Introduction

Superconducting quantum circuits based on Josephson junctions are macroscopic circuits, but they can behave quantum mechanically, like artificial atoms, allowing the observation of quantum entanglement and quantum coherence on a macroscopic scale1,2,3,4,5,6,7. These artificial atoms can be used to test the laws of quantum mechanics on macroscopic systems and also offer a promising way to implement quantum information technology. Superconducting qubits have been widely investigated theoretically and implemented experimentally in the last fifteen years due to their advantages, such as scalability, design flexibility and tunability, for solid-state quantum computation and quantum simulations1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18.

In practical applications, the existence of noise (including extrinsic and intrinsic), inaccuracies (e.g., inaccurate operation in the coupling between qubits) and fluctuations (e.g., fluctuations in magnetic fields and electric fields) in superconducting quantum circuits is unavoidable19,20,21,22,23,24,25. For simplicity, in this paper we use fluctuations to represent noise, inaccuracies and fluctuations. These fluctuations degrade the performance of robustness and reliability in superconducting quantum circuits. Hence, it is highly desirable, for the development of practical quantum technology, to design control fields that are robust against fluctuations26,27,28,29,30,31,32.

Robustness has been recognized as one of the key properties for a reliable quantum information processor. Several methods have been developed for enhancing the robustness of quantum systems33,34. One important paradigm is feedback control, where the signal obtained from the system is fed back to adjust input control fields aiming at achieving improved robustness as well as other measures of system performance (e.g., stability)35. A typical example of the feedback paradigm is quantum error correction, where possible errors are corrected based on detection outcomes36. Usually, feedback control is difficult to implement in practical quantum systems due to the fast time scale of quantum systems and measurement backaction in the quantum domain. A more feasible paradigm is open-loop control for improving robustness of quantum systems where no feedback signal is required. Dynamical decoupling37,38 and optimal control methods32,39,40,41,42,43 can be used to design robust control fields for manipulating quantum states or quantum gates. In this paper, we develop a “smart” open-loop control method (i.e., sampling-based learning control) to guide the design of robust control fields for superconducting quantum systems. The sampling-based learning control (SLC) method includes two steps of “training” and “testing”44. In the training step, we obtain some artificial samples by sampling the fluctuation parameters and construct an augmented system using these samples. Then we employ a gradient-flow-based learning algorithm to learn optimal fields for the augmented system. The robustness of the control fields is tested and evaluated using additional samples generated by sampling fluctuation parameters in the testing step. Here we apply the SLC method to several examples of superconducting qubits, including single-charge qubits, two coupled charge qubits and two coupled phase qubits with fluctuations. Our results show that the SLC method can efficiently learn “smart” fields that are insensitive to even 40% ~ 50% fluctuations. The superconducting quantum circuits with the “smart” fields can run more reliably.

Results

Single-charge qubits with fluctuations

In superconducting quantum circuits, the Josephson coupling energy EJ and the charging energy EC are two significant quantities. Their ratio determines whether the phase or the charge dominates the behaviour of the qubit, which can form flux qubits or charge qubits1. The simplest charge qubit is based on a small superconducting island (called a Cooper-pair box) coupled to the outside world through a weak Josephson junction and driven by a voltage source through a gate capacitance within the charge regime (i.e., )1. The Hamiltonian of the Cooper-pair box can be described as1

where the phase drop φ across the Josephson junction is conjugate to the number n of extra Cooper pairs in the box, ng = CgVg/2e is controlled by the gate voltage Vg (Cg is the gate capacitance and 2e is the charge of each Cooper pair). In most experiments, the Josephson junction in the charge qubit is replaced by a dc superconducting quantum interference device (SQUID) to make it easier to control the qubit. In a voltage range near a degeneracy point, the system can be approximated as a qubit with the following Hamiltonian

where f(Vg) is related to the charging energy EC and this term can be adjusted through external parameters, such as the voltage Vg and g(Φ) corresponds to a controllable term including different control parameters, such as the flux Φ in the SQUID.

