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Zero-field nuclear magnetic resonance (NMR) provides complementary analysismodalities to those of high-field NMR and allows for ultra-high-resolutionspectroscopy and measurement of untruncated spin-spin interactions. Unlike forthe high-field case, however, universal quantum control -- the ability toperform arbitrary unitary operations -- has not been experimentallydemonstrated in zero-field NMR. This is because the Larmor frequency for allspins is identically zero at zero field, making it challenging to individuallyaddress different spin species. We realize a composite-pulse technique forarbitrary independent rotations of $^1$H and $^{13}$C spins in a two-spinsystem. Quantum-information-inspired randomized benchmarking and statetomography are used to evaluate the quality of the control. We experimentallydemonstrate single-spin control for $^{13}$C with an average gate fidelity of$0.9960(2)$ and two-spin control via a controlled-not (CNOT) gate with anestimated fidelity of $0.99$. The combination of arbitrary single-spin gatesand a CNOT gate is sufficient for universal quantum control of the nuclear spinsystem. The realization of complete spin control in zero-field NMR is anessential step towards applications to quantum simulation,entangled-state-assisted quantum metrology, and zero-field NMR spectroscopy.

Experimental Benchmarking of Quantum Control in Zero-Field Nuclear Magnetic Resonance