Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present‐day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR‐based QIP capable of reaching 10—30 qubits.

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Extensions of average Hamiltonian theory to quantum computation permit the design of arbitrary Hamiltonians, allowing rotations throughout a large Hilbert space. In this way, the kinematics and dynamics of any quantum system may be simulated by a quantum computer. A basis mapping between the systems dictates the average Hamiltonian in the quantum computer needed to implement the desired Hamiltonian in the simulated system. The flexibility of the procedure is illustrated with NMR on 13 C labelled Alanine by creating the non-physical Hamiltonian σzσzσz corresponding to a three body interaction.

An extension of the product operator formalism of NMR is introduced, which uses the Hadamard matrix product to describe many simple spin 1/2 relaxation processes. The utility of this formalism is illustrated by deriving NMR gradient-diffusion experiments to simulate several decoherence models of interest in quantum information processing, along with their Lindblad and Kraus representations.

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