pith. machine review for the scientific record. sign in

arxiv: 2602.11457 · v2 · submitted 2026-02-12 · 🪐 quant-ph

Recognition: unknown

The Pinnacle Architecture: Reducing the cost of breaking RSA-2048 to 100 000 physical qubits using quantum LDPC codes

Paul Webster , Lucas Berent , Omprakash Chandra , Evan T. Hockings , Nou\'edyn Baspin , Felix Thomsen , Samuel C. Smith , Lawrence Z. Cohen
Authors on Pith no claims yet
classification 🪐 quant-ph
keywords quantumarchitecturephysicalqubitscodescomputingfault-tolerantfewer
0
0 comments X
read the original abstract

The realisation of utility-scale quantum computing inextricably depends on the design of practical, low-overhead fault-tolerant architectures. We introduce the Pinnacle Architecture, which uses quantum low-density parity check (QLDPC) codes to allow for universal, fault-tolerant quantum computation with a spacetime overhead significantly smaller than that of any competing architecture. With this architecture, we show that 2048-bit RSA integers can be factored with fewer than one hundred thousand physical qubits, given a physical error rate of $10^{-3}$, code cycle time of $1$ microsecond and a reaction time of $10$ microseconds. We thereby demonstrate the feasibility of utility-scale quantum computing with an order of magnitude fewer physical qubits than has previously been believed necessary.

This paper has not been read by Pith yet.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 17 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mitigating Classical Resource Costs in Quantum Error Correction via Generalized qLDPC Predecoding

    quant-ph 2026-05 unverdicted novelty 7.0

    An automated predecoder generator for arbitrary qLDPC codes cuts decoder utilization by up to 3963x and supports hardware scaling to tens or hundreds of thousands of logical qubits within power limits.

  2. Distributed Quantum Error Correction with Bivariate Bicycle Codes in a Modular Architecture

    quant-ph 2026-05 unverdicted novelty 6.0

    The [[144,12,12]] bivariate bicycle code is distributed across 4 to 12 processors in a star network, with simulations showing logical error rates under varying nonlocal noise scaling.

  3. Factoring $2048$ bit RSA integers with a half-million-qubit modular atomic processor

    quant-ph 2026-05 unverdicted novelty 6.0

    A modular atomic processor with 500,000 qubits factors 2048-bit RSA numbers in roughly the same time as a single large module when inter-module Bell-pair communication runs at 10^5 per second.

  4. Architecting Early Fault Tolerant Neutral Atoms Systems with Quantum Advantage

    quant-ph 2026-04 unverdicted novelty 6.0

    A teleportation-based parallelization architecture for neutral-atom quantum error correction delivers up to 3x speedup over extractor methods at fixed space cost and enables simulated quantum advantage at 11,495 atoms...

  5. Fault-Tolerant Quantum Computing with Trapped Ions: The Walking Cat Architecture

    quant-ph 2026-04 unverdicted novelty 6.0

    A trapped-ion architecture based on LDPC codes and cat-state factories achieves 110 logical qubits and one million T gates per day using 2514 physical qubits, with estimates for Heisenberg model simulation on 100 site...

  6. GreenPeas: Unlocking Adaptive Quantum Error Correction with Just-in-Time Decoding Hypergraphs

    quant-ph 2026-04 unverdicted novelty 6.0

    GreenPeas delivers a just-in-time GPU compiler for decoding hypergraphs that achieves >10x speedup on surface and bivariate bicycle codes, unlocking circuit-level decoding for adaptive quantum error correction.

  7. Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays

    quant-ph 2026-04 unverdicted novelty 6.0

    A family of quantum LDPC codes with encoding rates exceeding 1/2 achieves logical error rates of 10^{-13} per round on atom arrays under 0.1% circuit noise using hierarchical decoding.

