Summer Research Opportunities in Peter Love's Research Group, Summer 2008
We expect to hire one or two undergraduates for research this summer.
Students who express an interest should indicate which ONE of the projects (C1, C2, Q1, Q2) below they find most interesting.
CLASSICAL PHYSICS PROJECTS:
Fluid mechanics remains one of the most challenging problems faced by computational physics. Most numerical approaches solve the incompressible Navier-Stokes equations, describing an enormous range of flows in both two and three dimensions. Typically a spatial grid is used to discretize the equations. While many 2-D situations will use a Euclidean plane as the underlying geometry, it is possible that the grid may be a discretization of a sphere or other surface in which the geometry is non-Euclidean. In such geometries (as on a sphere), the angles of a triangle need not sum to pi. Imagine, for example, the triangle connecting the North pole and any two points on the equator. We may specialize to grids made up of equilateral triangles, as any 2-D surface may be discretized in this way [1]. In this case the geometry is defined locally by the number of triangles meeting at each grid point. If six triangles meet, the geometry is locally flat. If fewer than six triangles meet, the geometry has positive local curvature. If more than six triangles meet, the geometry has negative local curvature. If the properties of the triangulation, including the local curvature, are allowed to change we call the geometry dynamical. The simplest methods for simulating fluid dynamics are lattice-gases. In these models the grid is a regular lattice on which fictitious particles move. The average motion of the particles defines a flow which can be shown to obey the Navier-Stokes equations [2]. The proposed work addresses two questions. Firstly, can one create a simple model for two-dimensional fluid flow on an arbitrary fixed surface? Secondly, can one create a model in which the fluid flow can alter the shape of the surface, i.e. where the matter and geometry degrees of freedom are coupled dynamically?
Project C1 - Dynamical geometry in two dimensions without flow In this project the time dependence of the number of triangles will be investigated, for the case that the average particle velocity is zero ¬ i.e. there is no flow. The simplest case is to begin from an icosahedral starting configuration of fixed size and compute the average behavior over an ensemble of initial particle positions. In one dimension the average lattice size grows as t1/2 , as shown in [7, 8]. This project extends this work to two dimensions. Questions of interest are: How does the form of time dependence of the size depend on the rule used to change the geometry? Are there symmetry principles which could be used to classify different rules? Are there changes in the time dependence of the number of triangles for different rules? In the one-dimensional case sufficiently small initial conditions do not grow as a function of time, but oscillate instead. Can we find (numerically or analytically) such oscillating configurations in two dimensions? Project C2 - Simulation of the Navier Stokes equations on the sphere Many analytical solutions of the Navier-Stokes equations for special cases exist [9]. Such solutions are valuable as means for characterizing the accuracy of a numerical method. This project will use analytical solutions of the two-dimensional Navier-Stokes equations on a sphere to characterize the accuracy and efficiency of the lattice-gas model for a flow on a fixed geometry. What are suitable special case exact solutions for flow on a sphere? How well does the model reproduce such solutions? How seriously do small changes in discrete geometry affect the accuracy of the flow both locally and globally?
QUANTUM INFORMATION PROJECTS
Project Q1 - Lie decompositions of the unitary group and optimization of functions on the unitary group.
This work is a continuation of the thesis work of Byron Drury
The only quantum logic gates which can be directly experimentally implemented are one qubit local operations and a few very simple two qubit operations. As discouraging as this might seem for the prospects of building a useful quantum computer, this does not turn out to be an insurmountable obstacle to universal quantum computation: It was shown in 1995 by several research groups working independently that the set of one qubit operations and the controlled-NOT gate (CNOT) - a very simple two qubit gate whose action is simply to do nothing if the state of the first qubit is zero and to flip the state of the second qubit if the first is a one - was in fact universal in the sense that any unitary operation on any number of qubits could be built out of these elementary building blocks. However, the number of CNOT gates required for $n$ qubits was of order $n^3 4^n$. Since CNOT gates are time consuming and difficult to implement this enormous CNOT cost is a significant obstacle.
