Pseudoentanglement Ain't Cheap
Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang
TL;DR
The paper links pseudoentanglement to non-Clifford resource requirements by showing that a gap $t$ in entanglement across cuts enforces a linear lower bound $Ω(t)$ on the number of non-Clifford gates needed to prepare the ensemble. It introduces a polynomial-time quantum entanglement-estimation algorithm based on the Weyl stabilizer framework: for states stabilized by at least $2^{n-t}$ Pauli operators, it estimates the entropy across any bipartition within an additive error of $\frac{t}{2}$. The key technical contributions are (i) entropy bounds expressed through the dimensions of the Weyl stabilizer spaces, (ii) an efficient procedure using Bell difference sampling to learn a superset of $\mathrm{Weyl}(\ket\psi)$ and (iii) reductions showing local Clifford operations preserve entanglement across cuts, enabling distillation arguments. Under linear-time quantum-secure pseudorandom-function assumptions, these bounds are tight up to polylog factors, implying optimal or near-optimal non-Clifford costs for pseudoentangled state implementations and connecting to related results on pseudorandom quantum states. The work advances understanding of the interplay between state complexity, entanglement structure, and computational resources with potential implications for quantum cryptography and holography.
Abstract
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 $\frac{t}{2}$ bits of the true entanglement entropy.