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Hardware-Efficient, Fault-Tolerant Quantum Computation with Rydberg Atoms

Iris Cong, Harry Levine, Alexander Keesling, Dolev Bluvstein, Sheng-Tao Wang, Mikhail D. Lukin

arXiv:2105.13501v3quant-phphysics.atom-ph

TL;DR

Neutral-atom quantum computing needs fault-tolerant operations that address Rydberg decay, leakage, and correlated errors. This work characterizes these channels and designs hardware-efficient protocols that convert leading errors into Pauli-Z errors using tailored codes and hardware capabilities. The resulting schemes substantially reduce resource costs compared with general-purpose approaches.

  • Problem

    Rydberg-state decay and related processes create leakage and correlated errors that traditional fault-tolerant methods do not directly address, limiting error-corrected neutral-atom computation.

  • Method

    The paper characterizes Rydberg error channels and uses optical pumping, ancillas, blockade, selection rules, and seven-qubit or three-qubit codes to design fault-tolerant protocols.

  • Results

    The protocols convert leading errors to Pauli-Z errors and reduce resource overhead by an order of magnitude compared with some general-purpose fault-tolerant protocols.

  • Takeaways & Limitations

    Hardware-efficient fault-tolerance tailored to Rydberg error structure can support scalable neutral-atom quantum computation using alkali and alkaline-earth platforms.

Abstract

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Neutral atom arrays have recently emerged as a promising platform for quantum information processing. One important remaining roadblock for the large-scale application of these systems is the ability to perform error-corrected quantum operations. To entangle the qubits in these systems, atoms are typically excited to Rydberg states, which could decay or give rise to various correlated errors that cannot be addressed directly through traditional methods of fault-tolerant quantum computation. In this work, we provide the first complete characterization of these sources of error in a neutral-atom quantum computer and propose hardware-efficient, fault-tolerant quantum computation schemes that mitigate them. Notably, we develop a novel and distinctly efficient method to address the most important errors associated with the decay of atomic qubits to states outside of the computational subspace. These advances allow us to significantly reduce the resource cost for fault-tolerant quantum computation compared to existing, general-purpose schemes. Our protocols can be implemented in the near-term using state-of-the-art neutral atom platforms with qubits encoded in both alkali and alkaline-earth atoms.

I. INTRODUCTION

The paper characterizes Rydberg-mediated errors and develops hardware-tailored fault-tolerant protocols that convert leading errors into Pauli-Z errors. Seven-qubit and three-qubit encodings reduce resource requirements while supporting scalable, near-term implementations.

  • Motivation: Rydberg-state decay can cause leakage outside the computational subspace and high-weight correlated errors through undesired blockade effects.Relevant channels include blackbody radiation, radiative decay, and intermediate-state scattering.
  • Fault-tolerant codes: Nine atoms—seven data qubits and two ancillas—suffice for fault-tolerant encoding with the seven-qubit Steane code and universal operations.The seven-atom architecture uses triangular-lattice geometries.
  • Fault-tolerant codes: A three-atom repetition code provides a more compact leading-order fault-tolerant protocol for species with sufficiently large nuclear spin and high-fidelity ground-state operations.The protocol uses a bias-preserving CNOT and can be extended to a bias-preserving Toffoli operation.
  • Error reduction: Optical pumping, ancilla atoms, Rydberg blockade, and dipole selection rules convert leading leakage and correlated errors into Pauli-Z errors.This error bias prevents the converted errors from spreading across qubits during the designed protocols.
  • Resource efficiency: Ryd-7 requires 2 ancilla qubits instead of 72 and at most 60 two-qubit gates instead of 1416 for the compared highest-cost logical operation.The reduction exploits the Rydberg error structure and hardware capabilities.

C. Towards experimental implementation

The protocols are designed for experimentally accessible atom geometries and rely on characterizing dominant Rydberg-gate error channels. Their implementation requires blockade-range control, individual measurement or Rydberg detection, and coherence-preserving operations.

