Abstract
The engineering of quantum devices has reached the stage where we now have smallscale quantum processors containing multiple interacting qubits within them. Simple quantum circuits have been demonstrated and scaling up to larger numbers is underway. However, as the number of qubits in these processors increases, it becomes challenging to implement switchable or tunable coherent coupling among them. The typical approach has been to detune each qubit from others or the quantum bus it connected to, but as the number of qubits increases this becomes problematic to achieve in practice due to frequency crowding issues. Here, we demonstrate that by applying a fast longitudinal control field to the target qubit, we can turn off its couplings to other qubits or buses. This has important implications in superconducting circuits as it means we can keep the qubits at their optimal points, where the coherence properties are greatest, during coupling/decoupling process. Our approach suggests another way to control coupling among qubits and data buses that can be naturally scaled up to large quantum processors.
Introduction
Superconducting quantum circuits^{1,2,3,4} are promising candidates to realize quantum processors and simulators. They have been used to demonstrate various quantum algorithms and implement thousands of quantum operations within their coherence time,^{5} in which controllable couplings are inevitable. The typical way to couple/decouple two superconducting quantum elements with always on coupling is to tune their frequencies in or out of resonance,^{6,7,8,9,10,11,12} or with sideband transitions by parametric driven.^{13,14} These methods are widely adopted even for the most recent universal gate implementations.^{5,15,16,17,18,19,20,21,22,23,24,25,26} However, they suffer from several issues, namely, it is technically difficult to avoid the frequency crowding problem in largescale circuits; the qubits cannot always work at the coherent optimal point; and the fast tuning of the qubit frequency results in nonadiabatic information leakage. To overcome the above problems, significant effort both theoretically and experimentally^{27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47} has been devoted to develop coupling methods for parametrically tuning the coupling strength between two quantum components. These proposals or demonstrations either only achieved low fidelity or needs additional circuity to modify the qubit architecture, increasing the complexity of the circuity and bring in additional crosstalk problems, which in the process of scaling up, might be a major hinderance. Therefore, the implementation of a switch for coherent coupling between quantum elements is still a big challenge in scalable quantum circuits.
In this article, we propose and demonstrate a simple architectureindependent method to switch on/off the coupling between two quantum elements via a longitudinal control field.^{48–51} We demonstrate this on two different superconducting systems: transmon qubits^{52} of the Xmon variety,^{53} and flux qubit–resonator systems.^{54} In the first instance (Fig. 1a–c) two qubits Q_{1} and Q_{2} are coupled through the same coplanar wave guide resonator with coupling strength g_{1} ≈ g_{2} ≈ 20 MHz, respectively. The qubit frequencies ω_{1} and ω_{2} are lower than the resonator frequency ω_{c} = 5.456 GHz with detuning Δ_{1,2} = ω_{c} − ω_{1,2}. In this dispersive limit, the Hamiltonian of the qubit–qubit system can be written as
where g = g_{1}g_{2}(Δ_{1} + Δ_{2})/2Δ_{1}Δ_{2} is the effective coupling strength, and Δ_{1,2} = ω_{c} − ω_{1,2} is the detuning of the qubit frequencies from the resonator. Here σ_{x}, σ_{y}, and σ_{z} are the Pauli operators with σ_{±} = (σ_{x} ± iσ_{y})/2.
The second system (Fig. 1d–f) can be described by the flux qubit–resonator Hamiltonian as^{55}
where a^{†} (a) is the creation (annihilation) operator of the resonator field with the resonance frequency ω_{r}/2π, and ħΔ is the energy gap of the flux qubit with ħε being the energy difference between the two persistentcurrent states. The qubit frequency is given by \(\omega _1 = \sqrt {{\mathrm{\Delta }}^2 + \varepsilon ^2}\).
Results
Let us focus our attention on our switch for both quantum circuits beginning with the qubit–qubit system.
Qubit–qubit system configuration
We begin by fixing the the qubit frequency of Q_{2} at ω_{1} = 4.9174 GHz, denoted by the horizontal solid line in Fig. 2a. We then tune the qubit frequency of Q_{1} by its z bias line and probe its microwave excitation response. Figure 2a shows the spectroscopic measurement with an anticrossing due to the qubit–qubit coupling clearly observed at 4.9174 GHz. Fitting the spectrum lines (dashed blue curves in Fig. 2a) establishes the coupling strength as g/2π = 0.61 MHz.
