J.A. Jones/ Progress in Nuclear Magnetic Resonance Spectroscopy 38(2001)325-360 possible implementations of a quantum computer, as combined together to implement any other desired it is usually more sensible to use a larger and more gate. While many different sets of gates are possible convenient set of gates. As one qubit gates are usually a simple approach is to implement the set of all possi much simpler to perform than gates involving two or ble one qubit gates, together with one or more non- more qubits, it is often reasonable to assume that any trivial two qubit gates [33] one qubit gate(or, at least a reasonable approximation One qubit gates correspond to rotations of a sp to it) is available. The combination of this set of one within its own one-spin Hilbert space, which can be qubit gates with any single non-trivial two qubit gate, readily achieved using RF fields. Note that it is neces such as the controlled-NOT gate forms an adequate set sary to apply these rotations selectively to individual [33], from which any other gate may be built with qubits. In most other suggested implementations of relative ease quantum computation [2,3] this is easily achieved using some type of spatial localisation: the physical objects implementing the qubits are located at well 5. Building NMR quantum computers defined and distinct locations in space. This approach While it would in principle be possible to use a is not possible in NMR, as each qubit is implemented using an ensemble of nuclei. each of which is located wide range of different approaches to build a quantum at a different place in the NMR sample, and all of computer,all the main proposals to date [2,3] have which are undergoing rapid motion Instead different used broadly similar approaches, based on the quan- qubits are implemented using different nuclei in the tum circuit model outlined above. This model contains five major components, each of which must same molecule, and they are distinguished using the different resonance frequencies of each nucleus be implemented in order to construct a working computer [37]. Four central components can all be Two qubit gates, such as the controlled-NOT gate, implemented within NN are more complicated as they involve conditional below, while the fifth component, error correction evolution(that is, the evolution of one spin must is discussed in Section 12 depend on the state of the other spin), and thus require some interaction between the two qubits. The J- 5.1. Qubits coupling in NMR is well suited to this purp Note that all the different nuclei making up an NMR The first of these requirements, a set of qubits quantum computer must participa appears easy to achieve, as the two spin states of coupling network. It is not necessary(or even desir- spin-1/2 nuclei in a magnetic field provide a natural able)that all the nuclei are directly coupled together implementation. However, one important feature but they must be connected, directly or indirectly, by which distinguishes NMR quantum computers from some chain of resolved couplings. Since J-coupling other suggested implementations is that NMR studies only occurs within a molecule, and does not connect not a single isolated quantum system, but rather a very different molecules, we can treat an ensemble of large number (effectively an ensemble) of such molecules as an ensemble of identical mutually systems. Thus an NMR quantum computer is actually isolated computers an ensemble of indistinguishable computers, one on each molecule in the NMR sample. This has a number 5.3. Initialisation of subtle and important consequences as discussed below Quantum logic gates transform qubits from one state to another, but this is only useful if the qubits 5.2. Logic gates start off in some well defined initial state. In practice it is sufficient to have some method for reaching any one In order to perform an arbitrary computation it is initial state, and the obvious choice is to have all the necessary to implement arbitrary quantum logic qubits in the o)state, corresponding to a CLEAR opera circuits. This can be achieved as long as it is possible tion. Any other desired starting state can then be easily to implement an adequate set of gates, which can bepossible implementations of a quantum computer, as it is usually more sensible to use a larger and more convenient set of gates. As one qubit gates are usually much simpler to perform than gates involving two or more qubits, it is often reasonable to assume that any one qubit gate (or, at least a reasonable approximation to it) is available. The combination of this set of one qubit gates with any single non-trivial two qubit gate, such as the controlled-not gate forms an adequate set [33], from which any other gate may be built with relative ease. 5. Building NMR quantum computers While it would in principle be possible to use a wide range of different approaches to build a quantum computer, all the main proposals to date [2,3] have used broadly similar approaches, based on the quan￾tum circuit model outlined above. This model contains ®ve major components, each of which must be implemented in order to construct a working computer [37]. Four central components can all be implemented within NMR systems as described below, while the ®fth component, error correction, is discussed in Section 12. 5.1. Qubits The ®rst of these requirements, a set of qubits, appears easy to achieve, as the two spin states of spin-1/2 nuclei in a magnetic ®eld provide a natural implementation. However, one important feature which distinguishes NMR quantum computers from other suggested implementations is that NMR studies not a single isolated quantum system, but rather a very large number (effectively an ensemble) of such systems. Thus an NMR quantum computer is actually an ensemble of indistinguishable computers, one on each molecule in the NMR sample. This has a number of subtle and important consequences as discussed below. 5.2. Logic gates In order to perform an arbitrary computation it is necessary to implement arbitrary quantum logic circuits. This can be achieved as long as it is possible to implement an adequate set of gates, which can be combined together to implement any other desired gate. While many different sets of gates are possible, a simple approach is to implement the set of all possi￾ble one qubit gates, together with one or more non￾trivial two qubit gates [33]. One qubit gates correspond to rotations of a spin within its own one-spin Hilbert space, which can be readily achieved using RF ®elds. Note that it is neces￾sary to apply these rotations selectively to individual qubits. In most other suggested implementations of quantum computation [2,3] this is easily achieved using some type of spatial localisation: the physical objects implementing the qubits are located at well de®ned and distinct locations in space. This approach is not possible in NMR, as each qubit is implemented using an ensemble of nuclei, each of which is located at a different place in the NMR sample, and all of which are undergoing rapid motion. Instead different qubits are implemented using different nuclei in the same molecule, and they are distinguished using the different resonance frequencies of each nucleus. Two qubit gates, such as the controlled-not gate, are more complicated as they involve conditional evolution (that is, the evolution of one spin must depend on the state of the other spin), and thus require some interaction between the two qubits. The J￾coupling in NMR is well suited to this purpose. Note that all the different nuclei making up an NMR quantum computer must participate in a single coupling network. It is not necessary (or even desir￾able) that all the nuclei are directly coupled together, but they must be connected, directly or indirectly, by some chain of resolved couplings. Since J-coupling only occurs within a molecule, and does not connect different molecules, we can treat an ensemble of molecules as an ensemble of identical mutually isolated computers. 5.3. Initialisation Quantum logic gates transform qubits from one state to another, but this is only useful if the qubits start off in some well de®ned initial state. In practice it is suf®cient to have some method for reaching any one initial state, and the obvious choice is to have all the qubits in the u0l state, corresponding to a clear opera￾tion. Any other desired starting state can then be easily obtained. 334 J.A. Jones / Progress in Nuclear Magnetic Resonance Spectroscopy 38 (2001) 325±360