24 August 2004 Finding small two-qubit circuits
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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++.
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Vivek V. Shende, Vivek V. Shende, Igor L. Markov, Igor L. Markov, Stephen S. Bullock, Stephen S. Bullock, "Finding small two-qubit circuits", Proc. SPIE 5436, Quantum Information and Computation II, (24 August 2004); doi: 10.1117/12.542381; https://doi.org/10.1117/12.542381

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