On the Pauli Spectrum of QAC0

TL;DR

This work introduces the Pauli spectrum for quantum channels as a natural quantum analogue of the classical Fourier spectrum and studies it through the Choi representation. The authors prove a low-degree concentration bound for polynomial-size QAC^0 circuits with a limited number of ancilla qubits, showing that the high-weight Pauli components of the Choi representation are exponentially suppressed in k^{1/d}, up to an ancilla factor. As immediate consequences, they obtain average-case lower bounds against parity and majority for such circuits and provide a quasipolynomially sample-efficient learning algorithm for the corresponding low-degree channels. They also develop two parallel learning frameworks: Choi-state shadow tomography for Pauli coefficients and measurement-query (QPSQ) based learning, together with a convex rounding step to CPTP maps. Their results bolster the use of Pauli-analytic techniques in quantum circuit complexity and open questions about extending low-degree concentration to all polynomial-size QAC^0 circuits, with potential implications for state synthesis and pseudorandomness.

Abstract

The circuit class $\mathsf{QAC}^0$ 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 $\mathsf{QAC}^0$ 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 $\mathsf{QAC}^0$ circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical $\mathsf{AC}^0$ circuits. We conjecture that the Pauli spectrum of $\mathsf{QAC}^0$ circuits satisfies low-degree concentration, in analogy to the famous Linial, Nisan, Mansour theorem on the low-degree Fourier concentration of $\mathsf{AC}^0$ circuits. If true, this conjecture immediately implies that polynomial-size $\mathsf{QAC}^0$ circuits cannot compute parity. We prove this conjecture for the class of depth-$d$, polynomial-size $\mathsf{QAC}^0$ circuits with at most $n^{O(1/d)}$ 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 $(\frac{1}{2} + 2^{-Ω(n^{1/d})})$-fraction of inputs, and - the $n$-bit majority function on more than $(1 - Ω(n^{-1/2}))$-fraction of inputs. Additionally we show that this class of $\mathsf{QAC}^0$ circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for $\mathsf{QAC}^0$ circuits. More broadly, our results add evidence that "Pauli-analytic" techniques can be a powerful tool in studying quantum circuits.

Figures (7)

• Figure 1: A simple unitary that does not satisfy low-degree Pauli concentration in the naive sense.
• Figure 2: A (backwards) lightcone of a qubit.
• Figure 3: The channel $\mathcal{E}_{U, \psi}$ mapping $n$ qubits to $1$ qubit by first applying the unitary $U$ to $n$ input qubits as well as an $a$-qubit auxiliary state $\ket{\psi}$, then tracing out all but the last qubit --- in the "$\mathrm{out}$" register.
• Figure 4: Tensor network diagram proving alternative definition for $\Phi_{U}$ as stated in \ref{['fact:choirep-altdef']}. We adopt the standard convention that origin=c]180$U$$\,=U^\top$. Our register labelling follows \ref{['fig:circuit-register-labels']}.
• Figure 5: Simulating a Toffoli (i.e. $\mathsf{CNOT}$) gate using $\textsf{CZ}\xspace$ gates and Hadamards.

Theorems & Definitions (57)

• Theorem 1: Informal version of \ref{['thm:pauli-concentration']}
• Theorem 2: Informal version of \ref{['QAC-corr-bound']}
• Theorem 3: Informal version of \ref{['thm:learnqac0']}
• Conjecture 1: Spectral concentration for $\mathsf{QAC}^0$
• Definition 4
• Definition 6
• Definition 7: Choi representation, Choi state
• proof
• Proposition 10
• proof