On the Computational Power of QAC0 with Barely Superlinear Ancillae

TL;DR

This work investigates the computational capabilities of QAC0 circuits with barely superlinear ancillae, focusing on whether parity-like functions can be computed in constant depth. By developing a spectral-norm, low-degree operator approximation framework (Pauli analysis) and a layer-by-layer degree control, the authors obtain tight upper bounds on the approximate degree of Heisenberg-evolved measurements, leading to superlinear ancilla lower bounds for parity, majority, and MOD_k, and to sublinear approximate degree for QLC0. They extend these techniques to quantum state and channel synthesis, proving that high-degree states and channels cannot be efficiently produced or implemented by depth-d QAC0 with modest ancillae, and they provide bootstrapping methods to push ancilla requirements further toward parity not in QAC0. The results bridge quantum circuit complexity with classical LC0/AC0 techniques, offering new insights into the limits of shallow quantum computation and implications for agnostic learning and long-range entanglement generation. Overall, the paper advances our understanding of how ancilla resources interplay with depth to constrain the expressive power of quantum circuits beyond light-cone limitations.

Abstract

$\mathrm{QAC}^0$ is the family of constant-depth polynomial-size quantum circuits consisting of arbitrary single qubit unitaries and multi-qubit Toffoli gates. It was introduced by Moore [arXiv: 9903046] as a quantum counterpart of $\mathrm{AC}^0$, along with the conjecture that $\mathrm{QAC}^0$ circuits can not compute PARITY. In this work we make progress on this longstanding conjecture: we show that any depth-$d$ $\mathrm{QAC}^0$ circuit requires $n^{1+3^{-d}}$ ancillae to compute a function with approximate degree $Θ(n)$, which includes PARITY, MAJORITY and $\mathrm{MOD}_k$. We further establish superlinear lower bounds on quantum state synthesis and quantum channel synthesis. This is the first superlinear lower bound on the super-linear sized $\mathrm{QAC}^0$. Regarding PARITY, we show that any further improvement on the size of ancillae to $n^{1+\exp(-o(d))}$ would imply that PARITY $\not\in$ QAC0. These lower bounds are derived by giving low-degree approximations to $\mathrm{QAC}^0$ circuits. We show that a depth-$d$ $\mathrm{QAC}^0$ circuit with $a$ ancillae, when applied to low-degree operators, has a degree $(n+a)^{1-3^{-d}}$ polynomial approximation in the spectral norm. This implies that the class $\mathrm{QLC}^0$, corresponding to linear size $\mathrm{QAC}^0$ circuits, has approximate degree $o(n)$. This is a quantum generalization of the result that $\mathrm{LC}^0$ circuits have approximate degree $o(n)$ by Bun, Robin, and Thaler [SODA 2019]. Our result also implies that $\mathrm{QLC}^0\neq\mathrm{NC}^1$.

Figures (5)

• Figure 1: $\mathbf{QAC}^{0}$ Circuit Example
• Figure 2: Circuit computing $\operatorname{Parity}_{m}$ for $m=2n$
• Figure 3: Circuit for $\operatorname{Parity}_{n_1n_2}$
• Figure 4: Step 1, $n^c\to n^2$ ancillae
• Figure 5: Step 2, $n^2 \to n^{1+\exp \left( -o(d) \right) }$ ancillae

Theorems & Definitions (84)

• Theorem 1.1: informal of \ref{['cor:qac0-whole']}
• Theorem 1.2: informal of \ref{['thm:main:WorstCase']}
• Theorem 1.3: informal of \ref{['thm:main:AverageCase']}
• Theorem 1.4: informal of \ref{['cor:arbitrary']}
• Theorem 1.5: informal of \ref{['thm:state-synthesis']}
• Theorem 1.6: informal of \ref{['thm:qchannel-degree']}
• Theorem 2.1: Parseval's theorem
• Definition 2.2
• Definition 2.3: Approximate Degree
• Example 2.5