Resource overheads and attainable rates for trapped-ion lattice surgery

I. INTRODUCTION

• Reliable two-qubit operations between separate modules require entanglement distribution, whose probabilistic and noisy nature creates communication-qubit overhead and purification demands.Entanglement purification non-deterministically converts many weakly entangled pairs into fewer strongly entangled pairs.Source: §I · p. 1
• The paper estimates ion resources and attainable lattice-surgery rates across cycle times, using trapped-ion technology surveys to propose three feasible surgery-time paradigms.The analysis focuses on lattice surgery between surface-code encoded qubits and a fixed number of communication ions.Source: §I · p. 1

A. Surface code and lattice surgery

• Surface codes combine high error tolerance with nearest-neighbor interactions and support universal quantum computation on encoded qubits.This work uses the rotated surface code with parameters [[d^2, 1, d]].Source: A. Surface code and lattice surgery · p. 2
• Lattice surgery couples two code patches through joint syndrome extraction, requiring at least d rounds to address measurement-induced errors.Unlike transversal gates, lattice surgery uses nearest-neighbor interactions but is slower because measurements can introduce undetected errors.Source: A. Surface code and lattice surgery · p. 2
• For spatially separated distance-d surface codes, lattice surgery requires d entangled pairs per time-step, while d^2 pairs are needed over the full operation.The operation performs an XX parity check between encoded qubits.Source: A. Surface code and lattice surgery · p. 2

B. Trapped-ion architectures

• Elementary Logical Units contain linear Coulomb crystals with communication ions for photonic interconnects and may also include sympathetic-cooling ions.The communication ions create entangled pairs between ELUs.Source: B. Trapped-ion architectures · p. 2
• Linear crystals offer all-to-all connectivity without rearrangement, but their effective length limits the number of ions they can support.Monroe et al. estimated a limit near 100 ions, while later work questioned whether that estimate was optimistic.Source: B. Trapped-ion architectures · pp. 2–3
• Lattice surgery between separated surface-code qubits uses shared entanglement and needs only d pairs per time-step, despite requiring d^2 pairs in total.The lower per-step entanglement rate distinguishes lattice surgery from transversal gates.Source: Figure 2 · p. 3
• Segmented linear traps increase capacity by chaining crystals, but lose all-to-all connectivity and require routing whose average swap count scales linearly with modules and qubits per segment.Monroe et al. estimated that segmented traps might support up to 1000 ions.Source: B. Trapped-ion architectures · p. 3
• QCCD architectures route ion chains through fixed corridors and junctions, offering theoretical connectivity and speed advantages but posing fabrication challenges.The largest cited QCCD is Quantinuum’s 32-qubit racetrack computer, while proposals target around 1000 ions and beyond.Source: B. Trapped-ion architectures · p. 3

C. Trapped-ion surface code implementations

• Trapped-ion surface-code work includes distance-two teleportation, Bacon-Shor implementations, software tools, simulations, segmented-trap studies, and monolithic proposals.The cited implementations and proposals vary substantially in scope and reported performance measures.Source: C. Trapped-ion surface code… · pp. 3–4
• Reported syndrome-cycle times range from around 3–8 ms in simulations to approximately 300 µs in a monolithic surface-code proposal.The simulation result is associated with a distance-3 surface code and a pseudothreshold of 3 × 10^-3.Source: C. Trapped-ion surface code… · pp. 3–4

III. ESTIMATING SURFACE CODE CYCLE TIMES

• Lattice-surgery rates are upper bounded by syndrome-extraction cycle rates, which depend on gates, measurements, shuttling, cooling, and architecture.The paper proposes three feasible cycle-time paradigms conditioned on technological milestones.Source: §III · p. 4

A. Trapped-ion gates

• Typical single-qubit gates above the surface-code threshold take 2–12 µs, while the assumed surface-code gate sequence takes around 10 µs.The estimate assumes operational two-qubit error rates up to 0.1%.Source: A. Trapped-ion gates · p. 4

B. Trapped-ion measurements

• Fluorescence measurements take about 10 µs but scatter photons that can decohere nearby unmeasured qubits during error-correction circuits.The paper considers decoherence suppression and spatially separating ions before measurement.Source: B. Trapped-ion measurements · p. 4

1. Techniques for suppressing decoherence

• Shorter illumination reduces measurement-induced decoherence but lowers fidelity because fewer photons are scattered and detected.Improved photon collection and detector efficiency could support measurements that are both faster and higher fidelity.Source: §1 · p. 4
• Quantum logic spectroscopy transfers information to another ion species, reducing disturbance from measurement photons but complicating control and requiring additional ions.The approach can reuse preexisting cooling ions and may at most double the ions for one-to-one pairing.Source: §1 · p. 4
• Tailored micromotion has reduced measurement cross-talk errors by an order of magnitude and may reduce or eliminate the need for spectroscopy or shuttling.This is presented as an alternative using ions of the same species encoded in different energy levels.Source: §1 · p. 4

