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Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays

Yue Wu, Shimon Kolkowitz, Shruti Puri, Jeff D Thompson

arXiv:2201.03540v1quant-phphysics.atom-ph

TL;DR

Error-corrected quantum computing demands qubit numbers and physical error rates that challenge current hardware, while matter-based error locations have not been extensively studied for erasure detection. The paper encodes 171Yb qubits in a metastable level and monitors other subspaces to convert gate errors into erasures. It reports a 4.4-fold increase in the surface-code threshold and faster logical-error scaling below threshold.

  • Problem

    Physical qubit operations are error-prone, and techniques for detecting error locations in matter-based qubits have not been extensively studied.

    Source: §I · p. 1

  • Method

    The approach encodes 171Yb qubits in a metastable level and monitors transitions to other subspaces to detect Rydberg-gate errors as erasures.

    Source: §II · pp. 2–3

  • Results

    The circuit-level surface-code threshold increases 4.4-fold, with a steeper logical-error scaling below threshold.

    Source: §V · p. 6

  • Takeaways & Limitations

    The authors conclude that the approach is promising for demonstrating fault-tolerant logical operations with near-term experimental hardware.

    Source: §V · p. 6

Abstract

from arXiv · show

Executing quantum algorithms on error-corrected logical qubits is a critical step for scalable quantum computing, but the requisite numbers of qubits and physical error rates are demanding for current experimental hardware. Recently, the development of error correcting codes tailored to particular physical noise models has helped relax these requirements. In this work, we propose a qubit encoding and gate protocol for ${}^{171}$Yb neutral atom qubits that converts the dominant physical errors into erasures, that is, errors in known locations. The key idea is to encode qubits in a metastable electronic level, such that gate errors predominantly result in transitions to disjoint subspaces whose populations can be continuously monitored via fluorescence. We estimate that 98% of errors can be converted into erasures. We quantify the benefit of this approach via circuit-level simulations of the surface code, finding a threshold increase from 0.937% to 4.15%. We also observe a larger code distance near the threshold, leading to a faster decrease in the logical error rate for the same number of physical qubits, which is important for near-term implementations. Erasure conversion should benefit any error correcting code, and may also be applied to design new gates and encodings in other qubit platforms.

I. INTRODUCTION

Because physical noise varies by platform, matching error-correcting codes to that noise can improve fault-tolerance prospects; leakage outside the computational space can instead lower thresholds. This work proposes encoding 171Yb qubits to convert many gate errors into detectable erasures, and reports higher surface-code thresholds and faster logical-error reduction.

I. INTRODUCTION

  • Erasure conversion addresses the difficulty of detecting errors in matter-based qubits, where leakage outside the computational space can require extra gates and lower the threshold.Erasure errors have known locations and are described as significantly easier to correct than depolarizing errors.Source: §I · p. 1
  • The proposed 171Yb encoding uses a metastable electronic level so most Rydberg-gate decays leave the computational space for monitored levels, revealing their locations as erasures.The approach is presented as requiring no additional gates or ancilla qubits for erasure conversion.Source: §I · p. 1
  • 4.15% versus 0.937%: the predicted surface-code threshold rises with 98% erasure conversion, alongside faster logical-error reduction below threshold.The authors describe the results as relevant to near-term experiments and report that the threshold is within the range of current experimental neutral-atom gate fidelities.Source: §I · p. 2
  • The paper describes its protocol as complementary to a Rydberg-array proposal based on biased noise, handling leakage without additional gates and without restricting circuits to bias-preserving gates.Source: §I · p. 2

II. ERASURE CONVERSION IN 171YB QUBITS

The proposed 171Yb encoding places the qubit in a metastable level and uses Rydberg-mediated gates, whose dominant decay events often leave the computational subspace. Monitoring those other subspaces detects erasures; the estimated conversion fraction is 98%.