For superconducting qubits in laboratories, the existence of fluctuations is unavoidable (e.g., fluctuations in the Josephson coupling energy and the charging energy, or inaccuracies in the magnetic flux). We assume that possible fluctuations exist in both f(Vg) and g(Φ). Suppose that the factors f(Vg) and g(Φ) can be controlled by adjusting external parameters. Since EJ could be around ten GHz and EC could be around one hundred GHz (e.g., the experiment in8 used EJ1 = 13.4 GHz, EJ2 = 9.1 GHz, EC1 = 117 GHz and EC2 = 152 GHz), we assume f(Vg)/ħ∈[0, 40] GHz and g(Φ)/ħ∈[0, 9.1] GHz. We could have used 10, instead of 9.1, but we chose 9.1 simply because it was the number used in one experiment. The practical control terms are and (where the fluctuation parameters θz∈[1 − Θz, 1 + Θz] and θx∈[1 − Θx, 1 + Θx]) due to possible multiplicative fluctuations. Here the bounds of fluctuations Θz and Θx characterize the maximum ranges of fluctuations in θz and θx, respectively. The fluctuations can originate from the fluctuations in the magnetic flux Φ, the voltage Vg, the Josephson coupling energy EJ and the charging energy EC.

As an example, we assume that the initial state is |ψ0〉 = |g〉 and the target state is either |ψtarget〉 = |e〉 or . Let the operation time be T = 1 ns. Now we employ the sampling-based learning control method (see the methods Section) to learn an optimal control field for manipulating the charge qubit system from the initial state to a target state. The time interval t ∈ [0, 1] ns is equally divided into 100 smaller time intervals. Without loss of generality, we assume θx and θz to have uniform distributions and have the same bound of fluctuations (i.e., Θx = Θz). An augmented system is constructed by selecting Nx = 5 values for θx and Nz = 5 values for θz. The initial control fields are f(Vg)/ħ = sin t+ cos t+ 20 GHz and g(Φ)/ħ = sin t+ cos t+ 5 GHz. The learning algorithm runs for about 7000 iterations for |ψtarget〉 = |e〉 (4000 iterations for ) before it converges to optimal control fields. After the optimal control fields are learned for the augmented system, they are applied to 5000 samples generated by stochastically selecting the values of the fluctuation parameters for evaluating the performance. The fidelity between the final state |ψ(T)〉 and the target state |ψtarget〉 is defined as F(|ψ(T)〉, |ψtarget〉) = |〈ψ(T)|ψtarget〉|50. The relationship between the average fidelity and the bounds of the fluctuations is shown in Fig. 1. Although the performance decreases when increasing the bounds on the fluctuations, the “smart” fields can still drive the system from the initial state |ψ0〉 = |g〉 to a given target state with high fidelity (the average fidelity is for |ψtarget〉 = |e〉 and for ) even though the bound on the fluctuations is 25% (i.e., 50% fluctuations relative to the nominal value).

Figure 1
figure 1

Average fidelity versus the bounds of fluctuations Θz and Θx for charge qubits.

The fluctuation parameters θz and θx have uniform distributions in [1 − Θ, 1 + Θ] (i.e., we assume Θz = Θx = Θ). Here we consider |ψ0〉 = |g〉 and |ψtarget〉 = |e〉 (Case 1) or (Case 2). Every average fidelity is calculated using 5000 tested samples.

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We also test the relationship between the number of values Nf for θx and θz (Nx = Nz = Nf) and the average fidelity for bounds on the fluctuations Θz = Θx = 15%. The performance is shown in Fig. 2. It is clear that the performance is excellent for Nf = 5 or 7. Although it is possible to improve the performance through using more samples, too many samples will cost more computation resources and spend too much time for learning a set of optimal fields. For example, the laptop Thinkpad T440p, with a CPU of 2.5 GHz, needs about 13 minutes to find the optimal fields for Nf = 5; while this laptop requires about 42 minutes for Nf = 11. When increasing the number of fluctuation parameters, the time consuming quickly increases with the increasing of Nf. Hence, we choose Nf = 5 for each fluctuation parameter in all of the following numerical results.

