† Corresponding author. E-mail:
‡ Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11374344, 11404386, and 91321208), the National Basic Research Program of China (Grant No. 2014CB921401), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB07010300).
In the preparations of superconducting qubits, circuit design is a vital process because the parameters and layout of the circuit not only determine the way we address the qubits, but also strongly affect the qubit coherence properties. One of the most important circuit parameters, which needs to be carefully designed, is the mutual inductance among different parts of a superconducting circuit. In this paper we demonstrate how to design a gap-tunable flux qubit by layout design and inductance extraction using a fast field solver FastHenry. The energy spectrum of the gap-tunable flux qubit shows that the measured parameters are close to the design values.
Superconducting quantum circuits based on Josephson junctions[1] are promising candidates for the implementation of quantum computers.[2] This is due to the fact that the complex superconducting quantum circuits can be freely constructed by using well-established micro-fabrication technology in the semiconductor industry. Also the superconducting quantum circuits have advantages, such as being easy to control, scalability, and strong inter-coupling as compared with other quantum systems such as ion traps,[3] nuclear spins,[4,5] electron spins,[6] neutral atoms,[7] and photons.[8] The basic component of a quantum computer is the qubit, which is a two-level system. For a superconducting circuit to function as a qubit, its circuit parameters must be well designed. One parameter which can play an important role in affecting the characteristics of a superconducting qubit is the inductance. In this paper, we demonstrate how to choose proper values for the inductances in designing a gap-tunable flux qubit.[9,10]
A superconducting flux qubit loop is interrupted by three Josephson junctions,[11] the critical current of one of the junctions is α times smaller than those of the other two junctions, which are identical and characterized by the Josephson energy EJ and the charging energy Ec. When the magnetic flux threading the qubit loop Φɛ is around half flux quantum Φ0/2, the qubit ground and excited states correspond to clockwise and anticlockwise persistent currents ±Ip in the qubit loop. When the qubit is biased at the symmetric point or the optimal point Φɛ = Φ0/2, the persistent current states are degenerate, quantum tunneling lifts this degeneracy and forms an energy gap Δ. The Hamiltonian of a flux qubit is given by:
One of the biggest challenges of realizing quantum computers with superconducting qubits is coherence. A qubit in a quantum computer needs to perform a huge number of quantum logic operations to solve a practical problem. Like any other quantum system, a qubit undergoes decoherence due to coupling to noise in its environment. Each noise source with a qubit coupled can be represented by a fluctuation term of some parameter λ in its Hamiltonian δ λ and characterized by its quantum spectral density sλ(ω) = 1/(2π) ∫ dt〈δλ(0)δλ(t)〉 e−iωt.[12]
The effect of decoherence can be separated into two effects, i.e., energy relaxation and dephasing, and characterized by relaxation rate Γ1 and dephasing rate Γϕ respectively,[13]
![](cpb_25_12_120305/cpb160648ieqn1.gif)
The readout of a flux qubit state utilizes the factor that the two persistent current states |±Ip〉 form a basis for the qubit states. Measuring the persistent current in the qubit loop corresponds to a projective measurement of the qubit in the persistent current state basis. Since the SQUID loop is inductively coupled to the qubit loop, the qubit state influences the critical current of the SQUID. Any change of the persistent current is reflected in the SQUID flux bias Φsq and modulates its critical current Isq. We apply pulses to the readout SQUID and detect its switching. The switching is a stochastic process due to quantum tunneling. The pulse amplitude at which the readout SQUID switching takes place has a distribution with typical spread about 1% of Isq, and the center of the distribution is slightly smaller than Isq. In order to obtain sufficient visibility in the readout signal, we need to have about 1% of change in Isq in a change of qubit state from |−Ip〉 to |+Ip〉, that is 2Msq Ip ≈ 10mΦ0. For a persistent current Ip = 500 nA, we have Msq ≈ 20 pH. While we want the mutual inductance Msq to be as small as possible for minimizing qubit decoherence, the requirement for the readout to be efficient sets a lower limit for the mutual inductance Msq, which as shown above is typically several tens of pico-Henry.
Just like the mutual inductance between the qubit and the readout SQUID, mutual inductances between the qubit and control lines also have their lower limits. The mw/flux bias line is used for providing fast flux bias to shift the qubit frequency
We develop a dedicated layout computer-aided design (CAD) tool to model the qubit circuit, figure
We use the fast field solver FastHenry[15] to extract the inductances numerically. The FastHenry solver is one of the most popular techniques to extract circuit inductances. It uses the fast multipole method (FMM) to accelerate matrix-vector production. The algorithm used in FastHenry is an acceleration of the mesh formulism approach. The linear system resulting from the mesh formulation is solved by using a generalized minimal residual (GMRES) algorithm with a fast multiple algorithm to efficiently compute iteration. The digital layout file is used as the input of FastHenry to extract the inductances. After the calculation, the result is compared with the desired inductance value and the layout is adjusted accordingly. After a few iterations we obtain a layout with inductances close to the desired values. Table
![]() | Table 1. Mutual inductances extrated by FastHenry for qubit a and qubit b. Here for qubit b we only extract the mutual inductance between the qubit and the resonator Mqr. . |
After the qubit circuit is properly designed the sample is then fabricated and measured. The details about the fabrication and measurement procedures can be found in Refs. [16] and [17]. Figure
In this paper, we demonstrate how to design a gap-tunable flux qubit with proper values of mutual inductances between different circuit parts. It is valuable for ensuring long coherence time. It provides enough control freedom over the qubit state and has sufficient readout efficiency. The method demonstrated here can be generalized to other types of superconducting circuits.
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