Liquid-State NMR Quantum Computer: Hamiltonian Formalism and Experiments

Yasushi Kondo, Mikio Nakahara, Shogo Tanimura

Denis‐Alexandre Trottier

Quantum computers offer the possibility of solving some problems more efficiently than their classical counterparts. The current forerunner in the experimental demonstration of quantum algorithms is Nuclear Magnetic Resonance (NMR). Known for its implementations at liquid state, NMR quantum computing consists of computing on nuclear spins. In the liquid crystal state, dipolar couplings are available, offering an increased clock frequency and a faster recycling of algorithms. Here investigated is the cost at which this comes, namely, a more complicated internal Hamiltonian, making the system harder to characterize and harder to control. In this thesis I present new methods for characterizing the Hamiltonian of dipolar coupled spin systems, and I report experimental results of characterizing an oriented 6-spin system. I then present methods and results concerning the quantum optimal control of this same spin system. Finally, I present experiments and simulations regarding the certification of computational quantum gates implemented in that same dipolar coupled spin system.

Yasushi Kondo

Dave Bacon

Quantum error correction (QEC) is all for naught if it cannot be implemented experimentally in the laboratory. Further, even if it can be implemented in small laboratory experiments, these experiments need not lead to a technology that is scalable to large quantum computers (where large is defined as “big enough to contemporaneously outperform the best classical computer on some problem”). In this chapter we will survey some of the experiments that have been performed to implement QEC. These experiments are all a long way from demonstrating viable QEC, but demonstrate the proof-of-principle methods that will need to be implemented in the future if quantum computation is to be made viable. This chapter deals with experimental implementations whose goal is to implement QEC, and not experiments using passive or open-loop methods, which are covered in Chapter 22. Experiments in liquid-state NMR The first experiments that attempted to perform QEC were performed using room-temperature liquid-state NMR. Liquid-state NMR [CFH97, GC97] is a testbed for quantum computing ideas in which one uses the internal states of coupled nuclear spins from a molecule as the qubits in a quantum computer. In its original and experimentally implemented form, liquid-state NMR is not generally thought to be scalable: the signal used to read out the result of the quantum computation decays exponentially [W97] in the number of qubits in the system and the mixed states produced in these experiments can be shown to possess no quantum entanglement.

Raymond Laflamme

Raymond Laflamme, Emanuel H. Knill, David G. Cory +11 more

After a general introduction to nuclear magnetic resonance (NMR), we give the basics of implementing quantum algorithms. We describe how qubits are realized and controlled with RF pulses, their internal interactions, and gradient fields. A peculiarity of NMR is that the internal interactions (given by the internal Hamiltonian) are always on. We discuss how they can be effectively turned off with the help of a standard NMR method called ``refocusing''. Liquid state NMR experiments are done at room temperature, leading to an extremely mixed (that is, nearly random) initial state. Despite this high degree of randomness, it is possible to investigate QIP because the relaxation time (the time scale over which useful signal from a computation is lost) is sufficiently long. We explain how this feature leads to the crucial ability of simulating a pure (non-random) state by using ``pseudopure'' states. We discuss how the ``answer'' provided by a computation is obtained by measurement and how this measurement differs from the ideal, projective measurement of QIP. We then give implementations of some simple quantum algorithms with a typical experimental result. We conclude with a discussion of what we have learned from NMR QIP so far and what the prospects for future NMR QIP experiments are.