Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems

Estimating the fidelity between a desired target quantum state and an actual prepared state is essential for assessing the success of experiments. For pure target states, we use functional representations that can be measured directly and determine the number of copies of the prepared state needed for fidelity estimation. In continuous variable (CV) systems, we use the Wigner function, which can be measured via displaced parity measurements. We provide upper and lower bounds on the sample complexity required for fidelity estimation, considering the worst-case scenario across all possible prepared states. For target states of particular interest, such as Fock and Gaussian states, we find that this sample complexity is characterized by the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>L</a:mi> <a:mn>1</a:mn> </a:msup> </a:math> -norm of the Wigner function, a measure of Wigner negativity widely studied in the literature, in particular in resource theories of quantum computation. For discrete variable systems consisting of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>n</c:mi> </c:math> qubits, we explore fidelity estimation protocols using Pauli string measurements. Similarly as for the CV approach, the sample complexity is shown to be characterized by the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>L</e:mi> <e:mn>1</e:mn> </e:msup> </e:math> -norm of the characteristic function of the target state for both Haar random states and stabilizer states. Furthermore, in a general black box model, we prove that, for any target state, the optimal sample complexity for fidelity estimation is characterized by the smoothed <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mi>L</g:mi> <g:mn>1</g:mn> </g:msup> </g:math> -norm of the target state. To the best of our knowledge, this is the first time the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>L</i:mi> <i:mn>1</i:mn> </i:msup> </i:math> -norm of the Wigner function provides a lower bound on the cost of some information processing task.