  8. dqc_simulator: an easy-to-use distributed quantum computing simulator

    quant-ph 2026-04 unverdicted novelty 6.0

    dqc_simulator is a new Python toolkit for automating realistic simulations of both hardware and software in distributed quantum computing systems.

  9. Demonstrating Record Fidelity for the Quantum Fourier Transform

    quant-ph 2026-04 unverdicted novelty 6.0

    Parity Architecture delivers record ~0.01 fidelity for 50-qubit QFT on IBM hardware with super-exponential scaling improvement.

  10. Optimising Quantum Error Correction Using Morphing Circuits

    quant-ph 2026-04 unverdicted novelty 6.0

    Morphing circuits optimize syndrome extraction for Abelian 2BGA and other QEC codes, yielding new circuits with improved parameters, connectivity, and stability against measurement errors.

  11. Heterogeneous architectures enable a 138x reduction in physical qubit requirements for fault-tolerant quantum computing under detailed accounting

    quant-ph 2026-04 unverdicted novelty 6.0

    Heterogeneous quantum architectures with task-specific hardware and QEC encodings deliver up to 138x lower physical-qubit overhead than monolithic baselines for fault-tolerant algorithms, including RSA-2048 factoring ...

  12. Securing Elliptic Curve Cryptocurrencies against Quantum Vulnerabilities: Resource Estimates and Mitigations

    quant-ph 2026-03 conditional novelty 6.0

    Resource estimates show Shor's algorithm can break 256-bit ECDLP with fewer than 1450 logical qubits and 90 million Toffoli gates on fast-clock quantum hardware, enabling on-spend attacks on cryptocurrency mempools.

  13. Constant depth magic state cultivation with Clifford measurements by gauging

    quant-ph 2026-03 unverdicted novelty 6.0

    Gauging enables constant-depth logical XS dagger measurements for color-code magic state cultivation, achieving 10^{-12} logical error rates at 0.05% physical error for distance-7 codes while retaining over 1% of shot...

  14. Understanding oxide-thickness-dependent variability in dense Si-MOS quantum dot arrays

    quant-ph 2026-05 unverdicted novelty 5.0

    A 17 nm SiO2 gate oxide thickness minimizes threshold voltage variability below 63 mV standard deviation in dense silicon quantum dot arrays.

  15. Understanding oxide-thickness-dependent variability in dense Si-MOS quantum dot arrays

    quant-ph 2026-05 unverdicted novelty 5.0

    17 nm SiO2 oxide thickness minimizes threshold voltage variability below 63 mV standard deviation in dense 7x7 silicon quantum dot arrays fabricated via 300 mm CMOS and EUV lithography.

  16. Tolerating Device Failure in Distributed Quantum Computing

    quant-ph 2026-05 unverdicted novelty 5.0

    Distributed toric and hyperbolic Floquet codes maintain logical error suppression when entire nodes fail at low rates, with the toric code outperforming a monolithic device below 0.05% physical error rate for node fai...

  17. Space-Time Tradeoffs of Pauli-Based Computation in Distributed qLDPC Architectures

    quant-ph 2026-05 unverdicted novelty 5.0

    Large qLDPC blocks in distributed quantum computing enable Pauli-based computation to run up to 10x faster than surface codes for optimization algorithms by using spare nodes to bypass serialization bottlenecks.

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · cited by 16 Pith papers · 7 internal anchors

  1. [1]

    This is op- tional, but it is useful in cases where a large number of logi- cal qubits must be stored but not processed

    Memory The architecture can also include memory. This is op- tional, but it is useful in cases where a large number of logi- cal qubits must be stored but not processed. It consists ofν code blocks of an Jnm, km, dmK quantum error-correcting code encoding µ=νk m logical qubits. Since these log- ical qubits are not processed, full processing blocks are not...

    work page

  2. [2]

    In this mode, there is a single processing unit with κ logical qubits (and, for simplicity, we assume there is no memory)

