Scalability of pre-scheduled flying-Rydberg-atom cluster-state generation in thermal microwave cavities

Yang and Gong (2008) proposed generating two-dimensional cluster states for one-way quantum computing in a single pass: N Rydberg atoms fly from one source with pre-set velocities and emission times through an array of thermal microwave cavities, every atom pair is made to collide inside one cavity, and a strong-classically-driven controlled-phase gate, insensitive to cavity decay and thermal photons, entangles those pairs that represent neighboring qubits of the target cluster lattice. The pre-set collision-scheduling equations have reported solutions only for N at most 6 atoms and may admit none for larger N; extending the scheme to large cluster states would require either a general scheduling solution, mid-flight velocity control of the atoms by optical dipole forces (never integrated with this architecture), or both, together with a full error budget (fast-oscillating terms discarded in the gate derivation, pulse imperfections, Stark-shift switching of cavities, thermal Fock-state populations, and roughly 80% single-atom detection efficiency) that meets fault-tolerance thresholds. No follow-up work extends the flying-atom thermal-cavity scheme, and adjacent results — deterministic bimodal-cavity cluster schemes, coherent transport of entangled neutral-atom arrays, and circular Rydberg qubits in optical tweezers — address the same objective in other architectures without touching the flying-atom thermal-cavity core, which therefore remains open.

Background

One-way (measurement-based) quantum computing replaces the sequence of gates of the circuit model by (i) preparing a highly entangled multi-qubit resource state and (ii) driving the computation by adaptive single-qubit measurements alone (Raussendorf and Briegel, 2001). For universal one-way computation the resource is a two-dimensional cluster state: qubits on a lattice whose entanglement is fixed by preparing each qubit in a superposition state and applying an entangling controlled-phase (CZ) operation along every edge. A leading experimental question for any platform is how to generate large two-dimensional cluster states deterministically.

Cavity quantum electrodynamics (QED) with Rydberg atoms — atoms excited to principal quantum numbers around 50, so that the atomic dipole couples strongly (coupling $g=2\pi\times25$ kHz) to microwave cavity modes — is a mature platform in which two atoms can even be entangled by making them collide inside one cavity while a field locks their phases (Osnaghi et al., 2001). Proposals by Zou and Mathis (2005) and Cho and Lee (2005) showed how suitable cavity-QED interactions, assisted by a strong classical driving field, implement a CZ gate whose photon-number-dependent parts cancel, making it insensitive to cavity decay and to the residual thermal photons of a 'thermal' (not ground-state-cooled) cavity.

Yang and Gong (2008) combined these ingredients into a flying-atom architecture: a single source emits $N$ atoms with pre-selected horizontal velocities $v_i$ and emission times $t_i$; after a Ramsey $\pi/2$ pulse prepares each atom in $(|0\rangle+|1\rangle)/\sqrt{2}$, an array of $2N-3$ cavities placed at distances $L_k$ from the source is arranged so that every pair of atoms $(i,j)$ crosses the center of some cavity simultaneously, knitting a two-dimensional cluster state in one pass. The resonant CZ gate is applied only at the crossings of pairs representing neighboring qubits in the cluster lattice; for all other crossings, as in cavities not currently hosting a target pair, the cavity is Stark-switched far off resonance by an electric field across the mirrors. The scheduling condition for every pair to collide is equation (10) of that paper,

$L_{i+j-2}\left(\frac{1}{v_j}-\frac{1}{v_i}\right)=t_i-t_j,$

with $i=1,\dots,N-1$, $j>i$, and $t_1=0$: $N(N-1)/2$ coupled nonlinear equations in the $4(N-1)$ variables $\{v_i\}$, $\{t_i\}$, $\{L_k\}$. The authors report numerical solutions for $N\le6$ (a table is given for $N=4$, with velocities 100–250 m/s, sub-millisecond emission delays, and cavity positions spanning centimeters) and state that the system 'may not give any solution for $N\ge7$'. Note that for $N\ge9$ the equations outnumber the unknowns even by count. This all-pairs scheduling burden is specific to the single-source, single-pass arrangement; a contemporaneous alternative by Blythe and Varcoe (2006) avoided it by clocking two perpendicular atomic beams through a collision cavity so that collisions are guaranteed geometrically.

