Unconditionally separating noisy $mathsf{QNC}^0$ from bounded polynomial threshold circuits of constant depth

2024-08-29

AI Summary

Establishing an unconditional separation between constant-depth quantum circuits ($mathsf{QNC}^0$) and bounded polynomial-threshold constant-depth classical circuits ($mathsf{bPTFC}^0[k]$) under realistic local stochastic noise, to demonstrate the intrinsic advantage of quantum parallelism in noisy settings. Method: We introduce a family of modulo-$p$ relation problems, construct fault-tolerant non-Clifford $mathsf{QNC}^0$ circuits using logical magic-state encoding and a local random noise model, and leverage nonlocal games together with high-dimensional qudits (qupits). Contribution/Results: We achieve the first unconditional separation for arbitrary $k = O(n^{1/(5d)})$: $mathsf{QNC}^0$ circuits solve parity halving and related tasks exactly, whereas $mathsf{bPTFC}^0[k]$ circuits fail on average. The result holds for both qubits and qupits, significantly extending the theoretical frontier of quantum advantage under physically motivated noise models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

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 $ extit{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.

Problem

• Unconditionally separate noisy quantum circuits from classical threshold circuits
• Demonstrate quantum computational advantage without classical assumptions
• Identify problems solvable by constant-depth quantum but not classical circuits

Innovation

• Unconditionally separate noisy quantum circuits
• Constant-depth quantum circuits solve problems
• Robust quantum advantages across prime dimensions