  Noise within solid-state systems at low temperatures, where many of the
degrees of freedom of the host material are frozen out, can typically be traced
back to material defects that support low-energy excitations. These defects can
take a wide variety of microscopic forms, and for amorphous materials are
broadly described using generic models such as the tunneling two-level systems
(TLS) model. Although the details of TLS, and their impact on the
low-temperature behavior of materials have been studied since the 1970s, these
states have recently taken on further relevance in the field of quantum
computing, where the limits to the coherence of superconducting microwave
quantum circuits are dominated by TLS. Efforts to mitigate the impact of TLS
have thus far focused on circuit design, material selection, and material
surface treatment. In this work, we take a new approach that seeks to directly
modify the properties of TLS through nanoscale-engineering. This is achieved by
periodically structuring the host material, forming an acoustic bandgap that
suppresses all microwave-frequency phonons in a GHz-wide frequency band around
the operating frequency of a transmon qubit superconducting quantum circuit.
For embedded TLS that are strongly coupled to the electric qubit, we measure a
pronounced increase in relaxation time by two orders of magnitude when the TLS
transition frequency lies within the acoustic bandgap, with the longest $T_1$
time exceeding $5$ milliseconds. Our work paves the way for in-depth
investigation and coherent control of TLS, which is essential for deepening our
understanding of noise in amorphous materials and advancing solid-state quantum
devices.
