Detection Loss Bounds the Assembly and Parallelism Bounds the Loop: The Vacancy Map of a Moving-Tweezer Atom Array

1. Introduction

Rubidium tweezer arrays at cryogenic temperature now hold single atoms for thousands of seconds [1] and are assembled defect-free at the thousand-atom scale by a single moving tweezer that transports one atom per move [2,3]. The same architecture is a candidate for the sustained operation that error correction requires, where a lost atom is carried by the code as a located erasure until a rearrangement pass refills its site [4,5,6], and where continuous refilling has been demonstrated on other platforms with deflector-pair rearrangement [7,8] and lattice reloading [9].

Assembly and refilling are usually treated as different problems with different figures of merit: assembly as a budget of trap lifetime against assembly time, refilling as a rate of loss against a rate of reload. They are the same process. In both, a count of vacancies is served by passes whose duration depends on how many atoms they move, while the array keeps losing atoms during the pass. The companion paper [10] writes this process as a queue with state-dependent service and studies it for the deflector-pair architecture; here we specialize it to the moving tweezer, where the per-atom cost is a fixed millisecond and the pass duration is set by the planner, and we calibrate every coefficient to the cryogenic experiments.

The two readings of the map give opposite design levers. In assembly the residual vacancy count is set by the losses during one cycle, which at cryogenic lifetimes are the per-image detection loss and almost nothing else; the number of moving beams does not enter, and the defect-free size limit is a statement about imaging. In a sustained loop the standing vacancy density is set by the detection-to-repair latency, again independent of the tweezer; but the array size the loop can serve at all is set by the tweezer’s throughput, that is, by its parallelism, and by how the pass duration grows with the number of moves. Section 2 states the map and its calibration, Sec. Section 3 checks it against the published assembly, Sec. Section 4 derives the assembly size limit, Sec. Section 5 gives the loop’s size limit and its scaling with parallelism, and Sec. Section 7 names the two measurements that fix both.

2. One Map Serves Assembly and Reload

2.1. Assembly and Reload Differ Only in Whether the Reservoir Refills

An array has N target sites and D vacancies. Time is measured in units of 10 ms, the syndrome round of a sustained loop, and we call that unit a round even in the assembly reading. Each round every atom present is lost with probability p, so the array loses a Poisson number with mean $p(N-D)$. A pass images the array (a per-atom loss $p_d$), plans, and moves n atoms from a reservoir into the vacant sites, lasting

$$t\left(n\right)=c_0+c_1{n}^{\gamma }\mathrm{rounds},$$

(1)

with $c_0$ the fixed overhead of imaging, planning and transfer, $c_1$ the per-move cost and $\gamma$ the exponent with which the pass grows in its content. A moved atom arrives with probability $1-p_t$; the site of a lost atom stays vacant. The greedy pass takes every vacancy present, and at pass boundaries, with losses at their mean, the vacancy count obeys

$$D^{\prime }=p_{t}D+(N-D)\left(p(c_0+c_1{D}^{\gamma })+p_d\right).$$

(2)

The first term is the vacancies the pass failed to fill; the second is the loss during the pass, in flight and at the image.

Assembly is the map iterated from the stochastic load: $D_0=(1-f)N$ for a fill fraction f, a reservoir equal to the surplus of the load that is not refilled, and $D_k$ after k cycles; for Poisson holes the defect-free probability is $e^{-D_k}$. The sustained loop is the same map with a reservoir that is refilled and a pass that starts whenever a vacancy is present; its working point is the stable fixed point $D_{\mathrm{lo}}$ of Eq. (2), and for $\gamma >1$ the map can also have an unstable fixed point $D_{\mathrm{hi}}$ above which each pass takes longer than the losses it repairs [10]. Nothing in the map distinguishes the two readings except whether the reservoir refills.

2.2. Every Coefficient Comes from the Cryogenic Experiments

The moving-tweezer coefficients come from the two published assemblies in 2,000-site cryogenic arrays. The 4 K experiment [3] assembled 1,024 atoms in a 2,070-trap two-wavelength array and reports a vacuum lifetime of about 5,000 s, a detection loss of $0.15\%$ per image, transport survival of $98.7\%$ and move success of $99.05\%$, a first cycle of about 2 s with about 1,014 moves and a second of about 1.5 s with 135 moves, an average defect rate of $0.3\%$, and defect-free registers in more than ten percent of realizations. Its predecessor at 6 K [2] assembled 828 atoms in 2,088 sites, from a $53\%$ load, in cycles of about 1.65 s (about 150 ms of planning, 500 ms of transfer, and under a second of execution for about 800 moves), with a lifetime of about 100 s at the time, and reached $95\%$ occupancy after two cycles. Trap lifetimes of 6,000 s have been measured in the same kind of cryostat [1].

