We are currently developing a superconducting circuit realization of our protocol. For further details, see our research page on Tunable environments for superconducting circuits. In the future theoretical work, we will study the protocol when the coupling to the environment is strong and the approximations of the Markovian master equations do not hold. Also, we will use optimal control to improve our protocol.

## Numerically exact methods for open quantum systems

Ideally, the quantum devices of the future consist of quantum systems that can be controlled but are otherwise isolated from the rest of the world. In practice, there always exists decoherence caused by a coupling between the system and its surroundings which is typically referred to as the environment. The environment is often modelled with a bosonic bath which consists of harmonic oscillators at thermal equilibrium and is coupled bilinearly to the system. This coupling perturbs the coherent time evolution of the system and, thus, cause errors in the fidelity of the device.

Such errors are conventionally estimated theoretically with Markovian master equations for the reduced density operator of the system. These equations typically rely on a set of approximations, i.e. Born, Markov and secular approximations, which all become inaccurate at low temperatures and when the coupling to the environment is strong. However, the time evolution can also be solved exactly in the path-integral formalism which can be transformed into the form of the stochastic Liouville—von Neumann equation. In addition to accurate description of the decoherence processes, this allows the estimation of errors caused by the above-mentioned approximations. Such estimations will most likely become important in the future information processing devices that operate at the quantum level and require high precision.

We are currently developing a numerical solver for the stochastic Liouville–von Neumann equation. This allows us to study the effects typically neglected or unattainable with master equations, such as the Lamb shift and entanglement with the bath, as a function of the bath coupling strength. The magnitudes of these corrections are especially interesting in the contexts of the single-qubit gate operations, decay rate, and fidelity. We will also study the influence of the bath coupling strength on a two-qubit system, e.g. on the entanglement between two qubits.

## Energy-efficient qubit control

Precise single-qubit gates are an essential part of a quantum computer. We have derived [npj Quant. Inf. 3, 17 (2017)] from the quantum-mechanical photon–qubit interactions, for the first time, the greatest lower bound for the error of a single-qubit gate implemented with a pulse of given energy. This result implies that even with a flawless physical implementation, a large-scale quantum processor necessarily faces serious challenges in the heat management. We present a practical scheme how the control pulse can be reused in the quantum processor, leading to orders of magnitude lower heat dissipation.

The above-mentioned derivation also allowed us to solve the quantum state of the controlling mode that minimizes the error of a given single-qubit gate. Specifically, we have shown that the optimal way to perform a rotation on a superconducting qubit is to use so-called squeezed coherent states or squeezed cat states, depending on the desired rotation angle.