To begin our investigation of switching on/off the qubit–qubit coupling, we apply a control field with frequency ω_{z}/2π = 20 MHz to Q_{1} by the z bias line. This results in a longitudinal interaction Hamiltonian H_{L} = ħλ_{z} cos(ω_{z}t)σ_{z,1}, where λ_{z} is determined by the amplitude of the applied rf driving current I_{α} in the z bias line. By performing an unitary transform U = exp[−i2λ_{z} sin(ω_{z}t)σ_{z}/ω_{z}], the total Hamiltonian H = H_{JC} + H_{L} of the system reduces to the effective Hamiltonian of the form^{48,49}
where we have neglected the small fastoscillating terms assuming \(\omega _z \gg g\). In this case g_{eff} = gJ_{0}(2λ_{z}/ω_{z}) is the effective coupling strength under longitudinal control, where J_{0}(x) is the zerothorder Bessel function of the first kind. Our effective Hamiltonian clearly shows that g_{eff} vanishes when 2λ_{z}/ω_{z} is a zero point of the Bessel function J_{0}(2λ_{z}/ω_{z}), that is when λ_{z} ≈ 1.2ω_{z}. At this point the qubit–qubit coupling is switched off. Moreover, the coupling strength g_{eff} can be continuously tuned between two values with opposite signs by changing the ratio 2λ_{z}/ω_{z}. We can define the switch on/off ratio R as the ratio between vacuum Rabi frequencies with and without the control field. Henceforth, we will call λ_{zoff} the switchoff point and a longitudinal control pulse with such an amplitude a switchoff pulse. Here, we have neglected the fastoscillating terms to give a simplified description to illustrate the idea of this scheme, in methods we have done numerical simulations with the full Hamiltonian (see Methods). In fact, experiment^{56} and theory^{49} have shown that transparency to the transverse classical field can be induced with a longitudinal control pulse. Here, we replace the transverse classical field with a quantum element (qubit or resonator), which becomes decoupled from all transverse interactions. This is the governing principle of our controlled coupling scheme.
For the qubit–qubit case, in view of the qubits small anharmonicities around 250 MHz, we included one extra level in our numerical simulations. Results are consistent with that of the analytical twolevel case described by Eq. (3). The driving amplitude used for the qubitqubit system in this work is well below the qubit anharmonicity and leakage to higher levels is well below 10^{−4}. Note that if the amplitude approaches the anharmonicity, leakage will become nonnegligible.
Next, to test our longitudinal, control fieldbased switch we perform spectroscopic measurements on the qubit–resonator system under different amplitudes λ_{z} of the longitudinal control fields. These are shown in Fig. 2b where it can be seen that the amplitude of the anticrossing gap Δ_{g} = 2g_{eff} decreases to zero as λ_{z} → 1.2ω_{z}, and then opens up again as λ_{z} further increases. At the switchoff point λ_{zoff} ≈ 24 MHz, the amplitude of the anticrossing Δ_{g} becomes undetectable. This is definite evidence that the coupling can be tuned and switched off by the longitudinal control field.
Qubit–resonator system configuration
Here, we tune the qubit energy gap Δ/2π to be equal to the resonator frequency ω_{r}/2π = 2.417 GHz by applying a long dc bias in the α bias line (see Fig. 1c). Spectroscopic measurements are then performed. Figure 2c shows the measurement results with an anticrossing due to the qubit–resonator coupling clearly observed at 2.417 GHz. Fitting the spectrum lines (dashed blue curves) allows us to determine the coupling strength as g/2π = 9.14 MHz.