  • Geometry: Both Ryd-7 and Ryd-3 fit naturally on triangular lattices when the relevant blockade radius exceeds 3d.This interaction range has already been demonstrated, and atom movement could reduce the requirement further.
  • Experimental capabilities: Near-deterministic loading into two- and three-dimensional lattice structures and high-fidelity control have already been demonstrated for neutral-atom systems.These capabilities support near-term implementation of the proposed architectures.
  • Readout: Quantum-error correction requires individual-qubit measurement or Rydberg-population detection while preserving coherence in nearby atoms for feed-forward correction.Selected atoms may be moved into a readout zone for rapid photon-scattering measurements.
  • Atomic species: Alkaline-earth atoms offer high-fidelity clock-state encodings, while large nuclear spin in fermionic species is advantageous for these protocols.The schemes are discussed as generalizable beyond the primarily considered alkali-atom systems.
  • Error channels: The dominant entangling-operation errors are BBR transitions, spontaneous radiative decay, and intermediate-state scattering for specified excitation schemes.The analysis treats these contributions to leading order in total error probability.
  • Error channels: BBR errors can produce correlated Kraus-map events conditioned on the control-qubit state during multiqubit Rydberg gates.For the illustrated C^aZ^b gates, a BBR map on one qubit may accompany Z errors on other participating qubits.

B. Error modeling for RD transitions

RD transitions produce leakage, Pauli-Z errors, and correlated errors whose structure depends on decay pathways and branching ratios. Their relative importance shifts with Rydberg level and temperature.

  • RD decay pathways: RD spontaneous emission can return atoms to |1⟩ or other ground-state sublevels, creating distinct error channels.Decay to |1⟩ can leave a superposition of |1⟩ and |r⟩, while decay to other sublevels causes leakage from the computational space.
  • RD decay pathways: RD decay to |1⟩ is modeled as a combination of Pauli-Z errors and leakage into |r⟩.The corresponding Kraus operators include maps for Rydberg leakage and computational-state effects, with coefficients determined by transition probability and pulse choice.
  • RD decay pathways: RD decay to other ground-state sublevels causes computational-subspace leakage without influencing neighboring Rydberg operations.Dipole selection rules reduce the number of significant channels when a stretched Rydberg state is used; decay into |0⟩ is negligible to leading order.
  • Correlated errors: RD events during primitive entangling gates can produce correlated errors across multiple involved qubits.A decay-related Kraus map on one qubit can combine with Pauli-Z and |r⟩⟨1| errors on other qubits.
  • Rate dependence: Γ0 ∼ 1/n_eff^3, and RD processes dominate at smaller n or very low temperature while BBR processes dominate at larger n.The total RD rate is temperature-independent, whereas BBR rates depend on temperature and effective principal quantum number.

C. Errors from intermediate state scattering

Intermediate-state scattering forms a restricted subset of Rydberg decay channels, while technical and hyperfine imperfections add mainly Pauli-Z errors and leakage. The proposed fault-tolerant construction converts the relevant errors into correctable forms while enforcing no residual Rydberg population.

  • Intermediate-state scattering: Intermediate-state scattering channels are a subset of RD channels, producing decay into |1⟩ or two other hyperfine ground states.The restriction follows from σ+-polarized excitation and choosing a low-n P3/2 intermediate state.
  • Experimental imperfections: Laser fluctuations are modeled as Pauli-Z errors and leakage into |r⟩, while hyperfine-operation imperfections primarily cause Pauli-Z errors and leakage to other hyperfine states.Single-qubit errors are generally smaller than two-qubit-gate errors in the cited experimental setting.
  • Experimental imperfections: Tweezer-induced dephasing can be alleviated to achieve T2 ∼ 1 s, while Raman-scattering leakage can be suppressed to timescales > 10 s.The passage attributes these improvements to dynamical decoupling and sufficient detuning of the tweezer light.
  • Protocol scope: Certain hyperfine-rotation imperfections are not directly corrected by the protocol but can be minimized with composite pulse sequences.BB1 suppresses pulse-amplitude errors to sixth order, and other frequency errors can be further reduced through cooling, source stabilization, or composite pulses.
  • Fault-tolerant construction: The error model includes high-weight correlated errors, but Rydberg gates introduce no Pauli-X or Pauli-Y errors.The construction uses ancillas, blockade effects, dipole selection rules, and optical pumping to convert the relevant errors into Pauli-Z-type errors.
  • Fault-tolerant construction: After correction, code preparation, error-correction rounds, and logical gates leave at most one physical-qubit error per logical qubit and no Rydberg population.These properties are stated to leading order in ptot for the distance-3 construction.

