One Qubit Gates

The second step for realizing QIP is to give a means for controlling the qubits so that quantum algorithms can be implemented. The qubits are controlled with carefully modulated external fields to realize specific unitary evolutions called ``gates''. Each such evolution can be described by a unitary operator applied to one or more qubits. The simplest method for demonstrating that sufficient control is available is to show how to realize a set of one- and two-qubit gates that is ``universal'' in the sense that in principle, every unitary operator can be implemented as a composition of gates [15,16,17].

One-qubit gates can be thought of as rotations of the Bloch sphere and can be implemented in NMR with electromagnetic pulses. In general, the effect of a magnetic field on a nuclear spin is to cause a rotation around the direction of the field. In terms of the quantum state of the spin, the effect is described by an internal Hamiltonian of the form $H=(\omega_x\sigma_x+\omega_y\sigma_y+\omega_z\sigma_z)/2$. The coefficients of the Pauli matrices depend on the magnetic field according to $\vec{\omega} = (\omega_x,\omega_y,\omega_z)= -\mu \mathbf{B}$, where $\mu$ is the nuclear magnetic moment and $\mathbf{B}$ is the magnetic field vector. In terms of the Hamiltonian, the evolution of the spin's quantum state in the presence of the magnetic field $\mathbf{B}$ is therefore given by $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_t}\mbox{$\rangle\hspa... ...t}\vert$}{\psi_0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, so that the spin direction in the Bloch sphere rotates around $\vec{\omega}$ with angular frequency $\omega=\vert\vec{\omega}\vert$.

In the case of liquid-state NMR, there is an external, strong magnetic field along the $z$-axis and the applied electromagnetic pulses add to this field. One can think of these pulses as contributing a relatively weak magnetic field (typically less than $.001$ of the external field) whose orientation is in the $xy$-plane. One use of such a pulse is to tip the nuclear spin from the $z$-axis to the $xy$-plane. To see how that can be done, assume that the spin starts in the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, which points up along the $z$-axis in the Bloch sphere representation. Because this state is aligned with the external field, it does not precess. To tip the spin, one can start by applying a pulse field along the $x$-axis. Because the pulse field is weak compared to the external field, the net field is still almost along the $z$-axis. The spin now rotates around the net field. Because it started along $z$, it moves only in a small circle near the $z$-axis. To force the spin to tip further, one changes the orientation of the pulse field at the same frequency as the precession caused by the external field. This is called a ``resonant'' pulse. Because typical precession frequencies are hundreds of ${\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,$, such a pulse consists of radio-frequency (RF) electromagnetic fields.

To better understand how resonant pulses work, it is convenient to use the ``rotating frame''. In this frame, we imagine that our apparatus rotates at the precession frequency of the nuclear spin. In this way, the effect of the external field is removed. In particular, in the rotating frame the nuclear spin does not precess, and a resonant pulse's magnetic field looks like a constant magnetic field applied, for example, along the $(-x)$-axis of the rotating frame. The nuclear spin responds to the pulse by rotating around the $x$-axis as expected: If the spin starts along the $z$-axis, it tips toward the $(-y)$-axis, then goes to tthe $(-z)$-, the $y$-, and finally back to the $z$-axis, all in the rotating frame. See Fig. 4.

FIG. 4: Single bit rotation around the $x$-axis in the rotating frame. An applied magnetic field along the rotating frame's $(-x)$-axis due to a resonant RF pulse moves the nuclear spin direction from the $z$-axis toward the $(-y)$-axis. The initial and final states for the nuclear spin are shown for a $90^\circ$ rotation. If the strength of the applied magnetic field is such that the spin evolves according to the Hamiltonian $\omega_x\sigma_x/2$, then it has to be turned on for a time $t=\pi/(2\omega_x)$ to cause the rotation shown.

The rotating frame makes it possible to define the state of the qubit realized by a nuclear spin as the state with respect to this frame. As a result, the qubit's state does not change unless RF pulses are applied. In the context of the qubit realized by a nuclear spin, the rotating frame is called the ``logical frame''. In the following, references to the Bloch sphere axes and associated observables are understood to be with respect to an appropriate, usually rotating, frame. Different frames can be chosen for each nuclear spin of interest, so we often use multiple independently rotating frames and refer each spin's state to the appropriate frame.