From these: the per-move cost is about a millisecond, $c_1=0.1$ round, and it does not depend on N; the cycle overhead is what remains of a cycle once its moves are subtracted at a millisecond each, about 1.3 s from the second cycle (1.5 s less 135 ms) and about 1 s from the first (2 s less 1,014 ms), and we take $c_0=130$ rounds for assembly; the background loss is $p=2\times {10}^{-6}$ per round at 5,000 s; the per-image loss is $p_d=0.15\%$, which we read as the loss per image (if the published figure is a survival per rearrangement sequence rather than per image, the per-image loss is smaller and the assembly limit of Sec. Section 4 is pessimistic by the same factor); the transport loss is $p_t=0.013$ from the transport survival alone and $0.02$ with the failed moves stacked on, and we carry both. Stacking the failed moves onto the loss is conservative, since a failed move usually leaves the atom in its origin trap, where it is still in the reservoir for the next cycle. For the sustained loop of Sec. Section 5 the overhead $c_0$ is the detection-to-repair latency rather than today’s planning and transfer time, and we take $c_0=10$ rounds (100 ms), the value of the companion paper, so that the two architectures share a cadence; the loss rate there is operational, readout loss on the atoms imaged and Rydberg loss on the atoms gated, and at the measured $0.15\%$ per image with half the atoms imaged every round it is $p\approx 7\times {10}^{-4}$, which we round to the bracket $p={10}^{-3}$, with $p={10}^{-2}$ as the gate-loss bracket. The 10 ms round, the half-array mid-circuit imaging and the 100 ms latency are design targets like $c_0$ itself, not measurements; the ${10}^{-2}$ bracket is an atom that survives ten rounds and is a bound rather than a scenario. The exponent $\gamma$ is the one number no experiment reports, and Sec. Section 6 says why it is a property of the planner.

2.3. A Discrete-Event Run Checks the Map Under Conservation Laws

The map is deterministic and at pass boundaries. The stochastic process behind it was run as a discrete-event world on the verification-first engine of the companion paper [10]: one state machine whose every transition publishes a projection checked against nine runtime invariants, two of which are conservation laws (every vacancy was made by a loss and unmade only by a repair; every atom taken from the reservoir either filled a site or was lost in transit). Randomness enters through two uniforms per round and one per pass, converted to counts by inverse CDF, so paired arms consume the same draws and an audit confirms it. The per-image loss is folded into the per-round rate as $p_d/c_0$ in the engine and kept separate in the map. Error bars are 95% half-widths; every run reported here completed with no invariant finding.

3. The Map Reproduces the Published 1,024-Atom Assembly

The first cycle of a shot-based assembly is not a pass over vacancies. The loaded atoms sit at random positions in a 2,000-site array, and the first cycle relocates nearly all of them into the target region: about 1,014 moves for the 1,024-site target and about 800 for the 828-site one, of order N rather than of order the vacancy count, with a reservoir that is the margin of the load over the target (a $50\%$ load of 2,070 traps is about 1,035 atoms for 1,024 sites). That compaction is outside the map, whose pass moves one atom per vacancy, and it is why the experiment entered its second cycle with 135 vacancies where a vacancy pass from a half-filled target would leave about ten. The map describes the second cycle and every cycle after it, which is the regime a sustained loop lives in, and that is the check made here. Table 1 runs the assembly reading at the calibration of Sec. Section 2.2 from the measured start of the experiment’s second cycle ($D_0=135$, one pass), then iterates the map’s own passes from a half-filled target ($D_0=500$, two passes) to reach its residual floor, with the room-temperature-lifetime predecessor beside it.

From 135 holes, one pass leaves 3.2 to 4.2 against the measured average of 3.1. The map’s own passes converge to 1.9 to 2.0 holes, the floor of Sec. Section 4, where the engine’s defect-free fraction is 10 to 11 percent against the published “more than ten”. The engine and the map agree to three percent in every cell, which is the statement that shot-based assembly is the vacancy map with the reservoir not refilled. The 828-atom assembly is reproduced to within the uncertainty of its lifetime: the map puts the residual at $1.8\%$ at the reported 100 s and at $4\%$ at 40 s against $5\%$ measured, and the lifetime during that run is the least certain of the inputs. Two differences are informative. The map’s Poisson defect-free fraction from the floor of Sec. Section 4 is $e^{-1.84}=16\%$ where the engine measures 10 to 11 percent; the engine’s holes after the second pass have a variance above the mean because the transport failures of the first pass propagate, and the published fraction sits between the two. And the direct comparison from 135 holes gives a defect-free fraction of 1 to 3 percent, below the published ten, at the same mean: the published hole distribution is narrower than Poisson, which a scalar model cannot represent and which the experiment’s per-shot hole counts would settle.

