Recognition: 2 theorem links
Shor's algorithm is possible with as few as 10,000 reconfigurable atomic qubits
Pith reviewed 2026-05-16 04:29 UTC · model grok-4.3
The pith
Shor's algorithm can be executed at cryptographically relevant scales with as few as 10,000 reconfigurable atomic qubits.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By leveraging advances in high-rate quantum error-correcting codes, efficient logical instruction sets, and circuit design, Shor's algorithm can be executed at cryptographically relevant scales with as few as 10,000 reconfigurable atomic qubits. Increasing the number of physical qubits improves time efficiency by enabling greater parallelism; under plausible assumptions, the runtime for discrete logarithms on the P-256 elliptic curve could be just a few days for a system with 26,000 physical qubits, while the runtime for factoring RSA-2048 integers is one to two orders of magnitude longer.
What carries the argument
high-rate quantum error-correcting codes on reconfigurable neutral-atom arrays
If this is right
- Discrete-logarithm instances on the P-256 curve could finish in a few days using 26,000 physical qubits.
- Factoring RSA-2048 integers would require one to two orders of magnitude more runtime than the P-256 case on the same hardware.
- Neutral-atom platforms with a few tens of thousands of qubits could support cryptographically relevant quantum computation once fault tolerance is reached.
Where Pith is reading between the lines
- Reconfigurability appears to be a key advantage for keeping physical-qubit counts low compared with fixed-lattice architectures.
- The same code and layout techniques could lower resource estimates for other quantum algorithms that rely on similar arithmetic primitives.
- If neutral-atom systems demonstrate the assumed error rates at scale, the timeline for testing Shor's algorithm on relevant sizes shortens considerably.
Load-bearing premise
Neutral-atom hardware can achieve universal fault-tolerant operations below the error-correction threshold at the required scale and with sufficient reconfigurability.
What would settle it
An experiment that measures logical error rates above the threshold needed for the high-rate codes when scaled to thousands of atoms, or a revised overhead calculation showing that the codes require substantially more than 10,000 physical qubits.
read the original abstract
Quantum computers have the potential to perform computational tasks beyond the reach of classical machines. A prominent example is Shor's algorithm for integer factorization and discrete logarithms, which is of both fundamental importance and practical relevance to cryptography. However, due to the high overhead of quantum error correction, optimized resource estimates for cryptographically relevant instances of Shor's algorithm require millions of physical qubits. Here, by leveraging advances in high-rate quantum error-correcting codes, efficient logical instruction sets, and circuit design, we show that Shor's algorithm can be executed at cryptographically relevant scales with as few as 10,000 reconfigurable atomic qubits. Increasing the number of physical qubits improves time efficiency by enabling greater parallelism; under plausible assumptions, the runtime for discrete logarithms on the P-256 elliptic curve could be just a few days for a system with 26,000 physical qubits, while the runtime for factoring RSA-2048 integers is one to two orders of magnitude longer. Recent neutral-atom experiments have demonstrated universal fault-tolerant operations below the error-correction threshold, computation on arrays of hundreds of qubits, and trapping arrays with more than 6,000 highly coherent qubits. Although substantial engineering challenges remain, our theoretical analysis indicates that an appropriately designed neutral-atom architecture could support quantum computation at cryptographically relevant scales. More broadly, these results highlight the capability of neutral atoms for fault-tolerant quantum computing with wide-ranging scientific and technological applications.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a theoretical resource estimate for executing Shor's algorithm at cryptographically relevant scales (e.g., RSA-2048 factoring and P-256 discrete logarithms) using high-rate quantum error-correcting codes, optimized logical gate sets, and reconfigurable neutral-atom hardware, claiming that as few as 10,000 physical qubits suffice under stated error models, with runtimes ranging from days to months depending on qubit count and parallelism.
Significance. If the modeling assumptions hold, the result substantially lowers prior qubit-overhead estimates for fault-tolerant Shor's algorithm from millions to ~10k physical qubits, providing concrete targets for neutral-atom platforms and highlighting how reconfigurability and high-rate codes can reduce resources for cryptographic applications.
major comments (2)
- [Resource Estimation] The central 10,000-qubit claim (abstract and resource-estimate section) is derived from external code-threshold literature and hardware-performance assumptions rather than a fully self-contained derivation with explicit error-bar propagation; a sensitivity analysis varying the logical error rate and code rate parameters would be required to support the exact figure.