A. FTQC with BBR errors

The seven-qubit Steane code is adapted to Rydberg-specific errors by detecting leakage, converting errors to Pauli-Z types, and implementing logical operations fault-tolerantly. Stabilizer measurements and universal logical gates are designed to leave at most single-qubit errors per logical qubit.

  • Steane-code error correction: The seven-qubit Steane code identifies and corrects single-qubit X and Z errors through six stabilizer measurements.The code is a CSS code with three X-type and three Z-type stabilizers.
  • Leakage detection: Rydberg leakage can be detected with an ancilla using the blockade effect and converted into atom loss or Z-type errors.A nearby ancilla distinguishes the presence or absence of Rydberg population after a 2π pulse.
  • Fault-tolerant stabilizer measurements: After each stabilizer-measurement round, the correct eigenvalues are obtained to leading order in p_tot while introducing at most one physical X or Z error.Resetting and repeating the ancilla measurement removes the effect of ancilla errors.
  • Logical operations: The Steane code implements logical Hadamard, Pauli, and S gates transversally, while controlled-phase operations use seven physical gates with leakage probing.The logical CCZ is decomposed into 27 physical CCZ operations and reordered into nine groups to reduce detection steps.
  • Logical operations: The Hadamard and CCZ gates form a universal set, yielding fault-tolerant quantum operations on the code space against BBR errors.The logical CCZ construction uses reordered physical gates and intermediate leakage detection.

4. Logical state preparation

Logical-state preparation and the full intrinsic error model are incorporated into the fault-tolerant architecture. Optical pumping converts non-Rydberg leakage into correctable Pauli-Z errors without measurement and feed-forward, while the protocols reduce resource requirements relative to general-purpose schemes.

  • Logical state preparation: The Steane Latin-rectangle circuit prepares the logical |0⟩L state with a Pauli error on at most one physical qubit.Controlled-NOT operations are replaced by Rydberg controlled-phase gates with target Hadamards.
  • Full error model: Non-Rydberg leakage is leakage into hyperfine ground states outside the computational subspace, distinct from leakage into the original Rydberg state.Rydberg-state leakage is grouped with BBR errors because it can be detected using an ancilla.
  • Leakage conversion: Optical pumping converts hyperfine-manifold leakage into Pauli-Z errors without affecting qubit coherence when no error occurs.For 87Rb, the procedure uses π pulses, σ+ excitation, decay, and repeated state transfer before restoring qubit populations.
  • Integration and scope: The leakage-correction procedure can be inserted between Rydberg entangling gates, making the protocols fault-tolerant against generic intrinsic Rydberg decay errors.Laser imperfections causing Pauli-Z errors or single-qubit Rydberg leakage are also covered by the framework.
  • Resource comparison: The protocol uses 2 ancilla qubits for logical CCZ, compared with 72 in Yoder, Takagi, and Chuang, and uses 60 rather than 1416 two-qubit gates in the cited error-correction scenario.The comparison concerns fault-tolerant logical CCZ implementations.

D. Scalable implementation

The protocols are organized for scalable neutral-atom implementations with configurable geometry, measurement–gate tradeoffs, and explicit fault-tolerant circuits. Their scope includes hardware-specific assumptions and a leading-order approximation for one Rydberg-decay channel.