    Serial Operation As a baseline, let us first consider a serial mode of op- eration. In this mode, there is a single processing unit with κ logical qubits (and, for simplicity, we assume there is no memory). During each logical cycle, a joint logical Pauli measurement is performed on the processing unit and magic engine. In parallel, the magic engine produ...

    work page

  3. [3]

    In this context, we can separate the architecture up into a separate process- ing unit for each independent circuit and perform all the circuits in parallel

    Fully Parallel Operation As a next step, we can consider the case of implement- ing a circuit that can be completely separated out into two or more independent circuits. In this context, we can separate the architecture up into a separate process- ing unit for each independent circuit and perform all the circuits in parallel. This reduces the number of lo...

    work page

  4. [4]

    In such a circuit, no subset of logical qubits is entirely separable from the rest, but significant parts of the circuit involve operations on disjoint registers of logical qubits

    Flexibly Parallel Operation More common and general is the case where a circuit can be implemented partially in parallel. In such a circuit, no subset of logical qubits is entirely separable from the rest, but significant parts of the circuit involve operations on disjoint registers of logical qubits. A conventional circuit implementation would allow such...

    work page

  5. [5]

    Recall that each processing unit accesses memory via a port

    General Operation The final step to our fully general operation is to option- ally incorporate the memory. Recall that each processing unit accesses memory via a port. We allow for read-only memory access, which requires only gates that act as a con- trol on the port and a target on the processing unit [8, 35]. This means that the access can be provided b...

    work page

  6. [6]

    Specifically, with a code distance of d= 16 , we can encode 14β logical qubits in 860β physical qubits

    Processing Units Using β processing blocks constructed from the GB code family introduced above (for any β∈N ), we can encode κ=βk logical qubits in βnpb physical qubits. Specifically, with a code distance of d= 16 , we can encode 14β logical qubits in 860β physical qubits. For better protection, we can instead use a code distance of d= 24 and encode 16β ...

    work page

  7. [7]

    These blocks naturally have the required L and R logical sectors, with k/2>5 logical qubits in each sector whend≥10

    Magic Engines We construct each magic engine from code blocks of the same GB code family as those used for the processing blocks. These blocks naturally have the required L and R logical sectors, with k/2>5 logical qubits in each sector whend≥10. We use 15-to-1 magic state distillation on these code blocks to produce encoded | ¯T⟩ magic states [44]. Follo...

    work page 2048

  8. [8]

    For simplicity, we match the window size with the number of logical qubits in a logical sector, k/2

    Memory For the memory, we use the same code blocks as are used for the processing blocks. For simplicity, we match the window size with the number of logical qubits in a logical sector, k/2. Each port then corresponds to one of the Z- type gadgets used in the gadget system of the processing blocks, along with a bridge to connect to a processing unit. The ...

    work page

  9. [9]

    Algorithm The two-dimensional Fermi-Hubbard model represents a system of interacting fermions and has the Hamiltonian H=H h +H I = X ⟨i,j⟩ X σ∈{↑,↓} a† i,σaj,σ +a † j,σai,σ +u X i ˆni,↑ˆni,↓. (14) Here, i denotes the sites of an L×L lattice, ⟨i, j⟩ denotes pairs of nearest neighbours on this lattice, σ∈ {↑,↓} denotes spins states, a† and a represent creat...

    work page

  10. [10]

    In this regime, N≤2050 and we find numerically that the number of logical cycles satisfies T= 8×10 6, which also upper bounds theT count

    Implementation and Results Concretely, we consider the case of even L≤32 and u= 4 . In this regime, N≤2050 and we find numerically that the number of logical cycles satisfies T= 8×10 6, which also upper bounds theT count. This implies that the logical spacetime volume satisfies NT ≤2×10 10. Hence, the algorithm can be implemented with negligible failure p...

    work page 2050

  11. [11]

    [ 8], which uses techniques de- veloped by Ekerå and Håstad [58] and by Chevignard et al