2. Ion-shuttling speeds

• The fastest reported shuttling moved ions 210 µm in 6 µs with a maximum gain of 0.36 ± 0.08 quanta, while a slower 55-µs operation gained 0.1 quanta.Faster shuttling introduces more thermal noise, which can harm two-qubit gates.Source: §2 · p. 5

3. Estimates for shuttling times

• Assuming 300 µm separation is tolerable, the reported shuttling speed implies a two-way shuttling time of about 10 µs for suppressing decoherence.The cited cross-talk measurements include 3.5×10^-3–1.5×10^-2 at 110 µm and 2 × 10^-5 at 370 µm.Source: §3 · p. 5

C. Cooling trapped ions

• EIT cooling reaches near-ground-state temperatures for 40-ion chains in under 300µs and 100-ion Penning traps within 200µs.EIT has limited motional-frequency coverage and is slow for low-frequency excitations.Source: C. Cooling trapped ions · p. 5
• Sympathetic cooling is established but relatively slow and requires additional ions, with reported relaxation times between 10 and 100ms for chains up to length 28.Persistent cooling and proposals for one-ion sympathetic cooling on tens-of-microseconds timescales are discussed as possible improvements.Source: C. Cooling trapped ions · p. 5
• Rapid exchange cooling demonstrated 107.3µs round-trip shuttling for two calcium ions, described as an order of magnitude faster than typical sympathetic cooling.The approach shuttles coolant ions from a continuously chilled bank to computational ions.Source: C. Cooling trapped ions · pp. 5–6

D. Cycle time paradigms

• 1000µs cycles allow flexibility for routing and cooling, including one or more cooling stages per cycle and shuttling to avoid measurement-induced decoherence.This regime is especially suited to segmented-linear trap and QCCD architectures.Source: D. Cycle time paradigms · p. 6
• 100µs cycles require fewer than one dedicated cooling round per cycle, with limited routing or shuttling and likely mitigation of measurement-induced decoherence.Linear traps, Wigner crystals, and QCCDs are identified as viable architectures.Source: D. Cycle time paradigms · p. 6
• 10µs cycles permit no processes interrupting syndrome extraction, requiring persistent cooling and architectures without routing or shuttling.Linear traps, Wigner crystals, and micro-trap architectures are the likely candidates in this regime.Source: D. Cycle time paradigms · p. 6

A. The lattice surgery cycle

• Each lattice-surgery cycle must complete entanglement distribution, entanglement purification, and joint syndrome extraction within T.These stages correspond to the concurrent stages illustrated in Fig. 3.Source: A. The lattice surgery cycle · pp. 6–7, Figure 3 · p. 7
• Parallel pipelining lets entanglement collection, purification, and surgery each use the full cycle window T, unlike sequential scheduling.The analysis assumes a continuously running engine and excludes startup deadtime.Source: A. The lattice surgery cycle · p. 7

B. A heuristic for fault-tolerant lattice surgery

• Fault tolerance is defined heuristically by requiring all necessary lattice-surgery subroutines to succeed with high probability.The criteria are tied to the three operational subroutines shown in Fig. 4.Source: B. A heuristic for fault-tolerant… · p. 7, Figure 4 · p. 7
• The purification condition requires an n-to-1 protocol that converts input Bell pairs of fidelity F_in into an output pair meeting F_ideal with probability p.The multiplexing condition then raises the probability of obtaining at least one purified pair above P_pair.Source: B. A heuristic for fault-tolerant… · p. 7
• Entanglement collection must provide enough pairs for d multiplexed purification instances within T, with collection success at least P_LS.This ensures resources are available for the next lattice-surgery round.Source: B. A heuristic for fault-tolerant… · p. 7

• The required number of purification circuits follows from the probability 1 − (1 − p)^n of obtaining at least one successful purification in n trials.The circuit count is selected to exceed the target confidence P_pair.Source: §1 · p. 8
• The lattice surgery requires N_LS = dN_pK raw entangled pairs when d purification instances each use N_p raw pairs and K multiplexed circuits are deployed.This combines the purification input requirement with the number of surgery stitches.Source: §1 · p. 8
• With cycle time T and entanglement-attempt rate R, the collection process has A = TR attempts to obtain N_LS pairs.Vacant communication ions are pulsed independently with success probability p_e, while already entangled ions are not pulsed again.Source: §1 · p. 8
• The minimum communication-ion count is chosen so a binomial number of successes reaches at least N_LS with confidence P_LS.Each ion pair is entangled after A attempts with probability P_onepair.Source: §1 · p. 8

2. Maximum attainable rate given given number of communication ions

• A_min is the minimum number of attempts required to populate N_LS ions with overall confidence P_LS when N_ions is fixed.The resulting rate is limited by the entanglement-attempt rate R.Source: §2 · p. 8
• The maximum attainable cycle time for fixed N_ions is A_min/R, where A_min is the minimum attempts needed to populate the required lattice-surgery pairs with confidence P_LS.A_min is defined by the condition that enough communication ions become entangled for the operation.Source: §2 · p. 8