  • Geometrical considerations: Scalable architectures place logical qubits on a coarser lattice over physical atoms, with ancillas mediating error correction and logical gates.Nearest-neighbor logical entanglement imposes a minimum blockade-radius requirement.
  • Stabilizer-measurement implementation: Fault-tolerant X⊗4 stabilizer measurements initialize ancillas, apply Rydberg gates to four data qubits, detect leakage, optically pump it, and measure the ancilla.A −1 ancilla outcome triggers a repeated unprotected measurement procedure.
  • Logical-gate implementation: Fault-tolerant logical CZ applies seven physical Rydberg gates, probes leakage after each gate, converts detected leakage, and applies optical pumping.The protocol uses a second ancilla for leakage detection and correction.
  • Resource tradeoffs: Reducing entangling gates can require additional measurements, while reducing measurement shots can require additional operations.The preferred tradeoff depends on the relative timescales of measurements and gates.
  • Improvements: The leading-order protocol ignores the |0⟩⟨1| Rydberg-decay channel, whose branching ratio in 87Rb is approximately 10^-3.Shelving, higher magnetic fields, or higher-nuclear-spin species are proposed to suppress this channel further.
  • Logical-gate implementation: Logical CCZ applies physical gates in nine groups, detecting leakage after each group and correcting possible correlated errors through stabilizer measurements.The grouped schedule is defined by the reordered physical three-qubit gates.

V. LEADING-ORDER FAULT-TOLERANCE WITH A REPETITION CODE

A repetition code can support leading-order fault-tolerant computation only if every operation preserves the strongly biased error model. The paper develops Rydberg pulse protocols that implement the needed bias-preserving entangling gates while suppressing target and control decay-induced X errors.

  • Bias-preserving operations: All encoding, decoding, stabilizer-measurement, and logical gates must be bias-preserving to avoid introducing Pauli-X or Y errors.The repetition code is therefore difficult to use for fault-tolerant computation despite the biased noise model.
  • Bias-preserving operations: A standard Rydberg CNOT is not bias-preserving because a target Z error becomes an X error after the final Hadamard.The same implementation can also produce target X errors through control-atom decay during the controlled-phase gate.
  • Bias-preserving CNOT: The proposed pulse sequence directly conditionally swaps |0⟩ and |1⟩ without target Hadamard gates, using stretched Rydberg states available for species with I ≥5/2.For 85Rb, the protocol transfers qubit populations through stretched |d±⟩ states and uses blockade-dependent resonance to implement the conditional operation.
  • Bias-preserving CNOT: The direct pulse sequence restores either the original or opposite qubit state, while Rydberg errors arising during resonant pulses produce only Z-type Pauli errors.The mapping relies on the distinct magnetic-sublevel structure of the stretched Rydberg states.
  • Control-decay protection: The implementation requires two Rydberg-state pairs with blockade radii satisfying RB,1 > 2d and d < RB,2 < 2d.The two pairs support distinct blockade conditions for the control, ancilla, and target atoms.
  • Control-decay protection: Using control and ancilla atoms to maintain blockade makes the protocol robust to control decay errors to leading order, eliminating CNOT-induced X errors.Decay during the final step yields computational-basis projections expressible as Z errors, and the construction generalizes to a bias-preserving Toffoli gate.

B. Leading-order fault-tolerance with the repetition code

The repetition-code construction supports logical operations by combining bias-preserving physical gates with transversal or structured fault-tolerant procedures. Logical Hadamard, Toffoli, controlled-phase, and CCZ operations are assembled while converting leakage into Z-type errors and limiting their propagation.

