Scalable, single-site-verified cluster-state entanglement of neutral atoms in optical lattices by parallel controlled-collision gates and its use for one-way quantum computation

In 2003, Mandel, Greiner, Widera, Rom, Hänsch and Bloch demonstrated that controlled collisions between neighbouring neutral atoms in an optical lattice — produced by spin-dependent transport of a Mott insulator — act as a massively parallel array of entangling quantum gates, and observed coherent entangling–disentangling dynamics in a one-dimensional chain of roughly one hundred thousand atoms. The evidence was indirect: the disappearance and reappearance of Ramsey fringes and spatial interference patterns, without single-atom-resolved detection, so the size of the entangled cluster was never measured, the demonstration was strictly one-dimensional, and its extension to two and three dimensions was argued to require lattice fillings above a percolation threshold (31% for a 3D simple-cubic lattice) in the presence of defects. The paper left open how to turn this proof of principle into a scalable resource: single-site, single-atom correlation measurements that quantitatively verify the many-body entanglement, defect control sufficient for higher-dimensional cluster growth, and single-particle operations and measurements that would allow the cluster state to be consumed for one-way (measurement-based) quantum computation. Two decades later the detection and addressing primitives exist in the form of quantum gas microscopes and single-spin addressing, and measurement-based computation has been realized on photonic and Rydberg platforms; but no experiment has combined the controlled-collision gate array with single-site detection to generate and verify macroscopic, multi-dimensional cluster entanglement in an optical lattice, and scalable entanglement generation and verification in optical lattices remains at the stage of theoretical schemes. The open problem is to realize, and quantitatively verify with single-site resolution, large-scale cluster-type entanglement of neutral atoms in an optical lattice by parallel entangling gates, robust to lattice defects and extended beyond one dimension, and to demonstrate its use as a computational resource for measurement-based quantum computation.

Background

An optical lattice is a periodic array of microscopic light-shift traps formed by interfering laser beams; when a Bose–Einstein condensate is loaded into it and the lattice depth is increased, the gas undergoes a quantum phase transition to a Mott insulator, a state in which a fixed number of atoms (ideally exactly one) occupies each lattice site, realizing a vast register of individually trapped neutral-atom qubits (typically two long-lived hyperfine clock states of the $^{87}$Rb atom, which serve as the qubit). In 1999 Jaksch, Briegel, Cirac, Gardiner and Zoller proposed a scalable entangling mechanism for such a register: a spin-dependent optical lattice, whose two polarization components trap and transport the two qubit states in opposite directions, is used to split the wave packet of each atom and shift it half a lattice spacing onto the neighbouring site, where the atoms collide for a controlled time $t_{\text{hold}}$ and acquire a collisional phase shift $\varphi = U_{01}t_{\text{hold}}/\hbar$, with $U_{01}$ the onsite interaction energy between atoms in the two internal states. Run simultaneously across the lattice, this implements an array of parallel conditional-phase (Ising-type) quantum gates. Briegel and Raussendorf showed that a string or array of atoms coupled by such nearest-neighbour interactions evolves into a cluster state — a highly entangled graph-type state in which each qubit is entangled with its neighbours — and Raussendorf and Briegel showed that cluster states are a universal resource for the one-way quantum computer: computation is performed by sequences of single-qubit measurements alone, with the entanglement of the resource state supplying the computational power. Mandel and co-workers realized the gate array experimentally in 2003 in a one-dimensional geometry covering about one hundred thousand lattice sites, observing that the visibility of Ramsey fringes (oscillations of the spin population versus the phase of a final microwave pulse, an interferometric signature sensitive to entanglement) dropped to about 5% when the collisional phase reached $\varphi=\pi$ and revived at $\varphi\approx 2\pi$, for up to four entangling–disentangling cycles. As the authors emphasized, this fringe dynamics is indicative but not a proof of large-scale entanglement: a vanishing Ramsey fringe is insensitive to the number of entangled particles, single-atom-resolved correlation measurements were unavailable, vacancies (from imperfect Mott preparation, a surrounding superfluid shell, and roughly 5% spontaneous-emission excitations, plus about 5% microwave pulse-area errors) break the chain at defects, and in one dimension a single vacancy limits the cluster. The paper proposed that two additional state-dependent lattice shifts would extend the scheme to two and three dimensions, where a filling factor above the percolation threshold (31% for a 3D simple-cubic lattice, the point above which occupied sites form a system-spanning connected cluster) would still yield a large entangled cluster, and envisioned the resulting cluster states as resources for one-way quantum computing, which requires the capability to perform single-particle operations and measurements on individual sites. The relevant toolkit later emerged independently of this experiment: a quantum gas microscope is a high-numerical-aperture fluorescence imaging system that resolves individual atoms on individual lattice sites, and, because light-assisted collisions (pair loss driven by the imaging light, which removes both atoms of a doubly occupied site) convert the detected pattern into a definite atom-number measurement, it also projects the site occupations; it was first demonstrated for Hubbard-regime lattices in 2009–2010; single-spin addressing by tightly focused beams followed in 2011; and site-resolved correlation measurements of entanglement generated by superexchange (the second-order tunnelling interaction of the Bose– and Fermi–Hubbard models) were demonstrated in 2015–2016. What has not been achieved is the integration of the spin-dependent-transport controlled-collision gate array with this microscopy and addressing machinery to produce and verify the macroscopic, multi-dimensional cluster states envisaged in 2003, or to run a measurement-based computation on them.

Problem Statement

Realize, and quantitatively verify with single-site, single-atom resolution, the large-scale cluster-state entanglement of neutral atoms in an optical lattice that the 2003 controlled-collision experiment proposed but did not deliver, and demonstrate its use as a resource for measurement-based (one-way) quantum computation. Concretely: (1) operate the parallel entangling gate array — spin-dependent transport bringing neighbouring atoms into controlled collisions that imprint the collisional phase $\varphi=U_{01}t_{\text{hold}}/\hbar$ — beyond the original one-dimensional geometry to entangle clusters in two and three dimensions, with lattice preparation whose defect structure and filling factor are controlled well enough (above the 31% percolation threshold of the 3D simple-cubic geometry, or by defect removal) that system-spanning entangled clusters form rather than vacancy-limited fragments; (2) verify the resulting many-body entanglement directly with single-site resolution, determining a certified lower bound on the entangled cluster size and how the entanglement scales with dimension, filling and defect density, replacing the indirect Ramsey-fringe-visibility criterion (which is insensitive to the number of entangled particles) and quantifying the gate-fidelity limits (microwave pulse-area errors, spontaneous-emission excitations, residual dephasing from lattice inhomogeneities and differential light shifts) identified in the source; and (3) combine this with the now-available single-particle operations and site-resolved measurements to run a one-way quantum computing protocol on the lattice cluster state, establishing the entangling–disentangling toolbox of controlled collisions as a scalable, verifiable, computationally usable entanglement resource.

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