Given we now know g we can now test our switching protocol. Applying a longitudinal control field (similar to the qubit–qubit case) with frequency ω_{z} and amplitude λ_{z}, we observed in Fig. 2d that the amplitude of the anticrossing gap Δ_{g} = 2g_{eff} decreases to zero as λ_{z} → 1.2ω_{z}. The bright spot seen at Δ/2π = 2.417 GHz and near λ_{z}/2π ~ 180 MHz is the resonator’s resonance signature. When we drive the flux qubit, the resonator can be excited due to coupling with the qubit microwave driving control line. This excitation can be detected because there is coupling between the resonator and the qubit readout SQUID, which results in the bright spot at the resonator frequency in Fig. 2d.
Performance
Our exploration of the switch using spectroscopic measurements has qualitatively shown its operation in both the qubit–qubit and qubit–resonator configurations, yet it is difficult to quantify exactly how well it is operating from those measurements alone. However, timedomain vacuum Rabi oscillation measurements with the switch turned on and off separately will allow us to measure the time scale of the relaxation decay. Figure 3a for the qubitqubit system and (c) for the qubit–resonator system shows the vacuum Rabi oscillations when the amplitude λ_{z} of the longitudinal control field is increased from zero. The data clearly shows that the oscillation frequency ω_{c} = 2g_{eff} decreases with increasing λ_{z}. When λ_{z} reaches the switchoff point λ_{zoff}, the oscillation frequency reaches a minimum. Further, Fig. 3b for the qubit–qubit system and (d) for the qubit–resonator system shows the comparison of the dynamics without a control pulse (blue) and with switchoff pulse (orange). At the switchoff point the dynamic behavior is an exponential decay—the oscillation frequency is too small to be observed on the experimental data, indicating a very small effective coupling g_{eff}. The rate of exponential decay due to energy relation in the switchedoff state is very close to the decay rate without the switching pulse (shown explicitly in Fig. 3b for the qubit–qubit system). This indicates that the switching pulse does not influence the qubits relaxation time. More quantitatively, the characteristic times for the decays in Fig. 3b are

1.
Singlequbit energy relaxation rate: T = 15.58 ± 0.28 μs

2.
No switching pulse with maximal coupling: T = 9.57 ± 0.35 μs

3.
Switching pulse tuned to turn off the coupling: T = 15.25 ± 0.24 μs
We observed that the difference between the “No switching pulse with maximal coupling” and “Switching pulse tuned to turn off the coupling” situations is 9.57 compared with 15.25 for the “2” and “3” situations, this is because the relaxation time of Q_{2} is 7.0 μs, shorter than that of Q_{1} (15.58 μs). Further, it is quite interesting that the decay behavior between the “1” and “3” situations are almost the same, even if we attribute all the slight difference to a weak, residual coupling, it should be less than 2.7 kHz (see Methods). The lower limit of the on/off ratio is g/g_{r} = 0.61 MHz/2.7 kHz ~230, where g is the coupling strength between the two qubits. Further, the on/off ratio could be significantly enhanced by improving the qubit coherent properties and the precision of the control pulse, since the effective coupling g_{eff} = gJ_{0}(2λ_{z}/ω_{z}) can be tuned from positive to negative crossing the zero point. Also, as can be seen in the Methods section (Fig. 6) the numerical simulation shows an on–off ratio higher than 10^{5}, which hightlights the potential of the longitudinal control field as a highefficiency way to turn off the coupling.
While we have shown that our longitudinal pulse enables an effective switching operation, it is important to establish its effect on the quantum coherence of the systems. To this end we will now demonstrate the dynamical switching on and off of the coupling. In this case a target qubit is initially prepared in its excited state and brought into coherent resonance with its corresponding quantum element, after a delay, a switchoff pulse is applied over a period of time. In Fig. 4a, d, the coupling is switched off when the target qubit is in the excited state. In Fig. 4b, e, the coupling is switched off when the system is in an entangled state. In Fig. 4c, f, the coupling is switched off when the target qubit is on the ground state. We found that regardless of what state the system is in, when the switchoff pulse is applied, the coherent oscillation is paused (except for free evolution and decay during the switchoff time interval), and when the switchoff pulse is removed, the coherent oscillation resumes.