    Algorithm The algorithm we use is a generalisation of that pre- sented by Gidney in Ref. [ 8], which uses techniques de- veloped by Ekerå and Håstad [58] and by Chevignard et al. [59]. We refer to this algorithm as Gidney’s algorithm. This algorithm uses residue number system arithmetic to replace modular arithmetic over NRSA (the number being factored) w...

    work page 2048

  12. [12]

    These processing units can run in parallel throughout most of the computation

    Implementation on Pinnacle Architecture To implement the algorithm on the Pinnacle Architec- ture, we begin by allocating a processing unit for each working register. These processing units can run in parallel throughout most of the computation. The only exception is the relatively short periods when the accumulators of the working registers are being agg...

    work page

  13. [13]

    Physical Qubits:We now determine the number of physical qubits required

    Resource Analysis a. Physical Qubits:We now determine the number of physical qubits required. Each working register corre- sponds to a processing unit with κ(f, ℓ, m) logical qubits, as given in Eq. (23), along with a magic engine. The num- ber of physical qubits required for the ρ working registers is therefore nw =ρ npb κ(f, ℓ, m) k +n me ! .(24) Each m...

    work page

  14. [14]

    Following Ref

    Results We now consider the resources—both physical qubits and time—required to factor an RSA-2048 integer on the instantiation of the Pinnacle architecture presented in Sec- tion V, given different hardware parameters, namely the code cycle time and physical error rate. Following Ref. [8], we expect the required logical error rate per logical qubit per l...

    work page 2048

  15. [15]

    19 In the same formalism, an n-qubit Clifford operator U can be represented by a 2n×2n matrix MU that preserves the symplectic inner product (i.e., such that MU J MT U =J )

    The symplectic complement W ⊥ of subspace W is a subspace whose elements commute with all elements in W. 19 In the same formalism, an n-qubit Clifford operator U can be represented by a 2n×2n matrix MU that preserves the symplectic inner product (i.e., such that MU J MT U =J ). The action of U on Pv by conjugation corresponds to matrix multiplication on t...

    work page

  16. [16]

    Shor, Algorithms for quantum computation: Discrete loga- rithms and factoring, inProceedings 35th Annual Symposium on Foundations of Computer Science(1994) pp

    P. Shor, Algorithms for quantum computation: Discrete loga- rithms and factoring, inProceedings 35th Annual Symposium on Foundations of Computer Science(1994) pp. 124–134

    work page 1994

  17. [17]

    Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    work page 1996

  18. [18]

    Opportunities and challenges in fault-tolerant quantum computation.arXiv preprint arXiv:2210.15844, 2022

    D. Gottesman, Opportunities and challenges in fault- tolerant quantum computation (2022), arXiv:2210.15844

    work page arXiv 2022

  19. [19]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cle- land, Surface codes: Towards practical large-scale quan- tum computation, Physical Review A86, 032324 (2012), arXiv:1208.0928

    work page arXiv 2012

  20. [20]

    Horsman, A

    D. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New Journal of Physics14, 123011 (2012), arXiv:1111.4022

    work page arXiv 2012

  21. [21]

    A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

    D. Litinski, A game of surface codes: Large-scale quantum computing with lattice surgery, Quantum3, 128 (2019), arXiv:1808.02892

    work page Pith review arXiv 2019

  22. [22]

    How to Factor 2048 Bit RSA Integers in 8 Hours Using 20 Million Noisy Qubits,

    C. Gidney and M. Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum5, 433 (2021), arXiv:1905.09749

    work page arXiv 2048

  23. [23]

    How to factor 2048 bit RSA integers with less than a million noisy qubits

    C. Gidney, How to factor 2048 bit RSA integers with less than a million noisy qubits (2025), arXiv:2505.15917

    work page internal anchor Pith review arXiv 2048

  24. [24]