II. ERASURE CONVERSION IN 171YB QUBITS

  • The 171Yb qubit uses the hyperfine states of metastable 6s6p 3P0, with |1⟩ selectively coupled to a Rydberg state for two-qubit gates.The metastable level has an estimated lifetime of about 20 s; the example Rydberg state is 6s75s 3S1.Source: §II · p. 2
  • About 95% of Rydberg-state decays leave the qubit subspace, and fluorescence detection of ground-state atoms or autoionized Rydberg population identifies these errors.Ground-state atoms are detected on a cycling transition, while remaining Rydberg population is converted into Yb+ ions for fluorescence detection.Source: §II · p. 3
  • About 50 times less infidelity remains after conditioning on no detected erasure than the average gate infidelity; undetectable leakage is very small.The residual error reflects returns to Q and no-jump evolution, while the probability of undetectable leakage is described as very small.Source: Figure 2 · p. 3, §II · p. 3
  • 98% of spontaneous-decay errors are estimated to become erasures, exceeding the 95% suggested by branching ratios because some decays to Q are re-excited and then detected.Only 5% of decays directly return to the qubit subspace; 61% are blackbody transitions and 34% radiative decays to the ground state.Source: §II · p. 3
  • 4.15(2)% versus 0.937(4)%: the circuit-level surface-code threshold is higher for Re = 0.98 than for pure computational errors, Re = 0.Source: Figure 3 · p. 4

III. SURFACE CODE SIMULATIONS

Circuit-level XZZX surface-code simulations model Pauli errors and erasures, then decode syndrome data with a weighted Union Find decoder. At Re = 0.98, the threshold is 4.15%, and logical errors fall faster below threshold than with pure Pauli errors, though the exponent eventually returns to the Pauli value at very small physical error rates.

III. SURFACE CODE SIMULATIONS

  • The simulations use a d×d data-qubit array and d measurement rounds, with erasure locations and syndrome results decoded by weighted Union Find.The decoder is optimal for pure erasures and performs comparably to conventional matching decoders for Pauli errors.Source: §III · p. 4
  • 4.15% versus 0.937%: the circuit-level threshold rises by a factor of 4.4 when erasure conversion changes from Re = 0 to Re = 0.98.The threshold reaches 5.13% at Re = 1.Source: §III · p. 4
  • Below threshold, the logical error rate is approximated by Ap^ν, and a larger ν means faster suppression at fixed code distance.Source: §III · p. 4
  • At Re = 0.98, the fitted logical-error exponent is ν = 4.35(2), compared with νp = 3 for pure Pauli errors and νe = 5 for pure erasures.For d = 5, matching this exponent with pure Pauli errors would require d = 7, using nearly twice as many qubits.Source: §III · p. 4, Figure 4 · p. 5
  • At very small physical error rates, the exponent eventually returns to νp because the lowest-weight Pauli failure becomes dominant.For d = 5, the onset of this behavior is barely visible in the reported scaling.Source: §III · p. 4

IV. DISCUSSION

The discussion argues that erasure conversion brings surface-code thresholds within reach of current neutral-atom gate fidelities and may reduce resource costs. It also identifies the analysis’s focus on two-qubit gate errors and compares erasure conversion with biased-noise approaches.

IV. DISCUSSION

  • For Re = 0.98, the threshold error rate corresponds to a two-qubit gate fidelity of 95.9%, which is exceeded by current state-of-the-art.Entangled states with fidelity 97.4% were demonstrated for hyperfine qubits in Rb, and fidelity 99.1% was demonstrated for ground-Rydberg qubits in 88Sr.Source: §IV · p. 5
  • Hundreds of demonstrated neutral-atom qubits could support one surface-code logical qubit at d = 11 or five at d = 5.The authors expect erasure conversion to benefit other codes, particularly alongside neutral-atom arrays’ flexible connectivity.Source: §IV · p. 5
  • For the XZZX surface code, the threshold is 2.27% at bias η = 100 and 3.69% at infinite bias; erasure conversion is higher when Re ≥ 0.96.The comparison uses bias-preserving gates for biased noise; erasure conversion does not restrict circuits to such gates.Source: §IV · p. 5
  • The analysis focuses on two-qubit gate errors; adding initialization, measurement, and single-qubit errors at reasonable values does not significantly affect the threshold two-qubit gate error.The authors leave detailed analysis of these other errors and atom loss to future work.Source: §IV · p. 5
  • Rejecting magic states with detected erasures can reduce raw-state error from O(p) to O((1 − Re)p), potentially lowering distillation overhead.The authors estimate that 98% erasure conversion can reduce raw magic-state infidelity by over an order of magnitude.Source: §IV · p. 5

V. CONCLUSION

The paper proposes using 171Yb’s level structure to convert spontaneous Rydberg decay during two-qubit gates into detected erasures. Surface-code simulations predict a 4.4-fold threshold increase and steeper logical-error suppression below threshold.