Since 1995 a number of advances have been made towards the CNOT optimization of universal quantum circuits. In particular, M\"ott\"onen and Vartiainen have lowered the upper bound on the asymptotic CNOT cost of universal circuits to $\frac{23}{48}4^n$ in leading order using a decomposition known as the Cosine-Sine Decomposition or CSD, and Khaneja and Glaser have proposed a related decomposition (now known as the Khaneja Glaser Decomposition, or KGD) which may prove to perform better asymptotically than the CSD and has already been used to produce extremely efficient two and three qubit circuits. All of these improvements have exploited the structure of the Lie algebra of the group of allowable operators, $SU(2^n )$, and future improvements will be made possible by a better understanding of this underlying structure.
Work to date has improved the constructions in the literature somewhat for three qubits, and extended them to up to five qubits. This work has indicated that further improvements are likely, and this project would pursue these improvements in the decompositions, by producing code to explicitly decompose unitary matrices as efficiently as possible.
All the decompositions described above may be regarded as parameterizations of the unitary group and as such have many applications in quantum information theory beyond the optimization of quantum programs for small quantum computers. Some examples (at most ONE of these could be pursued during the term of the research) of these applications are:
1) These parameterizations may be used as part of an algorithm to minimize (or maximize) a function on the unitary group. For example one might combine generation of random unitary matrices (jumps in the group and very easy to produce) with a steepest-descent algorithm following a jump. This gives a picture of the optimization as randomly leaping followed by minimization to the nearest local minimum. The efficiency of this method would scale as the number of local optima - about as good as one can expect. This minimization algorithm could be used in the entanglement project defined below.
2) Evolution of open quantum systems - the decompositions described above should enable one to understand the evolution of a quantum system of $n$ qubits coupled to an environment in terms of the double cosets of a unitary group acting on 3n qubits. The simplest example would be a unique description of noise for a single qubit. Experimental data for this system is beginning to be available for superconducting qubits.
3) Elucidation of small examples of quantum algorithms. The smallest quantum calculations of scientific interest are simulations of e.g. Molecular properties. To produce experimental proposals for such calculations one must explicitly produce a gate factoring of the unitary operations generated by the system Hamiltonian. These unitaries may have symmetries specific to the system which would allow further optimization of the gate sequence.
All of these projects would involve a considerable amount of directed reading in the area of lie algebras and lie groups, the calculation of many simple examples and the implementation of the decompositions and possibly some applications thereof in python.
Project Q2 - Computation of entanglement in thermal mixed states
This project is a continuation of the senior thesis work of Kate Carlysle
A general technique for the calculation of mixed state quantum entanglement works by convex roof minimization over the mean entanglements of possible pure state ensembles. The multiplicity of available ensembles complicates entanglement measures for mixed states. Randomly-generated unitary matrices allow construction of these ensembles; the entanglement of the mixed state is the mean over the entanglements of the pure state decomposition. This numerical method reproduces analytical predictions for the entanglement of mixed Bell states in two qubits. Furthermore, the mixed state technique extends to cases of three or more qubits. Although an equal mixture of orthogonal Bell states results in zero entanglement, not all mixtures of orthogonal states can produce an unentangled state. Application of this entanglement measure to thermal mixed states of two and three qubits produces the surprising result that mixing of the ground state with the first excited state alone does not eliminate entanglement. The high temperature limit, though, is an equal mixture of all energy levels and must have zero entanglement.
Applying the result of Osborne, which allows exact calculation of mixed state entanglement for rank-two mixed states should allow solution of the case that the ground and first excited states dominate the thermal mixture. This case is of particular relevance for adiabatic quantum computation, and should provide an analytic test case for the numerical method described above. Analytical work would focus on calculations based on Osbornes work, while numerical work would involve the implementation of the above method (previously implemented in mathematica) in python, and extensive numerical investigation of few qubit cases of relevance to experimental work.