4. Detection Loss Sets the Assembly Size Limit

The fixed point of Eq. (2) in the assembly reading is the residual the cycles converge to,

$${\overline{D}}_{\mathrm{asm}}\approx {\displaystyle \frac{N(p{c}_0+p_d)}{1-p_t}},$$

(3)

since $c_{1}D$ is negligible against $c_0$ once D is small. At the cryogenic calibration the two terms in the numerator are $p{c}_0=2.6\times {10}^{-4}$ and $p_d=1.5\times {10}^{-3}$: the per-image loss is six times the background loss over the whole 1.3 s cycle, and the residual of 1.8 holes at $N=1024$ is 1.5 of them from the image. Three consequences follow.

The largest array that can be assembled defect-free with order-one probability is where the floor reaches one hole,

$$N_1\approx {\displaystyle \frac{1-p_t}{p{c}_0+p_d}},$$

(4)

which is 560 atoms at the measured $p_d=0.15\%$ and $c_0=130$, 1,300 at $p_d=0.05\%$, and 2,700 at $0.01\%$; at $0.01\%$ with the overhead cut to 100 ms it is 8,000 (Table 2, Figure 1). The defect-free probability at a given N is $e^{-N/N_1}$ for Poisson holes, so the 1,024-atom experiment sits at $N/N_1=1.8$ and a 2,048-atom target at the same imaging loss would be defect-free in about two percent of shots.

Parallelism does not enter. With m moving beams the per-move cost is $c_1/m$, the cycle is shorter by at most the execution time, and the residual after two cycles from a half-filled 1,024-site target is 2.0 holes for $m=1,2,4$ and 8 alike; at 2,048 sites it is 4.0 for every m. Faster moves shorten the cycle that the reservoir atoms wait through, which at a 5,000 s lifetime costs nothing, and do not touch the image. The lever for assembly is $p_d$ first and $c_0$ second, and Table 2 says by how much: at the measured imaging loss, cutting the overhead from 1.3 s to 100 ms buys 15 percent in $N_1$; at $0.01\%$ imaging loss it buys a factor of three.

The room-temperature ceiling is the same formula with the background term restored. At a 20 s lifetime, a 100 ms cycle and $0.5\%$ imaging loss, $N_1$ is about a hundred atoms; at 60 s and $0.15\%$ it is about three hundred. That is why room-temperature assembly saturated at a few hundred atoms and why the cryogenic lifetime, which removes the first term, was what allowed the thousand-atom milestone; and it is why the next factor of two is in the second term, which the cryostat does not touch.

5. Parallelism Sets the Reload-Loop Size Limit

In the sustained reading the reservoir refills and a pass starts whenever a vacancy is present. Two things then hold at every array size, and a third depends on the size.

At low move duty the standing vacancy density is $p{c}_0$, whatever the tweezer does. A vacancy detected at one image stands until the pass that follows the next one, so the working point is about $pN{c}_0$ vacancies plus the content of the pass in flight: at $p={10}^{-3}$ and $c_0=10$, one percent of the array, the same density a deflector pair leaves at the same cadence. The pass in flight grows with the move duty (Table 3: 1.1 percent of the array at a thousand sites, 4.4 percent at eight thousand), so the floor is the latency’s alone only while the beam is mostly idle. The remedy for that floor is a shorter detection-to-repair latency, not a faster tweezer.

The pass cannot be served one atom at a time: with $c_0=10$ a one-move pass costs ten rounds, so the loop batches, and the batch it can sustain is bounded by its throughput. At $\gamma =1$ the loop keeps up while the move duty $pN{c}_1$ is below $1-p_t$, that is below $N=9,800$ sites at $p={10}^{-3}$ and 980 at $p={10}^{-2}$, and the working point climbs toward that line.

The array size one beam can serve depends on the exponent (Table 3), and Sec. Section 6 says why $\gamma =1$ is the expectation for a moving tweezer at these vacancy counts and $\gamma >1$ the risk case. At $\gamma =1$ the working point is 11 vacancies at a thousand sites, 67 at four thousand and 355 at eight thousand, with the measured mean about 1.5 times the map’s value at every size (the pass in flight), no crossing in any replication, and the working point passing a quarter of the array near $1.3\times {10}^4$ sites. At $\gamma =1.5$ the loop has a basin at a thousand sites, a working point of 17 with an unstable boundary at 96, and a burst of 64 vacancies is recovered in every replication; at two thousand sites the working point is gone and every replication diverges from zero with no burst at all. At $\gamma =1.25$ the working point is gone between two and three thousand sites; at $\gamma =2$, near five hundred. At the gate-loss bracket the limits are 900, 310 and below 250 sites for $\gamma =1$, $1.25$ and $1.5$.

