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$\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)

Daniel Grier, Jackson Morris, Kewen Wu

TL;DR

We study the power of constant-depth quantum circuits $\mathsf{QAC}^0$, which combine arbitrary single-qubit gates with generalized Toffoli gates, for computing Boolean functions. The paper establishes a strong separation $\mathsf{QAC}^0 \nsubseteq \mathsf{AC}^0[p]$ and shows that providing polynomially many copies of the input allows $\mathsf{QAC}^0$ to simulate $\mathsf{TC}^0$ via $\mathsf{QAC}^0 \circ \mathsf{NC}^0$, thereby highlighting a substantial quantum advantage when input copies are available. Central to the results are the $\mathsf{W}$ test for Hamming weight and an exact amplitude amplification technique that converts approximate constant-depth constructions into exact ones, enabling exact parity, exact fanout, and exact computation of a broad class of symmetric functions. The work also proves that all symmetric $\mathsf{AC}^0$ functions can be computed exactly in $\mathsf{QAC}^0$, and it develops a framework around state–unitary duality to separate $\mathsf{QAC}^0$ from certain classical circuit classes, with implications for copy complexity and potential future refinements to near-term quantum architectures.

Abstract

$\mathsf{QAC}^0$ is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical $\mathsf{AC}^0$ lower bounds to $\mathsf{QAC}^0$ have failed. We give one possible explanation of this: $\mathsf{QAC}^0$ circuits are significantly more powerful than their classical counterparts. We show the unconditional separation $\mathsf{QAC}^0\not\subset\mathsf{AC}^0[p]$ for decision problems, which also resolves for the first time whether $\mathsf{AC}^0$ could be more powerful than $\mathsf{QAC}^0$. Moreover, we prove that $\mathsf{QAC}^0$ circuits can compute a wide range of Boolean functions if given multiple copies of the input: $\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$. Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.

$\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)

TL;DR

We study the power of constant-depth quantum circuits , which combine arbitrary single-qubit gates with generalized Toffoli gates, for computing Boolean functions. The paper establishes a strong separation and shows that providing polynomially many copies of the input allows to simulate via , thereby highlighting a substantial quantum advantage when input copies are available. Central to the results are the test for Hamming weight and an exact amplitude amplification technique that converts approximate constant-depth constructions into exact ones, enabling exact parity, exact fanout, and exact computation of a broad class of symmetric functions. The work also proves that all symmetric functions can be computed exactly in , and it develops a framework around state–unitary duality to separate from certain classical circuit classes, with implications for copy complexity and potential future refinements to near-term quantum architectures.

Abstract

is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical lower bounds to have failed. We give one possible explanation of this: circuits are significantly more powerful than their classical counterparts. We show the unconditional separation for decision problems, which also resolves for the first time whether could be more powerful than . Moreover, we prove that circuits can compute a wide range of Boolean functions if given multiple copies of the input: . Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.
Paper Structure (27 sections, 27 theorems, 44 equations, 2 figures)

This paper contains 27 sections, 27 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. The $W$ Test.
  4. Quantum-Classical Separations.
  5. Exact Preparation of $\ket W$ and Nekomata.
  6. Exact Computation in $\QAC^0$.
  7. Future Directions
  8. Preliminaries
  9. Symmetric Boolean Functions.
  10. Classical Circuit Complexity.
  11. (Multi-Qubit) Quantum Gates.
  12. Quantum Circuits.
  13. $\QAC$ and $\QAC^0$.
  14. Other Quantum Circuit Classes.
  15. Quantum States.
  16. ...and 12 more sections

Key Result

Theorem 1

There exists a language $L$ which can be decided by a $\QAC^0$ circuit with perfect completeness and soundness $2^{-\poly(n)}$ on inputs of size $n$. However, $L$ requires $\AC^0[p]$ circuits of size $2^{\poly(n)}$ for all primes $p > 1$. Thus, $\ComplexityFont{BQAC}^0 \not \subset \AC^0[p]$.

Figures (2)

  • Figure 1: The $W$ test circuit which weakly computes $\mathsf{EX}_{n/2}$, where the two-qubit gates in the second layer above are controlled $Z$-gates.
  • Figure 2: The circuit of truncated parallel repetition for $n=4$.

Theorems & Definitions (51)

  • Theorem 1: See also \ref{['thm:qac0_sep']}
  • Theorem 2: See also \ref{['thm:tc0_in_qac0_copy']}
  • Corollary 3: See also \ref{['cor:exact_qac0']}
  • Corollary 4: See also \ref{['cor:exact_qtc0']}
  • Theorem 5: See also \ref{['thm:exact_w_1']}
  • Theorem 6: See also \ref{['cor:sym_ac0_in_qac0']}
  • Theorem 7: grover1998quantumbrassard2002quantum
  • Corollary 8
  • Corollary 9
  • proof : Proof of \ref{['cor:exact_qac0', 'cor:exact_qtc0']}
  • ...and 41 more