  Simulating quantum circuits on classical hardware is a powerful and necessary
tool for developing and testing quantum algorithms and hardware as well as
evaluating claims of quantum supremacy in the Noisy Intermediate-Scale Quantum
(NISQ) regime. Schr\"odinger-style simulations are limited by the exponential
growth of the number of state amplitudes which need to be stored. In this work,
we apply scalar and vector quantization to Schr\"odinger-style quantum circuit
simulations as lossy compression schemes to reduce the number of bits needed to
simulate quantum circuits. Using quantization, we can maintain simulation
fidelities $>0.99$ when simulating the Quantum Fourier Transform, while using
only 7 significand bits in a floating-point number to characterize the real and
imaginary components of each amplitude. Furthermore, using vector quantization,
we propose a method to bound the number of bits/amplitude needed to store state
vectors in a simulation of a circuit that achieves a desired fidelity, and show
that for a 6 qubit simulation of the Quantum Fourier Transform, 15
bits/amplitude is sufficient to maintain fidelity $>0.9$ at $10^4$ depth.