Figure 2
figure 2

Average fidelity versus the number Nf of values for θx and θz (Nf = Nx = Nz).

Here, Θz = Θx = 15% (i.e., 30% fluctuations), |ψ0〉 = |g〉 and . Every average fidelity is calculated using 5000 samples.

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Two coupled qubits with fluctuations

We first consider the coupled qubit circuit in51 where a symmetric dc SQUID with two sufficiently large junctions is used to couple two charge qubits (see Fig. 3). Each qubit is realized by a Cooper-pair box with Josephson coupling energy EJj and capacitance CJj (j = 1, 2). Each Cooper-pair box is biased by an applied voltage Vj through the gate capacitance Cj (j = 1, 2). We apply a flux Φs inside the large-junction dc SQUID loop with two junctions of large EJ0. The Hamiltonian of the coupled charge qubits can be described as

Figure 3
figure 3

A coupled-qubit circuit with a biased-current source of impedance Z(ω)51.

Two charge qubits are coupled by the dc SQUID with two junctions with large EJ0.

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Due to possible fluctuations, we assume that the Hamiltonian for practical systems is

where the fluctuation parameters θj ∈ [1 − Θj, 1 + Θj] (j = 1, 2, 3, 4, 5).

We let g(Φ1)/ħ = g(Φ2)/ħ = 9.1 GHz, the control terms f(V1)/ħ∈[0, 40] GHz, f(V2)/ħ∈[0, 40] GHz, |χ(t)/ħ|≤ 0.5 GHz and θ5(t)≡ 1. The operation time T = 2 ns. We assume that the fluctuation parameters θj(j = 1, 2, 3, 4) may be time varying. Hence, θ3 and θ4 may correspond to time-varying additive fluctuations. The fluctuations in θ1 and θ2 may originate from the time-varying errors in the driving fields. As an illustrative example, we let θj = 1 − ϑj cos t, where each ϑj has a uniform distribution in the interval [− Θj, Θj]. For simplification, we assume θ1 = θ2, θ3 = θ4 and Θ1 = Θ2 = Θ3 = Θ4 = Θ. We now consider a controlled-phase-shift gate operation on an initial state |ψ0〉 = α1|g, g〉+α2|g, e〉+α3|e, g〉+α4|e, e〉; i.e., the target state is |ψtarget〉 = α1|g, g〉+α2|g, e〉+α3|e, g〉−α4|e, e〉. In particular, we let α1 = 0.7, α2 = 0.1, α3 = 0.7i and α4 = 0.1i. The time interval t∈[0, 2] ns is equally divided into 200 smaller time intervals. The control fields are initialized as: f(V1)/ħ = f(V2)/ħ = sin t+ cos t+ 5 GHz, χ(t)/ħ = 0.25 sin t GHz. The learning algorithm runs for about 5000 iterations before the optimal control fields are found. Then the learned fields are applied to 5000 samples that are generated by selecting the values of the fluctuation parameters according to a uniform distribution. The performance is shown in Fig. 4. Although the performance decreases when increasing the bounds on the fluctuations, the “smart” fields can still drive the system from the initial state |ψ0〉 = 0.7|g, g〉+ 0.1|g, e〉+ 0.7i|e, g〉+ 0.1i|e, e〉 to the target state |ψtarget〉 = 0.7|g, g〉+ 0.1|g, e〉+ 0.7i|e, g〉− 0.1i|e, e〉 with high fidelity (average fidelity 0.9941) even with 40% fluctuations.

Figure 4
figure 4

Average fidelity versus the bound on the fluctuations Θ for two coupled charge qubits with a biased-current source.

