Parametric coupling for superconducting qubits

We propose a scheme to couple two superconducting charge or flux qubits biased at their symmetry points with unequal energy splittings. Modulating the coupling constant between two qubits at the sum or difference of their two frequencies allows to bring them into resonance in the rotating frame. Switching on and off the modulation amounts to switching on and off the coupling which can be realized at a nanosecond speed. We discuss various physical implementations of this idea, and find that our scheme can lead to rapid operation of a two-qubit gate. DOI: 10.1103/PhysRevB.73. 064512 PACS numbers: 74.50. r, 73.40. Gk The high degree of control which has been achieved on microfabricated two-level systems based on Josephson tunnel junctions 1‐3 has raised hope that they can form the basis for a quantum computer. Two experiments, representing the most advanced quantum operations performed in a solidstate environment up to now, have already demonstrated that superconducting qubits can be entangled. 4,5 Both experiments implemented a fixed coupling between two qubits, mediated by a capacitor. The fixed-coupling strategy would be difficult to scale to a large number of qubits, and it is desirable to investigate more sophisticated schemes. Ideally, a good coupling scheme should allow fast two-qubit operations, with constants of order 100 MHz. It should be possible to switch it ON and OFF rapidly with a high ON/OFF ratio. It should also not introduce additional decoherence compared to a single-qubit operation. Charge and flux qubits can be biased at a symmetry point 2,6 where their coherence times are the longest because they are insensitive to first order to the main noise source, charge, and flux noise, respectively. It is therefore advantageous to try to keep all such quantum bits biased at this symmetry point during experiments where two or more are coupled. In that case, the resonance frequency of each qubit is set at a fixed value determined by the specific values of its parameters and cannot be tuned easily. The critical currents of Josephson junctions are controlled with a typical precision of only 5%. The charge qubit energy splitting at the symmetry point depends linearly on the junction parameters so that it can be predicted with a similar precision. The flux-qubit energy splitting called the gap and noted , on the other hand, depends exponentially on the junctions critical current 9 and it is to be expected that two flux qubits with nominally identical parameters have significantly different gaps. 7 Therefore the problem we would like to address is the following: how can we operate a quantum gate between qubits biased at the optimal point and having unequal resonance frequencies? We first discuss why the simplest fixed linear coupling scheme as was implemented in the two-qubit experiments 4,5 fails in that respect. Consider two flux qubits biased at their flux-noise insensitive point Q= Q being the total phase drop across the three junctions, and inductively coupled as shown in Fig. 1a. 7 The uncoupled energy states of each qubit are denoted 0i, 1ii=1,2 and their minimum energy separation hii. Throughout this paper, we will suppose that 12. As shown before, 7,8 the system Hamiltonian can be written as H=Hq1+Hq2+HI, with Hqi =h/2izi i=1,2 and HI=hg0x1x2=hg01 2 + 1 2 + 1 2 + 1 2 . Here we introduced the Pauli matrices x.z;i referring to each qubit subspace, the raising lowering operators i i and we wrote the Hamiltonian in the energy basis of each qubit. It is more convenient to rewrite the previous Hamiltonian in the interaction representation, resulting