V. CONCLUSION

  • The proposed 171Yb approach converts spontaneous Rydberg decay during two-qubit gates into directly detected erasure errors.The proposal leverages the alkaline-earth atom’s level structure.Source: §V · p. 6
  • The circuit-level surface-code threshold increases 4.4-fold, bringing it within the range of current experimental neutral-atom gate fidelities.The approach also yields steeper logical-error scaling below threshold.Source: §V · p. 6
  • The authors anticipate that erasure conversion will also apply to other error-correcting codes and physical platforms.The proposed approach is presented as promising for near-term demonstrations of fault-tolerant logical operations.Source: §V · p. 6

A. Error correcting code simulations

The simulations model two-qubit gate errors and erasures, then use a weighted union-find decoder to evaluate surface-code logical errors. Supplementary gate-operation details describe initialization, measurement, and single-qubit rotations for 171Yb qubits.

A. Error correcting code simulations

  • Two-qubit gates receive Pauli errors with probability pp and erasure errors with probability pe, with the erasure fraction set by pe/(pp + pe) = Re.Erasure events reinitialize both qubits in a mixed state; the model assumes native CZ and CNOT gates and neglects idle errors.Source: A. Error correcting code simulations · p. 8
  • The surface-code decoder graph connects circuit-generated space-time error points, weights edges by truncated ln(p′), and uses a weighted union-find decoder to identify errors consistent with syndromes.Simulations use d syndrome rounds followed by a perfect measurement round, and the implementation was benchmarked against.Source: A. Error correcting code simulations · p. 8
  • The biased-noise XZZX comparison assumes bias-preserving CNOT and CZ gates and reports thresholds in terms of total two-qubit gate infidelity for large η.The quoted threshold omits single-qubit preparation and measurement noise for direct comparison with erasure-conversion results.Source: A. Error correcting code simulations · p. 8

S1. 171YB GATE OPERATIONS

  • 171Yb qubits are initialized in the metastable 3P0 manifold by transferring atoms from 1S0 on the clock transition, and midcircuit measurement selectively transfers |1⟩ back for fluorescence detection.Optical pumping through intermediate S and D states is offered as an alternative to clock-transition driving.Source: S1. 171YB GATE OPERATIONS · p. 9
  • Single-qubit rotations can use Raman transitions and light shifts on the 6s7s 3S1 transition or the Rydberg state; photon-scattering errors can be converted at similar or greater levels than for two-qubit gates.The text refers to section S5 for the comparison.Source: S1. 171YB GATE OPERATIONS · p. 9

S2. YB BRANCHING RATIOS

The branching-ratio analysis estimates how often 171Yb Rydberg decays return to the qubit manifold, using approximate atomic calculations and decay pathways. It also estimates rapid, high-fidelity fluorescence detection of ground-state atoms.

S2. YB BRANCHING RATIOS

  • The branching estimates use a single-active-electron approximation and Numerov-computed wavefunctions for the 6s75s 3S1 F = 3/2 state.Doubly excited series perturbers are not quantitatively evaluated because spectroscopic data are insufficient; the authors expect their branching into 3P0 not to be worse than estimated values.Source: S2. YB BRANCHING RATIOS · pp. 9–10
  • Only 0.025 of radiative decays terminate directly in the J = 0 qubit manifold Q; other pathways include relaxation to 1S0 or repumping 3P2 without populating Q.For n = 75, estimated decay fractions are 0.39 radiative and 0.61 blackbody-induced; Fig. S5 illustrates the pathways and branching ratios.Source: S2. YB BRANCHING RATIOS · p. 9, Figure S5 · p. 9
  • The estimated branching ratios from 6s7s 3S1 to 3PJ levels are 0.15, 0.40, and 0.45 for J = 0, 1, and 2, respectively.The resulting estimate is that 0.15 of decays via 3S1 reach Q.Source: S2. YB BRANCHING RATIOS · p. 10
  • Approximately 0.16 of decays via 6s5d 3DJ states terminate in Q, based on estimated pathways and the population distribution among intermediate 6snp 3PJ′ levels.Only 3D1 can decay to Q; the 3D1-to-Q branching ratio uses theoretical matrix elements from Ref..Source: S2. YB BRANCHING RATIOS · p. 10
  • The combined estimate is that 0.051 of all Rydberg decays return to Q; the unknown S-versus-D pathway branching ratio is taken as 0.5 because their probabilities of reaching Q are similar.The analysis estimates that 14% of decays from 6snp levels with n > 6 return to Q.Source: S2. YB BRANCHING RATIOS · p. 10

