Experimental preparation of W states through many-body physics on a quantum simulator

TL;DR

This work addresses the challenge of deterministically generating high-quality $W$ states in many-body quantum systems by leveraging topological ring frustration in an odd-$L$ Rydberg-atom ring. The authors design an adiabatic protocol that maps the ground state to a superposition of antiferromagnetic kink states, effectively realizing a $W$-state-like entangled state, and validate it on the Aquila neutral-atom platform up to $L=11$. A Bayesian tomography framework, combined with a two-measurement fidelity estimator, enables robust fidelity assessment despite hardware limitations that hinder direct $x$-basis tomography. The experimental fidelity bound for $L=11$ is $\mathcal{F}_e \approx 0.774$, and simulations indicate scalable performance with near-term improvements; the approach demonstrates a practical path to leveraging $W$-state entanglement for quantum information tasks and potential quantum advantage in programmable quantum simulators.

Abstract

$W$ states are quantum correlated states possessing both bipartite and multipartite entanglement, which makes them useful for several quantum algorithms. We propose a protocol to generate these states by exploiting {\it topological ring frustration}, and implement it on a programmable Rydberg atom array up to 11 qubits, successfully generating many-body $W$ states of Rubidium atoms. Numerical simulations show promising scaling of the algorithm to tens of qubits with near-term achievable updates on the quantum machines. To validate our state preparation protocol and probe quantum entanglement, we devise a fidelity estimator that requires only two sets of measurements. To implement it, we develop a novel and efficient Bayesian state-tomography approach that takes advantage of accurate classical numerical simulations to overcome limitations in the experimental setup. Hence, a lower bound fidelity of around $77\%$ is certified for the experimentally prepared state of 11 qubits. This work provides a state-of-the-art procedure to generate high-quality quantum entangled $W$ states, demonstrating once more how principles of physics can overcome traditional barriers of computation, and be exploited for quantum advantage.

Figures (4)

• Figure 1: Experimental setup to generate $W$ state through frustration and expose the generated many-body entanglement(a) The setup geometry image taken by the Rydberg atom array Aquila Aquila where bright dots are atoms in their ground states forming a ring geometry with odd parity system size. Lattice distance is marked by $a$, and it is set to $a=7.1\mu$m in this specific image of $L=11$ atoms. An anti-ferromagnetic kink state is marked in red, where we point to the position of the ferromagnetic defect (see text). This is one viable experimental shot result after the main pulse sequence in (c) is applied. (b) The many-body energy gap decreases with system size as $L^{-2}$ in odd-parity chain sizes leading to a gapless spectrum in the thermodynamic limit, whereas for even-parity chain sizes, the spectrum is gapped. (c) Experimental pulse sequence to prepare a many-body $W$ state where ramp times $\tau_{\Omega,\rm ramp}$, $\tau_{\Delta,\rm ramp}$, Rabi frequency $\Omega$ and detuning $\Delta$ are optimized for best $W$ state populations. The rotation sequence is applied in the second set of experiments to measure the state preparation fidelity by introducing a Rabi frequency phase which changes the basis, green line, while setting the detuning to zero. Here, too, the evolution time $\tau_{\rm rot}$ and rotation Rabi frequency $\Omega_{\rm rot}$ are optimized.
• Figure 2: Numerical simulation of the $W$ state preparation protocol(a) The numerical fidelity of state preparation $\mathcal{F}_{\rm th}$ as a function of the system size, for atom spacing $a=6$$\mu$m, Rabi frequency $\Omega=15$ rad$/\mu$s and total evolution time of $t_{\rm F}=1, 2, 4$$\mu$s. (b) Infidelity (that it, $1-\mathcal{F}_{\rm th}$), of the state preparation for slower protocols, and (c) corresponding time needed to achieve an infidelity smaller than $10^{-3}$ as a function of the system size $L$. These numerical data have been obtained for $\Omega=15$ rad$/\mu$s and $a=6$$\mu$m.
• Figure 3: Experimental results of $W$ state preparation protocol with chain sizes $L=5-11$. The populations of the prepared state, where gray and red boxes are the raw and error-mitigated data Bravy2021, see Methods for information on utilized error mitigation. The black error bars are computed via bootstrapping the experimental bit strings efron1979efron1994. The dashed-black horizontal lines mark $1/L$, which is the population value for a uniform $W$ state. For each experiment, 1000 bit-strings are collected, and more than $900$ of them are used in data analysis after post-processing based on the correct initial state. See SM for details.
• Figure 4: Measured $W$ state fidelity conditioned on the theoretical simulations(a) Kullback-Leibler divergence between experimental and numerical distributions. The blue squares refer to the KL divergence between the bit-string distributions, i.e., the the outcomes of the projective measurement on the quantum state after the main pulse sequence. The yellow diamonds represent the KL divergence between the distributions of total magnetization, i.e., the frequencies of states with a given magnetization, regardless of the microscopic bit-string distribution. Since a perfect state preparation would yield always the same magnetization, the agreement between experimental and numerical distributions is an excellent indicator of the accuracy of our numerical modeling. (b) The fidelity of the prepared states, according to Eq. \ref{['eq:sm_fidelity_estimator']}: the shaded area correspond to the fidelity spread according to the numerical prior distribution, which is created using just the diagonal ($z$-basis) experimental data. The markers are the result of the Bayesian procedure in eq. \ref{['eq:BayesianUpdate']} and are computed with the posterior probabilities which take into account the results of the (off-diagonal) measurements after rotation. The generated states are found to be quantum correlated as the fidelity is much larger than $1/L$ for all system sizes.