Parity is not in QAC0 (Moore's parity conjecture)

Lineage and corrections

Lean solution module OAI/InformationTheory/QuantumCircuit/Parity.lean evidence for this event

Lean Comparator challenge RegularParity (polynomial-size 2/3 specialization) evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean Comparator challenge QACParity evidence for this event

Parity is not in QAC0 (Moore's parity conjecture) evidence for this event

Claims: for every fixed depth $d$, every $c\ge1$ and every $0<\varepsilon\le1/2$, for all large $n$ no depth-$d$ circuit of one-qubit unitaries and unbounded-arity Toffoli gates on at most $n^c$ qubits, ancillas starting in $|0\rangle$ and one output qubit measured, computes parity with probability at least $1/2+\varepsilon$ on every input. Taking $\varepsilon=1/6$ gives bounded error $2/3$. The companion paper gives a second proof via regular trajectories and pruning; via Xu-Li reductions the same obstruction holds for strict majority. It does not cover circuits with intermediate measurements, postselection or primitive fanout gates, and gives no explicit size-depth tradeoff beyond polynomial size. parent of this event

Parity is not in QAC0 (Moore's parity conjecture) parent of this event

Companion: Regular trajectories, pruning and quantum parity evidence for this event

Moore, Quantum circuits: fanout, parity, and counting (1999) evidence for this event

VibeMathed record: Parity is not in QAC0 (Moore's parity conjecture) evidence for this event