A. Detection of atoms in 1S0

  • For 171Yb atoms in 1S0, the authors estimate imaging fidelity above 99.9% in less than half the 6Li imaging time under the compared conditions.The estimated position spread is 120 nm for the same detected-photon count, attributed to a factor-of-81 smaller position-spread scaling quantity for 171Yb.Source: A. Detection of atoms in 1S0 · p. 10

B. Detection of ions

The ion-imaging analysis estimates high-fidelity detection from a small number of scattered photons while accounting for recoil and electric-field motion. It also compares alternate detection strategies and describes simulations of the LP two-qubit gate.

B. Detection of ions

  • Autoionization gives the Yb+ ion a recoil velocity of approximately 3.5 m/s, which contributes to its motion during fluorescence imaging.For the stated imaging parameters, the initial velocity is about 550 times the recoil velocity.Source: B. Detection of ions · p. 11
  • 99% detection fidelity is estimated with an average of 5 detected photons at η = 0.1, provided ion spread remains below 2.5 µm and dark counts are absent.The necessary imaging time is less than 2 µs.Source: B. Detection of ions · p. 11
  • The imaging estimate is pessimistic: only gate-participating atoms need resolving, recoil is three-dimensional, and some decay products have lower recoil.The assumed collection and detection efficiency is challenging but achievable with a 30% quantum-efficiency detector and NA=0.7 objectives on two sides.Source: B. Detection of ions · p. 11
  • At 30 µm separation, a neighboring ion produces a 16 mV/cm field and about 5 µm displacement during 200-photon scattering, potentially limiting ion resolution nearby.The authors note this may motivate fewer gates per array cycle.Source: B. Detection of ions · p. 11

C. Alternate detection strategies for population in Rydberg states

  • Waiting for more than 7/Γ, approximately 1 ms, lets over 99.9% of ions decay, but requires trapping intermediate Rydberg states and gives a crude erasure-conversion estimate of Re ≈ 0.9.Fluorescence detection is described as fast and compatible with existing experimental techniques.Source: C. Alternate detection strategies for… · p. 11

S4. GATE SIMULATIONS

  • The LP-gate simulations model a two-atom system with Rydberg blockade and a BBR-populated state, then classify fluorescence outcomes as detectable erasures or no erasure.The simulations focus on the protocol from Ref., applied to 171Yb in Ref..Source: S4. GATE SIMULATIONS · pp. 11–12
  • When Vrr ≫ Ω, the LP protocol uses a detuned pulse with a phase slip to return initial states with different accumulated phases, implementing a CZ gate.Figure S6 shows the pulse sequence, Rydberg population for different initial states, and individual erasure-channel probabilities.Source: S4. GATE SIMULATIONS · p. 12, Figure S6 · p. 12

A. Analytic error model

The analytic model tracks decay and re-excitation through detectable subspaces during two-qubit gates, then checks its predictions against a full two-atom simulation.

