Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 5

Take $|0⟩$ to have ${\sigma }_z=+1$ and use angular-frequency gyromagnetic ratios. Neglecting interspin coupling, the Hamiltonian of the Zeeman energy isFor ${\gamma }_1={\gamma }_2=\gamma$, define $ϵ=\hslash \gamma B/(k_{B}T)$. The exact thermal density matrix in the order $|00⟩,|01⟩,|10⟩,|11⟩$ isAt high temperature, $|ϵ|\ll 1$, expand both numerator and normalization to obtainFor positive $\gamma$, $|00⟩$ is the most populated state; a negative gyromagnetic ratio reverses that population preference under this basis convention.

For the lower-qubit controlled rotation, the control is the upper qubit and only the subspace $|10⟩,|11⟩$ rotates. Write $u=\cos (\theta /2)$ and $v=\sin (\theta /2)$. Unitary conjugation of its diagonal populations gives $c{u}^2+d{v}^2$ and $c{v}^2+d{u}^2$, together with off-diagonal coherences. The ideal crush gradient stipulated in the source removes those coherences without changing populations. ThereforeThe other two populations remain $a,b$. This controlled-transfer gate is not unitary as a complete operation because it includes dephasing.

For an upper-qubit target, the lower qubit is the control: the mixed subspace is $|01⟩,|11⟩$. ThusThe population $a$ of $|00⟩$ is unaffected in either direction because its control bit is zero. The dephasing step also leaves this diagonal population unchanged, so any sequence of these controlled-transfer gates preserves $a$.

Use the high-temperature state consistently to first order in $ϵ$. Applying the lower-target controlled-transfer gate with $\cos {\theta }_1=1/3$ givesThe upper-target gate with $\cos {\theta }_2=0$ then equalizes the second and fourth populations. The result isThis is a pseudo-pure state; the identity part is unchanged by unitary gates and does not contribute to traceless NMR signals.

For $n$ equal, weakly polarized spins, the high-temperature expansion is $\rho \simeq 2^{-n}[I+(ϵ/2){\sum }_j{Z}_j]$. The all-zero population is $a_0=2^{-n}(1+nϵ/2)$. Preserving this largest population and averaging the other $2^n-1$ populations givesThus the signal-bearing effective purity falls approximately as $n/2^n$ at fixed small per-spin polarization. The linear expansion requires $n|ϵ|\ll 1$. More generally, if the exact single-spin polarization is $r=\tanh (ϵ/2)$, the same population-preserving averaging gives $p_n=[(1+r{)}^n-1]/(2^n-1)$, which still decreases exponentially for fixed $0<r<1$. This scaling concerns high-temperature ensemble preparation, not an impossibility theorem for every initialization method.

In heteronuclear liquid-state NMR quantum computing, different gyromagnetic ratios give well-separated resonance frequencies. A resonant transverse radio-frequency pulse selects one nucleus; its phase sets the rotation axis in the transverse plane and its duration and amplitude set the rotation angle. Detuning, free precession and rotating-frame phase changes supply $z$ rotations. For homonuclear spins, differing chemical shifts can provide frequency selectivity, but selective pulses must be narrow enough not to rotate the other spin. Exactly magnetically equivalent spins cannot be independently addressed by a spatially uniform radio-frequency field merely because they are labelled as different qubits.

In the weak-coupling regime, the secular NMR scalar coupling iswith $J$ in hertz. Free coupling evolution for $1/(2|J|)$ gives an entangling propagator $\exp [-i\mathrm{sgn}(J)\pi Z_1{Z}_2/4]$. Suitable single-qubit $z$ rotations convert it to a Controlled-Z gate, and Hadamard conjugation on the target converts controlled-$Z$ to a controlled-NOT gate. Refocusing pulses remove unwanted Zeeman evolution and unwanted couplings. This construction applies to resolved, weakly coupled homonuclear spins as well as heteronuclear spins; strongly coupled or equivalent spins need a different control analysis.

These controls supply a universal gate set on a finite addressable connected register, but conventional liquid-state molecular NMR is not an architecture that scales straightforwardly to arbitrary register size. A finite chemical-shift bandwidth causes spectral crowding as more spins and coupling-split transitions are added; increasingly selective pulses become slow relative to coherence times, and complicated coupling networks require extensive refocusing. The exponentially small pseudo-pure-state signal further limits the complete ensemble scheme. Formal unitary universality therefore does not establish scalable physical addressability, gate speed or initialization.