Experimental Direct Quantum Fidelity Learning via a Data-Driven Approach

Fidelity estimation is an important technique for evaluating prepared quantum states in noisy quantum devices. A recent theoretical work proposed a frugal approach called neural quantum fidelity estimation (NQFE) [X. Zhang et al., Phys. Rev. Lett. 127, 130503 (2021).]. While this requires a much smaller number of measurement operators than full quantum state tomography, it uses a weight-based floating measurement strategy that predetermines the top global Pauli operators that contribute the most to the fidelity and uses discrete fidelity intervals as predictions. In this Letter, we develop a measurement-fixed NQFE based on a transformer model which requires less measurement cost and can output continuous estimates of fidelity. Here we further experimentally apply the NQFE in a realistic situation using a nuclear spin quantum processor. We prepare the ground states of local Hamiltonians and arbitrary states and investigate how to estimate their fidelity with reference states, and we compare the fidelity estimation strategy with our and the original NQFE to conventional tomography. It is shown that NQFE can estimate the fidelity with comparable accuracy to the tomography approach. In the future, NQFE will become an important tool for benchmarking quantum states ahead of the advent of well-trusted fault-tolerant quantum computers.

Figure 1

Comparison between the measurement-fixed and original NQFE in learning the fidelity. (a) and (c) present the fidelity estimation error $ϵ$ as a function of the total number of measurements $M_{\mathrm{tot}}=k{M}_s$ for $n=4$ and $n=10$, respectively. The dashed line is the ultimate precision of our NQFE without shot noise. (b) and (d) present the distribution of $ϵ$ at the 95% confidence level in [0.9, 1]. Although measurement-fixed NQFE inputs fewer expectation values, it achieves higher fidelity estimation precision under the same measurement cost.

Figure 2

Results of the measurement-fixed NQFE in learning fidelity of ground states with $n=80$. The main figure plots the distribution of test samples as a function of the error $ϵ$, and the inset figure shows the distribution of $ϵ$ at the 95% confidence level as a function of the fidelity.

Figure 3

Experimental qubits and quantum circuit for estimating fidelity. (a) Our four-qubit quantum processor (approximately a 1D chain) is encoded by four carbon nuclear spins and its coupling strength and chemical frequency. (b) The PCP is optimized to prepare the ground state from ${\rho }_{\mathrm{pps}}$, via minimizing the expectation value of ${\mathcal{H}}^{(2)}$, and then its fidelity is estimated by NQFE and FQST.

Figure 4

Experimental results of implementing NQFE for ground states [(a),(b)] and arbitrary states [(c),(d)]. The measurement-fixed NQFE can achieve higher fidelity estimation accuracy and more realistically mimic the FQST results for structural quantum states, while the original NQFE may be a better choice for arbitrary states.