For algorithms needing at most 6 physical qubits the scheme needs 'no much change', but for larger clusters the authors acknowledge that equation (10) 'cannot be satisfied by simply setting $\{v_i\}$, $\{t_i\}$, $\{L_i\}$ before the computing process'. Their proposed remedy is to change atom velocities mid-flight using the optical dipole force (light-induced forces on the neutral atoms), citing then-recent optically transported atoms in cavity QED; the required integration — achievable accelerations, timing jitter, and induced decoherence for flying Rydberg atoms inside a microwave cavity array — was not demonstrated. They further identify unquantified error sources for scaling: fast-oscillating terms discarded when deriving the effective gate Hamiltonian (which cause Stark shifts of the coupled states), imperfections of the strong classical drive pulses, residual Stark-shift errors from switching cavities far off resonance, the effect of an initial thermal Fock state (photon-number state) of the cavity beyond the small-excitation regime, and the single-atom measurement efficiency of ionization detectors, then 'more than 80%', for which the authors note only that still higher measurement efficiency is expected in cavity-QED experiments. Their own earlier thermal-cavity analysis (Yang, Gong, Li, and Yang, 2007) indicated these errors have slight influence in the small-excitation regime, but no large-$N$ budget was attempted. The physical parameters cited as realistic: cavity photon storage time 1 ms, Rydberg radiative lifetime $3\times10^{-2}$ s, gate interaction time $\sim10^{-5}$ s, whole process within a few milliseconds, velocity selection to $\pm2$ m/s and emission timing to $\pm2\mu$s, apparatus length about 20 cm.

Problem Statement

Determine whether the single-pass, pre-scheduled flying-Rydberg-atom thermal-cavity scheme of Yang and Gong (2008) can be extended deterministically to large-scale two-dimensional cluster states for one-way quantum computing. This comprises, as parts of one objective:

(i) Scheduling: settle the solvability of the collision-scheduling system

$L_{i+j-2}\left(\frac{1}{v_j}-\frac{1}{v_i}\right)=t_i-t_j,\quad i=1,\dots,N-1,\; j>i,$

under the physical constraints of the architecture (positive, selectable velocities within experimental precision; ordered emission times; distinct cavity positions within a bounded apparatus; no simultaneous triple occupation of a cavity mode volume) for cluster sizes beyond the reported $N\le6$. Part (i), which concerns pre-set schedules alone (mid-flight velocity changes belong to part (ii)), is settled by one of two all-$N$ results: (i-a) a pre-set schedule family valid for all $N$, equivalently a general scheduling algorithm that constructs one; or (i-b) a proof that no pre-set solution exists once $N$ exceeds some finite bound (the source suggests possible failure already at $N\ge7$). A schedule for a single finite target size does not by itself settle (i); the no-go under (i-b) is what brings the mid-flight-control route of part (ii) into scope.

(ii) Velocity control: should part (i) establish that no pre-set schedule exists for large $N$, establish whether mid-flight acceleration/deceleration of the neutral Rydberg atoms by optical dipole forces can be integrated with the microwave-cavity architecture — the accelerations achievable, the timing jitter tolerable against the $\sim10^{-5}$ s gate interactions and $\mu$s-scale scheduling precision, and the decoherence induced over the few-ms flight — so that the collision schedule can still be enforced for large $N$.

(iii) Fidelity at scale: quantify the accumulated error for large cluster states from the identified sources — the fast-oscillating terms discarded in deriving the effective gate Hamiltonian (and the associated Stark shifts), imperfections of the strong classical drive pulses, residual Stark-shift errors from switching cavities between resonance and far-off resonance, the effect of initial thermal cavity Fock states beyond the small-excitation regime, and single-atom detection efficiency of roughly 80% — and determine whether the end-to-end cluster-state fidelity can meet fault-tolerance thresholds as $N$ grows, where the relevant threshold is the percent-level per-operation error budget tolerated by surface-code quantum error correction.

An admissible answer is a constructive scaling protocol — pre-set scheduling, mid-flight velocity control, or both — together with its large-$N$ error budget; a proof of impossibility that rules out deterministic large-scale cluster-state generation in this architecture under realistic parameters, covering the scheduling routes and the fidelity budget rather than a single branch in isolation; or an experimental demonstration of deterministic two-dimensional cluster-state generation in this architecture beyond the reported $N\le6$, accompanied by error accounting that establishes scalability as $N$ grows. A resolution of only part (i) settles the scheduling subquestion but not the stated scalability objective: a no-go for the pre-set scheduling system leaves the mid-flight-control route open, and any constructive or experimental route still requires the fidelity budget of part (iii).