  • Fault-tolerant operations: Logical X-basis preparation and measurement are transversal, while stabilizers are measured with circuits whose CNOTs use the bias-preserving implementation.These operations provide the basic fault-tolerant components for the three-qubit repetition-code protocol.
  • Logical Toffoli gate: Logical Toffoli is implemented as a product of nine physical Toffoli gates using the repetition-code encoding.Each physical gate is implemented in a bias-preserving fashion.
  • Logical Toffoli gate: A physical gate contributes at most one physical Z error per logical qubit, but target-code errors can spread through later Toffoli gates.This distinguishes the leading-order fault-tolerance argument from a fully transversal construction.
  • Logical Hadamard gate: Logical Hadamard is implemented with a logical Toffoli and fault-tolerant X-basis measurements because the repetition code is not CSS and lacks a transversal Hadamard.Logical Hadamard together with logical Toffoli or CCZ provides a universal logical gate set.
  • Logical controlled-phase gate: Logical controlled-phase uses pairwise physical controlled-phase gates, followed by optical pumping that converts non-Rydberg leakage into possible Z errors.Stabilizers need only be measured after the entire logical operation because the resulting Z errors commute with the physical controlled-phase gates.
  • Logical CCZ gate: Logical CCZ is leading-order fault-tolerant because Z errors commute with remaining gates, and it uses fewer resources than the logical Toffoli.The resource advantage comes from using standard Rydberg gates rather than the more complicated bias-preserving CNOT pulse sequences.

C. Scalable implementation

The scalable repetition-code protocol uses lattice geometries, blockade-radius conditions, and near-term neutral-atom capabilities to support fault-tolerant operations. Its resource cost is reduced for stabilizer extraction, while bias preservation remains limited by stretched-Rydberg-state decay.

  • Geometrical layout: A triangular lattice places data and ancilla atoms on lattice vertices, with three data atoms forming each logical qubit.The smaller blockade radius must satisfy d < RB,2 < 2d for efficient bias-preserving CNOTs and stabilizer measurements.
  • Geometrical layout: A square-lattice alternative requires RB,1 > 3.61d and d < RB,2 < 2d.Atom rearrangement could reduce the larger blockade-radius requirement and remove the need for a second Rydberg-state set.
  • Resource comparison: The Ryd-3 protocol significantly reduces entangling gates for stabilizer extraction versus seven-qubit approaches without substantially increasing ancilla requirements.Its logical Hadamard costs more than in Ryd-7, while the logical CCZ cost is essentially the same.
  • Improvements: Bias preservation is limited by decay of the stretched Rydberg D state into qubit states.Higher-angular-momentum Rydberg states or atomic species with higher nuclear spin could further suppress these errors.
  • Improvements: The Ryd-3 approach inherently addresses Rydberg pulse imperfections within its radiative-decay error model and can be extended to protect against atom loss.Atom-loss protection requires additional physical operations between Rydberg operations.
  • Experimental implementation: Experiments have demonstrated near-deterministic loading, trapping, and rearrangement of tens to hundreds of neutral atoms into two-dimensional lattices relevant to the protocol.These capabilities include triangular-lattice geometries and support near-term implementation considerations.

A. Measurements and feed-forward corrections

Ancilla measurement, atom transport, optical pumping, and species-selective readout provide routes toward feed-forward correction and implementation of the proposed neutral-atom FTQC protocols. The paper also identifies unresolved extensions, including precise threshold studies and comparisons with larger-distance or topological codes.

  • Measurements and feed-forward corrections: Ancilla states can be measured by fluorescence, alternative atomic species, or Rydberg-EIT detection, with transport reducing readout-induced cross-talk.The transport estimate allows displacements D > 50 µm within 250 µs while keeping ΔN < 1 for typical traps.
  • Measurements and feed-forward corrections: For alkaline-earth atoms, stretched ground-state qubits use two-stage optical pumping to correct non-Rydberg leakage.The stages repump metastable 3P states and then pump non-stretched ground states into |1⟩.
  • Measurements and feed-forward corrections: The protocols convert complex Rydberg errors into Pauli-Z errors using ancillas, blockade, dipole selection rules, and optical pumping.The paper presents Steane-code and hardware-tailored three-qubit-code protocols with scalable layouts and near-term feasibility.
  • Measurements and feed-forward corrections: The work reports an order-of-magnitude reduction in physical-gate or ancilla overhead relative to some general-purpose FTQC protocols.The authors suggest that exploiting multilevel structure may transfer to other quantum platforms.
  • Measurements and feed-forward corrections: Precise error thresholds, larger-distance-code scalability, and detailed studies combining experimental imperfections with intrinsic decay remain open directions.The manuscript also notes related leakage-correction work based on detecting and replacing lost atoms.

