Fast logical operations in quantum LDPC codes using simple resource states

Source: arXiv:2607.16166 · Published 2026-07-17 · By Mark Webster, Nicolas Delfosse

TL;DR

This paper addresses the challenge of performing fast logical operations on multiple qubits simultaneously encoded in quantum LDPC codes, which offer high encoding rates and reduced qubit overhead compared to surface codes. Prior methods for logical measurement in LDPC codes rely on complex resource states that merge memory and measurement codes, imposing significant hardware complexity and practical limitations. The authors propose a new protocol that leverages only simple cat states to jointly measure ℓ commuting logical operators by introducing a novel scheduler code that determines the measurement sequence and enables joint decoding of logical outcomes. This approach significantly reduces the required cat-based measurements per operator and accelerates fault-tolerant quantum computation. Numerical simulations on quantum LDPC codes Q70 and Q102 demonstrate nearly a 3x speedup over prior Viterbi protocols for measuring 20 commuting operators.

Additionally, the authors adapt the protocol for use with a variant of the CliNR partial error correction scheme to speed up logical Clifford circuits by up to 74x for Q102 and logical Toffoli gates by 5x. This work sheds light on practical, near-term strategies for accelerating logical operations in quantum LDPC codes by directly exploiting simple resource states and error detection codes, rather than relying on complex homological constructions.

Key findings

• The multi-Pauli MECM protocol reduces the average number of cat-based measurements per logical Pauli to 1.71 for ℓ=20, compared to 5.06 measurements for the single-Pauli Viterbi protocol (∼3x speedup)
• The truncated MEDM protocol requires 1.81 measurements per operator for ℓ=20, slightly above MECM but still considerably faster than prior protocols
• Using the new fast joint measurement protocols within a logical CliNR scheme yields speedups of 18.5x (Q70) and 74.4x (Q102) for random Clifford circuits relative to gate-based Viterbi synthesis
• For logical Toffoli gates implemented via CliNR, the speedup over Viterbi is 4.2x (Q70) and 5.0x (Q102)
• Cat-state measurement error rates modeled as independent bit-flips with pF=6.3×10^-3 at physical error rate p=10^-4 and cat state size 30 were used in simulations
• Scheduler codes are chosen from best-known-distance binary linear codes to optimize error detection and decoding performance
• The truncated MECM protocol terminates fewer measurements once posterior probability >1–ε, exploiting joint outcomes to accelerate logical measurement
• Logical Clifford gates that cannot be implemented via frame tracking require significantly fewer resource state measurements using the proposed protocol

Threat model

Adversaries are modeled indirectly via random independent bit-flip errors on measurement outcomes of cat states, reflecting physical decoherence and gate errors with no active adversarial manipulation assumed. The protocols ensure logical error rates below target thresholds despite these errors. Adversaries cannot manipulate or delay measurements beyond this error model, nor manipulate classical decoding stages.

Methodology — deep read

1. Threat Model & Assumptions: The adversary is modeled via an independent bit-flip channel on cat-based measurement outcomes with rate pF, derived from underlying physical error rate p=10^-4 and cat state size w=30. The protocols are designed for fault-tolerance but target a specific logical error rate ε=10^-10 rather than full asymptotic fault-tolerance. The main adversarial effect considered is measurement errors causing incorrect logical outcome decoding. The code structure and syndrome extraction processes are assumed ideal except for these errors.

2. Data: No external datasets are used; the protocols are evaluated via numerical simulation on known quantum LDPC codes Q70 and Q102 from prior work [16]. These codes specify the number of logical qubits encoded and the cat state sizes relevant for measurement.

3. Architecture/Algorithm: The core innovation is the introduction of scheduler codes, linear binary codes with generator matrices G that define a measurement schedule combining ℓ commuting logical Pauli operators into m joint measurements (columns of G). Each measurement corresponds to a product of logical Paulis indexed over the rows.

Two primary protocols are defined:

• MEDM (multi-operator error-detected measurement): Repeated measurement attempts are performed with error detection via the parity check matrix H of the scheduler code; attempts restart upon detecting errors.
• MECM (multi-operator error-corrected measurement): Single attempts with decoding applied to measurement outcomes to correct errors and determine the most likely logical result. This includes a truncated variant that adaptively terminates measurements once sufficient posterior confidence is achieved.

The protocols require choosing scheduler codes with appropriate distance to meet target logical error rates. Classical coding theory tools like weight enumerators, lookup decoders, and heuristic random information set decoders are used to analyze the logical error rates and performance.

4. Training Regime: Not applicable; numerical evaluation uses simulation parameters. The simulation models the bit-flip error channel on cat state measurement outcomes and calculates expected measurement counts and error rates analytically or via Monte Carlo sampling with up to 4096 samples for larger ℓ.

5. Evaluation Protocol: Metrics include the average number of cat-state measurements needed to reach logical error rate ε=10^-10. Baselines are the single-Pauli Viterbi measurement protocol from prior work. Performance is evaluated for ℓ = 1 to 20 commuting Paulis. Additional metrics include speedup factors in logical gate execution when integrated with the CliNR partial error correction scheme. Ablations compare truncated and full protocols.