The fluctuation parameters θj = 1 − ϑj cos t (j = 1, 2, 3, 4), where each ϑj has a uniform distribution in [− Θ, Θ]. Here we assume θ1 = θ2 and θ3 = θ4. The initial state |ψ0〉 = 0.7|g, g〉 + 0.1|g, e〉 + 0.7i|e, g〉 + 0.1i|e, e〉 and the target state |ψtarget〉 = 0.7|g, g〉 + 0.1|g, e〉 + 0.7i|e, g〉 − 0.1i|e, e〉. Each average fidelity is calculated using 5000 samples.

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In the two numerical examples of single-charge qubits and two coupled charge qubits, we use some ideal parameter values to show the effectiveness and excellent performance of the proposed method. It is straightforward to extend our method to other systems. Indeed, our proposed method is very flexible in the selection of the operation time T and the target state and is also robust against fluctuations with different distributions. Here, we consider another example based on the two coupled phase qubits in Ref. 45. Each phase qubit is a nonlinear resonator built from an Al/AlOx/Al Josephson junction and two qubits are coupled via a modular four-terminal device (for detail, see Fig. 1 in45). This four-terminal device is constructed using two nontunable inductors, a fixed mutual inductance and a tunable inductance. The equivalent Hamiltonian can be described as

where N1 and N2 are the number of levels in the potentials of qubits 1 and 246. The typical values for N1 and N2 are N1 = N2 = 5. Due to possible fluctuations, we assume that the practical Hamiltonian has the following form

with θj ∈ [1 − Θ, 1 + Θ] (j = 1, 2, 3).

We assume that the frequencies ω1(t), ω2(t) ∈ [0, 5] GHz can be adjusted by changing the bias currents of two phase qubits and Ωc(t) ∈ [−100, 100] MHz can be adjusted by changing the bias current in the coupler. Let ω3 = ω4 = 2 GHz, the operation time T = 50 ns and each fluctuation parameter θj (j = 1, 2, 3) in (6) has a truncated Gaussian distribution in [1 − Θ, 1 + Θ]. Assume that the probability density function of the truncated Gaussian distribution is , where μ = 0, σ = Θ/3, l = − Θ, r = Θ, is the probability density function of the standard normal distribution and Φ(x) is its cumulative distribution function.

We now consider a CNOT operation. In particular, we let the initial state be and the target state be a maximum entangled state . In the training step, the fluctuations are uniformly sampled. However, in the testing step the samples are selected by sampling the fluctuation parameters with a truncated Gaussian distribution. For simplicity, we let θ1 = θ2. The initial vaules are ω1 = ω2 = sin t + cos t + 0.5 GHz and Ωc(t) = 50 + 50 sin t MHz. Other parameter settings are the same as those in the example of coupled charge qubits. The performance is shown in Fig. 5 and a set of “smart” fields is shown in Fig. 6 for Θ = 25% (i.e., 50% fluctuations). The “smart” fields can drive the system from |ψ0〉 to |ψtarget〉 with high fidelity (average fidelity 0.9970) even with 50% fluctuations.

Figure 5
figure 5

Average fidelity versus the bound on the fluctuations Θ for two coupled phase qubits.

Each θj (j = 1, 2, 3) has a truncated Gaussian distribution in [1 − Θ, 1 + Θ] and we assume θ1 = θ2. The initial state is and the target state is . Each average fidelity is calculated using 5000 samples.

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Figure 6
figure 6

A set of “smart” fields corresponding to ω1(t), ω2(t) and Ωc(t) for the problem of two coupled phase qubits when the bound on the fluctuations is very large, with Θ = 25% (i.e., 50% fluctuations).

