We show that it is theoretically possible to use higher energy levels for storing and controlling two qubits within a superconducting transmon. This is done by identifying energy levelsas product states between multiple effecitve qubits. As a proof of concept we realise a complete set of gates necessary for universal computing by numerically optimising control pulses for single qubit gates on each of the qubits, entangling gates between the two qubits in one transmon, and an entangling gate between two qubits from two coupled transmons. The optimisation considers parameters which could make it possible to validate this experimentally. With these control pulses it is in principle possible to double the number of available qubits without any overhead in hardware. The additional qubits could be used in algorithms which need many short-living qubits such as syndrom qubits in error correction or by embedding effecitve higher connectivity in qubit networks.

Efforts to scale-up quantum computation have reached a point where the principal limiting factor is not the number of qubits, but the entangling gate infidelity. However, a highly detailedsystem characterization required to understand the underlying errors is an arduous process and impractical with increasing chip size. Open-loop optimal control techniques allow for the improvement of gates but are limited by the models they are based on. To rectify the situation, we provide a new integrated open-source tool-set for Control, Calibration and Characterization (C3), capable of open-loop pulse optimization, model-free calibration, model fitting and refinement. We present a methodology to combine these tools to find a quantitatively accurate system model, high-fidelity gates and an approximate error budget, all based on a high-performance, feature-rich simulator. We illustrate our methods using fixed-frequency superconducting qubits for which we learn model parameters to an accuracy of <1% and derive a coherence limited cross-resonance (CR) gate that achieves 99.6% fidelity without need for calibration. [/expand]

Reaching high speed, high fidelity qubit operations requires precise control over the shape of the underlying pulses. For weakly anharmonic systems, such as superconducting transmonqubits, short gates lead to leakage to states outside of the computational subspace. Control pulses designed with open-loop optimal control may reduce such leakage. However, model inaccuracies can severely limit the usability of such pulses. We implemented a closed-loop optimization that simultaneously adapts all control parameters based on measurements of a cost function built from Clifford gates. By parameterizing pulses with a piecewise-constant representation that matches the capabilities of the control hardware we create a 4.16 ns single-qubit pulse with 99.76% fidelity and 0.044% leakage. This is a seven-fold reduction of the leakage rate of the best DRAG pulse we have calibrated at such short durations on the same system.

The cross-resonant gate is an entangling gate for fixed frequency superconducting qubits introduced for untunable qubits. While being simple and extensible, it suffers from long durationand limited fidelity. Using two different optimal control algorithms, we probe the quantum speed limit for a CNOT gate in this system. We show that the ability to approach this limit depends strongly on the ansatz used to describe the optimal control pulse. A piecewise constant ansatz with a single carrier leads to an experimentally feasible pulse shape, shorter than the one currently used in experiments, but that remains relatively far from the speed limit. On the other hand, an ansatz based on the two dominant frequencies involved in the optimal control problem allows to generate an optimal solution more than twice as fast, in under 30ns. This comes close to the theoretical quantum speed limit, which we estimate at 15ns for typical circuit-QED parameters, which is more than an order of magnitude faster than current experimental microwave-driven realizations, and more than twice as fast as tunable direct-coupling experimental realizations.