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arXiv:1204.2750 · Physical Review A

Minimal and robust composite two-qubit gates with Ising-type interaction

Tsubasa Ichikawa, Utkan Güngördü, Masamitsu Bando, Yasushi Kondo, Mikio Nakahara

Published Feb 19, 2013• 16 citations
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Abstract

We construct a minimal robust controlled-not gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number ($N=3$) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust swap gate cannot be constructed with $N=3$ but requires $N=6$ elementary two-qubit gates.

DOI
10.1103/physreva.87.022323

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Publisher’s Note: Minimal and robust composite two-qubit gates with Ising-type interaction [Phys. Rev. A87, 022323 (2013)]

Tsubasa Ichikawa, Utkan Güngördü, Masamitsu Bando +2 more

Received 26 February 2013DOI:https://doi.org/10.1103/PhysRevA.87.029908©2013 American Physical Society

Physical Review A•Feb 28, 2013•0 citations
We construct a minimal robust controlled-not gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number (N=3N=3N=3) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust swap gate cannot be constructed with N=3N=3N=3 but requires N=6N=6N=6 elementary two-qubit gates.
We address the problem of constructing quantum circuits to implement an arbitrary two-qubit quantum computation. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 171717 gates: 151515 one-qubit rotations and 222 controlled-NOT (CNOT) gates. We also constructively prove a worst-case upper bound of 232323 elementary gates, of which at most four (CNOT gates) entail multiqubit interactions. Our analysis shows that synthesis algorithms suggested in previous work, although more general, entail larger quantum circuits than ours in the special case of two qubits. One such algorithm has a worst case of 616161 gates, of which 181818 may be CNOT gates.
The two-qubit interaction Hamiltonian of a given physical implementation determines whether or not a two-qubit gate such as the controlled-NOT (CNOT) gate can be realized easily. It can be shown that, e.g., with the XY\mathrm{XY}XY interaction more than one two-qubit operation is required in order to realize the CNOT operation. Here we propose a two-qubit gate for the XY\mathrm{XY}XY interaction which combines the CNOT and SWAP operations. By using this gate quantum circuits can be implemented efficiently, even if only nearest-neighbor coupling between the qubits is available.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, yyy-rotation, zzz-rotation, and phase gates.