A collaboration involving myself, Adriano Barenco (Oxford), Timothy Spiller (Hewlett-Packard Labs, Bristol) and Rüdiger Schack (Royal Holloway College, London) studied the effects of noise on quantum computers, looking at proposed techniques of quantum error correction with both analytical arguments and numerical simulations.

In these simulations we compared the fidelity of states encoded in three- or five-qubit codes against noise, as a function of the strength of the environmental coupling and the time between error correction steps. We have allowed for the fact that quantum gates will operate in a finite amount of time, and that noise will affect the system during encoding and decoding. Errors that occur during encoding and decoding are not necessarily correctable, while if too long a time is allowed between error correcting steps, second-order errors can accumulate. There appears to be an optimal time between error-correcting steps to minimize both of these uncorrectable sources of error; for weak noise, this means that error correcting even in the presence of noise can still be beneficial over performing no error correction at all. The difficulties we examined have since been addressed by the development of fault-tolerant techniques.

Rüdiger Schack and I have also been studying how small quantum computers, with as few as three qubits, might be able to simulate interesting quantum chaotic systems, such as the quantum baker's map. This was actually done in 2002 using NMR quantum computers; in the near term it may also be possible with linear ion trap computers (proposed by Cirac and Zoller).

I have also worked on topics relating to multipartite entanglement, both on my own, and in collaboration with Oliver Cohen. We characterized all local unitary invariants of three quantum bits, and related them to operational quantities (such as the distillation yield of Greenberger-Horne-Zeilinger states).

More recently, I have collaborated with Andris Ambainis and Hilary Carteret on the quantum-to-classical transition of quantum random walks. We looked at the rate of change of the second position moment in the long-time limit; classically, this grows as the square root of the time, while in the quantum case it grows linearly with time. We were able to use this as an indicator of quantum versus classical behavior, and derive explicit expressions for two different approaches to the classical limit.

Some of my work on numerical simulation of quantum systems has also been applied to systems of interest in quantum computation.

Some interesting quantum computing sites:

- Centre for Quantum Computation
- Centre for Quantum Computer Technology, Australia
- Institute for Quantum Information, Caltech
- Samuel L. Braunstein's Tutorial on Quantum Computation

It is possible to ``unravel'' these master equations into stochastic differential equations for state vectors, rather than density operators. One realization of such an equation is called a ``quantum trajectory'' (i.e., a trajectory in Hilbert space), and averaging the state over all realizations with the appropriate weight reproduces the correct density operator.

This procedure is analogous to replacing a classical Fokker-Planck equation with a stochastic Langevin equation; but unlike the classical case, this relationship is not unique in quantum mechanics. There are an infinite number of different unravelings of a given master equation. Three of the most common are the quantum jumps or Monte Carlo Wavefunction unravelings (used in quantum optics), the quantum state diffusion equation of Gisin and Percival, and the ortho-jumps of Diosi.

I have applied quantum trajectories to a number of problems, including quantum dissipative chaos, the calculation of quantum optical spectra and time-correlation functions, and error correction in quantum computers. With Nicolas Gisin and Marco Rigo I have recently demonstrated that different unravelings exhibit similar behavior in the classical limit, providing some evidence that the existence of classical trajectories is predicted regardless of ambiguities at the quantum level.

I have also done some work on the connection between quantum trajectories and the Decoherent Histories formalism of Griffiths, Omnès, and Gell-Mann and Hartle.

More recently, I developed a toy model of quantum trajectories using quantum bits both for the system and the environment. This model was intended for pedagogical purposes, to encourage the use of quantum trajectories in the quantum information community. It also allows explicit calculations of many interesting quantities, such as the change of entropy, entanglement between the system and environment, randomness produced in measurement, and information gain.

The heart of the code involves two main classes: the State class, representing states in Hilbert space, and the Operator class representing the linear operators on those states. The State class permits representations of systems with arbitrary numbers of degrees of freedom, allowing them to be added, multiplied by real or complex scalars, renormalized, and taken in inner products. The Operator class includes a set of Primary Operators, such as the creation and annihilation operators, Pauli matrices, position and momentum, and atomic transitions; these can be added, multiplied, conjugated, multiplied by scalars or time-dependent functions and applied to different degrees of freedom to build up composite operators of arbitrary complexity.

The power and modularity of C++ means that these representations of quantum systems are quite independent of any particular choice of unraveling, and expressions can be coded in a form very close to the mathematics of the physical system.

While this software was originally developed to simulate problems in quantum optics and atomic physics, it has since then been extensively applied to problems in quantum information. Among these are: simulation of an NMR quantum computer simulating the quantum baker's map (collaboration with Rüdiger Schack); simulation of a quantum dot/optical microcavity hybrid system, in which the nanocrystal dots serve as quantum bits and the microcavity modes couple them together (collaboration with Hailin Wang); and simulation of magnetic resonance force microscopy measurements of single electron spins (collaboration with Hsi-Sheng Goan). It has also been used in studying quantum error correction and quantum random walks.

QSD references and software download

Since then, I have explored the connections between decoherent histories and quantum trajectories, including a demonstration that the common quantum jump trajectories of quantum optics correspond to a set of approximately decoherent histories. I have also worked (with Jonathan Halliwell) on the problem of hydrodynamics histories, that is, histories which avoid the usual system/environment split and instead look at collective variables describing the average of some physical quantity in a small volume. We developed a simple spin model which demonstrated some of the desired properties of hydrodynamic histories.

I have also studied various notions of classicality of histories, and the relationship between approximate and exact decoherence, in the hopes of throwing more light on the interpretation of quantum mechanics. In collaboration with James B. Hartle, I looked at a family of coarse-grainings for a linear chain of oscillators, and showed that the most ``classical'' description was also the most decoherent and predictable. As part of this work, we developed a new definition of entropy for sets of histories.

Decoherence bibliography (under construction)

I have looked at a class of chaotic models using the formalisms of Decoherent Histories and Quantum Trajectories. Classically, a dissipative chaotic system tends towards a strange attractor with a fractal structure; that is, the attractor exhibits substructure at all length scales.

Quantum mechanically this is impossible, as structure on a scale smaller than Planck's constant has no meaning. As one approaches the classical limit, more and more layers of structure appear; but there is always a limiting scale, at which the existence of quantum uncertainty and noise from the dissipative environment blurs out the fractal.

Because of this phenomenon, quantities used to characterize classical chaos (such as Lyapunov exponents and fractal dimension) are not well-defined for the equivalent quantum systems. It is possible, however, that extensions of these concepts may still be useful. One useful proposal, due to Schack and Caves, is hypersensitivity to perturbations. I am currently collaborating on studies of this in quantum computers.