Glossary

Bloch sphere.

A representation of the state space of a qubit using the unit sphere in three dimensions. See Fig. 3.

Crosstalk.

In using physical control to implement a gate, crosstalk refers to unintended effects on qubits not involved in the gate.

Decoupling.

A method for ``turning off'' the interactions between two sets of spins. In NMR, this task can be achieved if one applies a rapid sequence of refocusing pulses to one set of spins. The other set of spins can then be controlled and observed as if independent of the first set.

Deviation of a state.

If $\rho$ is a density matrix for a state and $\rho=\alpha{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}+ \beta\sigma$, then $\sigma$ is a deviation of $\rho$.

Ensemble computation.

Computation with a large ensemble of identical and independent computers. Each step of the computation is applied identically to the computers. At the end of the computation, the answer is determined from a noisy measurement of the fraction $p_{\mathfrak{1}}$ of the computers whose answer is ``$\mathfrak{1}$''. The amount of noise is important for resource accounting: To reduce the noise to below $\epsilon$ requires increasing the resources used by a factor of the order of $1/\epsilon^2$.

Equilibrium state.

The state of a quantum system in equilibrium with its environment. In the present context, the environment behaves like a heat bath at temperature $T$ and the equilibrium state can be written as $\rho = e^{-H/kT}/Z$, where $H$ is the effective internal Hamiltonian of the system and $Z$ is determined by the identity $\mbox{tr}\rho = 1$.

FID.

Free induction decay. To obtain a spectrum on an NMR spectrometer after having applied pulses to a sample, one measures the decaying planar magnetization induced by the nuclear spins as they precess. The $x$- and $y$-components $M_x(t)$ and $M_y(t)$ of the magnetization as a function of time are combined to form a complex signal $M(t)=M_x(t)+iM_y(t)$. The record of $M(t)$ over time is called the FID, which is Fourier-transformed to yield the spectrum.

Inversion.

A pulse that flips the component of the spin along the $z$-axis. Note that any $180^\circ$ rotation around an axis in the $xy$-plane has this effect.

$J$-coupling.

The type of coupling present between two nuclear spins in a molecule in the liquid state.

Labeled molecule.

A molecule in which some of the nuclei are substituted by less common isotopes. A common labeling for NMR QIP involves replacing the naturally abundant carbon isotope $^{12}$C, with the spin-${1\over 2}$ isotope $^{13}$C.

Larmor frequency.

The precession frequency of a nuclear spin in a magnetic field. It depends linearly on the spin's magnetic moment and the strength of the field.

Logical frame.

The current frame with respect to which the state of a qubit carried by a spin is defined. There is an absolute (laboratory) frame associated with the spin observables $\sigma_x,\sigma_y,$ and $\sigma_z$. The observables are spatially meaningful. For example, the magnetization induced along the $x$-axis is proportional to $\mbox{tr}(\sigma_x\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mb... ...ce*{-4.3pt}\langle$}{\psi}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$})$, where $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is the physical state of the spin. Suppose that the logical frame is obtained from the physical frame with a rotation by an angle of $\theta$ around the $z$-axis. The observables for the qubit are then given by ${\sigma_x}^{({\mathsf {L}})}=\cos(\theta)\sigma_x+\sin(\theta)\sigma_y$, ${\sigma_y}^{({\mathsf {L}})}=\cos(\theta)\sigma_y-\sin(\theta)\sigma_z$, and ${\sigma_z}^{({\mathsf {L}})}=\sigma_z$. As a result, the change to the logical frame transforms the physical state to a logical state according to $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi}\mbox{$\rangle\hspace... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. That is, the logical state is obtained from the physical state by a $-\theta$ rotation around the $z$-axis. A resonant logical frame is used in NMR to compensate for the precession induced by the strong external field.

Magnetization.

The magnetic field induced by an ensemble of magnetic spins. The magnitude of the magnetization depends on the number of spins, the extent of alignment and the magnetic moments.

Nuclear magnetic moment.

The magnetic moment of a nucleus determines the strength of the interaction between its nuclear spin and a magnetic field. The precession frequency $\omega$ of a spin ${1\over 2}$ nucleus is given by $\mu B$, where $\mu$ is the nuclear magnetic moment and $B$ the magnetic field strength. For example, for a proton, $\mu=42.7{\mathchoice{\mbox{Mhz}}{\mbox{Mhz}}{\mbox{\small Mhz}}{\mbox{\tiny Mhz}}}\,/{\mathchoice{\mbox{T}}{\mbox{T}}{\mbox{\small T}}{\mbox{\tiny T}}}\,$.

NMR spectrometer.

