Exponential Separations between Quantum Learning with and without Purification

TL;DR

This paper demonstrates that access to the purification of a mixed quantum state enables exponential reductions in sample complexity for various quantum learning tasks, compared to strategies without purification.

Contribution

It introduces the concept that purification access can replace large quantum memory, achieving exponential efficiency gains in quantum learning tasks.

Findings

Constant sample complexity for estimating properties with purification

Exponential sample complexity without purification for the same tasks

Implications for quantum cryptography and resource optimization

Abstract

In quantum learning tasks, quantum memory can offer exponential reductions in statistical complexity compared to any single-copy strategies, but this typically necessitates at least doubling the system size. We show that such exponential reductions can also be achieved by having access to the purification of the target mixed state. Specifically, for a low-rank mixed state, only a constant number of ancilla qubits is needed for estimating properties related to its purity, cooled form, principal component and quantum Fisher information with constant sample complexity, which utilizes single-copy measurements on the purification. Without access to the purification, we prove that these tasks require exponentially many copies of the target mixed state for any strategies utilizing a bounded number of ancilla qubits, even with the knowledge of the target state's rank. Our findings also lead to practical applications in areas such as quantum cryptography. With further discussions about the source and extent of the advantages brought by purification, our work uncovers a new resource with significant potential for quantum learning and other applications.