Appendix A: Numerical Computation of Branching Ratios and Transition Rates

The appendix computes blackbody-radiation and radiative-decay transition rates from a stretched 87Rb Rydberg state, then uses these rates to characterize decay-induced evolution. The calculations identify dominant transition channels and derive a master-equation example for a 2π pulse.

  • Blackbody-radiation transitions: Branching ratios are obtained by combining Planck-distributed blackbody photons with Einstein coefficients and normalizing transition rates by Γ_BBR.The calculation uses radial dipole matrix elements for transitions from the stretched 70S1/2 state of 87Rb.
  • Blackbody-radiation transitions: The calculated blackbody-radiation branching ratios are plotted for transitions to P states with mJ = 3/2 and mJ = 1/2.These channels are represented by empty orange circles and filled blue diamonds, respectively.
  • Radiative decay: Radiative decay from the stretched 70S1/2 state is almost entirely composed of two- or four-photon processes to five ground-manifold states.This structure supports converting Rydberg errors into Z-type errors for FTQC.
  • Radiative decay: The remaining three radiative-decay channels each have branching ratios on the order of 10^-3, substantially below the five dominant transitions.The decay to the stretched state with minimal mF is described as highly unlikely when the total error probability is already small.
  • Master-equation example: For a driven three-level system, solving the master equation for a 2π Rydberg pulse yields the final state and corresponding Kraus-operator parameters to leading order in γ/Ω.The example uses tπ = π/(2Ω) and neglects blackbody and other-hyperfine radiative transitions.

Appendix C: Atom Loss Errors

Atom loss can be detected and corrected within the FTQC framework, with additional ancilla and gate resources. Preventative repumping and leakage-to-loss conversion provide complementary ways to suppress or correct residual Rydberg population.

  • Atom-loss correction: Atom loss detection and correction require one ancilla qubit and extra gates per operation.The method applies when trapping is imperfect or trapping lasers are turned off during Rydberg excitation.
  • Atom-loss correction: A multi-step ancilla protocol requires two positive measurements to confirm atom loss and limits phase-flip propagation to one physical error per logical qubit.Leakage can be addressed by repeating the relevant repumping procedures.
  • Atom-loss correction: The atom-loss circuit also suppresses residual hyperfine leakage because it does not distinguish atom loss from leakage into other hyperfine states.It is applied after leakage correction using blockade and optical pumping.
  • Leakage conversion: Detected Rydberg leakage can be converted into atom loss, localized, and followed by replacement with a fresh atom prepared in |1⟩.Ejection can occur naturally through the tweezer’s anti-trapping potential or be accelerated with a weak ionizing field.
  • Preventative repumping: Preventative repumping converts a large fraction of Rydberg leakage errors into Z-type errors without requiring the atom-loss circuit.The procedure swaps populations, drives likely leaked states to a short-lived P state, and repeats the swap.

Appendix E: Fault-Tolerant Detection of Rydberg Leakage Errors

Fault-tolerant leakage detection combines repeated ancilla measurements with stabilizer syndromes that distinguish correlated errors caused by delayed detection. The same hardware-tailored approach supports leakage-aware CNOT and Toffoli constructions and explicit resource accounting.

