A quantum algorithm estimates Tr(ρ^k O) with O(√k) queries to a purification-preparing unitary, quadratically faster than sample-based methods, with a claimed matching lower bound.
Vir- tual distillation for quantum error mitigation
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Measuring Less to Learn More: Quadratic Speedup in learning Nonlinear Properties of Quantum Density Matrices
A quantum algorithm estimates Tr(ρ^k O) with O(√k) queries to a purification-preparing unitary, quadratically faster than sample-based methods, with a claimed matching lower bound.