- [Hardware Model] § on hardware feasibility: the extrapolation from current neutral-atom experiments (hundreds of qubits, >6,000 trapped) to universal fault-tolerant operation at 10k-qubit scale with the required reconfigurability overhead is presented as an assumption; the manuscript should quantify the reconfigurability cost in the logical-gate overhead calculation.
minor comments (2)
- [Results] Table or figure reporting runtime vs. qubit count should explicitly list the code parameters (rate, distance, threshold) used for each data point.
- [Abstract] The abstract's phrasing 'under plausible assumptions' for the few-day P-256 runtime would benefit from a one-sentence definition of those assumptions in the main text.
Simulated Author's Rebuttal
We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below and have made revisions to improve the clarity and robustness of the resource estimates and hardware model.
read point-by-point responses
-
Referee: [Resource Estimation] The central 10,000-qubit claim (abstract and resource-estimate section) is derived from external code-threshold literature and hardware-performance assumptions rather than a fully self-contained derivation with explicit error-bar propagation; a sensitivity analysis varying the logical error rate and code rate parameters would be required to support the exact figure.
Authors: We agree that additional sensitivity analysis strengthens the presentation of the 10,000-qubit figure. While the estimate is constructed from established code thresholds in the literature combined with our optimized logical gates and circuit design, we have added a dedicated sensitivity analysis subsection in the revised resource-estimate section. This varies the logical error rate by factors of 2–10 around the nominal value and explores code rates within the range of high-rate codes, showing the physical qubit requirement remains between approximately 8,000 and 15,000 qubits. The analysis is presented as a new figure with accompanying text to provide explicit bounds. revision: yes
-
Referee: [Hardware Model] § on hardware feasibility: the extrapolation from current neutral-atom experiments (hundreds of qubits, >6,000 trapped) to universal fault-tolerant operation at 10k-qubit scale with the required reconfigurability overhead is presented as an assumption; the manuscript should quantify the reconfigurability cost in the logical-gate overhead calculation.
Authors: The referee is correct that reconfigurability was previously treated as an assumption. In the revised manuscript we have quantified this cost by including an explicit overhead factor of 20–30% added to logical gate cycle times to account for atom transport and rearrangement, drawing on recent experimental demonstrations of reconfigurable neutral-atom arrays. This adjustment is now incorporated into the runtime calculations and discussed in detail in the hardware feasibility section, with supporting references. revision: yes
Circularity Check
Resource estimates are self-contained against external benchmarks with no internal reduction
full rationale
The paper's central claim is a resource estimate for executing Shor's algorithm at scale using high-rate QEC codes, logical gate overheads, and reconfigurable neutral-atom operations. All load-bearing calculations (qubit counts, runtimes for P-256 and RSA-2048) are derived from explicit circuit constructions, code performance models, and stated error thresholds drawn from the broader literature. No equation or step reduces by construction to a quantity defined inside the paper itself, nor does any uniqueness claim rest solely on self-citation. The hardware feasibility assumptions are explicitly flagged rather than derived internally. This is the standard case of an independent estimate against external benchmarks, warranting score 0.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption High-rate quantum error-correcting codes achieve sufficiently low overhead for Shor's algorithm at the stated scale
- domain assumption Neutral-atom hardware can perform universal fault-tolerant operations below the error-correction threshold with the required reconfigurability
Forward citations
Cited by 19 Pith papers
-
Multi-Qubit Stabilizer Readout on a Dual-Species Rydberg Array
Dual-species Na-Cs Rydberg array enables simultaneous non-destructive readout of multiple Pauli-Z stabilizers on four-qubit plaquettes using a single global pulse sequence after compensating geometric phase errors.
-
Real-time Krylov Diagonalisation for Open Quantum Systems
Real-time Krylov subspace methods are extended to Lindblad open quantum systems and demonstrated on a Kerr resonator for estimating the Liouvillian gap in cat qubit regimes.
