Experimental protection of arbitrary states in a two-qubit subspace by nested Uhrig dynamical decoupling

We experimentally demonstrate the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD) sequence to preserve arbitrary quantum states in a two-dimensional subspace of the four-dimensional two-qubit Hilbert space on a nuclear magnetic resonance quantum information processor. The effect of the state preservation is studied first on four known states, including two product states and two maximally entangled Bell states. Next, to evaluate the preservation capacity of the NUDD scheme, we apply it to eight randomly generated states in the subspace. Although, the preservation of different states varies, the scheme, on the average, performs very well. The complete tomographs of the states at different time points are used to compute fidelity. State fidelities using NUDD protection are compared with those obtained without using any protection. The nested pulse schemes are complex in nature and require careful experimental implementation.

Figure 1

(a) Circuit diagram for the three-layer NUDD sequence. The innermost UDD layer consists of $X_0$ control pulses, the middle layer comprises $X_1$ control pulses, and the outermost layer consists of $X_{\varphi }$ pulses. The entire NUDD sequence is repeated $M$ times; ${\mathrm{\Delta }}_i$ are time intervals. (b) NMR pulse sequence to implement the control pulses for $X_0$ and $X_1$ UDD sequences. The values of the rf pulse phases ${\varphi }_1$ and ${\varphi }_2$ are set to $x$ and $y$ for the $X_0$ and to $-x$ and $-y$ for the $X_1$ UDD sequence, respectively. (c) NMR pulse sequence to implement the control pulses for the $X_{\varphi }$ UDD sequence. The solid rectangles denote $\pi /2$ pulses while the empty rectangles denote $\pi$ pulses, respectively. The time period ${\tau }_{12}$ is set to the value ${\left(2J_{12}\right)}^{-1}$, where $J_{12}$ denotes the strength of the scalar coupling between the two qubits.

Figure 2

(a) Structure of isotopically enriched chloroform-${}^{13}\mathrm{C}$ molecule, with the ${}^1\mathrm{H}$ spin labeling the first qubit and the ${}^{13}\mathrm{C}$ spin labeling the second qubit. The system parameters are tabulated alongside with chemical shifts ${\nu }_i$ and scalar coupling $J_{12}$ (in Hz) and NMR spin-lattice and spin-spin relaxation times $T_1$ and $T_2$ (in seconds). (b) NMR spectrum obtained after a $\pi /2$ readout pulse on the thermal equilibrium state and (c) NMR spectrum of the pseudopure $|00\rangle$ state. The resonance lines of each qubit in the spectra are labeled by the corresponding logical states of the other qubit.

Figure 3

Plot of fidelity versus time for (a) the $|01\rangle$ state and (b) the $|10\rangle$ state, without any protection and after applying NUDD protection. The fidelity of both the states remains close to 1 for up to long times, after NUDD protection.

Figure 4

NMR pulse sequence for the preparation of random states. The sequence of pulses before the vertical dashed red line achieve state initialization into the $|00\rangle$ state. The values of flip angles $\theta$ and $\varphi$ of the rf pulses are the same as the $\theta$ and $\varphi$ values describing a general state in the two-dimensional subspace $\mathcal{P}=\left\{\right|01\rangle ,|10\rangle \}$. Solid and empty rectangles represent $\frac{\pi }{2}$ and $\pi$ pulses, respectively, while all other rf pulses are labeled with their respective flip angles and phases; the interval ${\tau }_{12}$ is set to ${\left(2J_{12}\right)}^{-1}$, where $J_{12}$ is the scalar coupling.

Figure 5

Real (left) and imaginary (right) parts of the experimental tomographs of the (a) $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ state, with a computed fidelity of 0.99. Panels (b)–(e) depict the state at $T=0.28,0.55,0.83,1.10$ s, with the tomographs on the left and the right representing the state without any protection and after applying NUDD protection, respectively. The rows and columns are labeled in the computational basis ordered from $|00\rangle$ to $|11\rangle$.

Figure 6

Plot of fidelity versus time for (a) the Bell singlet state and (b) the Bell triplet state, without any protection and after applying NUDD protection.

Figure 7

Geometrical representation of eight randomly generated states on a Bloch sphere belonging to the two-qubit subspace $\mathcal{P}=\left\{\right|01\rangle ,|10\rangle \}$. Each vector makes angles $\theta ,\varphi$ with the $z$ and $x$ axes, respectively. The state labels $\mathrm{RS}-i$ ($i=1,...,8$) are explained in the text.

Figure 8

Bar plots of fidelity versus time of eight randomly generated states (labeled $\mathrm{RS}-i, i=1,...,8$), without any protection (blue cross-hatched bars) and after applying NUDD protection (red solid bars): (a) RS-1, (b) RS-2, (c) RS-3, (d) RS-4, (e) RS-5, (f) RS-6, (g) RS-7, and (h) RS-8. (i) Bar plot showing average fidelity of all eight randomly generated states at each time point. The state labels are explained in the main text.