Parallelism is the lever here, and its return diminishes with the exponent (Table 4, Figure 2). With m beams the per-move cost is $c_1/m$; for $D\ll N$ the unstable boundary is $D_{\mathrm{hi}}\approx {(pN{c}_1/m)}^{-1/(\gamma -1)}$, and the size at which it meets the working point scales as $m^{1/\gamma }$: linearly for a linear pass, as $m^{2/3}$ at $\gamma =1.5$, as $m^{1/2}$ for a quadratic one. Sixteen beams carry a linear pass to $2\times {10}^5$ sites and a $\gamma =1.5$ pass to 7,400. A hybrid that uses a deflector pair for the bulk and the moving tweezer for the residue is, in the map, a large m for the scattered vacancies and a small one for the clustered ones, and which of the two governs is the exponent question of the next section.

6. The Exponent Is a Property of the Planner

For a deflector pair the pass duration is concave in the number of vacancies, because one step moves a whole row and the number of distinct rows hit saturates [10]. For a moving tweezer every move is one step, so the execution time is linear in the number of moves by construction, and the only place an exponent above one can enter is the planning: the assignment of reservoir atoms to vacancies is solved exactly (cubic in the count) or greedily (linear), and the ordering of the moves so that no move crosses an occupied site ranges from linear to quadratic in the number of moves depending on the algorithm. The published cycles planned about 800 to 1,000 moves in about 150 ms. A planner with that cost at a thousand moves, even if cubic, costs milliseconds or less at the tens to low hundreds of moves a sustained loop presents, well below the 100 ms latency $c_0$; so at loop vacancy counts the expectation for a moving tweezer is $\gamma =1$, a pass dominated by its fixed overhead, and the size limit is the linear row of Table 4. The $\gamma >1$ rows are the risk case: a pass that is replanned with a collision-avoidance step whose cost grows with the occupied density rather than with the move count, or a pass that waits on a reservoir refill that itself scales with the vacancy count. A planner quadratic in the count would give the loop of Table 3 at $\gamma =1.5$ to 2, with a basin of a few hundred atoms and a size limit of a thousand, where the linear case gives ${10}^4$. The exponent is therefore a software choice, and the map says which choice to make: the linear heuristic, replanned per pass, even at some cost in move count. The measurement that establishes the exponent is the pass duration against the number of moves, at operating density, with the planner held fixed and named in the report.

7. Two Measurements Fix the Roadmap

The assembly limit of Sec. Section 4 and the loop limit of Sec. Section 5 each rest on one number the experiments can measure directly.

The first is the per-image loss $p_d$, separately from the background loss. The published $0.15\%$ is the sum of the two over a cycle; at a 5,000 s lifetime the background part is $0.03\%$ and the rest is the image, and Eq. (4) says that each factor of three in $p_d$ is a factor of two to three in the defect-free size. The per-shot hole distribution, not only its mean, is the second half of this measurement, since the published defect-free fraction above ten percent at a mean of 3.1 holes says the distribution is narrower than Poisson, and the model’s $e^{-D}$ is then a lower bound.

The second is the pass duration against the number of moves, with the planner fixed, over the range of vacancy counts a sustained loop would present (tens to hundreds), at two array sizes. It gives $\gamma$ and $c_1$ together, and with p and $c_0$ it places the machine in Table 4: the number of moving beams the roadmap needs at each array size follows from the row.

8. Scope and Limitations

The model is a scalar vacancy count with a single per-round loss rate. It carries no geometry, so it cannot say where the residual holes of an assembly are or how a hole distribution narrower than Poisson arises; the sustained reading draws one loss rate for imaged and gated atoms alike where a real loop loses atoms per operation, and the companion paper’s distance-tax estimate, which depends on where the holes stand, is not repeated here for that reason. The assembly calibration reads its inputs from the published text of two experiments and is a check of the map’s form, not a fit: the cycle overhead is inferred from the difference of two cycle times, the per-image loss is folded into the engine’s per-round rate, and the 828-atom lifetime during its run is uncertain by a factor of two. The sustained loop’s $c_0=10$ rounds is a design target, not a measured value; today’s shot-based overhead is 130 rounds, and the standing density scales with whichever is achieved. The reservoir’s refill rate in the loop studies ($2pN$ per round) is an assumption with no measurement behind it, and a reservoir loaded from a magneto-optical trap or a conveyor is itself imaged and lost like the array. The exponent $\gamma$ is the declared unknown, and the parallelism table is a statement about the map, checked stochastically at one beam only. A reservoir that is itself refilled in batches, once per magneto-optical-trap cycle, would show the working point as a sawtooth at that period.