-
Mitigating Classical Resource Costs in Quantum Error Correction via Generalized qLDPC Predecoding
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.
-
An Algorithm for Fast Assembling Large-Scale Defect-Free Atom Arrays
A graph neural network path planner and phase-aware Gerchberg-Saxton algorithm enable defect-free assembly of 10,000-atom arrays in under 6 ms, faster than typical atom loss times.
-
Spatial overhead reduction for 2D hypergraph product codes
A qubit-reduction method for hypergraph product codes preserves dimension, distance, and fault-tolerance properties, producing smaller codes such as [[441,64,6]] from [[610,64,6]] with comparable noise performance and...
-
The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm
Shor's algorithm generates and consumes magic resources in direct proportion to the difficulty of the underlying factoring problem.
-
Factoring $2048$ bit RSA integers with a half-million-qubit modular atomic processor
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.
-
Adaptable Continuous Variable Quantum Network with Finite Size Security
Experimental demonstration of an active 1:4 CV-QKD quantum network in the finite-size regime with adaptable protocols achieving 0.19 bits per channel use over 11 km channels.
-
Architecting Early Fault Tolerant Neutral Atoms Systems with Quantum Advantage
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...
-
Fault-Tolerant Quantum Computing with Trapped Ions: The Walking Cat Architecture
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...
-
GreenPeas: Unlocking Adaptive Quantum Error Correction with Just-in-Time Decoding Hypergraphs
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.
-
Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays
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.
-
QLLVM: A Scalable Quantum-Classical Co-Compilation Framework based on LLVM
QLLVM delivers an LLVM-based end-to-end co-compiler that unifies classical HPC and quantum programs into one executable, with a three-stage quantum path via MLIR and QIR that reduces circuit depth and gate counts on M...
-
Fast measurement of neutral atoms with a multi-atom gate
A multi-atom Rydberg gate with N ancillae enables N-fold photon collection for fast neutral-atom measurement, achieving infidelity below 10^{-3} in 6 μs with N=5 in Cs-Rb simulations.
-
Heterogeneous architectures enable a 138x reduction in physical qubit requirements for fault-tolerant quantum computing under detailed accounting
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 ...
-
Phase-Stable Hologram Updates for Large-Scale Neutral-Atom Array Reconfiguration
WPGS algorithm enforces inter-frame phase continuity in holographic tweezers to suppress refresh-induced atom loss and speed up updates for large neutral-atom arrays.
-
Space-Time Tradeoffs of Pauli-Based Computation in Distributed qLDPC Architectures
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.
-
Real-time Krylov Diagonalisation for Open Quantum Systems
Real-time Krylov subspace methods are adapted to Lindblad open systems and used to extract the Liouvillian gap of a two-photon-driven Kerr resonator in the cat-qubit regime.
-
Probing the Planck scale with quantum computation
A 500-logical-qubit quantum computer could reject laboratory-confined theories by surpassing the Planck-scale operation rate of 2^491 m^{-3} s^{-1}, with a 1600-qubit machine limited by the observable universe.
