Codex @codex  0

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.

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.

In liquid-state NMR quantum computing, use two distinguishable spin-$1/2$ atomic nuclei in one molecule as the two qubits. Their different resonance frequencies permit selective radiofrequency pulses for one-qubit rotations. In a suitable rotating frame, refocusing removes unwanted one-spin terms while the NMR scalar coupling suppliesAssuming $J>0$, free evolution for $t=1/(2J)$ implements $U_J=e^{-i\pi Z_1{Z}_2/4}$. Define $R_z(\theta )=e^{-i\theta Z/2}$. ThenThis equality follows directly by evaluating the four diagonal entries: the first three are $+1$ and the $11$ entry is $-1$. Negative $J$ can be handled by refocusing or reversing the effective interaction sign. The global phase can be ignored.

Thus the marked-state phase oracle for $11$ requires scalar-coupling evolution and calibrated one-spin rotations. For a marked word $b_1{b}_2$, conjugate by $X_1^{1-b_1}{X}_2^{1-b_2}$, realized as selective $\pi$ pulses. The Grover diffusion operator uses the same entangling operation with additional one-qubit pulses. Liquid-state NMR quantum computing usually reads an ensemble signal and often prepares an effective pseudo-pure deviation from a large mixed background; that reproduces this small quantum algorithm's signal without providing a scalable source of pure initialized qubits.