Creating a Labeled Pseudopure State

One way to realize the standard pseudopure state starting from the equilibrium density matrix $\rho_{\mbox{\tiny thermal}}$ is to eliminate the observable contributions due to terms of $\rho_{\mbox{\tiny thermal}}$ different from $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\ldots\mathfra... ...hfrak{0}\ldots\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$. There are several different methods of accomplishing this. For example, one can perform multiple experiments with different pre-processing of the equilibrium state so that signals from unwanted terms average to zero (temporal averaging). Or one can use gradients to remove the unwanted terms in one experiment (spatial averaging).

In this section, we show how to use spatial averaging to prepare a so-called ``labeled'' pseudopure state on two nuclear spins. In general, instead of preparing the standard pseudopure state with deviation $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\ldots}\mbox{$... ...angle$}{\mathfrak{0}\ldots}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ on $n$ spin-$1\over 2$ nuclei, one can prepare a ``labeled'' pseudopure state with deviation ${\sigma_x}^{({\mathsf {1}})}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$... ...angle$}{\mathfrak{0}\ldots}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ on $n+1$ spins. This state is easily recognizable with an NMR observation of the first spin: Assuming that all the peaks arising from couplings to other spins are resolved, the first spin's peak group has $2^n$ peaks corresponding to which logical states the other spins are in. If the current state is the above labeled pseudopure state, then all the other spins are in the logical state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, which implies that in the spectrum, only one of the peaks of the first spin's peak group is visible. See Fig. 12.

FIG. 12: Relationship of a labeled pseudopure state spectrum to a peak group. The top spectrum shows the peak group of a simulated nuclear spin coupled to three other spins with coupling constants of $100{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$, $60{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$, and $24{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$. The simulation parameters are the same as in Fig. 7. Given above each peak is the part of the initial deviation that contributes to the peak. The spin labels have been omitted. Each contributing deviation consists of $\sigma_x$ on the observed nucleus followed by one of the logical (up or down) states (density matrices) for each of the other spins. The notation is as defined after Eq. 9. The bottom spectrum shows what is observed if the initial deviation is the standard labeled pseudopure state. This state contributes only to the right-most peak, as this peak is associated with the logical $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ states on the spins not observed.

The labeled pseudopure state can be used as a standard pseudopure state on $n$ qubits. Observation of the final answer of a computation is possible by observing spin $\mathsf {1}$, provided that the coupling to the answer-containing spin is sufficiently strong for the peaks corresponding to its two logical states to be well separated. For this purpose, the couplings to the other spins need not be resolved in the peak group. Specifically, to determine the answer of a computation, the peaks of the peak group of spin $\mathsf {1}$ are separated into two subgroups, the first (second) containing the peaks associated with the answer-containing spin being in state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ ( $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$), respectively. Comparing the total signal in each of the two peak subgroups gives the relative probabilities of the two answers ($\mathfrak{0}$ or $\mathfrak{1}$).

The labeled pseudopure state can also be used to investigate the effect of a process that manipulates the state of one qubit and requires $n$ additional initialized qubits. Examples include experimental verification of one-qubit error-correcting codes as explained in Sect. 3.3.

