Constant-depth QAC computation of parity

Determine whether polynomial-size, constant-depth QAC circuits can compute the parity function, equivalently implement an unbounded fan-out operation.

Background

The paper distinguishes QAC circuits, which allow unbounded fan-in but not unbounded fan-out, from QNC circuits. The construction targets QAC because it is believed to be substantially weaker than QNC, yet the computational power of polynomial-size constant-depth QAC remains poorly understood. Establishing whether such circuits can compute parity would resolve a fundamental question about the relationship between these shallow quantum circuit classes.

References

On the other hand, whether $\QACZ$ can implement the $\FANOUT$ operation, or equivalently the $\PARITY$ function, is a longstanding open question .

— QAC0 Can Prepare Every Logarithmic-Qubit State  (2609.17408 - Gretta et al., 15 Sep 2026) in Section 1, Introduction

Whether polynomial-size constant-depth $QAC$ circuits can compute parity---equivalently, fan-out---is a long-standing open problem: the best known circuits have exponential size.

— Verifiable quantum advantage in extremely low depth  (2609.01448 - Gheorghiu, 1 Sep 2026) in Section 1, Introduction

Whether \mathsf{QAC}0 can compute \mathtt{PARITY}_n remains open.

— Fanout Complexity of Symmetric Boolean Functions in $\mathsf{QAC}^0$  (2609.05153 - Xu et al., 4 Sep 2026) in Abstract

For instance, what is the power of shallow bosonic circuits whose total energy is bounded polynomially? Could such a model compute Parity in constant depth when supplied with long-range gates, in contrast with what is believed to be possible for qubit \mathsf{QAC}0 circuits?

— A physical and universal model of bosonic computations with Solovay-Kitaev theorem  (2609.40226 - Rudolph et al., 30 Sep 2026) in Section 1. Results, subsection “Open questions,” item “Restricted models of bosonic computation”