Methods, systems, and apparatus for training quantum evolutions using sub-logical controls. In one aspect, a method includes the actions of accessing quantum hardware, wherein the quantum hardware includes a quantum system comprising one or more multi-level quantum subsystems; one or more control devices that operate on the one or more multi-level quantum subsystems according to one or more respective control parameters that relate to a parameter of a physical environment in which the multi-level quantum subsystems are located; initializing the quantum system in an initial quantum state, wherein an initial set of control parameters form a parameterization that defines the initial quantum state; obtaining one or more quantum system observables and one or more target quantum states; and iteratively training until an occurrence of a completion event.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method comprising: accessing quantum hardware, wherein the quantum hardware comprises: a quantum system comprising one or more multi-level quantum subsystems; one or more control devices that operate on the one or more multi-level quantum subsystems according to one or more respective control parameters that relate to a parameter of a physical environment in which the multi-level quantum subsystems are located; initializing the quantum system in an initial quantum state, wherein an initial set of control parameters form a parameterization that defines the initial quantum state; obtaining one or more quantum system observables and one or more target quantum states; and iteratively training until an occurrence of a completion event. 2. The method of claim 1, wherein iteratively training comprises iteratively training changes from the initial quantum state to achieve the one or more target quantum states. 3. The method of claim 1, wherein iteratively training comprises iteratively training, until the occurrence of a completion event, evolutions of the initial quantum state and subsequent quantum states to realize the one or more target quantum states. 4. The method of claim 3, wherein the evolutions are analog evolutions. 5. The method of claim 1, wherein iteratively training comprises, for each iteration: determining a value of a cost function that is based on a current quantum state for the iteration and the one or more of the quantum system observables; minimizing the value of the cost function to determine updated values of the control parameters that define the current quantum state; and determining whether the completion event has occurred. 6. The method of claim 5, wherein minimizing the value of the cost function to determine updated values of the control parameters comprises adjusting the control parameters. 7. The method of claim 5, further comprising, in response to determining that the completion event has occurred, providing the one or more target quantum states for experimental probing. 8. The method of claim 7, wherein (i) at least one of the quantum system observables comprises a Hamiltonian of the quantum system, (ii) the one or more target quantum states comprise one or more eigenstates of the Hamiltonian, and (iii) experimental probing comprises measuring the energy of one or more of the eigenstates to determine corresponding energy eigenvalues of the eigenstates. 9. The method of claim 7, wherein (i) the system observable is a molecular electronic structure Hamiltonian, (ii) the one or more target quantum states comprise a ground state of the molecular electronic structure Hamiltonian, and (iii) experimental probing comprises measuring the target quantum state to determine the ground state energy. 10. The method of claim 5, wherein the value of the cost function that is based on the quantum state and one or more of the system observables is an expectation value of the quantum state and one or more of the system observables. 11. The method of claim 10, wherein determining an expectation value of the quantum state and one or more of the system observables comprises: repeatedly initializing the quantum system in an initial quantum state; for each initialized quantum state, measuring the one or more system observables to determine a set of measurement results; based on the set of measurement results, determining an expectation value of the quantum state and one or more of the system observables. 12. The method of claim 11, wherein each initialized quantum state is different from each other initialized quantum state. 13. The method of claim 11, wherein determining an expectation value of the quantum state and one or more of the system observables comprises determining an expectation value of a density operator and the one or more system observables. 14. The method of claim 5, wherein minimizing the value of the cost function to determine updated values of the control parameters comprises performing a gradient-free greedy minimization method. 15. The method of claim 5, wherein the completion event is that the determined value of the cost function that is based on the quantum state and one or more of the system observables converges. 16. The method of claim 1, wherein obtaining one or more target quantum states comprises encoding a solution to an optimization problem into the ground state of the quantum system. 17. The method of claim 16, further comprising obtaining a solution to the optimization problem from the experimental probing. 18. The method of claim 1, wherein the quantum hardware comprises a quantum circuit. 19. The method of claim 18, wherein the control devices include one or more quantum gates that operate on the quantum system through one or more respective control parameters. 20. The method of claim 19, further comprising calibrating one or more of the quantum gates, comprising, for each quantum gate that is to be calibrated: defining a correct action of the quantum gate on the quantum system; performing a measurement to determine the action of the quantum gate on the quantum system; and in response to determining that the action of the quantum gate on the quantum system is not correct, adjusting the corresponding control parameters for the quantum gate. 21. The method of claim 20, wherein iteratively training comprises combining iterative training with calibrating one or more of the quantum gates. 22. The method of claim 1, wherein (i) the initial quantum state encodes training data, (ii) the one or more system observables serve as a prediction function, and (iii) iteratively training comprises solving a machine learning problem. 23. The method of claim 1, wherein the initial quantum state is a state of a resonator that is coupled to a superconducting qubit. 24. An apparatus comprising: quantum hardware comprising: a quantum system comprising one or more multi-level quantum subsystems; one or more control devices that operate on the one or more multi-level quantum systems according to one or more respective control parameters that each relate to a parameter of a physical environment in which the multi-level quantum subsystems are located; one or more classical processors configured to: obtain one or more quantum system observables and one or more target quantum states; initialize the quantum system in an initial quantum state, wherein an initial set of control parameters form a parameterization that defines the initial quantum state; and iteratively train, until the occurrence of a completion event. 25. The apparatus of claim 24, wherein iteratively training comprises iteratively training changes from the initial quantum state to achieve the one or more target quantum states. 26. The apparatus of claim 24, wherein iteratively training comprises iteratively training, until the occurrence of a completion event, evolutions of the initial quantum state and subsequent quantum states to realize the one or more target quantum states. 27. The apparatus of claim 26, wherein the evolutions are analog evolutions. 28. The apparatus of claim 24, wherein the multi-level quantum subsystems are qubits. 29. The apparatus of claim 24, wherein the qubits are superconducting qubits. 30. The apparatus of claim 29, wherein the control devices that operate on the superconducting qubits include at least one digital to analog converter with respective voltage control parameters. 31. The apparatus of claim 24, wherein the quantum hardware comprises a quantum circuit. 32. The apparatus of claim 31, wherein the quantum circuit comprises one or more quantum logic gates.