For preparing the two-qubit labeled pseudopure state, consider the two carbon nuclei in labeled TCE with the proton spin decoupled so that its effect can be ignored. A ``transition'' in the density matrix for this system is an element of the density matrix of the form $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{ab}\mbox{$\rangle\hspace*{... ...space*{-4.3pt}\langle$}{cd}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$, where $a,b,c$, and $d$ are $\mathfrak{0}$ or $\mathfrak{1}$. Let $\Delta(ab,cd) = (a-c)+(b-d)$, where in the expression on the right, $a,b,c$, and $d$ are interpreted as the numbers $0$ or $1$ as appropriate. Applying a pulsed gradient along the $z$-axis evolves the transitions according to: $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{ab}\mbox{$\rangle\hspace*{... ...space*{-4.3pt}\langle$}{cd}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$, where $\nu$ is proportional to the product of the gradient power and pulse time, and $z$ is the molecule's position along the $z$-coordinate. For example, $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\mathfrak{1}}\... ...}{\mathfrak{1}\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ has $\Delta=0$ and is not affected, whereas $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\mathfrak{0}}\... ...}{\mathfrak{1}\mathfrak{1}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ acquires a phase of $e^{-i2\nu z}$. There are only two transitions, $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\mathfrak{0}}\... ...}{\mathfrak{1}\mathfrak{1}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}\mathfrak{1}}\... ...}{\mathfrak{0}\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$, whose acquired phase has a rate of $\Delta=\pm 2$ along the $z$ axis. These transitions are called ``two-coherences''. The idea is to first recognize that these transitions can be used to define a labeled pseudopure ``cat'' state (see below), then to exploit the two-coherences' unique behavior under the gradient in order to extract the pseudopure cat state, and finally to ``decode'' to a standard labeled pseudopure state. Note that the property that two-coherences' phases evolve at twice the basic rate is a uniquely quantum phenomenon for two spins. No such effect is observed for a pair of classical spins.

The standard two-qubit labeled pseudopure state's deviation can be written as $\rho_{\textrm{\footnotesize std}_x}={\sigma_x}^{({\mathsf {1}})}{1\over 2}\left... ... {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}+{\sigma_z}^{({\mathsf {2}})}\right)$. We can consider other deviations of this form where the two Pauli operators are replaced by a pair of different, commuting products of Pauli operators. An example is

$$\rho_{\textrm{\footnotesize cat}_x} = \left({\sigma_x}^{({\m... ...sigma_z}^{({\mathsf {1}})}{\sigma_z}^{({\mathsf {2}})}\right),$$ (22)

where we replaced ${\sigma_x}^{({\mathsf {1}})}$ by ${\sigma_x}^{({\mathsf {1}})}{\sigma_x}^{({\mathsf {2}})}$ and ${\sigma_z}^{({\mathsf {2}})}$ by ${\sigma_z}^{({\mathsf {1}})}{\sigma_z}^{({\mathsf {2}})}$, and as announced, the two Pauli products commute. We will show that there is a simple sequence of $90^\circ$ rotations whose effect is to ``decode'' the deviations ${\sigma_x}^{({\mathsf {1}})}{\sigma_x}^{({\mathsf {2}})}\rightarrow {\sigma_x}^{({\mathsf {1}})}$ and ${\sigma_z}^{({\mathsf {1}})}{\sigma_z}^{({\mathsf {2}})}\rightarrow {\sigma_z}^{({\mathsf {2}})}$, thus converting the state $\rho_{\textrm{\footnotesize cat}_x}$ to $\rho_{\textrm{\footnotesize std}_x}$. The state $\rho_{\textrm{\footnotesize cat}_x}$ can be expressed in terms of the transitions as follows:

$$\rho_{\textrm{\footnotesize cat}_x} = \mbox{$\vert\hspace*{-... ...hfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}.$$ (23)

It can be seen that $\rho_{\textrm{\footnotesize cat}_x}$ consists only of two-coherences. Another such state is

$$\rho_{\textrm{\footnotesize cat}_y} = \left({\sigma_x}^{({\mathsf {1}})}{\sigma_y}^{({\mathsf {2}})}\ri... ...1\mskip-5mu l}}+{\sigma_z}^{({\mathsf {1}})}{\sigma_z}^{({\mathsf {2}})}\right)$$ (24)

$$= -i\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{... ...{\mathfrak{0}\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}.$$ (25)

Suppose that one can create a state that has a deviation of the form $\rho=\alpha\rho_{\textrm{\footnotesize cat}_x}+\beta\rho_{\textrm{\footnotesize rest}}$ such that $\rho_{\textrm{\footnotesize rest}}$ contains no two-coherences or zero-coherences. After a gradient pulse is applied, the state becomes

$$\alpha\left(\cos(2\nu z)\rho_{\textrm{\footnotesize cat}_x}+... ...e cat}_y}\right) +\beta\rho_{\textrm{\footnotesize rest}}(z),$$ (26)

