Liquid-state NMR quantum computing

Liquid-state NMR quantum computing encodes qubits in distinguishable nuclear spins of molecules in solution. Selective radiofrequency pulses implement one-spin control; NMR scalar coupling supplies entangling evolution. Under $H_J=(\pi \hslash J/2)Z_1{Z}_2$, evolution for $1/(2J)$ with $J>0$, followed by $-\pi /2$ rotations about each $z$ axis, produces a Controlled-Z gate up to global phase. Ensemble detection and common pseudo-pure preparations make small demonstrations distinct from scalable pure-state initialization.

A mixture of the maximally mixed state and a chosen pure state. Unitary conjugation preserves its identity part; every traceless observable has expectation $p⟨\psi |O|\psi ⟩$. The coefficient $p$ is effective polarization rather than literal purity: $\mathrm{tr}{\rho }^2=1/d+(1-1/d)p^2$. Preserving the largest population of $n$ weakly polarized equal spins and averaging the rest gives $p\simeq nϵ/[2(2^n-1)]$, displaying the signal penalty of this high-temperature ensemble preparation.

A controlled rotation gate followed by computational-basis dephasing mixes the two target populations in the control-one subspace. If they are $c,d$, their sum is preserved and their difference is multiplied by $\cos \theta$. The operation is not unitary after dephasing. On a two-qubit diagonal state, the population of $|00⟩$ is unchanged when either qubit is the target, because the other control bit is zero.