  • Leakage detection: Repeated ancilla checks identify leakage while protecting the detection process against ancilla phase-flip and leakage errors.A second detection round is required after a positive leakage indication to reject outcomes caused by ancilla errors.
  • Steane-code detection: For Steane-code stabilizer measurement, delayed data leakage can produce X5X6X7, X6X7, or X7 errors that are distinguished through Z-type stabilizer measurements.Ancilla leakage does not produce correlated data errors in the considered circuit.
  • Steane-code detection: For the logical CCZ gate, stabilizer eigenvalues distinguish the two correlated-error patterns arising from delayed leakage detection.The relevant patterns are R(2A, 2B; 2C)R(3A, 3B; 3C) and R(3A, 3B; 3C).
  • Bias-preserving gates: A bias-preserving Toffoli uses two ancillas placed on opposite sides of the target atom to avoid unwanted ancilla-ancilla blockade during one gate.The protocol is designed to eliminate control-decay-induced X-type errors in the same spirit as the bias-preserving CNOT.
  • Resource costs: Ryd-7 stabilizer measurements use 24 two-qubit gates without errors, while Ryd-3 uses eight two-qubit and four three-qubit gates before any remeasurement.Ryd-3’s CCZ uses 27 physical three-qubit gates in a round-robin implementation.
  • Bias-preserving gates: The Ryd-3 Hadamard gate is built from a fault-tolerant bias-preserving Toffoli followed by single-qubit measurements and rotations.Geometric substitutions using the ancillas address cases where a direct physical Toffoli interaction is too distant.

Appendix I: Computing Rydberg Blockade Radius Requirements for Rydberg FTQC Protocols

The protocols derive blockade-radius requirements by mapping physical qubits to lattice atoms and identifying the maximum interaction distances needed for logical gates. Stabilizer-based circuit modifications reduce the required radius for both Ryd-7 and Ryd-3 implementations, with explicit trade-offs in some cases.

  • Ryd-7 protocol: Ryd-7 maps each physical qubit in a seven-qubit logical state to an atom and determines the radius from the longest required interactions in logical CCZ gates.The standard construction uses 27 physical CCZ gates between neighboring logical qubits.
  • Geometric construction: Figure 15 labels data-atom indices within logical qubits and uses those assignments to derive the gate-count and blockade-radius requirements.The Ryd-7 and Ryd-3 layouts identify which physical atoms participate in logical operations.
  • Ryd-7 protocol: 3.61d is sufficient for Ryd-7 after exploiting the requirement that only two of three atom-pair distances must lie within the blockade radius.A direct implementation would require RB > 4d, but the protocol relaxes this condition.
  • Ryd-3 protocol: Ryd-3 determines RB,1 from the interaction range needed for logical Toffoli gates, with the direct implementation requiring RB,1 > 3.61d.The relevant interaction connects specified physical atoms in neighboring logical qubits.
  • Ryd-3 protocol: 3d is sufficient for Ryd-3 at the cost of four additional two-atom entangling gates per logical Toffoli or Hadamard operation.This trades a shorter interaction range for additional gate operations.
  • Ryd-7 protocol: 3d is sufficient for Ryd-7 after replacing the original 27-gate logical CCZ decomposition with a stabilizer-equivalent decomposition.The reduction uses the seven-qubit code’s stabilizers and preserves the 27 physical CCZ-gate count.

Appendix K: Square Lattice Geometry for Ryd-3

The square-lattice Ryd-3 layout alternates data and ancilla atoms and supports the protocol with two blockade radii. Its denser ancilla arrangement reduces the entangling-operation count for logical Hadamard and Toffoli operations.

  • Lattice geometry: The square-lattice Ryd-3 geometry alternates data and ancilla atoms on lattice vertices, with three data atoms forming each logical qubit.The smaller blockade radius must satisfy d < RB,2 < 2d for stabilizer measurements.
  • Blockade radii: The logical CCZ operation sets the larger-radius requirement because it is implemented using 27 physical CCZ gates.Each physical gate requires the relevant atom pairs to lie within RB,1.
  • Gate implementation: The denser ancilla arrangement permits direct implementation of every physical Toffoli gate without additional ancilla atoms or CNOT gates.This arrangement changes the resource requirements relative to the alternative implementation discussed in the protocol.
  • Gate implementation: 4 fewer two-qubit entangling operations are required for logical Hadamard or Toffoli operations in the square-lattice geometry.The reduction is relative to the operation counts shown in Table 2.
  • Bias-preserving CNOT: The bias-preserving CNOT requires optical pumping that selectively maps mF > 0 states to |1⟩ and mF < 0 states to |0⟩.The stated selectivity is required to avoid generating X- or Y-type errors.
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