where $\rho_{\textrm{\footnotesize rest}}(z)$ depends periodically on $z$ with spatial frequencies of $\pm\nu$, not $\pm 2\nu$ or $0$. We can then decode this state to

$$\varrho(z) = \alpha\left(\cos(2\nu z)\rho_{\textrm{\footnotesize std}_x}+\sin(... ...extrm{\footnotesize std}_y}\right) +\beta\rho'_{\textrm{\footnotesize rest}}(z)$$ (27)

$$= \alpha\left(\cos(2\nu z){\sigma_x}^{({\mathsf {1}})}+\sin(2\nu z)... ...\sigma_z}^{({\mathsf {1}})}\right)+\beta\rho'_{\textrm{\footnotesize rest}}(z).$$ (28)

If one now applies a gradient pulse of twice the total strength and opposite orientation, the first term is restored to $\alpha\rho_{\textrm{\footnotesize std}_x}$, but the second term retains non-zero periodicities along $z$. Thus, if we no longer use any operations to distinguish among different molecules along the $z$-axis, or if we let diffusion erase the memory of the position along $z$, then the second term is eliminated from observability by being averaged to zero. The desired labeled pseudopure state is obtained. Zero-coherences during the initial gradient pulse are acceptable provided that the decoding transfers them to coherences different from zero or two during the final pulse in order to ensure that they also average to zero. A pulse sequence that realizes a version of the above procedure is shown in Fig. 13.

FIG. 13: Quantum network and pulse sequence to realize a two-qubit labeled pseudopure state. The network is shown above the pulse sequence realizing it. A coupling constant of $100{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$ is assumed. Gradients are indicated by spirals in the network. The gradient strength is given as the red line in the pulse sequence. The doubling of the integrated gradient strength required to achieve the desired ``echo'' is indicated by a doubling of the gradient pulse time. The numbers above the quantum network are checkpoints used in the discussion below. The input state's deviation is assumed to be ${\sigma_z}^{({\mathsf {1}})}$. This deviation can be obtained from the equilibrium state by applying a $90^\circ$ rotation to spin $\mathsf {2}$ followed by a gradient pulse along another axis to remove ${\sigma_z}^{({\mathsf {2}})}$. Instead of using a gradient pulse, one can use phase cycling, which involves performing two experiments, the second having the sign of the phase in the first $y$ pulse changed, and then subtracting the measured signals.

We can follow what happens to an initial deviation density matrix of ${\sigma_z}^{({\mathsf {1}})}$ as the network of Fig. 13 is executed. We use product operators with the abbreviations $I={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}},X=\sigma_x,Y=\sigma_y,Z=\sigma_z$, and, for example $XY={\sigma_x}^{({\mathsf {1}})}{\sigma_y}^{({\mathsf {2}})}$. At the checkpoints indicated in the figure the deviations are the following

$$\begin{array}{@{}llcl} (1) & ZI\\ (2) & XI\\ (3) & YZ\\ (... ... -X(I+Z) &+& -(\cos(-2\nu z)X+\sin(-2\nu z)Y)(I-Z). \end{array}$$ (29)

Except for a sign, the desired state is obtained. The right-most term is eliminated after integrating over the sample, or after diffusion erases memory of $z$.

This method for making a two-qubit labeled pseudopure state can be extended to arbitrarily many ($n$) qubits with the help of the two $n$-coherences, which are the transitions with $\Delta=\pm n$. An experiment implementing this method can be used to determine how good the available quantum control is. The quality of the control is determined by a comparison of two spectral signals: $I_p$, the intensity of the single peak that shows up in the peak group for spin $\mathsf {1}$ when observing the labeled pseudopure state; and $I_0$, the intensity of the same peak in an observation of the initial deviation after applying a $90^\circ$ pulse to rotate ${\sigma_z}^{({\mathsf {1}})}$ into the plane. We performed this experiment on a seven-spin system and determined that $I_p/I_0= .73\pm.02$. This result implies a total error of $27\pm 2\%$. Because the implementation has $12$ two-qubit gates, an error rate of about $2\%$ per two-qubit gate is achievable for nuclear spins in this setting [23].