    Mohseni, A

    M. Mohseni, A. Scherer, K. G. Johnson, O. Wertheim, M. Ot- ten, N. A. Aadit, Y. Alexeev, K. M. Bresniker, K. Y. Cam- sari, B. Chapman, S. Chatterjee, G. A. Dagnew, A. Espos- ito, F. Fahim, M. Fiorentino, A. Gajjar, A. Khalid, X. Kong, B. Kulchytskyy, E. Kyoseva, R. Li, P. A. Lott, I. L. Markov, R. F. McDermott, G. Pedretti, P. Rao, E. Rieffel, A. Silva, J...

    work page arXiv 2025

  25. [25]

    T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, Tour de gross: A modular quantum computer based on bivariate bicycle codes (2025), arXiv:2506.03094

    work page internal anchor Pith review arXiv 2025

  26. [26]

    Webster, S

    P. Webster, S. C. Smith, and L. Z. Cohen, Explicit construc- tion of low-overhead gadgets for gates on quantum LDPC codes (2025), arXiv:2511.15989

    work page arXiv 2025

  27. [27]

    Chamberland and E

    C. Chamberland and E. T. Campbell, Universal quantum computing with twist-free and temporally encoded lattice surgery, PRX Quantum3, 010331 (2022), arXiv:2109.02746

    work page arXiv 2022

  28. [28]

    I. D. Kivlichan, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, W. Sun, Z. Jiang, N. Rubin, A. Fowler, A. Aspuru-Guzik, H. Neven, and R. Babbush, Improved fault-tolerant quan- tum simulation of condensed-phase correlated electrons via Trotterization, Quantum4, 296 (2020), arXiv:1902.10673

    work page arXiv 2020

  29. [29]

    Bravyi, G

    S. Bravyi, G. Smith, and J. Smolin, Trading classical and quantum computational resources, Physical Review X6, 021043 (2016), arXiv:1506.01396

    work page arXiv 2016

  30. [30]

    E. T. Campbell, Early fault-tolerant simulations of the Hub- bard model, Quantum Science and Technology7, 015007 (2022), arXiv:2012.09238

    work page arXiv 2022

  31. [31]

    A. C. Hughes, R. Srinivas, C. M. Löschnauer, H. M. Knaack, R. Matt, C. J. Ballance, M. Malinowski, T. P. Harty, and R. T. Sutherland, Trapped-ion two-qubit gates with >99.99% fidelity without ground-state cooling (2025), arXiv:2510.17286

    work page arXiv 2025

  32. [32]

    H. Zhou, C. Duckering, C. Zhao, D. Bluvstein, M. Cain, A. Ku- bica, S.-T. Wang, and M. D. Lukin, Resource analysis of low-overhead transversal architectures for reconfigurable atom arrays, inProceedings of the 52nd Annual International Symposium on Computer Architecture(2025) pp. 1432–1448, arXiv:2505.15907

    work page arXiv 2025

  33. [33]

    D. J. C. MacKay, G. Mitchison, and P. L. McFadden, Sparse graph codes for quantum error-correction, IEEE Transac- tions on Information Theory50, 2315 (2004), arXiv:quant- ph/0304161

    work page arXiv 2004

  34. [34]

    N. P. Breuckmann and J. N. Eberhardt, Quantum low- density parity-check codes, PRX Quantum2, 040101 (2021), arXiv:2103.06309

    work page arXiv 2021

  35. [35]

    Yoneda, W

    J. Yoneda, W. Huang, M. Feng, C. H. Yang, K. W. Chan, T. Tanttu, W. Gilbert, R. C. C. Leon, F. E. Hudson, K. M. Itoh, A. Morello, S. D. Bartlett, A. Laucht, A. Saraiva, and A. S. Dzurak, Coherent spin qubit transport in silicon, Nature 21 Communications12, 4114 (2021), arXiv:2008.04020

    work page arXiv 2021

  36. [36]

