Cups and Gates I: Cohomology Invariants and Logical Quantum Operations Nikolas P. Breuckmann, Margarita Davydova, Jens N. Eberhardt, Nathanan Tantivasadakarn Communications in Mathematical Physics, 2026 We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $$\\Lambda $$ Λ -fold cup product that can produce a logical operator in the $$\\Lambda $$ Λ -th level of the Clifford hierarchy on $$\\Lambda $$ Λ copies of the same quantum code, which we call the copy-cup gate . For any desired $$\\Lambda $$ Λ , we can construct several families of quantum codes that support gates in the $$\\Lambda $$ Λ -th level with various asymptotic code parameters.
Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary Vincent Steffan, Shin Ho Choe, Nikolas P. Breuckmann, Francisco Revson Fernandes Pereira, Jens Niklas Eberhardt Physical Review Letters, 2025 We introduce , a simple yet powerful way of constructing quantum codes that are local on a planar 2D lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters [ [ 288 , 8 , 12 ] ] using weight-6 stabilizers and [ [ 288 , 8 , 14 ] ] using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher nonlocality, we find a [ [ 512 , 18 , 19 ] ] code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error-correction code design.
Low-Overhead Entangling Gates From Generalised Dehn Twists Ryan Tiew, Nikolas P. Breuckmann IEEE Transactions on Information Theory, 2025 We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead and $\\mathcal{O}(d)$ time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as $[[18q^2,8,2q]]_{q\\in \\mathbb{N}}$ and $[[18q,8,\\leq 2q]]_{q\\in \\mathbb{N}}$ respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd $q$. We also show that the $[[90,8,10]]$ bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to $q\\leq8$ through a numerical search.
Fault-tolerant connection of error-corrected qubits with noisy links Joshua Ramette, Josiah Sinclair, Nikolas P. Breuckmann, Vladan Vuletić Npj Quantum Information, 2024 One of the most promising routes toward scalable quantum computing is a modular approach. We show that distinct surface code patches can be connected in a fault-tolerant manner even in the presence of substantial noise along their connecting interface. We quantify analytically and numerically the combined effect of errors across the interface and bulk. We show that the system can tolerate 14 times higher noise at the interface compared to the bulk, with only a small effect on the code’s threshold and subthreshold behavior, reaching threshold with ~1% bulk errors and ~10% interface errors. This implies that fault-tolerant scaling of error-corrected modular devices is within reach using existing technology.
Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes Oscar Higgott, Nikolas P. Breuckmann Prx Quantum, 2024 We construct families of Floquet codes derived from color-code tilings of closed hyperbolic surfaces. These codes have weight-two check operators, a finite encoding rate and can be decoded efficiently with minimum-weight perfect matching. We also construct semi-hyperbolic Floquet codes, which have improved distance scaling, and are obtained via a fine-graining procedure. Using a circuit-based noise model that assumes direct two-qubit measurements, we show that semi-hyperbolic Floquet codes can be 48 times more efficient than planar honeycomb codes and therefore over 100 times more efficient than alternative compilations of the surface code to two-qubit measurements, even at physical error rates of 0.3% to 1%. We further demonstrate that semi-hyperbolic Floquet codes can have a teraquop footprint of only 32 physical qubits per logical qubit at a noise strength of 0.1%. For standard circuit-level depolarizing noise at p=0.1%, we find a 30 times improvement over planar honeycomb codes and a 5.6 times improvement over surface codes. Finally, we analyze small instances that are amenable to near-term experiments, including a Floquet code derived from the Bolza surface that encodes four logical qubits into 16 physical qubits. Published by the American Physical Society 2024
Circuit-to-Hamiltonian from Tensor Networks and Fault Tolerance Anurag Anshu, Nikolas P. Breuckmann, Quynh T. Nguyen Proceedings of the Annual ACM Symposium on Theory of Computing, 2024 We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary "clock register" to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any "combinatorial state" with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As an application, we show that contracting injective tensor networks to additive error is BQP-hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth.
Fold-Transversal Clifford Gates for Quantum Codes Nikolas P. Breuckmann, Simon Burton Quantum, 2024 We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal gates and qubit permutations only.To demonstrate our approach, we specifically consider a [[30, 8, 3]] hyperbolic quantum code called Bring's code. Further, we show that by restricting the logical subspace of Bring's code to four qubits, we can obtain the full Clifford group on that subspace.
NLTS Hamiltonians from Good Quantum Codes Anurag Anshu, Nikolas P. Breuckmann, Chinmay Nirkhe Proceedings of the Annual ACM Symposium on Theory of Computing, 2023 The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that a particular family of constant-rate and linear-distance qLDPC codes correspond to NLTS local Hamiltonians, although we believe this to be true for all current constructions of good qLDPC codes.
Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes Oscar Higgott, Nikolas P. Breuckmann Prx Quantum, 2023 In this work we study the single-shot performance of higher dimensional hypergraph product codes decoded using belief-propagation and ordered-statistics decoding [Panteleev and Kalachev, 2021]. We find that decoding data qubit and syndrome measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes. For the 3D toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for $Z$ errors, compared to the threshold of 2.90% previously found using a two-stage decoder~[Quintavalle et al., 2021]. For the 4D toric code, for which both $X$ and $Z$ error correction is single-shot, our results are consistent with a sustainable single-shot threshold of 4.3% which is even higher than the threshold of 2.93% for the 2D toric code for the same noise model but using $L$ rounds of stabiliser measurement. We also explore the performance of balanced product and 4D hypergraph product codes which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.
