The Hamiltonian of a quantum system with control fields takes the following form: Now we introduce our framework that unifies the above two scenarios, optimizing a system’s performance over the control parameters and the device parameters jointly as illustrated in Fig. In contrast, we may sometimes need to optimize the spectral or other properties of the device Hamiltonian itself, e.g., for determining tunable Z Z-interaction on–off ratios 24, 25 or for designing 4-local interaction couplers 1. When the device parameters are determined and we only need to optimize over the control parameters for better performance or robustness, this is precisely what quantum optimal control and robust control do. In this work, we mainly focus on the two stages of quantum processor design that involve finding optimal values of the device parameters in the system Hamiltonian and deriving optimal control pulses to maximize the performance of the target quantum processor. This makes our method feasible not only for optimizing complex pulse shapes but also for optimizing large processor device designs, and more interestingly, jointly optimizing the design and control. More accurately, the ratio of the time needed to compute the figure of merit and its gradient is independent of the number of control parameters. The gradients can be computed in a single computation similar to the GRAPE algorithm and the back-propagation algorithm 23. In this work, we showed that the figure of merit of a target goal such as the gate fidelity can be made differentiable with respect to the parameters from the device and the control together. One milestone in this direction is the GRadient Ascent Pulse Engineering (GRAPE) algorithm 5. A common strategy to handle large optimization problems is using efficiently computed gradients to help us traverse the optimization landscape. In scenarios where we cannot estimate the control performances with analytical formulas, the optimization problem of the qubit design inevitably becomes a joint optimization problem of design and control with an even larger parameter space. For example, complex control schemes are needed for bosonic code qubits, and often numerical optimization is required to obtain the best gate performances 20, 21, 22. As mentioned earlier, the optimization of the device design is further complicated by the fact that we need to consider the control schemes simultaneously, which has been noted in previous work 18, 19. While the design versatility allows for many qubit and coupling types 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, it also makes the quest of finding the best design harder due to the larger parameter space. We consider superconducting circuits, which are one of the most promising hardware platforms due to their device design versatility and fabrication scalability. In this work, we treat them as equal components in a single optimization problem and use gradient information to speed up optimization. Traditionally, the optimization of the design 1, 2, 3, 4 and the optimization of the control 5, 6, 7 are usually studied separately. At this level of abstraction, it is clear that we are dealing with optimization problems that consist of both Hamiltonian design and control. Concrete examples of such quantum information processing tasks include quantum computing, quantum simulation, quantum error correction, and quantum metrology. To achieve desired operations, we need to design systems with suitable Hamiltonians and dissipation as well as time-dependent external controls. Information is stored in quantum systems and the dynamics are described by the Schrödinger equation or the master equation. Quantum mechanics provides an entirely new way to think about information processing. We also demonstrate the viability of gradient-based joint optimization over the device and control parameters through a few examples based on the superconducting qubits. Therefore, our work extends the scope of the quantum optimal control to device design and provides an efficient optimization method. In addition, we can compute the gradient of the design objective efficiently in a similar manner to the back-propagation algorithm and then utilize the gradient to optimize the device and the control parameters jointly and efficiently. In this work, we demonstrate that the figure of merit reflecting a design goal can be made differentiable with respect to the device and control parameters. Thus, optimization becomes more and more challenging. As we continuously seek better alternative qubit platforms, we explore the increasingly large device and control design space. In a quantum processor, the device design and external controls together contribute to the quality of the target quantum operations.
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