Use of the rotating frame together with RF pulses makes it possible to implement all one-qubit gates on a qubit realized by a spin-${1\over 2}$ nucleus. To apply a rotation around the $x$-axis, a resonant RF pulse with effective field along the rotating frame's $(-x)$-axis is applied. This is called an ``$x$-pulse'', and $x$ is the ``axis'' of the pulse. While the RF pulse is on, the qubit's state evolves as $e^{-i\omega_x\sigma_x t/2}$. The strength (or ``power'') of the pulse is characterized by $\omega_x$, the ``nutation'' frequency. To implement a rotation by an angle of $\phi$, the pulse is turned on for a period $t=\phi/\omega_x$. Rotations around any axis in the plane can be implemented similarly. The angle of the pulse field with respect to the $(-x)$-axis is called the ``phase'' of the pulse. It is a fact that all rotations of the Bloch sphere can be decomposed into rotations around axes in the plane. For rotations around the $z$-axis, an easier technique is possible. The current absolute phase $\theta$ of the rotating frame's $x$-axis is given by $\theta_0+\omega t$, where $\omega$ is the precession frequency of the nuclear spin. Changing the angle $\theta_0$ by $-\phi$ is equivalent to rotating the qubit's state by $\phi$ around the $z$-axis. In this sense, $z$-pulses can be implemented exactly. In practice, this change of the rotating frame's phase means that the absolute phases of future pulses must be shifted accordingly. This implementation of rotations around the $z$-axis is possible because phase control in modern equipment is extremely reliable so that errors in the phase of applied pulses are negligible compared to other sources of errors.

So far, we have considered just one nuclear spin in a molecule. But the RF fields are experienced by the other nuclear spins as well. This side-effect is a problem if only one ``target'' nuclear spin's state is to be rotated. There are two cases to consider depending on the precession frequencies of the other, ``non-target'' spins. Spins of nuclei of different isotopes, such as those of other species of atoms, usually have precession frequencies that differ from the target's by many ${\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,$ at $11.7{\mathchoice{\mbox{T}}{\mbox{T}}{\mbox{\small T}}{\mbox{\tiny T}}}\,$. A pulse resonant for the target has little effect on such spins. This is because in the rotating frames of the non-target spins, the pulse's magnetic field is not constant but rotates rapidly. The power of a typical pulse is such that the effect during one rotation of the pulse's field direction is insignificant and averages to zero over many rotations. This is not the case for non-target spins of the same isotope. Although the variations in their chemical environments result in frequency differences, these differences are much smaller, often only a few ${\mathchoice{\mbox{kHz}}{\mbox{kHz}}{\mbox{\small kHz}}{\mbox{\tiny kHz}}}\,$. The period of a $1{\mathchoice{\mbox{kHz}}{\mbox{kHz}}{\mbox{\small kHz}}{\mbox{\tiny kHz}}}\,$ rotation is $1{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$, whereas so-called ``hard'' RF pulses require only $10$'s of $\mu{\mathchoice{\mbox{s}}{\mbox{s}}{\mbox{\small s}}{\mbox{\tiny s}}}\,$ ( $.001{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$) to complete the typical $90^\circ$ or $180^\circ$ rotations. Consequently, in the rotating frame of a non-target spin with a small frequency difference, a hard RF pulse's magnetic field is nearly constant for the duration of the pulse. As a result, such a spin experiences a rotation similar to the one intended for the target. To rotate a specific nuclear spin or spins within a narrow range of precession frequencies, one can use weaker, longer-lasting ``soft'' pulses instead. This approach leads to the following strategies for applying pulses: To rotate all the nuclear spins of a given species (such as the two ${}^{13}$C of TCE) by a desired angle, apply a hard RF pulse for as short a time as possible. To rotate just one spin having a distinct precession frequency, apply a soft RF pulse of sufficient duration to have little effect on other spins. The power of soft pulses is usually modulated in time (``shaped'') to reduce the time needed for a rotation while minimizing ``crosstalk'', a term that describes unintended effects on other nuclear spins.