Our work is in quantum information, ranging from `Models of quantum computation' and quantum fault-tolerance to entanglement theory and many-body physics.
Models of quantum computation: We study the one-way quantum computer, a scheme of quantum computation consisting of local measurements on an entangled universal resource state [Phys. Rev. Lett. 86, 5188 (2001)]. Unitary evolution and measurement are the two fundamental ways of evolving a quantum system in time, and they are very different. Unitary evolution is deterministic and reversible, measurement is probabilistic and irreversible. Universal quantum computation can be built on either unitary evolution or measurement. In the former case, one is led to the (standard) circuit model, in the latter to measurement-based quantum computation. The one-way quantum computer is one such measurement-based scheme. By shifting the focus from unitary evolution to measurement, it encourages us to reconsider what the fundamentals of quantum computation are. We thus ask: ``What are the elementary building blocks of the one-way quantum computer? What is their composition principle?'' The final goal of this line of research is to obtain clues for how to construct novel quantum algorithms.
Another computational model we study are quantum cellular automata (QCA). These devices are constrained by translation-invariance in space (and time), and usually have only short-range interaction. Despite these stringent constraints, computational universal QCA can be constructed. By implementing quantum cellular automata, quantum systems which have the advantage of long decoherence times but the drawback of limited local control (= small number of knobs to turn) can be used for quantum information processing.
Coding theory and fault-tolerant quantum computation: Error-correction is what a large-scale quantum computer, once built, will spend most of its computation time with. It is therefore important to devise error-correction methods which allow for a high error threshold at a moderate operational overhead. One focus of our work on fault-tolerance is in systems with a geometrical constraint, e.g. low-dimensional lattice systems, and in topological methods.
We have presented a fault-tolerant one-way quantum computer [arXiv:quant-ph/0510135], and have described a method for fault-tolerant quantum computation in a two-dimensional lattice of qubits requiring local and translation-invariant nearest-neighbor interaction only [arXiv:quant-ph/0610082], [arXiv:quant-ph/0703143] (joint work with Jim Harrington (formerly Los Alamos National Laboratory, now HRL Laboratories) and Kovid Goyal (Caltech)). The obtained error threshold is 0.75 percent per quantum gate, which is the highest known threshold for a two-dimensional architecture with nearest-neighbor interaction. A large value of the error threshold is important for realization of fault-tolerant quantum computation because it relaxes the accuracy requirements of the experiment. The imposed constraint of nearest-neighbor interaction in a two-dimensional qubit array is suggested by experimental reality: Many physical systems envisioned for the realization of a quantum computer are confined to two dimensions and prefer short-range interaction, for example optical lattices, arrays of superconducting qubits and quantum dots.
Quantum computation and foundations of quantum mechanics: David Deutsch, one of the founders of the field of quantum information, writes "[It is not] obvious a priori that any of the familiar recursive functions is in physical reality computable. The reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws of physics 'happen to' permit the existence of physical models for arithmetic such as addition, subtraction and multiplication. If they did not, these familiar operations would be non-computable functions. We might still know of them and invoke them in mathematical proofs (which would presumably be called 'nonconstructive') but we could not perform them." [D. Deutsch, Proc. Roy. Soc. A 400, 97 (1985).].
Now that quantum computation is based on the laws of quantum mechanics, we ask: "Which is the key feature of quantum mechanics that causes the quantum speed-up?" -- There is no shortage of candidates, for example: superposition and interference, entanglement, largeness of Hilbert space, and contextuality of quantum mechanics. They are, presumably, all part of the picture. However, a precise connection remains to be drawn.
Characterization and quantification of multipartite entanglement: Entanglement is one of the weird features that distinguishes quantum mechanics from classial theories. It is regarded as a resource for many quantum information processing tasks: quantum key distribution (e.g. Ekert's protocol), suerdense coding, and even quantum computation (e.g. one-way computers). There has been tremendous progress in the characterization and quantification of entanglement in both bipartite and multipartite settings using various methods. There have been many entanglement measures proposed: entanglement of creation/formation, entanglement of distillation, relative entropy of entanglement, squashed entanglement, negativity, generalized robustness of entanglement, and geometric measure of entanglement, just to name a few. Some of these are only defined in bipartite settings.
Despite these advances, the characterization and quantification of entanglement are far from complete. For example, even for multipartite pure states, it has been a long-standing question whether there exists a finite minimal reversible entanglement generating set (MREGS). The existence of a MREGS would enable the generalization of the entanglement of distillation and formation to multipartite systems and would also provide a better characterization of multipartite entanglement. In the case of bipartite settings, Bell states provide the unqiue unit/metric to measure the content of entanglement. However, in the three-qubit settings, there are, in addition to Bell-state-type entangled states, two inequivalent genuine three-partite entangled states: GHZ and W states. In the case of four qubits, there are nine different families. Conceivably, the number of inequivalent types of entanglement grows very fast as the number of parties increases. In the past, we have studied the geometric measure, the relative entropy of entanglement, and the generalized robustness, and have established connections between these, via several inequalities. We are interested in studying various entanglement measures and seeking the connections among them.
Entanglement in many-body systems: A natural and more physical arena where one can apply the notion of entanglement involves many-body systems of interacting particles, e.g. quantum spins, fermions or bosons. There have been quite a lot of works done along this direction, and many understandings can be uncovered. For example, the behavior of entanglement entropy in spin chains was established using the conformal field theory. The area law of entanglement in various systems, including spins, fermions, and bosons.
We have applied a global and multipartite, in contrast to bi-partite, measure of entanglement to a quantum spin model, namely, the one-dimensional XY model in a transverse field. We have found that the entanglement can be used to detect quantum phase transitions in this family of model and several critical features can be identified analytically. The general connection of quantum entanglement and quantum phase transitions in many-body physics is one subject that we are currently investigating.
Quantum computational complexity: Computational complexity classifies various types of problems. Probles in the class P (polynomial) can be solved efficiently on classical computers. However, problems that are NP (nondeterministic polynomial) hard seem unlikely to be efficiently solved. What about the use of quantum computers? The computational complexity class that a quantum computer can efficiently solve is called BQP, which is the analogue of P. The quantum version of NP is called QMA (Quantum Merlin-Arthur). A problem that is QMA-hard is unlikely to be solved efficiently by a quantum computer. We are interested in classifying various models and understanding their computational complexity, given the access to a quantum computer.