  With the increasing size of quantum processors, sub-modules that constitute
the processor hardware will become too large to accurately simulate on a
classical computer. Therefore, one would soon have to fabricate and test each
new design primitive and parameter choice in time-consuming coordination
between design, fabrication, and experimental validation. Here we show how one
can design and test the performance of next-generation quantum hardware -- by
using existing quantum computers. Focusing on superconducting transmon
processors as a prominent hardware platform, we compute the static and dynamic
properties of individual and coupled transmons. We show how the energy spectra
of transmons can be obtained by variational hybrid quantum-classical algorithms
that are well-suited for near-term noisy quantum computers. In addition,
single- and two-qubit gate simulations are demonstrated via Suzuki-Trotter
decomposition. Our methods pave a promising way towards designing candidate
quantum processors when the demands of calculating sub-module properties exceed
the capabilities of classical computing resources.