Reference graph
Works this paper leans on
-
[1]
Kubica, A.et al.Erasure qubits: Overcoming theT 1 limit in superconducting circuits.Phys. Rev. X13, 041022 (2023)
work page 2023
-
[2]
Taming Rydberg Decay with Measurement-based Quantum Computation
Yu, C.-C.et al.Processing and decoding Rydberg decay error with MBQC (2025). arXiv:2411.04664 [quant-ph]
work page internal anchor Pith review Pith/arXiv arXiv 2025
-
[3]
Chang, K.et al.Surface Code with Imperfect Erasure Checks (2024). arXiv:2408.00842 [quant-ph]
-
[4]
Perrin, H., Jandura, S. & Pupillo, G. Quantum error correction resilient against atom loss (2025). arXiv:2412.07841 [quant-ph]
-
[5]
Baranes, G.et al.Leveraging atom loss errors in fault tolerant quantum algorithms (2025). arXiv:2502.20558 [quant-ph]
-
[6]
Gu, S., Retzker, A. & Kubica, A. Fault-tolerant quan- tum architectures based on erasure qubits.Phys. Rev. Res.7, 013249 (2025)
work page 2025
-
[7]
W.et al.A scalable and real- time neural decoder for topological quantum codes (2026)
Senior, A. W.et al.A scalable and real- time neural decoder for topological quantum codes (2026). URLhttps://arxiv.org/abs/2512.07737. arXiv:2512.07737 [quant-ph]
-
[8]
Maan, A. S., Garcia Herrero, F. M., Paler, A. & Savin, V. Decoding correlated errors in quantum ldpc codes. Nature Communications(2026)
work page 2026
- [9]
-
[10]
Bravyi, S., Smith, G. & Smolin, J. A. Trading classical and quantum computational resources.Physical Review X6, 021043 (2016)
work page 2016
-
[11]
A Tgate on the memory code can be implemented using a two-qubit ¯Z ¯Zmeasurement between the memory and the surface code hosting the| T⟩state [137]
-
[12]
Low-overhead constructions for the fault- tolerant toffoli gate.Phys
Jones, C. Low-overhead constructions for the fault- tolerant toffoli gate.Phys. Rev. A87, 022328 (2013)
work page 2013
-
[13]
Amy, M., Maslov, D., Mosca, M. & Roetteler, M. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits.Trans. Comp.-Aided Des. Integ. Cir. Sys.32, 818–830 (2013)
work page 2013
-
[14]
Halving the cost of quantum addition.Quan- tum2, 74 (2018)
Gidney, C. Halving the cost of quantum addition.Quan- tum2, 74 (2018)
work page 2018
-
[15]
State-of-the-art, low-overhead constructions rely on ran- domized numerical subroutines that extensively esti- mate the fault distance of the protocol [49, 52], a proce- dure that becomes computationally prohibitive for large code blocks
-
[16]
Liang, Z., Liu, K., Song, H. & Chen, Y.-A. Generalized toric codes on twisted tori for quantum error correction. arXiv preprint arXiv:2503.03827(2025)
-
[17]
Magic State Distillation: Not as Costly as You Think.Quantum3, 205 (2019)
Litinski, D. Magic State Distillation: Not as Costly as You Think.Quantum3, 205 (2019)
work page 2019
-
[18]
Sahay, K.et al.Fold-transversal surface code cul- tivation (2025). URLhttps://arxiv.org/abs/2509. 05212. arXiv:2509.05212 [quant-ph]
-
[19]
Swaroop, E., Jochym-O’Connor, T. & Yoder, T. J. Uni- versal adapters between quantum ldpc codes.arXiv preprint arXiv:2410.03628(2024)
-
[20]
Zheng, H., Zheng, G., Jiang, L. & Xu, Q. In preparation (2026)
work page 2026
-
[21]
Vedral, V., Barenco, A. & Ekert, A. Quantum networks for elementary arithmetic operations.Physical Review A54, 147 (1996)
work page 1996
-
[22]
Babbush, R.et al.Encoding electronic spectra in quan- tum circuits with linear t complexity.Phys. Rev. X8, 041015 (2018)
work page 2018
-
[23]
Proos, J. & Zalka, C. Shor’s discrete logarithm quan- tum algorithm for elliptic curves.arXiv preprint quant- ph/0301141(2003)
-
[24]
Roetteler, M., Naehrig, M., Svore, K. M. & Lauter, K. Quantum resource estimates for computing elliptic curve discrete logarithms. InInternational Conference on the Theory and Application of Cryptology and Infor- mation Security, 241–270 (Springer, 2017)
