Initialization of NMR quantum registers using long-lived singlet states

An ensemble of nuclear spin pairs under certain conditions is known to exhibit singlet state lifetimes much longer than other nonequilibrium states. This property of singlet states can be exploited in quantum information processing for efficient initialization of quantum registers. Here we describe a general method of initialization and experimentally demonstrate it with two-, three-, and four-qubit nuclear spin registers.

Figure 1

The circuit diagrams for initializing (a) two-qubit register, (b) even-qubit register, and (c) odd-qubit register. Here SL and $G_z$ denote spin-lock and the pulsed field gradient. $U_1$ and $U_2$ are gates as described in the text. In (b) and (c) the c${}^{\prime }$not gates with open circles correspond to not operation if the control is 0 and identity if the control is 1. The h gate corresponds to pseudo-Hadamard: $|0\rangle \stackrel{\mathrm{h}}{\to }(|0\rangle -|1\rangle )/\sqrt{2}$ and $|1\rangle \stackrel{\mathrm{h}}{\to }(|0\rangle +|1\rangle )/\sqrt{2}$. The dashed boxes indicate optional spin-locks.

Figure 2

The molecular structure (a) and Hamiltonian parameters (b) of 5-bromothiophene-2-carbaldehyde (5 mg dissolved in 0.5 ml of dimethyl sulphoxide-d6), forming a homonuclear two-qubit register. In (b) diagonal and off-diagonal elements correspond to the chemical shifts and the scalar coupling constant, respectively (in Hz). The ${}^1\mathrm{H}$ spectra correspond to the pseudopure state (c) and the equilibrium mixed state (d). The bar plots (e)–(h) correspond to the real (e), (g) and imaginary (f), (h) parts of the theoretical (e), (f) and experimental (g), (h) pseudopure $|01\rangle$ state. The remaining bar plots correspond to the experimental real parts of $|S_0\rangle$ (i), $|{\psi }_{+}\rangle$ (j), $|{\varphi }_{+}\rangle$ (k), and $|{\varphi }_{-}\rangle$ (l).

Figure 3

The molecular structure (a) and Hamiltonian parameters (b) of acrylonitrile (5 mg dissolved in 0.5 ml of CDCl${}_3$), forming a homonuclear three-qubit register. In (b) diagonal elements are chemical shifts (in Hz) and the off-diagonal elements are the scalar coupling constants (in Hz). The ${}^1\mathrm{H}$ spectra correspond to the pseudopure state (c) and the equilibrium mixed state (d). The bar plots are showing the real (e), (g) and imaginary (f), (h) parts of theoretical (e), (f) and experimental (g), (h) pseudopure $|010\rangle \langle 010|$ state.

Figure 4

The molecular structure (a) and Hamiltonian parameters (b) of aspirin (5 mg dissolved in 0.5 ml of CD${}_3$OD), forming a homonuclear four-qubit register. In (b) diagonal elements are chemical shifts (in Hz) and the off-diagonal elements are the scalar coupling constants (in Hz). The bar plot in (c) displays the diagonal elements of the density matrix obtained by tomography of the pseudopure $|1001\rangle$ state. The ${}^1\mathrm{H}$ spectra correspond to the pseudopure state (d) and the equilibrium mixed state (e).