Dr. Igor L. Markov Profile

Dr. Igor L. Markov

at Univ of Michigan

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Proceedings Article | 24 August 2004 Paper

Finding small two-qubit circuits

Vivek Shende, Igor Markov, Stephen Bullock

Proceedings Volume 5436, (2004) https://doi.org/10.1117/12.542381

KEYWORDS: Device simulation, Binary data, Quantum communications, Computer simulations, Matrices, Fourier transforms, C++, Quantum circuits, Quantum information, Phase measurement

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An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.

Proceedings Article | 24 August 2004 Paper

Graph-based simulation of quantum computation in the density matrix representation

George Viamontes, Igor Markov, John Hayes

Proceedings Volume 5436, (2004) https://doi.org/10.1117/12.542767

KEYWORDS: Quantum communications, Computer simulations, Device simulation, Quantum circuits, Matrices, Logic, Quantum computing, Binary data, Radiofrequency ablation, Matrix multiplication

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Quantum-mechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena 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 interactions 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 abilities 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 circuits that are subject to errors. In particular, we describe new graph-based algorithms implemented in the simulator QuIDDPro/D. While previously reported graph-based simulators 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.

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