Daniel K. L. Oi

Geometric Phases for Mixed States in Interferometry

We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.

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.

Direct estimations of linear and nonlinear functionals of a quantum state.

We present a simple quantum network, based on the controlled-SWAP gate, that can extract certain properties of quantum states without recourse to quantum tomography. It can be used as a basic building block for direct quantum estimations of both linear and nonlinear functionals of any density operator. The network has many potential applications ranging from purity tests and eigenvalue estimations to direct characterization of some properties of quantum channels. Experimental realizations of the proposed network are within the reach of quantum technology that is currently being developed.We present a simple device based on the controlled-SWAP gate that performs quantum state tomography. It can also be used to determine maximum and minimum eigenvalues, expectation values of arbitrary observables, purity estimation as well as characterizing quantum channels. The advantage of this scheme is that the architecture is fixed and the task performed is determined by the input data.

Observation of Geometric Phases for Mixed States using NMR Interferometry

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires one to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well-understood, noiseless, pure-state case.

Fidelity of single qubit maps

We describe a simple way of characterizing the average fidelity between a unitary (or anti-unitary) operator and a general operation on a single qubit, which only involves calculating the fidelities for a few pure input states, and discuss possible applications to experimental techniques including nuclear magnetic resonance (NMR).

Identifying an experimental two-state Hamiltonian to arbitrary accuracy

Precision control of a quantum system requires accurate determination of the effective system Hamiltonian. We develop a method for estimating the Hamiltonian parameters for some unknown two-state system and providing uncertainty bounds on these parameters. This method requires only one measurement basis and the ability to initialize the system in some arbitrary state which is not an eigenstate of the Hamiltonian in question. The scaling of the uncertainty is studied for large numbers of measurements and found to be proportional to the reciprocal of the square root of the number of measurements.

Generalization of the geometric phase to completely positive maps

We generalize the notion of relative phase to completely positive (CP) maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map.

Quantum optics for space platforms

The idea of building a space-based quantum network is appealing, but there are many challenges. A new model for low-cost satellites is bringing global quantum communication a step closer to reality.

Scalable error correction in distributed ion trap computers

A major challenge for quantum computation in ion trap systems is scalable integration of error correction and fault tolerance. We analyze a distributed architecture with rapid high-fidelity local control within nodes and entangled links between nodes alleviating long-distance transport. We demonstrate fault-tolerant operator measurements which are used for error correction and nonlocal gates. This scheme is readily applied to linear ion traps which cannot be scaled up beyond a few ions per individual trap but which have access to a probabilistic entanglement mechanism. A proof-of-concept system is presented which is within the reach of current experiment.

Two-qubit Hamiltonian tomography by Bayesian analysis of noisy data

We present an empirical strategy to determine the Hamiltonian dynamics of a two-qubit system using only initialization and measurement in a single fixed basis. Signal parameters are estimated from measurement data using Bayesian methods from which the underlying Hamiltonian is reconstructed, up to three unobservable phase factors. We extend the method to achieve full control Hamiltonian tomography for controllable systems via a multistep approach. The technique is demonstrated and evaluated by analyzing data from simulated experiments including projection noise.