    Malinowski, D

    M. Malinowski, D. T. C. Allcock, and C. J. Ballance, How to wire a 1000-qubit trapped ion quantum computer, PRX Quantum4, 040313 (2023), arXiv:2305.12773

    work page arXiv 2023

  37. [37]

    Bluvsteinet al., A quantum processor based on co- herent transport of entangled atom arrays, Nature604, 451 (2022), arXiv:2112.03923 [quant-ph]

    D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletic, and M. D. Lukin, A quantum proces- sor based on coherent transport of entangled atom arrays, Nature604, 451 (2022), arXiv:2112.03923

    work page arXiv 2022

  38. [38]

    Interleaving: Modular architectures for fault-tolerant photonic quantum computing

    H. Bombin, I. H. Kim, D. Litinski, N. Nickerson, M. Pant, F. Pastawski, S. Roberts, and T. Rudolph, Interleaving: Mod- ular architectures for fault-tolerant photonic quantum com- puting (2021), arXiv:2103.08612

    work page arXiv 2021

  39. [39]

    High-threshold and low- overhead fault-tolerant quantum memory

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature627, 778 (2024), arXiv:2308.07915

    work page arXiv 2024

  40. [40]

    L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Low- overhead fault-tolerant quantum computing using long- range connectivity, Science Advances8, eabn1717 (2022), arXiv:2110.10794

    work page arXiv 2022

  41. [41]

    D. J. Williamson and T. J. Yoder, Low-overhead fault-tolerant quantum computation by gauging logical operators (2024), arXiv:2410.02213

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  42. [42]

    B. Ide, M. G. Gowda, P. J. Nadkarni, and G. Dauphi- nais, Fault-tolerant logical measurements via homologi- cal measurement, Physical Review X15, 021088 (2025), arXiv:2410.02753

    work page arXiv 2025

  43. [43]

    Following Refs [6, 10], we assume that the time required between logical measurements is negligible compared with the timescale of a logical cycle

    work page

  44. [44]

    A. W. Cross, Z. He, P. J. Rall, and T. J. Yoder, Improved QLDPC surgery: Logical measurements and bridging codes (2025), arXiv:2407.18393

    work page arXiv 2025

  45. [45]

    Universal adapters between quantum LDPC codes,

    E. Swaroop, T. Jochym-O’Connor, and T. J. Yoder, Universal adapters between quantum low-density parity check codes, PRX Quantum7, 010324 (2026), arXiv:2410.03628

    work page arXiv 2026

  46. [46]

    Z. He, A. Cowtan, D. J. Williamson, and T. J. Yoder, Extrac- tors: QLDPC architectures for efficient Pauli-based compu- tation (2025), arXiv:2503.10390

    work page arXiv 2025

  47. [47]

    Quantum error correction below the surface code threshold.arXiv preprint arXiv:2408.13687, 2024

    R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev, J. Atalaya, R. Babbush, D. Bacon, B. Ballard, J. C. Bardin, J. Bausch, A. Bengtsson, A. Bilmes, S. Blackwell, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, D. A. Browne, B. Buchea, B. B. Buckley, D. A. Buell, T. B...

    work page arXiv 2025

  48. [48]

    Pataki and A

    D. Pataki and A. Pályi, Compiling the surface code to cross- bar spin qubit architectures, Physical Review B111, 115307 (2025), arXiv:2412.05425

    work page arXiv 2025

  49. [49]

    Leone, T

    H. Leone, T. Le, S. Srikara, and S. Devitt, Resource overheads and attainable rates for trapped-ion lattice surgery, Physical Review Research7, 023088 (2025), arXiv:2406.18764

    work page arXiv 2025

  50. [50]

    Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity,

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spec- tra in quantum circuits with linear T complexity, Physical Review X8, 041015 (2018), arXiv:1805.03662

    work page arXiv 2018

  51. [51]

    Architecting Distributed Quantum Computers: Design Insights from Resource Estimation

