A controlled-SWAP circuit with n copies of a quantum state can simultaneously estimate tr(ρ^k) for k=2..n using simple parity sums of its measurement probabilities.
Vedral, Quantum entanglemen
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Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities
A controlled-SWAP circuit with n copies of a quantum state can simultaneously estimate tr(ρ^k) for k=2..n using simple parity sums of its measurement probabilities.