1. Initial state |01⟩(or |10⟩)

  • 0.532 is α for the LP gate; for |01⟩ or |10⟩, R01 is the decay-time-weighted fraction of decays to |01⟩ that are re-excited.The model assumes equal probabilities for decay to |00⟩ and |01⟩, although actual branching favors |01⟩, and neglects intermediate-state time when tg > 100 ns.Source: §1 · pp. 12–13

2. Initial state |11⟩

  • 0.467 is β for the LP gate from |11⟩; BB can arise from re-excitation or |rr⟩ decay, both strongly suppressed by blockade.A second radiative decay can also populate RR after an initial radiative decay.Source: §2 · pp. 13–14

3. Summary

  • The model weights |00⟩, |01⟩/|10⟩, and |11⟩ by their input probabilities; |00⟩ has no Rydberg excitation, and PBB is undetectable leakage.The total erasure probability is the sum of the first five error-channel terms.Source: A. Analytic error model · p. 12, §3 · p. 14
  • The first three error channels scale as Γtg and dominate, while PBB grows as gates shorten and causes undetected atom loss.PBB decreases with Γ but increases as tg decreases; blockade must remain strong to maintain high-fidelity gates.Source: §3 · p. 14
  • No-erasure monitoring biases the inferred state toward |00⟩, but its average gate-infidelity contribution is approximately (pe/4)^2 and is not significant when pe ≪ 0.32.The erasure probabilities are approximately equal across the remaining computational states.Source: §3 · p. 14

B. Comparison to numerical simulations

  • The analytic predictions agree excellently with full two-atom master-equation simulation; gate error mainly depends on Γtg and also on Vrr/Γ at high fidelity.The comparison sets Vrp = Vrr, although Vrp is actually larger.Source: B. Comparison to numerical simulations · pp. 14–15

S5. ERASURE CONVERSION FOR OTHER ERRORS

The metastable-state encoding may extend erasure conversion beyond two-qubit gates to idle-qubit errors and single-qubit operations, while technical noise presents a practical trade-off.

S5. ERASURE CONVERSION FOR OTHER ERRORS

  • Idle-qubit spontaneous decay and photon scattering in 3P0 are erasures with very high probability.Spontaneous decay to 1S0 is detectable; Raman and Rayleigh scattering can avoid Pauli errors when tweezer detuning is large relative to the hyperfine splitting.Source: S5. ERASURE CONVERSION FOR OTHER ERRORS · p. 15
  • Single-qubit Raman gates via 6s7s 3S1 can use the same conversion when detuning exceeds that state's hyperfine splitting; Rydberg gates follow the two-qubit analysis.Source: S5. ERASURE CONVERSION FOR OTHER ERRORS · p. 15
  • Composite pulses can cancel slow technical noise, but their longer duration increases Rydberg decay probability; erasure conversion may make this trade-off more favorable.The cited noise sources include Doppler shifts and driving-laser frequency and intensity fluctuations.Source: S5. ERASURE CONVERSION FOR OTHER ERRORS · p. 15

S6. IMPACT OF ERRORS IN INITIALIZATION, MEASUREMENT AND SINGLE-QUBIT GATES

The simulations test sensitivity to single-qubit and ancilla errors, while the estimated stabilizer-cycle time is dominated by ancilla measurement.

S6. IMPACT OF ERRORS IN INITIALIZATION, MEASUREMENT AND SINGLE-QUBIT GATES

  • The simulations assume native Hadamard-basis ancilla initialization and measurement, while a Z-basis circuit requires four H gates whose errors can be pessimistically counted as two-qubit-gate errors.Source: S6. IMPACT OF ERRORS IN… · p. 15
  • 3.80(2)% is the threshold for Re = 0.98 at pm = 0.005, versus 2.85(1)% when pm = p; pm = 0.001 is indistinguishable from zero.Source: Figure S8 · p. 15, S6. IMPACT OF ERRORS IN… · pp. 15–16
  • When single-qubit errors have erasure fraction Re(1) = 0, Re is reduced by 1/(1 + p(1)/p), and is not significantly affected if p(1)/p < 1 − Re.Extending erasure detection to single-qubit gates would further relax this requirement.Source: S6. IMPACT OF ERRORS IN… · p. 15

S7. IMPACT OF ERASURE CONVERSION ON OPERATION SPEED

  • About 100 µs is the estimated cycle time for gates plus erasure detection, but ancilla measurement takes at least 20 ms and dominates total cycle duration.Erasure conversion will not significantly affect total repetition time unless measurement time is reduced by about two orders of magnitude.Source: S7. IMPACT OF ERASURE CONVERSION ON… · p. 16

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