Sabee Grewal

We show that any pseudoentangled state ensemble with a gap of t bits of entropy requires Ω(t) non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an n-qubit state that is stabilized by at least 2^n−t Pauli operators, our algorithm produces an estimate that is within an additive factor of t/2 bits of the true entanglement entropy.