The equipment used to apply RF pulses to and observe precessing magnetization from nuclear spins. Typical spectrometers consist of a strong, cylindrical magnet with a central bore in which there is a ``probe'' that contains coils and a sample holder. The probe is connected to electronic equipment for applying RF currents to the coils and for detecting weak oscillating currents induced by the nuclear magnetization.

Nuclear spin.

The quantum spin degree of freedom of a nucleus. It is characterized by its total spin quantum number, which is a multiple of ${1\over 2}$. Nuclear spins with spin ${1\over 2}$ are two-state quantum systems and can therefore be used as qubits immediately.

Nutation.

The motion of a spin in a strong $z$-axis field caused by a resonant pulse.

Nutation frequency.

The angular rate at which a resonant pulse causes nutation of a precessing spin around an axis in the plane.

One-qubit quantum computing.

The model of computation in which one can initialize any number of qubits in the state where qubit $\mathsf {1}$ is in the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangl... ...e*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!\scriptstyle{\mathsf {1}}}}$ and all the other qubits are in a random state. One can then apply one- and two-qubit unitary quantum gates and make one final measurement of the state of qubit $\mathsf {1}$ after which the system is reinitialized. The model can be used to determine properties of the spectral density function of a Hamiltonian which can be emulated by a quantum computer [37].

Peak group.

The spectrum of an isolated nuclear spin consists of one peak at its precession frequency. If the nuclear spin is coupled to others, this peak ``splits'' and multiple peaks are observed near the precession frequency. The nuclear spin's peak group consists of these peaks.

Precession.

An isolated nuclear spin's state can be associated with a spatial direction using the Bloch sphere representation. If the direction rotates around the $z$-axis at a constant rate, we say that it precesses around the $z$-axis. The motion corresponds to that of a classical top experiencing a torque perpendicular to both the $z$-axis and the spin axis. For a nuclear spin, the torque can be caused by a magnetic field along the $z$-axis.

Projective measurement.

A measurement of a quantum system determined by a complete set of orthogonal projections whose effect is to apply one of the projections to the system (``wave function collapse'') with a probability determined by the amplitude squared of the projected state. Which projection occurred is known after the measurement. The simplest example is that of measuring qubit $\mathsf {q}$ in the logical basis. In this case, there are two projections, namely, $P_{\mathfrak{0}}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak... ...3pt}\langle$}{\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ and $P_{\mathfrak{1}}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak... ...3pt}\langle$}{\mathfrak{1}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$. If the initial state of all the qubits is $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, then the probabilities of the two measurement outcomes $\mathfrak{0}$ and $\mathfrak{1}$ are $p_{\mathfrak{0}}=\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $p_{\mathfrak{1}}=\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, respectively. The state after the measurement is $P_\mathfrak{0}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}/\sqrt{p_{\mathfrak{0}}}$ for outcome $\mathfrak{0}$ and $P_\mathfrak{1}=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}/\sqrt{p_{\mathfrak{1}}}$ for outcome $\mathfrak{1}$.

Pseudopure state.

A state with deviation given by a pure state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace... ...ace*{-4.3pt}\langle$}{\psi}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$.

Pulse.

A transient field applied to a quantum system. In the case of NMR QIP, pulses are rotating magnetic fields (RF pulses) whose effects are designed to cause specific rotations of the qubit states carried by the nuclear spins.

Refocusing pulse.

A pulse that causes a $180^\circ$ rotation around an axis in the plane. A typical example of such a rotation is $e^{-i\sigma_x\pi/2} = -i\sigma_x$, which is a $180^\circ$ $x$-rotation.

Resonant RF pulse.

A pulse whose field oscillates at the same frequency as the precession frequency of a target nuclear spin. Ideally, the field is in the plane, rotating at the same frequency and in the same direction as the precession. However, as long as the pulse field is weak compared to the precession frequency (that is, by comparison, its nutation frequency is small), the nuclear spin is affected only by the co-rotating component of the field. As a result, other planar components can be neglected, and a field oscillating in a constant direction in the plane has the same effect as an ideal resonant field.

RF pulse.

A pulse resonant at radio frequencies. Typical frequencies used in NMR are in this range.

Rotating frame.

A frame rotating at the same frequency as the precession frequency of a spin.

Rotation.

In the context of spins and qubits, a rotation around $\sigma_u$ by an angle $\theta$ is an operation of the form $e^{-i\sigma_u\theta/2}$. The operator $\sigma_u$ may be any unit combination of Pauli matrices. This defines an axis in three-space, and in the Bloch sphere representation, the operation has the effect suggested by the terminology.

Spectrum.

In the context of NMR, the Fourier transform of an FID.

Weak measurement.

A measurement involving only a weak interaction with the measured quantum system. Typically, the measurement is ineffective unless an ensemble of these quantum systems is available so that the effects of the interaction add up to a signal detectable above the noise. The measurement of nuclear magnetization used in NMR is weak in this sense.