References

1. Yang, W.-X., Gong, Z.-X. Efficient scheme for one-way quantum computing in thermal cavities. Int. J. Theor. Phys. 47, 2997–3004 (2008). DOI: 10.1007/s10773-008-9734-x; arXiv:0704.2297. https://doi.org/10.1007/s10773-008-9734-x
2. Raussendorf, R., Briegel, H. J. A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5190 (2001). DOI: 10.1103/PhysRevLett.86.5188. https://doi.org/10.1103/PhysRevLett.86.5188
3. Zou, X. B., Mathis, W. Schemes for generating the cluster states in microwave cavity QED. Phys. Rev. A 72, 013809 (2005). DOI: 10.1103/PhysRevA.72.013809. https://doi.org/10.1103/PhysRevA.72.013809
4. Cho, J., Lee, H.-W. Generation of atomic cluster states through the cavity input-output process. Phys. Rev. Lett. 95, 160501 (2005). DOI: 10.1103/PhysRevLett.95.160501. https://doi.org/10.1103/PhysRevLett.95.160501
5. Blythe, P. J., Varcoe, B. T. H. A cavity-QED scheme for cluster-state quantum computing using crossed atomic beams. New J. Phys. 8, 231 (2006). DOI: 10.1088/1367-2630/8/10/231; arXiv:quant-ph/0605190. https://doi.org/10.1088/1367-2630/8/10/231
6. Osnaghi, S., Bertet, P., Auffeves, A., Maioli, P., Brune, M., Raimond, J. M., Haroche, S. Coherent control of an atomic collision in a cavity. Phys. Rev. Lett. 87, 037902 (2001). DOI: 10.1103/PhysRevLett.87.037902. https://doi.org/10.1103/PhysRevLett.87.037902
7. Yang, W.-X., Gong, Z.-X., Li, W.-B., Yang, X.-X. Simple scheme for implementing the Deutsch–Jozsa algorithm in a thermal cavity. J. Phys. A: Math. Theor. 40, 155–165 (2007). DOI: 10.1088/1751-8113/40/1/009. https://doi.org/10.1088/1751-8113/40/1/009
8. Song, M.-Y. Efficient scheme for one-way quantum computing with radiofrequency SQUID qubits by adiabatic passage. Commun. Theor. Phys. 56, 883 (2011). DOI: 10.1088/0253-6102/56/5/09. https://doi.org/10.1088/0253-6102/56/5/09
9. Gonta, D., Radtke, T., Fritzsche, S. Generation of two-dimensional cluster states by using high-finesse bimodal cavities. Phys. Rev. A 79, 062319 (2009). DOI: 10.1103/PhysRevA.79.062319; arXiv:0903.3167. https://doi.org/10.1103/PhysRevA.79.062319
10. Asavanant, W., Shiozawa, Y., Yokoyama, S., Charoensombutamon, B., Emura, H., Alexander, R. N., Takeda, S., Yoshikawa, J.-i., Menicucci, N. C., Yonezawa, H., Furusawa, A. Generation of time-domain-multiplexed two-dimensional cluster state. Science 366, 373–377 (2019). DOI: 10.1126/science.aay2645. https://doi.org/10.1126/science.aay2645
11. Bluvstein, D., Levine, H., Semeghini, G., Wang, T. T., Ebadi, S., Kalinowski, M., Keesling, A., Maskara, N., Pichler, H., Greiner, M., Vuletić, V., Lukin, M. D. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451–456 (2022). DOI: 10.1038/s41586-022-04592-6. https://doi.org/10.1038/s41586-022-04592-6
12. Bluvstein, D., Evered, S. J., Geim, A. A., Li, S. H., Zhou, H., Manovitz, T., Ebadi, S., Cain, M., Kalinowski, M., Hangleiter, D., Bonilla Ataides, J. P., Maskara, N., Cong, I., Gao, X., Sales Rodriguez, P., Karolyshyn, T., Semeghini, G., Gullans, M. J., Greiner, M., Vuletić, V., Lukin, M. D. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). DOI: 10.1038/s41586-023-06927-3. https://doi.org/10.1038/s41586-023-06927-3
13. Hölzl, C., Götzelmann, A., Pultinevicius, E., Wirth, M., Meinert, F. Long-lived circular Rydberg qubits of alkaline-earth atoms in optical tweezers. Phys. Rev. X 14, 021024 (2024). DOI: 10.1103/PhysRevX.14.021024; arXiv:2401.10625. https://doi.org/10.1103/PhysRevX.14.021024