Quantum-inspired Algorithm for the Collatz Conjecture

ABSTRACT
We develop and benchmark a quantum-inspired algorithmic framework that maps the classical 3n+ 1 Collatz function onto a quantum circuit intended for execution on a qubit array. Starting from a uniform superposition of all 2L binary seeds, the circuit deterministically appliesthe transformation n ↦ 3n + 1 followed by pointer-mediated division by 2k sequence that removes every trailing power of two. The algorithm uses Clifford+Toffoli primitives and is tailored for efficient classical emulation via matrix product state techniques that each elementary layer is expressed as a uniform matrix product operator whose bond dimension is bounded by four, while a greedy SVD compression routine controls entanglement growth. Our results illustrate the potential of quantum parallelism to explore challenging problems in discrete dynamics, while remaining compatible with classical device constraints.