Senrui Chen

The optimal quantum measurements for estimating different unknown parameters in a parameterized quantum state are usually incompatible with each other. Traditional approaches to addressing the measurement incompatibility issue, such as the Holevo Cram\'{e}r--Rao bound, suffer from multiple difficulties towards practical applicability, as the optimal measurement strategies are usually state-dependent, difficult to implement and also take complex analyses to determine. Here we study randomized measurements as a new approach for multi-parameter quantum metrology. We show quantum measurements on single copies of quantum states given by $3$-designs perform near-optimally when estimating an arbitrary number of parameters in pure states and more generally, {approximately low-rank well-conditioned states}, whose metrological information is largely concentrated in a low-dimensional subspace. The near-optimality is also shown in estimating the maximal number of parameters for three types of mixed states that are well-conditioned on their supports. Examples of fidelity estimation and Hamiltonian estimation are explicitly provided to demonstrate the power and limitation of randomized measurements in multi-parameter quantum metrology.

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today's quantum devices. A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable and what is not, which has been unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates, showing that learnable information corresponds to the cycle space of the pattern transfer graph of the gate set, while unlearnable information corresponds to the cut space. This implies the optimality of cycle benchmarking, in the sense that it can learn all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM's CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we give an attempt to characterize the unlearnable information by assuming perfect initial state preparation. However, based on the experimental data, we conclude that this assumption is inaccurate as it yields unphysical estimates, and we obtain a lower bound on state preparation noise.