Quantum Information and Computation, Vol. 5, No. 2 (2005) 113–130 c  Rinton Press GRAPH-BASED SIMULATION OF QUANTUM COMPUTATION IN THE DENSITY MATRIX REPRESENTATION GEORGE F. VIAMONTES a Department of Electrical Engineering and Computer Science, University of Michigan 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA IGOR L. MARKOV b Department of Electrical Engineering and Computer Science, University of Michigan 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA JOHN P. HAYES c Department of Electrical Engineering and Computer Science, University of Michigan 1301 Beal Avenue, Ann Arbor, Michigan 48109-2122, USA Received June 11, 2004 Revised December 17, 2004 Quantum-mechanical phenomena are playing an increasing role in information process- ing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phe- nomena efficiently is exceedingly difficult because of the vast size of the quantum state space involved. A major complication is caused by errors (noise) due to unwanted in- teractions between the quantum states and the environment. Consequently, simulating quantum circuits and their associated errors using the density matrix representation is potentially significant in many applications, but is well beyond the computational abili- ties of most classical simulation techniques in both time and memory resources. The size of a density matrix grows exponentially with the number of qubits simulated, rendering array-based simulation techniques that explicitly store the density matrix intractable. In this work, we propose a new technique aimed at efficiently simulating quantum cir- cuits that are subject to errors. In particular, we describe new graph-based algorithms implemented in the simulator QuIDDPro/D. While previously reported graph-based sim- ulators operate in terms of the state-vector representation, these new algorithms use the density matrix representation. To gauge the improvements offered by QuIDDPro/D, we compare its simulation performance with an optimized array-based simulator called QCSim. Empirical results, generated by both simulators on a set of quantum circuit benchmarks involving error correction, reversible logic, communication, and quantum search, show that the graph-based approach far outperforms the array-based approach for these circuits. Keywords : Quantum circuits, quantum algorithms, simulation, density matrices, quan- tum errors, graph data structures, decision diagrams, QuIDDs Communicated by : D. Wineland & B. Terhal a E-mail: gviamont@eecs.umich.edu b E-mail: imarkov@eecs.umich.edu c E-mail: jhayes@eecs.umich.edu 113