Experimental protection against evolution of states in a subspace via a super-Zeno scheme on an NMR quantum information processor

We experimentally demonstrate the freezing of evolution of quantum states in one- and two-dimensional subspaces of two qubits, on an NMR quantum information processor. State evolution was frozen and leakage of the state from its subspace to an orthogonal subspace was successfully prevented using super-Zeno sequences [Phys. Rev. Lett. 96, 100405 (2006)], comprising a set of radio frequency (rf) pulses punctuated by pre-selected time intervals. We demonstrate the efficacy of the scheme by preserving different types of states, including separable and maximally entangled states in one- and two-dimensional subspaces of two qubits. The change in the experimental density matrices was tracked by carrying out full state tomography at several time points. We use the fidelity measure for the one-dimensional case and the leakage (fraction) into the orthogonal subspace for the two-dimensional case, as qualitative indicators to estimate the resemblance of the density matrix at a later time to the initially prepared density matrix. For the case of entangled states, we additionally compute an entanglement parameter to indicate the presence of entanglement in the state at different times. We experimentally demonstrate that the super-Zeno scheme is able to successfully confine state evolution to the one- or two-dimensional subspace being protected.

Figure 1

(a) Molecular structure of cytosine with the two qubits labeled as $H^1$ and $H^2$ and tabulated system parameters with chemical shifts ${\nu }_i$ and scalar coupling $J_{12}$ (in Hz) and relaxation times $T_1$ and $T_2$ (in seconds). (b) NMR spectrum 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 qubit. (c) NMR spectrum of the pseudopure $|00\rangle$ state.

Figure 2

(a) Quantum circuit for preservation of the state $|11\rangle$ using the super-Zeno scheme. ${\Delta }_i=x_{i}t,(i=1...5)$ denote time intervals punctuating the unitary operation blocks. Each unitary operation block contains a controlled-phase gate ($Z$), with the first (top) qubit as the control and the second (bottom) qubit as the target. The entire scheme is repeated $N$ times before measurement (for our experiments $N=30$). (b) Block-wise depiction of the corresponding NMR pulse sequence. A $z$ gradient is applied just before the super-Zeno pulses, to clean up undesired residual magnetization. The unfilled and black rectangles represent hard ${180}^0$ and ${90}^0$ pulses, respectively, while the unfilled and gray-shaded conical shapes represent ${180}^0$ and ${90}^0$ pulses (numerically optimized using grape), respectively; ${\tau }_{12}$ is the evolution period under the $J_{12}$ coupling. Pulses are labeled with their respective phases and unless explicitly labeled, the phase of the pulses on the second (bottom) qubit are the same as those on the first (top) qubit.

Figure 3

Real (left) and imaginary (right) parts of the experimental tomographs of the (a) $|11\rangle$ state, with a computed fidelity of 0.99; (b)–(e) depict the state at $T=0.61,3.03,5.46,7.28$ s, with the tomographs on the left and the right representing the state without and after applying the super-Zeno preserving scheme, respectively. The rows and columns are labeled in the computational basis ordered from $|00\rangle$ to $|11\rangle$.

Figure 4

(a) Quantum circuit for preservation of the singlet state using the super-Zeno scheme. ${\Delta }_i,(i=1...5)$ denote time intervals punctuating the unitary operation blocks. The entire scheme is repeated $N$ times before measurement (for our experiments $N=10$). (b) NMR pulse sequence corresponding to one unitary block of the circuit in (a). A $z$ gradient is applied just before the super-Zeno pulses to clean up undesired residual magnetization. The unfilled rectangles represent hard ${180}^0$ pulses, the black filled rectangles representing hard ${90}^0$ pulses, while the shaded shapes represent numerically optimized (using grape) pulses and the gray-shaded shapes representing ${90}^0$ pulses, respectively; ${\tau }_{12}$ is the evolution period under the $J_{12}$ coupling. Pulses are labeled with their respective phases and unless explicitly labeled, the phase of the pulses on the second (bottom) qubit are the same as those on the first (top) qubit.

Figure 5

Real (left) and imaginary (right) parts of the experimental tomographs of the (a) $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ (singlet) state, with a computed fidelity of 0.99; (b)–(e) depict the state at $T=0.85,2.54,4.24,5.93$ s, with the tomographs on the left and the right representing the state without and after applying the super-Zeno preserving scheme, 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 of (a) the $|11\rangle$ state and (b) the $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ (singlet) state, without any preserving scheme and after the super-Zeno preserving sequence. The fidelity of the state with the super-Zeno preservation remains close to 1.

Figure 7

(a) Quantum circuit for preservation of the $\{01,10\}$ subspace using the super-Zeno scheme. ${\Delta }_i,(i=1...5)$ denote time intervals punctuating the unitary operation blocks. The entire scheme is repeated $N$ times before measurement (for our experiments $N=30$). (b) NMR pulse sequence corresponding to the circuit in (a). A $z$ gradient is applied just before the super-Zeno pulses, to clean up undesired residual magnetization. The unfilled rectangles represent hard ${180}^0$ pulses; ${\tau }_{12}$ is the evolution period under the $J_{12}$ coupling. Pulses are labeled with their respective phases.

Figure 8

Real (left) and imaginary (right) parts of the experimental tomographs of the (a) $|10\rangle$ state in the two-dimensional subspace $\{01,10\}$, with a computed fidelity of 0.98; (b)–(e) depict the state at $T=1.15,3.45,5.75,7.48$ s, with the tomographs on the left and the right representing the state without and after applying the super-Zeno preserving scheme, respectively. The rows and columns are labeled in the computational basis ordered from $|00\rangle$ to $|11\rangle$.

Figure 9

Real (left) and imaginary (right) parts of the experimental tomographs of the (a) $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ (singlet) state in the two-dimensional subspace $\{01,10\}$, with a computed fidelity of 0.98; (b)–(e) depict the state at $T=1.15,3.46,5.77,7.50$ s, with the tomographs on the left and the right representing the state without and after applying the super-Zeno preserving scheme, respectively. The rows and columns are labeled in the computational basis ordered from $|00\rangle$ to $|11\rangle$.

Figure 10

Plot of leakage fraction from the $\left\{\right|01\rangle ,|10\rangle \}$ subspace to its orthogonal subspace $\left\{\right|00\rangle ,|11\rangle \}$ of (a) the $|10\rangle$ state and (b) the $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ (singlet) state, without any preservation and after applying the super-Zeno sequence. The leakage to the orthogonal subspace is minimal (remains close to zero) after applying the super-Zeno scheme.

Figure 11

Plot of entanglement parameter $\eta$ with time, with and without applying the super-Zeno sequence, computed for (a) the $\frac{1}{\sqrt{2}}\left(\right|01\rangle -|10\rangle )$ (singlet) state, and (b) the same singlet state when embedded in the subspace $\left\{\right|01\rangle ,|10\rangle \}$ being preserved.