Low-degree approximation of QAC$^0$ circuits (2411.00976v2)

Abstract: QAC$^0$ is the class of constant-depth quantum circuits with polynomially many ancillary qubits, where Toffoli gates on arbitrarily many qubits are allowed. In this work, we show that the parity function cannot be computed in QAC$^0$, resolving a long-standing open problem in quantum circuit complexity more than twenty years old. As a result, this proves ${\rm QAC}^0 \subsetneqq {\rm QAC}{\rm wf}^0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2^{\widetilde{\Omega}(n^{1/d})}$ ancillary qubits, which is close to tight. This implies a similar lower bound on approximately preparing cat states using QAC circuits. Finally, we prove a quantum analog of the Linial-Mansour-Nisan theorem for QAC$^0$. This implies that, for any QAC$^0$ circuit $U$ with $a={\rm poly}(n)$ ancillary qubits, and for any $x\in{0,1}^n$, the correlation between $Q(x)$ and the parity function is bounded by ${1}/{2} + 2^{-\widetilde{\Omega}(n^{1/d})}$, where $Q(x)$ denotes the output of measuring the output qubit of $U|x,0^a\rangle$. All the above consequences rely on the following technical result. If $U$ is a QAC$^0$ circuit with $a={\rm poly}(n)$ ancillary qubits, then there is a distribution $\mathcal{D}$ of bounded polynomials of degree polylog$(n)$ such that with high probability, a random polynomial from $\mathcal{D}$ approximates the function $\langle x,0^a| U^\dag Z{n+1} U |x,0^a\rangle$ for a large fraction of $x\in {0,1}^n$. This result is analogous to the Razborov-Smolensky result on the approximation of AC$^0$ circuits by random low-degree polynomials.

• The paper demonstrates that the parity function cannot be computed by QAC0 circuits, distinguishing them from models with unbounded fanout.
• It establishes that approximating parity with depth-d QAC circuits requires 2^(Ω(n^(1/d)/log n)) ancillary qubits, highlighting steep resource demands.
• It develops a quantum analog of the LMN theorem, showing that the Fourier coefficients of QAC0 circuit functions are concentrated on low-degree terms.

Low-Degree Approximation of QAC$^0$ Circuits: Insights and Implications

The paper at hand addresses a pivotal challenge in quantum computational complexity, specifically solving a long-standing problem concerning quantum circuits, known as QAC$^0$. This class of circuits encompasses constant-depth quantum circuits with polynomially many ancillary qubits and the capability to employ Toffoli gates on an arbitrary number of qubits. The primary accomplishment of this work is demonstrating that the parity function cannot be computed in QAC$^0$, a question that has persisted unsolved for over two decades.

Principal Contributions

1. Parity and QAC$^0$: The research establishes that the parity function is not computable by QAC$^0$ circuits. This decisive conclusion effectively delineates QAC$^0$ as a distinct class within the hierarchy of quantum computational models, particularly highlighting that QAC$^0 \neq$ QAC$^0_{\rm wf}$, where the latter permits unbounded fanout.
2. Lower Bounds and Ancillary Qubits: The authors delineate that approximating parity with QAC circuits of depth $d$ necessitates an exponential number of ancillary qubits, specifically $2^{\Omega(n^{1/d}/\log n)}$. This lower bound is nearly tight, suggesting substantial resource demands for such approximations.
3. Analog of the Linial-Mansour-Nisan (LMN) Theorem: The paper pioneers a quantum analog of the LMN theorem, applying it to QAC$^0$ circuits. It demonstrates that the Fourier coefficients of functions corresponding to QAC$^0$ circuits are predominantly concentrated on low-degree parts, providing a key insight into the structure and behavior of these circuits.

Technical Approach

A significant technical advancement in this paper is the development of a low-degree polynomial approximation for the operators corresponding to QAC$^0$ circuits. Inspired by the classical Razborov-Smolensky theorem which addresses AC$^0$ circuits, the authors craft a quantum variant that comes with its unique challenges.

The standard classical method involves approximating boolean functions by low-degree polynomials that could yield exponentially large values. In contrast, the quantum approach ensures bounded operator norms (restricted between -1 and 1) at all inputs, facilitating a more refined analysis of quantum circuits.

Implications and Future Directions

The results deploy a rigorous foundation establishing limits on the computational power of QAC$^0$ circuits. Practically, these findings imply significant resource requirements for quantum circuits tasked with executing complex operations like parity. Theoretically, this work enriches the landscape of quantum circuit complexity, providing a framework for further distinctions and classifications within quantum computational classes.

Moreover, this paper anticipates advancements in quantum learning and testing algorithms for QAC$^0$ circuits. By eliminating restrictions on ancillary qubits, the results may inspire new algorithmic strategies and optimizations. Future research could extend these findings to broader classes of quantum circuits or investigate their implications on other computational problems sustainably within the quantum domain.

In conclusion, this paper delivers a comprehensive examination of QAC$^0$ circuits, resolving critical open issues and establishing a robust analytical framework. Its implications traverse both theoretical and applied quantum computing landscapes, signifying a substantial progression in understanding the capabilities and limitations of quantum circuits.