C. Device parameters and assumptions

• The analysis assumes 1MHz ion pulsing, 2.18×10^-4 entanglement success per attempt, and 0.94 pair fidelity.It also assumes 0.1% surface-code operational error, negligible purification-stage time, specified gate and measurement error channels, and no ion-loss or leakage errors.Source: C. Device parameters and assumptions · p. 8
• The resource model does not include ion loss, replacement, or leakage accumulation and assumes arbitrary simultaneous two-qubit gates in a linear crystal.The text notes that this connectivity assumption may not be practically achievable.Source: C. Device parameters and assumptions · p. 8

D. Optimizing entanglement purification with device level noise

• F = 0.9904 fidelity is achieved with probability p = 0.819 by a circuit using three Stephenson pairs as input.The protocol returns one output pair from three partially entangled inputs and succeeds under specified measurement-outcome conditions.Source: D. Optimizing entanglement… · p. 9, Figure 5 · p. 9

V. RESULTS

• Communication-ion requirements scale approximately linearly with code distance and cycle time, increasing by orders of magnitude as cycle time decreases.With roughly 1000 ions available per ELU, d = 9 is theoretically supported at 1000µs, whereas considerably faster cycles appear out of reach.Source: §V · p. 9, Figure 6 · p. 9
• 100 communication ions support distance-5 or distance-6 codes at around 100Hz, while 1000 and 10,000 ions enable larger distances at around 1KHz and 10KHz.At higher code distances, 100-ion configurations can become impossible because required entangled pairs exceed available communication ions.Source: §V · pp. 9–10, Figure 7 · p. 10
• 40,000 communication ions per ELU may be required for modest surface codes at a 10µs cycle, creating a major scalability concern.The stated overhead is driven primarily by the low inter-ELU coupling probability pc = 2.18 × 10^-4 [64].Source: §V · p. 10
• Communication-ion counts decrease with coupling probability before reaching plateaus of 46, 91, and 136 ions for code distances d = 3, 6, and 9.The plateaus represent the minimum unpurified-pair counts required by the purification protocol; convergence occurs near pc ≈ 10^-2, 0.1, and 0.5 for 1000µs, 100µs, and 10µs cycles.Source: §V · p. 10, Figure 8 · p. 11
• For 200 communication ions per trap and d = 9, the required coupling probabilities are pc ≈ 1.5 × 10^-3, 1.5 × 10^-2, and 1.5 × 10^-1 for 1000µs, 100µs, and 10µs cycles.These estimates imply improvements of one or several orders of magnitude in ion-coupling rates, with photon-collection improvements suggested as one possible route.Source: §V · p. 10
• The 3 →1 purification protocol has an 81.9% success rate but requires K = 5 circuits per lattice-surgery stitch at PPair = 0.999.This creates an effective 15 →1 overhead, motivating higher-fidelity distributed pairs to reduce or eliminate purification.Source: §V · p. 10

VI. CONCLUSION

• Hundreds, thousands, and tens of thousands of communication ions are required at cycle times of 1000µs, 100µs, and 100µs, respectively.The reported fastest-case cycle time appears as 100µs in the supplied passage.Source: §VI · p. 11
• The analysis estimates communication ions for lattice surgery between two trapped-ion surface-code qubits.Source: §VI · p. 11
• Three syndrome-extraction cycle-time paradigms are based on different technological milestones and support a heuristic for fault-tolerant lattice surgery.Source: §VI · p. 11
• Poor ion-coupling rates are identified as the primary factor behind the prohibitive overheads.Source: §VI · p. 11

Appendix: Noisy entanglement distillation

• The genetic algorithm searches n→k purification circuits using average marginal fidelity, optionally incorporating depolarizing gate noise and measurement errors.Source: Appendix: Noisy entanglement… · p. 16
• The search was restricted to k = 1 because k > 1 protocols produced mutually entangled output pairs, conflicting with the independent-pair assumption for lattice surgery.Source: Appendix: Noisy entanglement… · p. 16
• As input size increased, success probability decreased while average output fidelity increased, and no identified protocol exceeded Freq = 0.999.Source: Appendix: Noisy entanglement… · p. 16

Appendix: The Stephenson pair

• The Stephenson pair is formed by rotating an experimentally reported state with (I ⊗XZ)ρ(I ⊗XZ) to obtain ρ′ predominantly aligned with |ϕ+⟩⟨ϕ+|.Source: Appendix: The Stephenson pair · p. 16

Appendix: Tables of constants

• Figure 9 plots success probabilities against output pair fidelities for noisy n→1 protocols and highlights the highest-yield protocol above Fideal = 0.99.Source: Figure 9 · p. 17
• Figure 5 depicts the selected high-yield purification circuit, while Figure 10 shows the unrotated Stephenson state from supplementary material [64].Source: Figure 10 · p. 17
• Figure 11 shows a rotated version of Eq. A.1 obtained with ρ′ = (I ⊗XZ)ρ(I ⊗XZ).Source: Figure 11 · p. 17
• Table II summarizes the free parameters, and Table III catalogs the numerical constants used throughout the analysis.Source: Appendix: Tables of constants · p. 16, Table II · p. 17, Table III · p. 18