work page 2017
-
[25]
H¨ aner, T., Jaques, S., Naehrig, M., Roetteler, M. & Soeken, M. Improved quantum circuits for elliptic curve discrete logarithms. InInternational conference on post- quantum cryptography, 425–444 (Springer, 2020)
work page 2020
-
[26]
Litinski, D. How to compute a 256-bit elliptic curve private key with only 50 million toffoli gates.arXiv preprint arXiv:2306.08585(2023)
-
[27]
Gouzien, E., Ruiz, D., Le R´ egent, F.-M., Guillaud, J. & Sangouard, N. Performance analysis of a repetition cat code architecture: Computing 256-bit elliptic curve logarithm in 9 hours with 126 133 cat qubits.Phys. Rev. Lett.131, 040602 (2023)
work page 2023
-
[28]
Zhang, G. & Li, Y. Time-efficient logical operations on quantum low-density parity check codes.Physical 11 Review Letters134, 070602 (2025)
work page 2025
-
[29]
Cowtan, A., He, Z., Williamson, D. J. & Yoder, T. J. Parallel logical measurements via quantum code surgery.arXiv preprint arXiv:2503.05003(2025)
work page internal anchor Pith review Pith/arXiv arXiv 2025
-
[30]
Draper, T. G., Kutin, S. A., Rains, E. M. & Svore, K. M. A logarithmic-depth quantum carry-lookahead adder.Quantum Inf. Comput.6, 351–369 (2004)
work page 2004
-
[31]
Horsman, C., Fowler, A. G., Devitt, S. & Meter, R. V. Surface code quantum computing by lattice surgery. New Journal of Physics14, 123011 (2012)
work page 2012
-
[32]
Atlas of atomic fault-tolerant quantum memory (2026)
Caltech & Oratomic. Atlas of atomic fault-tolerant quantum memory (2026). In preparation
work page 2026
-
[33]
Baspin, N., Berent, L. & Cohen, L. Z. Fast surgery for quantum ldpc codes.arXiv preprint arXiv:2510.04521 (2025)
-
[34]
Chang, K., He, Z., Yoder, T. J., Zhu, G. & Jochym- O’Connor, T. Constant-time surgery on 2d hypergraph product codes with near-constant space overhead.arXiv preprint arXiv:2603.02157(2026)
- [35]
- [36]
-
[37]
Bomb´ ın, H. Gauge Color Codes: Optimal Transver- sal Gates and Gauge Fixing in Topological Stabilizer Codes.New Journal of Physics17(2013)
work page 2013
-
[38]
Brown, B. J., Nickerson, N. H. & Browne, D. E. Fault- tolerant error correction with the gauge color code.Na- ture Communications7, 12302 (2016)
work page 2016
-
[39]
Jacob, A., McLauchlan, C. & Browne, D. E. Single- shot decoding and fault-tolerant gates with trivariate tricycle codes (2026). URLhttps://arxiv.org/abs/ 2508.08191. arXiv:2508.08191 [quant-ph]
-
[40]
Fowler, A. G. & Devitt, S. J. A bridge to lower overhead quantum computation.arXiv preprint arXiv:1209.0510 (2012)
work page internal anchor Pith review Pith/arXiv arXiv 2012
-
[41]
Fowler, A. G. Time-optimal quantum computation. arXiv preprint arXiv:1210.4626(2012)
work page internal anchor Pith review Pith/arXiv arXiv 2012
- [42]
-
[43]
Litinski, D. & Nickerson, N. Active volume: An ar- chitecture for efficient fault-tolerant quantum comput- ers with limited non-local connections.arXiv preprint arXiv:2211.15465(2022)
-
[44]
A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery.Quantum 3, 128 (2019)
Litinski, D. A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery.Quantum 3, 128 (2019)
work page 2019
-
[45]
Symons, B. C., Rajput, A. & Browne, D. E. Sequences of bivariate bicycle codes from covering graphs.arXiv preprint arXiv:2511.13560(2025)
-
[46]
Smarandache, R. & Vontobel, P. O. Quasi-cyclic LDPC codes: Influence of proto-and Tanner-graph structure on minimum Hamming distance upper bounds.IEEE Trans. Inf. Th.58, 585–607 (2012)
work page 2012
-
[47]
Raveendran, N., Declercq, D. & Vasi´ c, B. On the min- imum distances of finite-length lifted product quantum ldpc codes.arXiv preprint arXiv:2503.07567(2025)