    D. Filippov, P. Yang, and P. Murali, Architecting distributed quantum computers: Design insights from resource estima- tion (2025), arXiv:2508.19160

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  52. [52]

    A. A. Kovalev and L. P. Pryadko, Quantum Kronecker sum- product low-density parity-check codes with finite rate, Physical Review A88, 012311 (2013), arXiv:1212.6703

    work page arXiv 2013

  53. [53]

    Panteleev and G

    P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum5, 585 (2021), arXiv:1904.02703

    work page arXiv 2021

  54. [54]

    du Crest, M

    J. du Crest, M. Mhalla, and V. Savin, Stabilizer inactiva- tion for message-passing decoding of quantum LDPC codes (2023), arXiv:2205.06125

    work page arXiv 2023

  55. [55]

    H.-K. Lin, X. Liu, P. K. Lim, and L. P. Pryadko, Single-shot and two-shot decoding with generalized bicycle codes (2025), arXiv:2502.19406

    work page arXiv 2025

  56. [56]

    Wang and F

    M. Wang and F. Mueller, Coprime bivariate bicycle codes and their layouts on cold atoms, Quantum10, 2009 (2026), arXiv:2408.10001

    work page arXiv 2009

  57. [57]

    F. J. MacWilliams and N. J. A. Sloane,The Theory of Error- Correcting Codes, North-Holland Mathematical Library, Vol. 16 (Elsevier, 1977)

    work page 1977

  58. [58]

    Williamson, and Theodore J

    A. Cowtan, Z. He, D. J. Williamson, and T. J. Yoder, Paral- lel logical measurements via quantum code surgery (2026), arXiv:2503.05003

    work page internal anchor Pith review arXiv 2026

  59. [59]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computation 22 with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005), arXiv:quant-ph/0403025

    work page arXiv 2005

  60. [60]

    Magic State Distillation: Not as Costly as You Think,

    D. Litinski, Magic state distillation: Not as costly as you think, Quantum3, 205 (2019), arXiv:1905.06903

    work page arXiv 2019

  61. [61]

    Goto, Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code, Scientific Reports6, 19578 (2016)

    H. Goto, Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code, Scientific Reports6, 19578 (2016)

    work page 2016

  62. [62]

    Even more efficient magic state distillation by zero-level distillation , publisher =

    T. Itogawa, Y. Takada, Y. Hirano, and K. Fujii, Efficient magic state distillation by zero-level distillation, PRX Quantum6, 020356 (2025), arXiv:2403.03991

    work page arXiv 2025

  63. [63]

    Magic state cultivation: growing T states as cheap as CNOT gates

    C. Gidney, N. Shutty, and C. Jones, Magic state cultiva- tion: Growing T states as cheap as CNOT gates (2024), arXiv:2409.17595

    work page internal anchor Pith review arXiv 2024

  64. [64]

    Sahay, P.-K

    K. Sahay, P.-K. Tsai, K. Chang, Q. Su, T. B. Smith, S. Singh, and S. Puri, Fold-transversal surface code cultivation (2025), arXiv:2509.05212

    work page arXiv 2025

  65. [65]

    [49], extrapolating under the assumption that the failure rate is linear in p for the case of p= 10 −4

    We estimate these reject rates using data on expected at- tempts (which we denote γ) from Ref. [49], extrapolating under the assumption that the failure rate is linear in p for the case of p= 10 −4. Specifically, the overall reject rate is then given by pr = (1−(1−1/γ) ξ)15, where ξ is the number of attempts that are possible in the available time and wit...

    work page

  66. [66]

    A. R. O’Rourke and S. Devitt, Compare the pair: Rotated ver- sus unrotated surface codes at equal logical error rates, Phys- ical Review Research7, 033074 (2025), arXiv:2409.14765

    work page arXiv 2025

  67. [67]

