Polynomial-time classical simulability of Hadamard–Toffoli circuits under limited Hadamard-on-|1⟩ usage

Establish the equivalence that quantum circuits composed solely of Hadamard and Toffoli gates are classically simulable in polynomial time if and only if there exists an equivalent representation of the circuit in which the Hadamard gates act on computational-basis |1⟩ states at most a logarithmic number of times.

Background

The paper implements a quantum-circuit version of the first level of DOOM using only Hadamard and Toffoli gates, and discusses how its specific structure enables efficient classical simulation. By resetting the ‘random’ qubits before each run, Hadamards are applied only to |0⟩ states, preventing negative phases and amplitude cancellations and yielding an equiprobable mixture over outcomes that can be efficiently sampled.

Motivated by this observation, the work proposes a broader conjecture delineating when Hadamard+Toffoli circuits are efficiently classically simulable—namely, when one can find an equivalent representation in which Hadamards act on |1⟩ states only a logarithmic number of times, thereby hypothesizing a structural condition that characterizes the tractable subset of such universal gate-set circuits.