Current quantum computing architectures lack the size and fidelity required for universal fault-tolerant operation, limiting the practical implementation of key quantum algorithms to all but the smallest problem sizes. In this work we propose an alternative method for general-purpose quantum computation that is ideally suited for such ``prethreshold'' superconducting hardware. Computations are performed in the
n-dimensional single-excitation subspace (SES) of a system of
n tunably coupled superconducting qubits. The approach is not scalable, but allows many operations in the unitary group SU(
n) to be implemented by a single application of the Hamiltonian, bypassing the need to decompose a desired unitary into elementary gates. This feature makes large, nontrivial quantum computations possible within the available coherence time. We show how to use a programmable SES chip to perform fast amplitude amplification and phase estimation, two versatile quantum subalgorithms. We also show that an SES processor is well suited for Hamiltonian simulation, specifically simulation of the Schr\"odinger equation with a real but otherwise arbitrary $n\ifmmode\times\else\texttimes\fi{}n$ Hamiltonian matrix. We discuss the utility and practicality of such a universal quantum simulator, and propose its application to the study of realistic atomic and molecular collisions.