Resources Required for Preparing Graph States
Graph states have become a key class of states within quantum computation. They form a basis for universal quantum computation, capture key properties of entanglement, are related to quantum error correction, establish links to graph theory, violate Bell inequalities, and have elegant and short graph-theoretical descriptions. We give here a rigorous analysis of the resources required for producing graph states. Using a novel graph-contraction procedure, we show that any graph state can be prepared by a linear-size constant-depth quantum circuit, and we establish trade-offs between depth and width. We show that any minimal-width quantum circuit requires gates that acts on several qubits, regardless of the depth. We relate the complexity of preparing graph states to a new graph-theoretical concept, the local minimum degree, and show that it captures basic properties of graph states.
KeywordsQuantum Computing Algorithms Foundations of computing
Unable to display preview. Download preview PDF.
- 3.Bouchet, A.: Connectivity of isotropic systems. In: Combinatorial Mathematics: Proc. of the Third International Conference. Ann. New York Acad. Sci., vol. 555, pp. 81–93 (1989)Google Scholar
- 4.Bouchet, A.: κ-transformations, local complementations and switching. In: Cycles and rays: Basic structures in finite and infinite graphs. NATO Adv. Sci. Inst. Ser., vol. C 301, pp. 41–50. Kluwer Acad. Publ., Dordrecht (1990)Google Scholar
- 7.Eisert, J., Gross, D.: Multi-particle entanglement. In: Lectures on Quantum Information, Wiley-VCH, Berlin (2006)Google Scholar
- 9.Geelen, J.F.: Matchings, Matroids and Unimodular Matrices. PhD thesis, Univ. Waterloo (1995)Google Scholar
- 10.Goyal, K., McCauley, A., Raussendorf, R.: Purification of large bi-colorable graph states (May 2006) quant-ph/0605228Google Scholar
- 11.Hein, M., Dür, W., Eisert, J., Raussendorf, R., Van den Nest, M., Briegel, H.J.: Entanglement in graph states and its applications. In: Proc. of the Int. School of Physics Enrico Fermi on Quantum Computers, Algorithms and Chaos (July 2005) quant-ph/0602096 Google Scholar
- 12.Markov, I., Shi, Y.: Simulating quantum computation by contracting tensor networks. In: Ninth Workshop on Quantum Information Processing (January 2006) (No proceedings) Google Scholar
- 16.Shi, Y., Duan, L.M., Vidal, G.: Classical simulation of quantum many-body systems with a tree tensor network (in completion) (February 2006)Google Scholar
- 17.Van den Nest, M.: Local equivalence of stabilizer states and codes. PhD thesis, Faculty of Engineering, K.U. Leuven, Belgium (May 2005)Google Scholar
- 18.Van den Nest, M., Miyake, A., Dür, W., Briegel, H.J.: Universal resources for measurement–based quantum computation (April 2006) quant-ph/0604010Google Scholar