A gradient estimator for Lie-symmetric PQCs expresses the gradient as a linear combination of Hadamard-test expectations whose coefficients are estimated via shadow tomography, yielding logarithmic shot scaling.
Learning k-qubit quantum operators via pauli decomposition
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Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries
A gradient estimator for Lie-symmetric PQCs expresses the gradient as a linear combination of Hadamard-test expectations whose coefficients are estimated via shadow tomography, yielding logarithmic shot scaling.