins in the molecule are marked as the corresponding qubits. The table summarizes the relevant NMR parameters, i.e., resonance frequencies ${\nu }_i$ and $J$-coupling constants. The ${}^{19}\mathrm{F}$ spectrum is obtained after a $\pi /2$ readout pulse on the thermal equilibrium state. The resonance lines of each qubit are labeled by the corresponding logical states of the other two qubits in the computational basis.

Figure 2

(a) Quantum circuit showing the specific sequence of implementation of the controlled-rotation, controlled-not, controlled-controlled-not, and controlled-controlled-phase gates required to construct a generic state and (b) NMR pulse sequence to implement a general three-qubit generic state; ${\tau }_{ij}$ is the evolution period under the $J_{ij}$ coupling. The ${180}^{\circ }$ pulses are represented by unfilled rectangles. The other pulses are labeled with their specific flip angles and phases. The last pulse (gray shaded) on the third qubit is a transition-selective ${180}^{\circ }$ pulse on the $|110\rangle$ to $|111\rangle$ transition about an arbitrary axis $\widehat{\mathbf{n}}$ which is inclined at angle $(\varphi +90)$ with the $x$ axis. The last two rectangular pulses on the first and second qubits are ${90}^{\circ }z$-rotations, to compensate the extra phases acquired (as described in the text).

Figure 3

The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the three-qubit generic state, reconstructed using full state tomography. The values of the parameters are $\alpha ={45}^{\circ },\beta ={55}^{\circ },\gamma ={60}^{\circ },\delta ={58}^{\circ },\varphi ={125}^{\circ }$. The rows and columns encode the computational basis in binary order, from $|000\rangle$ to $|111\rangle$. The experimentally tomographed state has a fidelity of $0.92$.

Figure 4

(a) Quantum circuit to implement a generalized GHZ state, derived from the general circuit for generic state construction given in Fig. 2. (b) Simplified circuit for experimental implementation of the GHZ state. (c) NMR pulse sequence corresponding to the circuit in (b). The ${\tau }_d=\frac{{\tau }_{13}-{\tau }_{12}}{2}$ period is tailored such that the system evolves solely under the $J_{13}$ coupling term.

Figure 5

The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the GHZ state, reconstructed using full state tomography. The rows and columns encode the computational basis in binary order, from $|000\rangle$ to $|111\rangle$. The experimentally tomographed state has a fidelity of 0.97.

Figure 6

(a) Quantum circuit to implement the $W$ state, derived from the general circuit for generic state construction given in Fig. 2. (b) Simplified circuit for experimental implementation of the $W$ state. (c) NMR pulse sequence to experimentally implement the $W$ state, starting from the initial pseudopure state $|100\rangle$. The first pulse on the second qubit implements a $U_{2\beta }^2$ rotation, with $2\beta =2{sin}^{-1}(1/\sqrt{3})\equiv 70.{53}^{\circ }$.

Figure 7

The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the $W$ state, reconstructed using full state tomography. The rows and columns encode the computational basis in binary order, from $|000\rangle$ to $|111\rangle$. The experimentally tomographed state has a fidelity of 0.96.

Figure 8

The real (Re) and imaginary (Im) parts of the density matrix for the $W$ state. (a) The two-qubit reduced density matrix ${\rho }_{AB}$. (b) The two-qubit reduced density matrix ${\rho }_{BC}$. (c) The entire three-qubit density matrix ${\rho }_{ABC}$, reconstructed from the corresponding two-qubit reduced density matrices. The rows and columns encode the computational basis in binary order, from $|00\rangle$ to $|11\rangle$ for two qubits and from $|000\rangle$ to $|111\rangle$ for three qubits. The tomographed state has a fidelity of 0.97.

Figure 9

The real (Re) and imaginary (Im) parts of the density matrix for the generic state. (a) The two-qubit reduced density matrix ${\rho }_{AB}$. (b) The two-qubit reduced density matrix ${\rho }_{BC}$. (c) The entire three-qubit density matrix ${\rho }_{ABC}$, reconstructed from the corresponding two-qubit reduced density matrices. The parameter set includes $\alpha ={45}^{\circ },\beta ={55}^{\circ },\gamma ={60}^{\circ },\delta ={58}^{\circ },\varphi ={125}^{\circ }$. The rows and columns encode the computational basis in binary order, from $|00\rangle$ to $|11\rangle$ for two qubits and from $|000\rangle$ to $|111\rangle$ for three qubits. The tomographed state has a fidelity of 0.90.