Abstract
The canonical decomposition for two-qubit operators has proven very useful for applications in quantum computing. This decomposition generates equivalence classes up to local quantum gates. We provide a variety of complete, explicit decompositions of given two-qubit operators in terms of single, double, and triple controlled-NOT (CNOT) gates. By analytically addressing the needed pre- and post-tensor product factors, we demonstrate that exact results are possible, even when a parameter is included. The examples given are of interest to superconducting qubit, spin-based, dipolar molecule, and other quantum information processing systems.
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References
Shende V.V., Bullock S.S., Markov I.L.: Recognizing small-circuit structure in two qubit operators. Phys. Rev. A 70, 012310 (2004)
Bullock S.S., Markov I.L.: Arbitrary two-qubit computation in 23 elementary gates. Phys. Rev. A 68, 012318 (2003)
Vidal G., Dawson C.M.: Universal quantum circuit for two-qubit transformations with three controlled-NOT gates. Phys. Rev. A 69, 010301(R) (2004)
Coffey M.W., Deiotte R., Semi T.: Comment on “universal quantum circuit for two-qubit transformations with three controlled-NOT gates” and “recognizing small-circuit structure in two-qubit operators”. Phys. Rev. A 77, 066301 (2008)
Vatan F., Williams C.: Optimal quantum circuits for general two-qubit gates. Phys. Rev. A 69, 032315 (2004)
Childs A.M., Haselgrove H.L., Nielsen M.A.: Lower bounds on the complexity of simulating quantum gates. Phys. Rev. A 68, 052311 (2003)
Zhang J., Vala J., Whaley K.B., Sastry S.: Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 67, 042313 (2003)
Makhlin Y.: Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations. Quant. Info. Proc. 1, 243 (2002)
Rezakhani A.T.: Characterization of two-qubit perfect entanglers. Phys. Rev. A 70, 052313 (2004)
Coffey M.W., Colburn G.G.: Feasibility of the controlled-NOT gate from certain model Hamiltonians. J. Phys. A 40, 9463 (2007)
Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Shende V.V., Markov I.L., Bullock S.S.: Finding small two-qubit circuits. Proc. SPIE 5436, 348 (2004)
Grajcar M. et al.: Switchable resonant coupling of flux qubits. Phys. Rev. B 74, 172505 (2006) cond-mat/0605484 (2006)
Niskanen A.O., Nakamura Y., Tsai J.-S.: Tunable coupling scheme for flux qubits at the optimal point. Phys. Rev. B 73, 094506 (2006)
Niskanen A.O., Vartiainen J.J., Salomaa M.M.: Optimal multiqubit operations for Josephson charge qubits. Phys. Rev. Lett. 90, 197901 (2007)
Zhang, J., Whaley, K. B.: Generation of quantum logic operations from physical Hamiltonians. Phys. Rev. A 71, 052317 (2005); quant-ph/0412169 (2004)
Kuznetsova E. et al.: Analysis of experimental feasibility of polar-molecule-based phase gates. Phys. Rev. A 78, 012313 (2008)
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Coffey, M.W., Deiotte, R. Exact canonical decomposition of two-qubit operators in terms of CNOT. Quantum Inf Process 9, 681–691 (2010). https://doi.org/10.1007/s11128-009-0156-3
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DOI: https://doi.org/10.1007/s11128-009-0156-3