    This suffices since we can choose the eight logical measure- ments in the first batch to act trivially on one of the four logical qubits involved in post-selection

    work page

  68. [68]

    Vittal, A

    S. Vittal, A. Javadi-Abhari, A. W. Cross, L. S. Bishop, and M. Qureshi, Flag-proxy networks: Overcoming the archi- tectural, scheduling and decoding obstacles of quantum LDPC codes, in2024 57th IEEE/ACM International Sym- posium on Microarchitecture (MICRO)(2024) pp. 718–734, arXiv:2409.14283

    work page arXiv 2024

  69. [69]

    Raveendran and B

    N. Raveendran and B. Vasić, Trapping sets of quantum LDPC codes, Quantum5, 562 (2021), arXiv:2012.15297

    work page arXiv 2021

  70. [70]

    Paetznick and K

    A. Paetznick and K. M. Svore, Repeat-until-success: Non- deterministic decomposition of single-qubit unitaries, Quantum Information and Computation14, 1277 (2014), arXiv:1311.1074

    work page arXiv 2014

  71. [71]

    arXiv preprint , author=

    A. Bocharov, M. Roetteler, and K. M. Svore, Efficient syn- thesis of universal repeat-until-success circuits, Physical Review Letters114, 080502 (2015), arXiv:1404.5320

    work page arXiv 2015

  72. [72]

    [ 15] in using this ap- proach

    For simplicity, we here follow Ref. [ 15] in using this ap- proach. We note that more recent advances in rotation syn- thesis have since been presented [65], which may allow for reduced runtimes if incorporated in future

    work page

  73. [73]

    Ekerå and J

    M. Ekerå and J. Håstad, Quantum algorithms for computing short discrete logarithms and factoring RSA integers, in Post-Quantum Cryptography, Vol. 10346 (2017) pp. 347–363, arXiv:1702.00249

    work page arXiv 2017

  74. [74]

    Chevignard, P.-A

    C. Chevignard, P.-A. Fouque, and A. Schrottenloher, Reduc- ing the number of qubits in quantum factoring, inAdvances in Cryptology – CRYPTO 2025, edited by Y. Tauman Kalai and S. F. Kamara (2025) pp. 384–415

    work page 2025

  75. [75]

    Jones, Low-overhead constructions for the fault- tolerant Toffoli gate, Physical Review A87, 022328 (2013), arXiv:1212.5069

    C. Jones, Low-overhead constructions for the fault- tolerant Toffoli gate, Physical Review A87, 022328 (2013), arXiv:1212.5069

    work page arXiv 2013

  76. [76]

    Litinski, How to compute a 256-bit elliptic curve private key with only 50 million toffoli gates, arXiv:2306.08585 (2023)

    D. Litinski, How to compute a 256-bit elliptic curve private key with only 50 million Toffoli gates (2023), arXiv:2306.08585

    work page arXiv 2023

  77. [77]

    M. E. Beverland, P. Murali, M. Troyer, K. M. Svore, T. Hoefler, V. Kliuchnikov, G. H. Low, M. Soeken, A. Sundaram, and A. Vaschillo, Assessing requirements to scale to practical quantum advantage (2022), arXiv:2211.07629

    work page internal anchor Pith review arXiv 2022

  78. [78]

    Baspin, L

    N. Baspin, L. Berent, and L. Z. Cohen, Fast surgery for quan- tum LDPC codes (2025), arXiv:2510.04521

    work page arXiv 2025

  79. [79]

    Improved simulation of stabilizer circuits

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Physical Review A70, 052328 (2004), arXiv:quant-ph/0406196

    work page arXiv 2004

  80. [80]

    Kliuchnikov, K

    V. Kliuchnikov, K. Lauter, R. Minko, A. Paetznick, and C. Pe- tit, Shorter quantum circuits via single-qubit gate approxi- mation, Quantum7, 1208 (2023), arXiv:2203.10064

    work page arXiv 2023