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 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 in hertz. Free coupling evolution for gives an entangling propagator . Suitable single-qubit rotations convert it to a Controlled-Z gate, and Hadamard conjugation on the target converts controlled- 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.
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