Parham, Natalie2022-09-012022-09-012022-09-012022-08-20http://hdl.handle.net/10012/18702Constant-depth quantum circuits, or shallow quantum circuits, have been shown to exhibit behavior that is uniquely quantum. This thesis explores the power and limitations of constant-depth quantum circuits, in particular as they compare to constant-depth classical circuits. We start with a gentle introduction to shallow quantum and classical circuit complexity, and we review the hardness of sampling from the output distribution of a constant-depth quantum circuit. We then give an overview of the shallow circuit advantage from the 1D Magic Square Problem from [Bravyi, Gosset, Koenig, Tomamichel 2020]. The first novel contribution is an investigation into the limitations of shallow quantum circuits for local optimization problems. We prove that if a shallow quantum circuit's input/output relation is exactly that of a local optimization problem, then we can construct a shallow classical circuit that also solves the optimization problem. We also prove an approximate version of this statement. Finally, we introduce a novel sampling task over an n-bit distribution D_n such that there exists a shallow quantum circuit that takes as input the state \ket{\GHZ_n} = \frac{1}{\sqrt{2}}(\ket{0^n} + \ket{1^n}) and produces a distribution close to D_n whereas, any constant-depth classical circuit with bounded fan-in and n + n^\delta random input bits for some \delta<1, will produce a distribution that is not close to D_n.enquantumshallow quantum circuitscomplexity theorymathMaster Thesis