Quantum advantages for Pauli channel estimation

We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an n-qubit Pauli channel to +/-epsilon precision. We give an estimation protocol with an n-qubit ancilla that succeeds with high probability using only O(n/epsilon(2)) copies of the Pauli channel, while proving that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least Omega(2(n)(/3)) rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a k-qubit ancilla (k <= n) is available, we obtain a sample complexity lower bound of Omega(2(()(n-k)()/3)) for any nonconcatenating protocol, and a stronger lower bound of Omega(n2(n-k)) for any nonadaptive, nonconcatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically interesting example for quantum advantages in learning and also bring insights for quantum benchmarking.