Jonathan A. Jones

Implementation of a quantum search algorithm on a quantum computer

In 1982 Feynman observed that quantum-mechanical systems have an information-processing capability much greater than that of corresponding classical systems, and could thus potentially be used to implement a new type of powerful computer. Three years later Deutsch described a quantum-mechanical Turing machine, showing that quantum computers could indeed be constructed. Since then there has been extensive research in this field, but although the theory is fairly well understood, actually building a quantum computer has proved extremely difficult. Only two methods have been used to demonstrate quantum logic gates: ion traps, and nuclear magnetic resonance (NMR),. NMR quantum computers have recently been used to solve a simple quantum algorithm—the two-bit Deutsch problem,. Here we show experimentally that such a computer can be used to implement a non-trivial fast quantum search algorithm initially developed by Grover,, which can be conducted faster than a comparable search on a classical computer.

Geometric quantum computation using nuclear magnetic resonance

A significant development in computing has been the discovery that the computational power of quantum computers exceeds that of Turing machines. Central to the experimental realization of quantum information processing is the construction of fault-tolerant quantum logic gates. Their operation requires conditional quantum dynamics, in which one sub-system undergoes a coherent evolution that depends on the quantum state of another sub-system; in particular, the evolving sub-system may acquire a conditional phase shift. Although conventionally dynamic in origin, phase shifts can also be geometric. Conditional geometric (or ‘Berry’) phases depend only on the geometry of the path executed, and are therefore resilient to certain types of errors; this suggests the possibility of an intrinsically fault-tolerant way of performing quantum gate operations. Nuclear magnetic resonance techniques have already been used to demonstrate both simple quantum information processing and geometric phase shifts. Here we combine these ideas by performing a nuclear magnetic resonance experiment in which a conditional Berry phase is implemented, demonstrating a controlled phase shift gate.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 dynam-ical 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 Berrys 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 1

Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer

Quantum computing shows great promise for the solution of many difficult problems, such as the simulation of quantum systems and the factorization of large numbers. While the theory of quantum computing is fairly well understood, it has proved difficult to implement quantum computers in real physical systems. It has recently been shown that nuclear magnetic resonance (NMR) can be used to implement small quantum computers using the spin states of nuclei in carefully chosen small molecules. Here we demonstrate the use of a NMR quantum computer based on the pyrimidine base cytosine, and the implementation of a quantum algorithm to solve Deutsch’s problem (distinguishing between constant and balanced functions). This is the first successful implementation of a quantum algorithm on any physical system.

Characterisation of protein unfolding by NMR diffusion measurements

The characterisation of non-native states of proteins is a key problem instudies of protein folding. Complete characterisation of these states requiresa description of both local and global properties, including moleculardimensions. Here we present results from pulsed field gradient experimentsdesigned to compare the effective hydrodynamic radii of a protein in nativeand non-native states. Measurements performed on lysozyme indicate that theeffective hydrodynamic radius increases by 38±1% on unfolding in urea,a result completely consistent with a recent study by small-angle X-rayscattering.

Geometric quantum computation

Abstract We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation.

NMR quantum computation

In this article I will describe how NMR techniques may be used to build simple quantum information processing devices, such as small quantum computers, and show how these techniques are related to more conventional NMR experiments.

Tackling systematic errors in quantum logic gates with composite rotations

We describe the use of composite rotations to combat systematic errors in single-qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of nuclear magnetic resonance quantum computing, these sequences should be applicable to any implementation of quantum computation.

