Experimentally Realizable Controlled-NOT Gate in a Flux-qubit / Resonator System
Shiro Saito, Todd Tilma*, Simon J. Devitt*, Kae Nemoto*, and Kouichi Semba
Physical Science Laboratory, *National Institute of Informatics
Superconducting qubits have attracted increasing interest in the context of quantum computing and quantum information processing . We are focusing on a superconducting flux qubit/resonator system [see Fig. 1(a)] where the resonator works as a quantum bus (qubus). Using the qubus concept we can perform any two-qubit operation between any two qubits that are coupled to the quantum bus without using multiple swap gates, which are required in other systems that use direct qubit-qubit coupling. We present an experimentally realizable microwave pulse sequence that effects the controlled-NOT (C-NOT) gate operation in a qubit/resonator system . From numerical simulations, we obtained a process fidelity FP  of 98.8 % (98.0 %) for a two-(three-)qubit/resonator system under ideal conditions.
Figure 1(a) is a schematic of our system. Each qubit couples to the resonator through a mutual inductance M. Because of the fixed coupling, the qubit transition frequency depends slightly on the resonator state. We utilize this frequency difference to realize the C-NOT gate in cooperation with a two-photon blue side band (BSB) transition, which can create an entanglement between the control qubit and the resonator. Figure 1(b) shows a basic operational sequence of pulses on our two-qubit/resonator system. The pulse characteristics are represented by a frequency (C: carrier frequency, BSB: BSB frequency), a length (π/2, π) and a phase (0, π, φ) from the top to the bottom. We obtained an FP of 98.8 % by optimizing the free evolution time T and the phase of the second BSB pulse φ. Here we set the frequencies of the control, target qubit and resonator at 6, 5 and 10 GHz, respectively, and the coupling between the qubit and the resonator at 0.1 GHz. Moreover, we obtained a high FP of 98.0 % even when we added a third qubit with a frequency of 7 GHz, which means that our proposal is scalable. The total gate time is 200 ns, which is much shorter than the coherence time of a flux qubit (4 µs). We estimated FP under decoherence (see Fig. 2). We obtained an FP of 90.3 % under the best conditions yet achieved experimentally: Q=106 and Γ1=Γ2=0.25 MHz. Even with this realistic decoherence model, FP still exceeds 90 %, showing that our gate remains robust against this type of loss.
This work was supported by KAKENHI.
 J. Clarke and F. K. Wilhelm, Nature 453 (2008) 1031.
 S. Saito, T. Tilma, S. J. Devitt, K. Nemoto, and K. Semba, Phys. Rev. B 80 (2009) 224509.
 I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44 (1997) 2455.
Fig. 1. (a) Superconducting flux-qubit/resonator system.
(b) Pulse sequence of C-NOT gate.
Fig. 2. Process fidelity under decoherence.
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