On the Pauli Spectrum of QAC0

The circuit class QAC0 was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size QAC0 cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the Pauli spectrum of QAC0 circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical AC0 circuits. We conjecture that the Pauli spectrum of QAC0 circuits satisfies low-degree concentration, in analogy to the famous Linial, Mansour, Nisan (LMN) theorem on the low-degree Fourier concentration of AC0 circuits. If true, this conjecture immediately implies that polynomial-size QAC0 circuits cannot compute parity. We prove this conjecture for the class of depth-d, polynomial-size QAC0 circuits with at most nO(1/d) auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute the n-bit parity function on more than (1/2 + 2−Ω(n1/d))-fraction of inputs, and the n-bit majority function on more than (1/2 + O(n−1/4))-fraction of inputs. Additionally we show that this class of QAC0 circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for QAC0 circuits. More broadly, our results add evidence that "Pauli-analytic" techniques can be a powerful tool in studying quantum circuits.