  We benchmark the performances of Qrack, an open-source software library for
the high-performance classical simulation of (gate-model) quantum computers.
Qrack simulates, in the Schr\"odinger picture, the exact quantum state of $n$
qubits evolving under the application of a circuit composed of elementary
quantum gates. Moreover, Qrack can also run approximate simulations in which a
tunable reduction of the quantum state fidelity is traded for a significant
reduction of the execution time and memory footprint. In this work, we give an
overview of both simulation methods (exact and approximate), highlighting the
main physics-based and software-based techniques. Moreover, we run
computationally heavy benchmarks on a single GPU, executing large quantum
Fourier transform circuits and large random circuits. Compared with other
classical simulators, we report competitive execution times for the exact
simulation of Fourier transform circuits with up to 27 qubits. We also
demonstrate the approximate simulation of all amplitudes of random circuits
acting on 54 qubits with 7 layers at average fidelity higher than $4\%$, a task
commonly considered hard without super-computing resources.
