Geometric Quantum Computation with NMR

Jonathan A. Jones, Vlatko Vedral, Artur Ekert and Giuseppe Castagnoli Centre for Quantum Computation, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK OCMS, New Chemistry Laboratory, South Parks Road, Oxford OX1 3QT, UK Elsag,Via Puccini 2, 1615 Genova, Italy (February 29, 2008) An exciting recent development has been the discovery that the computational power of quantum computers exceeds that of Turing machines [1]. The experimental realisation of the basic constituents of quantum information processing devices, namely fault-tolerant quantum logic gates, is a central issue. This requires conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem [2]. In particular, the subsystem may acquire a conditional phase shift. Here we consider a novel scenario in which this phase is of geometric rather than dynamical origin [3,4]. As the conditional geometric (Berry) phase depends only on the geometry of the path executed it is resilient to certain types of errors, and offers the potential of an intrinsically fault-tolerant way of performing quantum gates. Nuclear Magnetic Resonance (NMR) has already been used to demonstrate both simple quantum information processing [5–9] and Berry’s phase [10–12]. Here we report an NMR experiment which implements a conditional Berry phase, and thus a controlled phase shift gate. This constitutes the first elementary geometric quantum computation. Any quantum computation can be build out of simple operations involving only one or two quantum bits (qubits) [13]. A particularly simple two qubit gate in many experimental implementations, such as NMR [14], is the controlled phase shift. This may be achieved using a conditional Berry phase, and thus quantum geometrical phases can form the basis of quantum computation. We will use spin half nuclei as an example to demonstrate the