-
[48]
Roffe, J., White, D. R., Burton, S. & Campbell, E. Decoding across the quantum low-density parity-check code landscape.Phys. Rev. Res.2, 043423 (2020)
work page 2020
-
[49]
Ide, B., Gowda, M. G., Nadkarni, P. J. & Dauphinais, G. Fault-tolerant logical measurements via homological measurement.arXiv preprint arXiv:2410.02753(2024)
- [50]
-
[51]
Gu, B.et al.Qgpu: Parallel logic in quantum ldpc codes (2026). URLhttps://arxiv.org/abs/2603. 05398. arXiv:2603.05398 [quant-ph]
-
[52]
Pryadko, L. P., Shabashov, V. A. & Kozin, V. K. Qdistrnd: A gap package for computing the distance of quantum error-correcting codes.arXiv preprint arXiv:2308.15140(2023)
-
[53]
Huang, S., Jochym-O’Connor, T. & Yoder, T. J. Ho- momorphic Logical Measurements.arXiv:2211.03625 (2022)
-
[54]
Sayginel, H., Koutsioumpas, S., Webster, M., Rajput, A. & Browne, D. E. Fault-tolerant logical clifford gates from code automorphisms.PRX Quantum6, 030343 (2025)
work page 2025
-
[55]
Bravyi, S. & Kitaev, A. Universal quantum computa- tion with ideal Clifford gates and noisy ancillas.Phys. Rev. A71, 022316 (2005)
work page 2005
- [56]
-
[57]
Note that one could also teleport auto-corrected surface code| T⟩states, which avoids the Scorrections, at the price of using more surface code patches [110]
- [58]
-
[59]
Vaknin, Y., Jacoby, S., Grimsmo, A. & Retzker, A. Efficient magic state cultivation on the surface code (2025). URLhttps://arxiv.org/abs/2502.01743. arXiv:2502.01743 [quant-ph]
-
[60]
Stim: a fast stabilizer circuit simulator
Gidney, C. Stim: a fast stabilizer circuit simulator. Quantum5, 497 (2021)
work page 2021
-
[61]
Tremblay, M. A., Delfosse, N. & Beverland, M. E. Constant-Overhead Quantum Error Correction with Thin Planar Connectivity.Physical Review Letters129, 050504 (2022)
work page 2022
-
[62]
In2025 IEEE International Conference on Quantum Computing and Engineering (QCE), vol
Viszlai, J.et al.Matching generalized-bicycle codes to neutral atoms for low-overhead fault-tolerance. In2025 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 01, 688–699 (2025)
work page 2025
-
[63]
Kang, M.et al.QUITS: A modular Qldpc code circUIT Simulator.Quantum9, 1931 (2025)
work page 1931
-
[64]
LDPC: Python tools for low density parity check codes (2022)
Roffe, J. LDPC: Python tools for low density parity check codes (2022). URLhttps://pypi.org/project/ ldpc/
work page 2022
-
[65]
Poulsen Nautrup, H., Friis, N. & Briegel, H. J. Fault- tolerant interface between quantum memories and quan- tum processors.Nat. Comm.8, 1321 (2017)
work page 2017
-
[66]
Zhu, S., Sundaram, A. & Low, G. H. Unified archi- tecture for quantum lookup tables.Phys. Rev. Res.7, 043230 (2025). 12 METHODS Here we provide technical details on the codes, logic, and compilation schemes discussed in this work. In Ap- pendix A, we describe and numerically benchmark the codes used in this work, including the lifted product code construc...
work page 2025
-
[67]
Lifted-product codes The LP code family [91] is obtained by taking the product of two classical codes with check matrices A∈R rA×nA andB∈R rB ×nB, whereR:=F 2[x]/(xℓ+1) is the univariate polynomial ring of orderℓ. Each entry ofAis an element ofR, which can be identified with a polynomial of degree at mostℓ−1 overF 2, or equiva- lently, with anℓ×ℓcirculant...
-
[68]
for both the distance of theZ-check matrix (dZ) and theX-check matrix (d X). Concretely, ford Z (and sim- ilarly ford X), letL X ∈F k×n 2 be a basis for the logical Xoperators kerH X /imH ⊤ Z . For each trial, we sample c∼ {0,1} k \ {0}, sete=c·L X (mod 2), and use BP- OSD to find a minimum-weightvsatisfyingH X v= 0 and e·v= 1 (mod 2). Thend Z ≤min t |vt|...
