Solid-state qubits have recently advanced to the level that enables them, in principle, to be scaled up into fault-tolerant quantum computers. As these physical qubits continue to advance, meeting the challenge of realizing a quantum machine will also require the development of new supporting devices and control architectures with complexity far beyond the systems used in today’s few-qubit experiments. Here, we report a microarchitecture for controlling and reading out qubits during the execution of a quantum algorithm such as an error-correcting code. We demonstrate the basic principles of this architecture using a cryogenic switch matrix implemented via high-electron-mobility transistors and a new kind of semiconductor device based on gate-switchable capacitance. The switch matrix is used to route microwave waveforms to qubits under the control of a field-programmable gate array, also operating at cryogenic temperatures. Taken together, these results suggest a viable approach for controlling large-scale quantum systems using semiconductor technology.