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Chemical Physics Letters

Time-optimized quantum gates on linear three-qubit systems with indirect Ising coupling

Daxiu Wei, Andreas K. Spörl, Yan Zuo Chang, Navin Khaneja, Xiaodong Yang, Steffen J. Glaser

Published Aug 14, 2014• 2 citations
Publisher
DOI
10.1016/j.cplett.2014.08.007

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What is the time-optimal way of realizing quantum operations? Here, we show how important instances of this problem can be related to the study of shortest paths on the surface of a sphere under a special metric. Specifically, we provide an efficient synthesis of a controlled NOT (CNOT) gate between qubits coupled indirectly via Ising-type couplings to a third spin. Our implementation of the CNOT gate is more than twice as fast as conventional approaches. The pulse sequences for efficient manipulation of our coupled spin system are obtained by explicit computation of geodesics on a sphere under the special metric. These methods are also used for the efficient synthesis of indirect couplings and of the Toffoli gate. We provide experimental realizations of the presented methods on a linear three-spin chain with Ising couplings.
We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with a symmetric coupling J plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the CNOT(1, 3) between the indirectly coupled qubits 1 and 3 is , i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh–Hadamard rotation in the Hilbert space of qubit 3 shows that the time-optimal synthesis of the CNOT ± (1, 3) (which acts as the identity when the control qubit 1 is in the state |0⟩, while if the control qubit is in the state |1⟩, the target qubit 3 is flipped as | ± ⟩ → |∓⟩) also requires the same time T .