Discussion
We have demonstrated a highly efficient switch with an rf control field, the coupling strength can be in principle tuned to zero in both a qubit–qubit system and a qubit–resonator system. Dynamically switching on and off the coupling is also shown. The on–off ratio was measured to be above 230, and could be significantly improved in the future. In principle, the coulping could also be modified from g to any value in the range [−0.4g, g], which allows the system dynamics to be retarded or reversed. This could be useful in spin–echolike refocusing appliactions. This type of switch scheme can be applied to any qubit system with longitudinal control field and scales easily since no auxiliary circuit is needed. For applications to N qubits, we have done a preliminary investigation for cases with nearestneighboring coupling. For the case of 1D chain qubits with nearestneighboring coupling, our numerical simulations show that by applying driving fields on every other qubit, couplings can be switched off when all N qubits are at the same qubit frequency. Scaling up to 2D configurations is also straightforward. We also point out that besides as a tunable coupling scheme between quantum elements, the longitudinal coupling and control between electromagnetic fields and quantum devices itself is of great interest.^{56–63}
Methods
The qubit–qubit sample
This sample consists of six qubits fabricated with aluminum evaporation and liftoff techniques. The Josephson junctions are formed by using the Dolan bridge technique. The two qubits on which we perform the experiments are interconnected by a coplanar resonator, which couples them dispersively, the resonator has a resonance frequency about 5.45 GHz, well above the qubit frequencies. The T_{1} of the two measured qubits are approximately 14 and 7 μs, while \(T_2^ \ast\) is ~7 μs for both qubits.
The qubit–resonator system
The sample is a gaptunable flux qubit inductively coupled to a λ/4 coplanar resonator. A flux qubit is a superconducting loop interrupted by three Josephson junctions. Two of them are of the same size, and the other is smaller by a factor of α (α < 1). The larger two are characterized by Josephson energy E_{J} and charge energy E_{c}.The qubit states correspond to clockwise and counterclockwise persistent currents ±I_{p} in the qubit loop, I_{p} ≈ 400 nA in our sample. When the magnetic flux bias in the qubit loop Φ_{ε} = Φ_{0}/2, called the optimal point, the two persistentcurrent states are degenerate, quantum tunneling lifts this degeneracy, forming an energy gap ħΔ. Once E_{J} and E_{c} are fixed, the energy gap of a flux qubit is determined by α, the ratio of critical current between the smaller and larger junctions. Since the resonator frequency is fixed, and since there exists an optimal point of coupling between the qubit and resonator where the coherence properities of the system are the best, it is desirable to have the energy gap tunable. In our sample, the smaller junction has been replaced by a dcSQUID, called the the αloop, making the energy gap tunable in a wide range. The longitudinal control pulse for switching on/off the coupling is also applied to the qubit through the αloop. To achieve high magnetic flux bias stability, we use a gradiometric design for the qubit loop, making the qubit insensitive to global flux fluctuations. The qubit state is detected by a readout dcSQUID inductively coupled to the qubit. The sample is fabricated on a silicon wafer using electron beam lithography patterning, aluminum evaporation, and liftoff techniques. Josephson junctions are formed by using the Dolan bridge technique. The relaxation times of the qubit and the resonator are 0.45 and 4.6 μs, respectively.
Determination of the measured switch efficiency
In the case of the “Singlequbit energy relaxation” of Fig. 3b (Case 1), the relaxation time was measured as T = 15.58 ± 0.28 μs, while in the case of the “switching pulse tuned to turn off the coupling” (Case 3), the relaxation time was T = 15.25 ± 0.24 μs. Using the worst case after 14 μs decay, the 95% confidence upper bound of case 1 and the lower bound of case 3, the population ratio between case 3 and case 1 is 0.89. If we attribute the entire population drop to the residual coupling, the upper limit of oscillation frequency induced by the residual coupling is 2g_{r} = arccos(0.89)/(14 μs/2π) = 2.7 kHz.
Model of the qubit–qubit system
Our full system can be described by the Hamiltonian
with
Here, H_{0} describes the free Hamiltonian of the two qubits with \(\sigma _{z,i} = \left {e_i} \right\rangle \left\langle {e_i} \right  \left {g_i} \right\rangle \left\langle {g_i} \right\) being the Pauli matrix with the excited \(\left {e_i} \right\rangle\) and ground \(\left {g_i} \right\rangle\) states of the ith qubit. The interaction between the qubits is given by H_{I}, where g is the qubit–qubit coupling strength. Finally, the longitudinal driving field applied to qubit 1 is given by the Hamiltonian H_{L} where λ_{z} is the amplitude of that field. We will also assume the coupling strength g is much smaller than the frequency of the control field (\(g \ll \omega _z\)).
