Great strides have recently been made in developing stable and coherent quantum systems. One such system is the superconducting gmon qubit architecture developed by Martinis et. al [1]. Martinis et. al presents a simple coherent system with highly tuneable qubit-qubit couplings, making it a great candidate for use in quantum computing. Of specific interest to the field of quantum computing is the controlled evolution of these qubit systems to process information. To achieve this type of evolution we worked within the gmon qubit architecture and found optimal solutions for evolving a three qubit system from an easily achieved product state to a more difficult to achieve maximally entangled state. These solutions were found using a simulated annealing algorithm that minimized the error between the maximally entangled state and the final state of the system with respect to the protocols of the coupling parameters that compose the Hamiltonian of the system. These protocols, confirmed using Pontryagin's minimum principle, turned out to be simple bang-bang protocols that greatly shortcut adiabatic evolution and offer a structure that can be used to help employ quantum gates or regulate the energy of a quantum computer.
Great strides have recently been made in developing stable and coherent quantum systems. One such system is the superconducting gmon qubit architecture developed by Martinis et. al [1]. Martinis et. al presents a simple coherent system with highly tuneable qubit-qubit couplings, making it a great candidate for use in quantum computing. Of specific interest to the field of quantum computing is the controlled evolution of these qubit systems to process information. To achieve this type of evolution we worked within the gmon qubit architecture and found optimal solutions for evolving a three qubit system from an easily achieved product state to a more difficult to achieve maximally entangled state. These solutions were found using a simulated annealing algorithm that minimized the error between the maximally entangled state and the final state of the system with respect to the protocols of the coupling parameters that compose the Hamiltonian of the system. These protocols, confirmed using Pontryagin's minimum principle, turned out to be simple bang-bang protocols that greatly shortcut adiabatic evolution and offer a structure that can be used to help employ quantum gates or regulate the energy of a quantum computer.
We apply the theory of optimal control to the dynamics of two ``gmon'' qubits, with the goal of preparing a desired entangled ground state from an initial unentangled one. Given an initial state, a target state, and a Hamiltonian with a set of permissible controls, can we reach the target state with coherent quantum evolution, and, in that case, what is the minimum time required? The adiabatic theorem provides a far from optimal solution in the presence of a spectral gap. Optimal control yields the fastest possible way of reaching the target state and helps identify unreachable states. In the context of a simple quantum system, we provide examples of both reachable and unreachable target ground states, and show that the unreachability is due to a symmetry. We find the optimal protocol in the reachable case using three different approaches: (i) a brute-force numerical minimization, (ii) an efficient numerical minimization using the bang-bang ansatz expected from the Pontryagin minimum principle, and (iii) direct solution of the Pontryagin boundary value problem, which yields an analytical understanding of the numerically obtained optimal protocols. Interestingly, our system provides an example of singular control, where the Pontryagin theorem does not guarantee bang-bang protocols. Nevertheless, all three approaches give the same bang-bang protocol.
We apply the theory of optimal control to the dynamics of two Gmon qubits, with the goal of preparing a desired entangled ground state from an initial unentangled one. Given an initial state, a target state, and a Hamiltonian with a set of permissible controls, can we reach the target state with coherent quantum evolution and, in that case, what is the minimum time required? The adiabatic theorem provides a far from optimal solution in the presence of a spectral gap. Optimal control yields the fastest possible way of reaching the target state and helps identify unreachable states. In the context of a simple quantum system, we provide examples of both reachable and unreachable target ground states and show that the unreachability is due to a symmetry. We find the optimal protocol in the reachable case using three different approaches: (i) a brute-force numerical minimization (ii) an efficient numerical minimization using the bang-bang ansatz expected from the Pontryagin minimum principle, and (iii) direct solution of the Pontryagin boundary value problem, which yields an analytical understanding of the numerically obtained optimal protocols. Interestingly, our system provides an example of singular control, where the Pontryagin theorem does not guarantee bang-bang protocols. Nevertheless, all three approaches give the same bang-bang protocol.
Few technological ideas in recent memory have stirred as much excitement—or skepticism—as quantum computing. The promise is genuine: for certain computational tasks, a quantum machine can do in seconds what would take a classical supercomputer longer than the age of the universe. At the heart of realising that promise is the qubit, the quantum analogue of a classical bit, and among the many physical systems that can host a qubit, superconducting circuits have quietly become the frontrunner. They can be fabricated in the same cleanrooms used to make conventional chips, controlled with ordinary microwave electronics, and scaled up in ways that other approaches cannot yet match. This thesis traces the full arc from first principles to working code. We start with the quantum mechanics a reader needs—superposition, entanglement, measurement—and build from therethrough the physics of superconductivity to the engineering of real qubit circuits. A key character in that story is the Josephson junction: a thin sandwich of metal and insulator whose quirky nonlinear behaviour is what makes the whole enterprise of superconducting qubits possible. A major focus of this work is the surface code, a topological quantum error-correcting code widely regarded as the most promising approach for achieving fault-tolerant quantum computation. We provide a rigorous analysis of the surface code’s lattice structure, stabilizer formalism, syndrome extraction, and decoding algorithms—in particular the minimum-weight perfect matching (MWPM) algorithm. A full Python/Qiskit implementation of a distance-3 surface code is presented, including qubit initialization, syndrome measurement circuits, error simulation, and decoding logic. We close by looking ahead. Modular processors, 3D circuit integration, and the long road to fault-tolerant machines are all on the horizon, and we try to give an honest assessment of where things stand and what the real obstacles are. Throughout, the goal has been to bridge theory and practice—to be precise enough to be useful, but grounded enough to stay connected to what is actually happening in laboratories today.
A key step toward demonstrating a quantum system that can address difficult problems in physics and chemistry will be performing a computation beyond the capabilities of any classical computer, thus achieving so-called quantum supremacy. In this study, we used nine superconducting qubits to demonstrate a promising path toward quantum supremacy. By individually tuning the qubit parameters, we were able to generate thousands of distinct Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert space. As the number of qubits increases, the system continues to explore the exponentially growing number of states. Extending these results to a system of 50 qubits has the potential to address scientific questions that are beyond the capabilities of any classical computer.
Abstract Using simulated annealing, we find optimal protocols that evolve a simple product state into a three-qubit W state with a Hamiltonian that describes XY coupling and single-qubit gates, and determine the associated quantum speed limit. Applying Pontryagin's minimum principle, we fully characterize the optimal “bang-bang” protocols. While leakage affects performance, the protocols remain robust to implementation errors and operate well within relaxation and decoherence times. Our findings highlight Pontryagin's principle as a powerful tool for designing pulse shapes that directly link device interactions to specific quantum gates and target states.