Unconditionally separating noisy $\mathsfQNC^0$ from bounded polynomial threshold circuits of constant depth
<|reference_start|>Unconditionally separating noisy $\mathsfQNC^0$ from bounded polynomial threshold circuits of constant depth: We study classes of constant-depth circuits with gates that compute restricted polynomial threshold functions, recently introduced by [Kum23] as a family that strictly generalizes $\mathsf{AC}^0$. Denoting these circuit families $\mathsf{bPTFC}^0[k]$ for $\textit{bounded polynomial threshold circuits}$ parameterized by an integer-valued degree-bound $k$, we prove three hardness results separating these classes from constant-depth quantum circuits ($\mathsf{QNC}^0$).   $\hspace{2em}$ - We prove that the parity halving problem [WKS+19], which $\mathsf{QNC}^0$ over qubits can solve with certainty, remains average-case hard for polynomial size $\mathsf{bPTFC}^0[k]$ circuits for all $k=\mathcal{O}(n^{1/(5d)})$.   $\hspace{2em}$ - We construct a new family of relation problems based on computing $\mathsf{mod}\ p$ for each prime $p>2$, and prove a separation of $\mathsf{QNC}^0$ circuits over higher dimensional quantum systems (`qupits') against $\mathsf{bPTFC}^0[k]$ circuits for the same degree-bound parameter as above.   $\hspace{2em}$ - We prove that both foregoing results are noise-robust under the local stochastic noise model, by introducing fault-tolerant implementations of non-Clifford $\mathsf{QNC}^0/|\overline{T^{1/p}}>$ circuits, that use logical magic states as advice.   $\mathsf{bPTFC}^0[k]$ circuits can compute certain classes of Polynomial Threshold Functions (PTFs), which in turn serve as a natural model for neural networks and exhibit enhanced expressivity and computational capabilities. Furthermore, for large enough values of $k$, $\mathsf{bPTFC}^0[k]$ contains $\mathsf{TC}^0$ as a subclass. The main challenges we overcome include establishing classical average-case lower bounds, designing non-local games with quantum-classical gaps in winning probabilities and developing noise-resilient non-Clifford quantum circuits necessary to extend beyond qubits to higher dimensions.<|reference_end|>
@article{hsieh2024unconditionally,
    title={Unconditionally separating noisy $\mathsf{QNC}^0$ from bounded
  polynomial threshold circuits of constant depth},
    author={Min-Hsiu Hsieh, Leandro Mendes, Michael de Oliveira and
  Sathyawageeswar Subramanian},
    journal={arXiv preprint arXiv:2408.16378},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.16378},
    primaryClass={quant-ph cs.CC}
}