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Discussion

In numerical examples, a small number of samples for each possible fluctuation parameter is used to construct an augmented system. It is possible to achieve improved performance by using more samples. However, an increase in the number of samples in the training step will consume more computation resources. The tradeoff between resource consumption and performance that can be achieved should be investigated for specific tasks. In the SLC method, we employ a general gradient-flow-based algorithm to learn “smart” fields and the algorithm is usually much more efficient than other stochastic search algorithms (e.g., genetic algorithms) for control design of quantum systems39. The “smart” fields are “optimal” to the control landscapes of different samples since they are found by simultaneously optimizing the fields for these samples. It may be possible to use a similar theory to the quantum control landscape theory developed in39 to analyze these optimal properties. As examples, we only consider that each possible fluctuation parameter has several specific distributions in the testing step. However, the proposed method also works well for other time-varying or time-invariant distributions. Numerical results show that, in the training step, sampling fluctuation parameters according to simple uniform distributions can achieve excellent performance for these cases where the fluctuation parameters have other distributions. In our numerical examples, we only consider three classes of superconducting quantum systems with several specific parameter settings. Our method is also applicable to other superconducting qubits, such as the “Xmon” and “gmon” qubits47,48,49, since its performance is insensitive to the parameter settings and possible fluctuations.

In conclusion, we use a sampling-based learning control method to design robust fields that are insensitive to possible fluctuations. Numerical results show that the method can efficiently find “smart” fields for superconducting qubits even in the presence of 40% ~ 50% fluctuations in different parameters. The proposed method has potential for robust quantum information processing and can contribute to the design of more reliable quantum devices.

Methods

Sampling-based learning control (SLC)

The SLC method was first proposed for the control design of inhomogeneous quantum ensembles44. In this method, several artificial samples, generated through sampling possible inhomogeneous parameters, are used to learn optimal control fields and then these fields are applied to additional samples to test their performance. In this paper, we develop an SLC method for guiding the design of robust control fields for superconducting quantum systems with fluctuations.

Consider a quantum system with Hamiltonian H(u, θ, t) and the evolution of its state |ψ(t)〉 is described by the following Schodinger equation:

where u represents the control field and θ characterizes possible fluctuations. In the SLC method, we first generate N artificial samples by selecting different values of θ (e.g., the N samples correspond to . Using these samples, an augmented system is constructed as follows

The performance function J(u) for the augmented system is defined as

where |ψtarget〉 is the target state and is the final state for one sample (corresponding to θ(n)) at the time T. The task in the training step is to find an optimal control field u* that maximizes the performance function defined in Eq. (9).

In the testing step, we apply the optimized field u* to additional samples generated by randomly sampling the fluctuation parameters and evaluate the performance in terms of the fidelity. If the average fidelity for the tested samples are good enough, we accept the designed field and complete the design process. Otherwise, we should go back to the training step and learn another optimized control field (e.g., restarting the training step with a new initial field or a new set of samples).

Sampling

In order to construct an augmented system, we need to generate N artificial samples. We assume that there are two fluctuation parameters θx and θz. We may choose some equally-spaced samples in the θx – θz space. For example, the intervals [1 − Θx, 1 + Θx] and [1 − Θz, 1 + Θz] are divided into Nx + 1 and Nz + 1 subintervals, respectively, where Nx and Nz are usually positive odd numbers. Then, the number of samples N = NxNz, where and can be chosen from the combination of (, ) as follows

Gradient-flow-based learning algorithm

In order to find an optimal control field u* for the augmented system (8), a good choice is to follow the direction of the gradient of J(u) as an ascent direction. Assume that the performance function is J(u0) with an initial field u0. We can apply the gradient flow method to approximate an optimal control field u*. This can be achieved by iterative learning using the following updating (for details, see, e.g.,44)

where ηk is the updating stepsize for the kth iteration and ∇J(u) denotes the gradient of J(u) with respect to the control u. The calculation of ∇J(u) is described in44,52. For practical implementations, we usually divide the time interval [0, T] equally into a number of smaller time intervals Δt and assume that the control fields are constant within each time interval Δt. In the algorithm, we assume u(t) ∈ [V−, V+]. If uk+1 ≤ V−, we let uk+1 = V−. If uk+1 ≥ V+, we let uk+1 = V+. In numerical computations, if the change of the performance function for 100 consecutive training steps is less than a small threshold (i.e., for some k), then the algorithm converges and we end the training step. In this paper, we let for all numerical results.