Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 4

A universal quantum gate set can approximate any unitary operation on any finite register, to arbitrary requested accuracy, using a finite sequence of its gates on the register. Unobservable overall global phase is ignored. Approximation matters: a fixed finite alphabet has only countably many finite circuits, whereas there are uncountably many exact unitary matrices.

The Hadamard gate $H$ and T gate generate a dense set of single-qubit operations, but cannot themselves entangle different qubits. Adding the controlled-NOT gate gives a universal multi-qubit set. More generally, an available two-qubit gate must be genuinely entangling: it must take at least one product input to an entangled output. A product of local unitaries, or a SWAP gate composed with local unitaries, cannot provide this resource. Together with arbitrary single-qubit control, an entangling two-qubit gate on a connected interaction network supplies universal register control.

Since ${\sigma }_{\varphi }^2=I$, the rotation gate isFor two chronological $\pi$ pulses, later operations multiply on the left:Thus equatorial pulses generate arbitrary rotations about the $z$-axis. The minus sign is an overall global phase.

For the other chronological sequence, $T^4=Z$, andThis equality has no additional overall phase under the standard definitions of $H$ and $T$. Rotations about two nonparallel axes, such as $x$ and $y$, generate all single-qubit operations by an Euler angles decomposition. Equivalently, the available equatorial rotations include $R_y(\pi /2)$ and arbitrary $R_z$ rotations constructed above; for example $H=R_y(\pi /2)Z$ and $T=e^{i\pi /8}{R}_z(\pi /4)$. Hence these rotations provide a universal single-qubit set. Overall phases do not affect any probability or isolated-register observable; a phase controlled by an additional qubit would instead be relative and could not be discarded without accounting for it.

The minimum number of distinct fixed rotation gates for approximate universality is two. One fixed gate and its powers generate a commuting subgroup and cannot approximate all noncommuting single-qubit operations. Two suitable equatorial gates suffice: for example $R_x(\pi /\sqrt{2})$ and $R_y(\pi /\sqrt{3})$. Each has an angle irrational relative to $2\pi$, so its powers densely approximate rotations about its axis; combining the two supplies the Euler decomposition. Noncommutation alone is insufficient: two Clifford quarter-turns can generate only a finite subgroup. If continuous angles are regarded as available controls, two nonparallel rotation axes likewise suffice, now for exact Euler synthesis.

For $d$-dimensional unitaries, $\mathrm{tr}(U^{†}U)=d$. The relative propagator $W=U^{†}V$ is unitary, and its eigenvalues lie on the unit circle. The triangle inequality gives $|\mathrm{tr}W|\le d$, with equality precisely when all eigenvalues have the same phase, or $V=e^{i\gamma }U$. Therefore the specified trace-overlap gate fidelity lies between zero and one, is insensitive to global phase, and equals one exactly when the implemented gate has the same physical action as the target. It compares propagators independently of any particular input state; it is not the identically defined quantity as average state or average gate fidelity.

For target $R_x(\pi )$ and actual $R_x[\pi (1+ϵ)]$, the relative propagator is $R_x(\pi ϵ)$. Its trace is $2\cos (\pi ϵ/2)$, soThe absolute value is unnecessary in a sufficiently small neighbourhood of zero error, but belongs in the exact expression.

An ideal $\pi$ pulse is $-i{\sigma }_{\varphi }$. Since ${\sigma }_{\varphi }{\sigma }_{2\varphi }{\sigma }_{\varphi }={\sigma }_x$, the chronological three-pulse sequence has propagatorIt implements the Pauli X gate up to global phase, for every $\varphi$.

For this three-pulse amplitude-error compensation, the leading error in the supplied fidelity expansion vanishes when $1+2\cos \varphi =0$, givingThese choices minimize the leading small-angle-error infidelity. Substituting $\cos \varphi =\cos 2\varphi =-1/2$ into the fourth-order term givesThe fourth-order coefficient is nonzero, so this three-pulse sequence improves the single pulse's quadratic infidelity to quartic infidelity, rather than sixth-order infidelity.

An exact check is available. With $x=\pi ϵ/2$, multiplication of the three imperfect rotation gates at either optimum givesIts Taylor expansion has precisely the fourth-order coefficient above. The improvement concerns the same systematic fractional angle error on all three pulses; other errors need not cancel.