Fermionic quantum processing with programmable neutral atom arrays

Abstract

Simulating the properties of many-body fermionic systems is an outstanding computational challenge relevant to material science, quantum chemistry, and particle physics.-5.4pc]Please note that the spelling of the following author names in the manuscript differs from the spelling provided in the article metadata: D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, Jun Ye, and P. Zoller. The spelling provided in the manuscript has been retained; please confirm. Although qubit-based quantum computers can potentially tackle this problem more efficiently than classical devices, encoding nonlocal fermionic statistics introduces an overhead in the required resources, limiting their applicability on near-term architectures. In this work, we present a fermionic quantum processor, where fermionic models are locally encoded in a fermionic register and simulated in a hardware-efficient manner using fermionic gates. We consider in particular fermionic atoms in programmable tweezer arrays and develop different protocols to implement nonlocal gates, guaranteeing Fermi statistics at the hardware level. We use this gate set, together with Rydberg-mediated interaction gates, to find efficient circuit decompositions for digital and variational quantum simulation algorithms, illustrated here for molecular energy estimation. Finally, we consider a combined fermion-qubit architecture, where both the motional and internal degrees of freedom of the atoms are harnessed to efficiently implement quantum phase estimation as well as to simulate lattice gauge theory dynamics.

Keywords: digital quantum simulation; fermionic quantum processor; lattice gauge theories; quantum chemistry; tweezer arrays.

Fig. 1.

Fermionic quantum processor. (A) We consider a fermionic register based on fermionic atoms trapped in optical tweezers, where quantum information is encoded in the atomic occupation and processed using fermionic gates. The latter includes tunneling processes, delocalizing atoms between different tweezers (lighter spheres), as well as interaction gates, based on the Rydberg blockade mechanism. (B) We use these gates to construct fermionic quantum circuits, where certain subroutines are first precompiled to minimize circuit depths (as detailed below). (C) Fermionic circuits are particularly suited for quantum simulation of fermionic models, avoiding nonlocal overheads. Here, we consider the ground-state energy estimation of molecules using variational algorithms, as well as Trotter time evolution of LGTs.

Fig. 2.

Fermion-qubit register and tunneling gates. (A and B) show the sequence of laser pulses and tweezer moves used to implement a MERGE and a SHUTTLE gate between a pair of sites $(i,j)$, respectively, where the atomic superposition between two different tweezers is achieved either through direct tunneling (dashed line) or internal rotations. In the figure, tweezers of different colors correspond to different wavelengths. While the illustration shows the case of a single initially localized atom, we emphasize that the protocols also apply to situations where both tweezers contain contributions from many-body superpositions of several atoms delocalized over the whole system. (C) Level structure of ${}^{87}\mathrm{Sr}$. A fermionic register (F) is built by encoding quantum information into the presence/absence of an atom trapped by a given storage tweezer (red) in one of the hyperfine states of the ground-state manifold ${}^1{S}_0$. The latter is laser-coupled to the meta-stable excited state ${}^3{P}_0$, with Rabi frequency ${\mathrm{\Omega }}_c$ and detuning ${\mathrm{\Delta }}_c$, trapped by a second transport tweezer (green). Interactions between pairs of atoms are turned on by exciting the atom to the Rydberg state ${}^3{S}_1$, using a Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}$, where ${\mathrm{\Delta }}_{\mathrm{R}}$ is the corresponding detuning. Other hyperfine levels, energy resolved using a magnetic field and coupled through a microwave frequency ${\mathrm{\Omega }}_F$, serve as a qubit register (Q).

Fig. 3.

Fermionic subroutines. In (A), we indicate the elementary fermionic gates, namely the tunneling and interaction gate. Using these elementary gates, we can construct more complex gates like a density-dependent tunneling gate (B) and a pair-tunneling gate (C) where a pair of fermions can tunnel together from the hatched to the unhatched sites and vice versa. The specific angles required for an exact circuit decomposition of these compiled gates are provided in Eqs. 7 and 9.

Fig. 4.

Variational circuit for VQE. (A) A variational circuit used to prepare the ground-state of the LiH molecule. (B) Average energy difference $\overline{\delta E}$ in the presence of fluctuations in the trapping frequency and tweezer positions, characterized by standard deviations ${\mathrm{\Delta }}_{{\omega }_r}$ and ${\mathrm{\Delta }}_r$, respectively. $V_0$, ${\omega }_r$, and $r_{\mathit{zp}}$ denote the depth, radial frequency, and zero-point fluctuations of the harmonic trap, and $\tau$ is the transfer pulse time (Methods). The horizontal line signals chemical accuracy, $\delta E^{\ast }\approx 1.59$ mHa.

Fig. 5.

Fermion-qubit quantum circuits. (A) Decomposition of $C_i{\mathcal{U}}_{j,k,l}^{(\mathrm{dt})}$$({\theta }_1,{\theta }_2)$, a density-dependent fermionic hopping controlled by ancillary qubit $(i)$ into fermionic gates and the three-body fermion-qubit gate $C_i{\mathcal{U}}_{j,k}^{(\mathrm{int})}$. The angles of the interaction and the controlled interaction gate are $\pi$. The angles of the first and third tunneling gates are $({\theta }_1,{\theta }_2,0)$, whole for the second and fourth are $({\theta }_1,-{\theta }_2,0)$. (B) Trotter step required to time-evolve one plaquette under the ${\mathbb{Z}}_2$ LGT Hamiltonian Eq. 12), where atoms $0$ to $4$ and $5$ to $7$ encode local matter and gauge fields, respectively. The unitary circuits ${\mathcal{U}}^{(p)}$, with $p\in \{m,E,J,B\}$, implement the exponential of each term in the Hamiltonian, and the single-qubit rotations are given by ${\tilde{R}}^x(\theta )=e^{-i(\theta /2){\sigma }^x}$ (green hexagons) acting on the $\{|\tilde{1}\rangle ,|1\rangle \}$ subspace, while ${\mathcal{U}}_i^{(\mathrm{ph})}(\theta )=e^{-i\theta n_i}$ (orange box) is a local phase shift if a particle is present in tweezer $i$. We further use standard notations for qubit operations, namely, H indicates a Hadamard gate $H=(-|1\rangle +|\tilde{1}\rangle )\langle 1|/\sqrt{2}+(|1\rangle +|\tilde{1}\rangle )\langle \tilde{1}|/\sqrt{2}$, a dot denotes a control on the state $\left|1\right.〉$, and two connected dots correspond to a CZ gate. The angles of the interaction gates are $\pi$, whereas the angles of the tunneling gates are $({\lambda }_J\delta t,0,0)$.

Fig. 6.

Robustness to trap inhomogeneity. (A) The achievable physical time in digital simulation is extended by two orders of magnitude when utilizing the positional echo procedure in one dimension. There, it is especially effective as the relative phase noise becomes bounded (Materials and Methods). The simulation was performed for a simple hopping Hamiltonian with periodic boundary conditions, a hundred sites, and a Gaussian phase disorder with ${\sigma }_{\theta }$=0.035 applied between tunneling events (see main text and Materials and Methods). (B) For a generic many-body system, the phase noise is no longer constant, but the echo procedure still improves coherence of the system. For a 10 $\times$ 10 square lattice, under the same conditions as the one-dimensional case, the available simulation time is extended by an order of magnitude.