6. Reproducibility: The scheduler codes are selected from best-known classical codes tabulated in external coding references [46]. Decoder implementations and weight enumerator calculations are described in detail, including deterministic and heuristic approaches. No explicit code release or frozen weights are indicated. The quantum LDPC codes Q70 and Q102 are publicly described in prior literature [16].

End-to-end Example (ℓ=20, MECM): Given 20 commuting logical Paulis, a scheduler matrix G is constructed from concatenated codes G0 and G1 to meet undetectable error rate and logical error rate constraints. Cat-state measurements correspond to columns of G, each acting on products of logical Paulis. Measurement outcomes form a vector v corrupted by independent bit-flip errors. A decoding procedure (lookup or heuristic) finds the most likely logical outcome û minimizing error weight. The protocol adapts measurement length to reach posterior confidence >1–ε, minimizing average total cat measurements to ~1.7 per logical operator, representing ~3x speedup compared to single operator Viterbi. This result reflects a joint quantum-classical coding and decoding strategy enabled by the scheduler code.

Technical innovations

• Introduction of scheduler codes encoding measurement schedules as classical linear codes to jointly measure multiple commuting logical operators in quantum LDPC codes
• Extension of single-operator cat-state measurement protocols to multi-operator error-detected (MEDM) and error-corrected (MECM) protocols leveraging joint decoding
• Adaptive truncated measurement termination based on posterior probability thresholds to minimize average measurement rounds
• Combination of fast multi-Pauli measurement protocols with a logical-level variant of the CliNR partial error correction scheme to accelerate logical Clifford and Toffoli gates
• Use of classical coding theory tools (weight enumerators, lookup decoders, random information set decoders) to optimize error rates and measurement scheduling

Datasets

• Q70 quantum LDPC code — 6 logical qubits — public from prior work [16]
• Q102 quantum LDPC code — 22 logical qubits — public from prior work [16]

Baselines vs proposed

• Viterbi single-Pauli protocol for ℓ=20: average 5.06 cat-based measurements per operator vs MECM multi-Pauli: 1.71 measurements/operator (~3x speedup)
• Viterbi single-Pauli protocol for ℓ=10: 5.06 vs MECM multi-Pauli: 2.41 measurements/operator (~2.1x speedup)
• MEDM vs MECM for ℓ=20: 1.81 vs 1.71 measurements/operator
• CliNR Toffoli gate Q70 code: 4.2x faster than Viterbi-based gate synthesis
• CliNR Toffoli gate Q102 code: 5.0x faster than Viterbi
• CliNR random Clifford circuits Q70 code: 18.5x faster
• CliNR random Clifford circuits Q102 code: 74.4x faster

Limitations

• Assumes measurement errors modeled solely as independent bit-flip noise on cat state measurements; does not model correlated or more complex error channels
• Evaluation is purely simulation-based; lacks experimental or hardware validation of walking cat architecture and cat state preparation overheads
• Scheduler codes are selected from known classical codes rather than explicitly designed scheduler codes optimized for quantum measurement constraints
• Logical Clifford operations sometimes rely on frame tracking, restricting exact generalization to arbitrarily complex logical operations
• The approach assumes accessibility of cat states of sufficient size, which may limit applicability depending on hardware constraints
• Adversarial or adaptive noise models are not evaluated; robustness under active attacks or biased noise remains untested

Open questions / follow-ons

• What specific properties of quantum LDPC codes guarantee independence of cat-based measurement errors necessary for the protocols’ assumptions?
• Can dedicated scheduler codes be explicitly constructed to further reduce measurement overhead beyond using best-known classical codes?
• How does the protocol perform under different or correlated noise models, particularly in realistic hardware with non-Pauli noise?
• What are practical constraints and error budgets when implementing these protocols on physical architectures like trapped ions or superconducting qubits?

Why it matters for bot defense

From a bot-defense or CAPTCHA design perspective, this paper is tangentially relevant as it advances fault-tolerant quantum computation methods rather than classical challenge-response systems. However, it provides insights into how to reduce overhead and latency in fault-tolerant operations on encoded logical qubits, which could inform future post-quantum cryptographic hardware implementations. The scheduler code technique exemplifies how classical coding theory can optimize joint detection/measurement protocols to minimize computation time and errors. Practitioners interested in the future of quantum-resistant security may benefit from understanding these advances in logical operation efficiency within complex quantum error correction codes. Yet, direct application to CAPTCHAs or bot mitigation systems remains limited.

Cite

@article{arxiv2607_16166,
  title={ Fast logical operations in quantum LDPC codes using simple resource states },
  author={ Mark Webster and Nicolas Delfosse },
  journal={arXiv preprint arXiv:2607.16166},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.16166}
}

Read the full paper

• arXiv:2607.16166 (abstract)
• arXiv:2607.16166 (PDF)