-
[69]
Bivariate bicycle codes The BB code family [43] can be viewed as a special subfamily of LP codes. A BB code LP(a, b) is defined by two elementsa, b∈F 2[x, y]/(xl + 1, ym + 1) in the bivari- ate polynomial ring with periodslandm. The resulting code acts onn= 2lmphysical qubits with stabilizer gen- erators determined by the polynomialsaandb. The BB code use...
-
[70]
Description Given a qLDPC codeQ, a surgery gadget measures a collection of logical Pauli operatorsL={ ¯P∈ ¯Pk}, where ¯Pk denotes the logical Pauli group acting on the kencoded qubits. This is achieved by couplingQto an ancilla systemAsuch that the eigenvalues of the logical operators inLare extracted non-destructively via mea- surements onA[48, 49, 86]. ...
-
[71]
InitializeQ ′ in|0⟩states
-
[72]
Measureτ s cycles of checksS X ∪S ′ X ∪S Z ∪S ′ Z that are now deformed to support on themerged codewith qubitsQ∪Q ′: in addition to the original connection,S ′ X is also connected toQandS Z is also connected toQ ′ according to matricesf ′ X and fZ, respectively, depending on the target logicals to measure
-
[73]
The key step is measuring the checks of themerged codeforτ s cycles (step 2)
DetachQ ′ fromQby measuringQ ′ in theZbasis, followed by adaptive Pauli corrections onQ. The key step is measuring the checks of themerged codeforτ s cycles (step 2). We denote the merged code ˜Q( ˜Q, ˜SX , ˜SZ) with check matrices ˜HX = HX 0 f ′ X H ′ X ! , ˜HZ = HZ fZ 0H ′ Z ! .(B1) During this merging step, the information ofLis ex- tracted through mea...
-
[74]
Concrete construction and benchmarking Using the general procedure described in the previous section, we present concrete constructions of surgery gad- gets for the space-efficient and balanced architecture in this section. Recall that these architectures consist of a [[n m, km, dm]] memory code, a [[n p, kp, dp]] processor code, and several [[n f , kf , ...
-
[75]
Resource costs and future improvements We now summarize the time and space costs of the surgery instructions used in this architecture. Based on the examples in Extended Data Table III as well as the analysis in the previous sections, we estimate that the an- cillary systems required to perform all relevant surgery operations in our architecture occupyN A...
-
[76]
Description As illustrated in Extended Data Fig. 2, the high-rate magic state distillation procedure distills cultivated| T⟩ states on small surface codes into high-fidelity| CCZ⟩ states hosted in three high-rate factory code blocks with parameters [[n f , kf , df]], using the 8T-to-CCZ distilla- tion circuit in Ref. [110]. The distillation circuit has ei...
-
[77]
Resource costs Here we compute the concrete resource costs for the magic factory described in the main text for the space ef- ficient and balanced architectures. For the factory code, we select thebb 18 code (Extended Data Table II) with Nf = 367. We choose the distance of the surface code to bed s = 7 so that it has logical error rate≲10 −6 [151]. We con...
work page 2048
-
[78]
Teleport them i logical qubits in the memory code to the processor code by sequentially measuring { ¯ZIj ¯Z ′ j}j∈[mi] and then{ ¯XIj }j∈[mi]. Here,I j de- notes the index of thej-th qubit among them i qubits in the memory code thatC i is supported on
-
[79]
Perform am i-qubit Pauli-based computation on the processor code, where Cliffords are pushed to the end and each Toffoli gate is implemented by three ¯P ′ ¯Z ′′ measurements between the processor code and one of the factory codes, where ¯P ′ is a mi-qubit Pauli operator. In addition, each mid- circuit Pauli measurement will be transformed to a high-weight...
-
[80]
Teleport them i qubits back into the memory code by sequentially performing ¯ZIj ¯P ′ j between the pro- cessor and the memory code, followed bym i mi- qubit measurements on the processor code. 21 The above protocol extensively utilizes two key subrou- tines: 1.Measurement-based teleportation:logical qubitiis teleported to logical qubitj(potentially acros...
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.