Magnetic Field Sensing Beyond the Standard Quantum Limit Using 10-Spin NOON States

Quantum-Enhanced Measurement The single electron spin in a molecule, atom, or quantum dot precesses in a magnetic field and so can be used as a magnetic field sensor. As the number of spins in a sensor increases, so too does the sensitivity. Quantum mechanical entanglement of the spin ensemble should then allow the sensitivity to increase much more than would be expected from a simple increase in the number of individual spins in the ensemble. Using the highly symmetric molecule, trimethyl phosphite, a molecule containing a central P atom surrounded by nine hydrogen atoms, Jones et al. (p. 1166, published online 23 April) quantum mechanically entangled the 10 spins (or qubits) to generate a nearly 10-fold enhancement in the magnetic field sensitivity. The results pave the way for the further development of quantum sensors. Quantum mechanical entanglement of nuclei in a single molecule results in an enhancement of the magnetic field sensitivity. Quantum entangled states can be very delicate and easily perturbed by their external environment. This sensitivity can be harnessed in measurement technology to create a quantum sensor with a capability of outperforming conventional devices at a fundamental level. We compared the magnetic field sensitivity of a classical (unentangled) system with that of a 10-qubit entangled state, realized by nuclei in a highly symmetric molecule. We observed a 9.4-fold quantum enhancement in the sensitivity to an applied field for the entangled system and show that this spin-based approach can scale favorably as compared with approaches in which qubit loss is prevalent. This result demonstrates a method for practical quantum field sensing technology.

The effects of guanidine hydrochloride on the 'random coil' conformations and NMR chemical shifts of the peptide series GGXGG.

The effects of the commonly used denaturant guanidine hydrochloride(GuHCl) on the random coil conformations and NMR chemical shifts of theproteogenic amino acids have been characterized using the peptide seriesAc-Gly-Gly-X-Gly-Gly-NH2. The φ angle-sensitive couplingconstants, ROESY cross peak intensities and proline cis–trans isomerratios of a representative subset of these peptides are unaffected by GuHCl,which suggests that the denaturant does not significantly perturb intrinsicbackbone conformational preferences. A set of3JHNHα values is presented which agreewell with predictions of recently developed models of the random coil. Wehave also measured the chemical shifts of all 20 proteogenic amino acids inthese peptides over a range of GuHCl concentrations. The shifts exhibit alinear dependence on denaturant concentration and we report here correctionfactors for the calculation of ‘random coil’ 1H chemicalshifts at any arbitrary denaturant concentration. Studies of arepresentative subset of peptides indicate that 13C and15N chemical shifts are also perturbed by the denaturant.These results should facilitate the application of chemical shift-basedanalytical techniques to the study of polypeptides in solution with GuHCl.The effects of the denaturant on the quality of NMR spectra and on chemicalshift referencing are also addressed.

Quantum Computing with NMR

reference frames simplify the implementation of single qubit gates by avoiding the necessity of correcting phase shifts (z-rotations), but also provide a simple way of implementing desired z-rotations. Rotating the frame instead of the spin is simply a matter of computational book keeping, and so can in principle be done instantaneously and perfectly. For this reason it is often sensible to decompose single-qubit gates into circuits involving z-rotations wherever possible. One important example is the Hadamard gate, Eq. 19, which can be implemented using the pulse sequence 90−y 180z or equivalently 180z 90y. The z-rotations can be absorbed into the reference frame, and so Hadamard gates can be replaced by h = 90y, sometimes called the pseudo-Hadamard gate [12,19], or its inverse, h−1 = 90−y. As Hadamard gates frequently occur in pairs, these pairs can be replaced by one h−1 and one h gate, with the 180z rotations cancelling out; when they do not occur in pairs it is still possible to replace them with pseudo-Hadamard gates as long as the 180z rotations are absorbed into the reference frames. An alternative approach which has grown in popularity is to use methods from optimal control theory to develop shaped pulses which both provide selective excitation and refocus undesirable interactions. This approach is explored in more detail in Section 3.7 below. 3.3 Basic methods: controlled gates Next I turn to non-trivial two-qubit gates, and will begin by considering twospin (two-qubit) systems. It is in principle only necessary to construct a single gate of this kind, and the traditional gate used in most theoretical descriptions is the controlled-not gate, or controlled-X gate, Eq. 23. In NMR experiments, however, the key two-qubit gate is the controlled-Z gate