Moving to an interaction picture with respect to H_{0} + H_{L}, our resulting Hamiltonian can be written in the form
where we have used the Jacobi–Anger expansion and J_{n}(x) is the Bessel function of the first kind. As \(gJ_{n \ne 0}\left( {2\lambda _z{\mathrm{/}}\omega _z} \right) \ll \omega _z\), the nonresonant terms (n ≠ 0) can be neglected and the interaction between the qubits is reduced to
with the effective coupling strength g_{eff} = gJ_{0}(2λ_{z}/ω_{z}). When the ratio 2λ_{z}/ω_{z} reaches the first zero point of the Bessel function J_{0}(x) near x ≈ 2.4, the qubits are effectively decoupled. We call this point λ_{z,off} the switchoff point. The fast oscillation terms neglected above result in a small oscillation on the qubit–qubit state when the qubit qubit coupling is switched off. The amplitude of this oscillation decreases rapidly with increasing ω_{z}.
Model for the flux qubit–resonator system
The derivation of an effective Hamiltonian of the form
follows in a similar fashion to the qubit–qubit systems derivation where one simply replaces the second qubit \(\frac{\hbar }{2}\omega _2\sigma _{z,2}\) with a harmonic oscillator \(\hbar \omega _ra^\dagger a\) and that qubit’s raising and lower operators with the bosonic creation/destruction operators.
Numerical simulations
The time evolution of our system including dissipation can be described by the Lindblad Markov master equation
where
where γ_{qb} and γ_{r} are the decay rates of the qubit and the resonator, respectively. N(ω) = 1/[exp(βħω) − 1] is the average occupation number of the bath mode with the frequency ω at the temperature T with β = 1/(k_{B}T). In the experiment, measurements are taken in a dilution refrigerator at temperature T ≲ 20 mK. For ω_{qb}/2π = ω_{r}/2π = 2.417 GHz, the temperature of the bath can taken as 0, since N(ω_{qb}) and N(ω_{r}) are less than 0.003.
In the experiment, the system is initialized in the ground state \(\left {g,0} \right\rangle\) and a weak, transverse probe field with frequency ω is applied to detect the spectrum of the coupled system. The detected spectrum intensity is proportional to the integral of the probability P(t) of the qubit being in the excited state over a fixed time interval:
where
The schematic of the four lowest energy levels of the coupled system is shown Fig. 5a. Due to the coupling between the qubit and the resonator, the lowest two excited states of the system \(\left \pm \right\rangle = \left( {\left {e,0} \right\rangle \pm \left {g,1} \right\rangle } \right){\mathrm{/}}\sqrt 2\) are separated with a gap ~2g_{eff}. As shown in Fig. 5b when the amplitude of the control field λ_{z} = 0, the resonance absorption peaks corresponding to the transitions [denoted by the blue arrows in Fig. 5a] between the ground state \(\left {g,0} \right\rangle\) and the lowest two excited states \(\left \pm \right\rangle\) locate at δ = ω_{qb} − ω = ±g. As the amplitude λ_{z} of the control field increases, the gap first decreases and then vanishes at the switch point (λ_{z} ≈ 1.2ω_{z}), since the coupling between the qubit and the resonator is almost switched off. When λ_{z} > 1.2ω_{z}, the coupling is switched on again and the gap increases with λ_{z}. The coupling between the qubit and the resonator is switched off again when x = 2λ_{z}/ω_{z} reaches the second zero point of the Bessel function J_{0}(x).
