Thermal equilibrium as an initial state for quantum computation by NMR

An experimental implementation of an ensemble quantum algorithm is presented in which the thermal state is used as an initial state of the algorithm. More specifically, we use four active qubits to demonstrate the solution of the Deutsch-Jozsa problem using a nuclear magnetic resonance quantum computer. For the implemented algorithm the number of molecules in the NMR sample is irrelevant to the number of qubits available to an NMR quantum computer, and the initial state is chosen to be the state of thermal equilibrium, thereby avoiding the preparation of pseudopure states and the resulting exponential loss of signal as the number of qubits increases. As expected, measured spectra demonstrate a clear distinction between constant and balanced functions.

Figure 1

Schematic representation of the decomposition of $c{U}_{f_b}$ given in Eq. (9). The symbols are defined in the inset. Up to local spin operations (see main text for details), the propagators A–E correspond to the pulse sequence elements A–E of the streamlined pulse sequence shown in Fig. 2.

Figure 2

Streamlined pulse sequence for the implementation of the unitary transformation $c{U}_{f_b}$. The durations ${\tau }_{kl}=1∕\left(2J_{kl}\right)$ and ${\tau }_{kl}∕2=1∕\left(4J_{kl}\right)$ were chosen to be integer multiples of $\Delta =1∕\mid {\nu }_1-{\nu }_2\mid =81.75\mu \mathrm{s}$ [24]. Filled rectangles represent 180° pulses. The bell-shaped symbols labeled “e” and “g” represent selective e-SNOB 90° pulses [28] and selective Gaussian 180° pulses [29], respectively. During the selective e-SNOB 90° pulses, qubit 3 was actively decoupled by applying an MLEV-4 [31, 32, 33] expanded sequence of 180° pulses to qubit 3 (boxes labeled “M”). Dotted vertical lines represent local $z$ rotations. The overall duration of the pulse sequence is $84\mathrm{ms}$.

Figure 3

Experimental results for (A) the constant function $f_0$ and (B) the balanced function $f_b$. The spectra show the signal of the carbonyl ${}^{13}\mathrm{C}^{\prime }$ spin of BOC-(${}^{13}\mathrm{C}_2\text{−}{}^{15}\mathrm{N}{}^2\mathrm{D}_2^{\alpha }$-glycine)-fluoride [24] corresponding to qubit 1. Our NMR experiments were performed using a BRUKER AVANCE 400 spectrometer with five independent rf channels and a QXI probe (H,C-F,N). The lock coil was also used for deuterium decoupling utilizing a lock switch. The spectra with a spectral width of $900\mathrm{Hz}$ were acquired (128 scans) after applying the corresponding unitary transformations $c{U}_{f_0}$ and $c{U}_{f_b}$ to ${\rho }_1=I_{1x}$.

Figure 4

Set of control spectra (128 scans) of qubit 1 after application of a ${\mathrm{CNOT}}_{42}$ gate (with a duration of $37\mathrm{ms}$) to ${\rho }_b^{\prime }$ (A) and application of the pulse sequence of $c{U}_{f_b}$ (cf. Fig. 2) to $2I_{1x}{I}_{4z}$ (B). The additional control spectra result if the pulse sequence corresponding to $c{U}_{f_b}$ is applied to ${\rho }_0=2I_{1z}{I}_{3z}$ (C) and to ${\rho }_0=4I_{1z}{I}_{2z}{I}_{3z}$ (D), followed by a ${90}_y^{°}$ read out pulse.