When the decay rates of the qubit and the resonator satisfy \(\gamma _{q(r)} < g \ll \omega _{q(r)}\), the first two excited states \(\left \pm \right\rangle\) form a quasiinvariant subspace. If the system is initially in the state \(e0\rangle _{}^{}\), a Rabi oscillation induced by the qubit–resonator coupling is observed as shown in Fig. 5c. When the amplitude of the control field is zero, the period of the Rabi oscillation is T_{R} = 2π/g. When the strength of the control field increases, the effective coupling strength decreases and the period T_{R,eff} of the Rabi oscillation becomes longer. At the switch point, the coupling between the qubit and resonator is effectively switched off and the Rabi oscillation disappears.
Mathematical model example
Assume the system to be initialized in the state \(\left {\psi (0)} \right\rangle = \left e \right\rangle \otimes \left 0 \right\rangle\) with the qubit in its excited state and the resonator is in the vacuum state.
If no longitudinal control field is applied to the qubit (i.e., λ_{z} = 0), the Rabi oscillation between the qubit and the resonator takes place in the singleexcitation subspace \(\left\{ {\left {e,0} \right\rangle ,\left {g,1} \right\rangle } \right\}\). The period of this Rabi oscillation is T_{R} = π/g. After the control field is applied, the effective coupling between the qubit and the resonator is suppressed and almost eliminated at the switchoff point λ_{z}/ω_{z} ≈ 1.2. Correspondingly, the period of the effective Rabi oscillation T_{R,eff} around the switchoff point is prolonged greatly by the control field, as shown in Fig. 6a. Now, we define the on/off ratio as
Numerical simulations show that the period of the effective Rabi oscillation changes drastically near the switchoff point [see Fig. 6a]. The on/off ratio R as a function of the amplitude λ_{z} of the control field is displayed in Fig. 6b. We find that R decreases drastically when λ_{z} approaches the switchoff point λ_{z} ≈ 1.2ω_{z}, which is slightly smaller than the one that matches the zero point of the Bessel function J_{0}(2λ_{z}/ω_{z}). So if the amplitude λ_{z} of the control field can be precisely manipulated, the coupling between the qubit and the resonator can be effectively switched off by the longitudinal field, i.e., R < 10^{−5}.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. The datasets are also available in the figshare.com repository, [https://figshare.com/articles/An_efficient_and_compact_switch_for_quantum_circuits_data/6726965].
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Acknowledgments
We thank the Laboratory of Microfabrication, Institute of Physics CAS, and National Center for Nanoscience and Technology for the support of the sample fabrication. X.B.Z., Y.X.L., and D.N.Z. are supported by the National Basic Research Program (973) of China under Grant no. 2017YFA0304300. Y.X.L., D.N.Z., and C.P.S. are also supported under Grant No. 2014CB921400. X.B.Z. is supported by NSFC under Grant no. 11574380, the Chinese Academy of Science, Alibaba Cloud and Science and Technology Committee of Shanghai Municipality. K.N. acknowledges support from the Japanese MEXT GrantinAid for Scientific Research on Innovative Areas “Science of hybrid quantum systems” (Grant no. 2703). Y.X.L. and D.N.Z. are also supported by NSFC under Grant no. 91321208. We acknowledge Haohua Wang and Kai Xu at Zhejiang University for providing the technical support on the qubit–qubit device and its measurement setup in this experiment.
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Y.X.L. and X.B.Z. conceived the study. X.B.Z. and Y.W. designed the experiment. Y.W., X.B.Z., and Y.R.Z. designed the sample. H.D., X.B.Z., Z.G.Y., and Y.J.Z. fabricated the sample. Y.W., M.G., and Y.R.Z. carried out the measurements and data analysis with X.B.Z. and Y.X.L. providing supervision. Y.X.L. developed the theoretical mode. L.P.Y. performed theoretical simulations and wrote the supplementary material under the guide of Y.X.L. and C.P.S. D.N.Z. and L.L. advised on the experiment. W.J.M. and K.N. provided theoretical support. Y.X.L., Y.W., X.B.Z., A.D.C., and W.J.M. wrote the manuscript with contributions from all authors. X.B.Z. and Y.X.L. designed and supervised the project.
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Wu, Y., Yang, LP., Gong, M. et al. An efficient and compact switch for quantum circuits. npj Quantum Inf 4, 50 (2018). https://doi.org/10.1038/s4153401800996
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DOI: https://doi.org/10.1038/s4153401800996
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