Random-bond Ising model and its dual in hyperbolic spaces Benedikt Placke, Nikolas P. Breuckmann Physical Review E, 2023 We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the euclidean case. We explain this anomaly by a careful re-derivation of the Kramers--Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature $T$ and the fraction of antiferromagnetic bonds $p$. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.
The small stellated dodecahedron code and friends J. Conrad, C. Chamberland, N. P. Breuckmann, B. M. Terhal Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences, 2018
Cups and Gates I: Cohomology Invariants and Logical Quantum Operations NP Breuckmann, M Davydova, JN Eberhardt, N Tantivasadakarn Communications in Mathematical Physics 407 (5), 86 , 2026 2026 Citations: 30
Optimal Decoding with the Worm Z Tobias, NP Breuckmann, B Placke arXiv preprint arXiv:2603.05428 , 2026 2026
Copy-cup Gates in Tensor Products of Group Algebra Codes| Quantum Research Paper| Quantum Intelligence Network R Tiew, NP Breuckmann 2026
Copy-cup Gates in Tensor Products of Group Algebra Codes R Tiew, NP Breuckmann arXiv preprint arXiv:2602.23307 , 2026 2026
Logical Operators and Derived Automorphisms of Tile Codes NP Breuckmann, SH Choe, JN Eberhardt, FRF Pereira, V Steffan arXiv preprint arXiv:2511.14589 , 2025 2025
Tile codes: High-efficiency quantum codes on a lattice with boundary V Steffan, SH Choe, NP Breuckmann, FRF Pereira, JN Eberhardt Physical Review Letters 135 (17), 170601 , 2025 2025 Citations: 13
Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes B Placke, GM Sommers, NP Breuckmann, T Rakovszky, V Khemani arXiv preprint arXiv:2507.13342 , 2025 2025 Citations: 5
Classical and quantum glasses based on expanders B Placke, T Rakovszky, NP Breuckmann, V Khemani, GM Sommers arXiv preprint arXiv:2507.13342 , 2025 2025
Low-overhead entangling gates from generalised dehn twists R Tiew, NP Breuckmann IEEE Transactions on Information Theory , 2025 2025 Citations: 9
Topological Quantum Spin Glass Order and its realization in qLDPC codes V Khemani, B Placke, T Rakovszky, NP Breuckmann SMT 2025 , 2025 2025
Topological quantum spin glass order and its realization in qLDPC codes B Placke, T Rakovszky, NP Breuckmann, V Khemani arXiv preprint arXiv:2412.13248 , 2024 2024 Citations: 29
Bottlenecks in quantum channels and finite temperature phases of matter T Rakovszky, B Placke, NP Breuckmann, V Khemani arXiv preprint arXiv:2412.09598 , 2024 2024 Citations: 13
Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes O Higgott, NP Breuckmann PRX Quantum 5 (4), 040327 , 2024 2024 Citations: 64
Fold-transversal Clifford gates for quantum codes NP Breuckmann, S Burton Quantum 8, 1372 , 2024 2024 Citations: 96
Circuit-to-Hamiltonian from tensor networks and fault tolerance A Anshu, NP Breuckmann, QT Nguyen Proceedings of the 56th Annual ACM Symposium on Theory of Computing, 585-595 , 2024 2024 Citations: 16
Fault-tolerant connection of error-corrected qubits with noisy links J Ramette, J Sinclair, NP Breuckmann, V Vuletić npj Quantum Information 10 (1), 58 , 2024 2024 Citations: 69
Quantum computing error correction method, code, and system O HIGGOTT, NP Breuckmann US Patent 11,831,336 , 2023 2023 Citations: 9
NLTS Hamiltonians from good quantum codes A Anshu, NP Breuckmann, C Nirkhe Proceedings of the 55th Annual ACM Symposium on Theory of Computing, 1090-1096 , 2023 2023 Citations: 132
Balanced product quantum codes NP Breuckmann, JN Eberhardt IEEE Transactions on Information Theory 67 (10), 6653 - 6674 , 2021 2021 Citations: 323
Constructions and noise threshold of hyperbolic surface codes NP Breuckmann, BM Terhal IEEE transactions on Information Theory 62 (6), 3731-3744 , 2016 2016 Citations: 173
Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage NP Breuckmann, C Vuillot, E Campbell, A Krishna, BM Terhal Quantum Science and Technology 2 (3) , 2017 2017 Citations: 155
Multi-stage group testing improves efficiency of large-scale COVID-19 screening JN Eberhardt, NP Breuckmann, CS Eberhardt Journal of Clinical Virology 128, 104382 , 2020 2020 Citations: 143
NLTS Hamiltonians from good quantum codes A Anshu, NP Breuckmann, C Nirkhe Proceedings of the 55th Annual ACM Symposium on Theory of Computing, 1090-1096 , 2023 2023 Citations: 132
Fold-transversal Clifford gates for quantum codes NP Breuckmann, S Burton Quantum 8, 1372 , 2024 2024 Citations: 96
Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead O Higgott, NP Breuckmann Physical Review X 11 (3), 031039 , 2021 2021 Citations: 95
Local Decoders for the 2D and 4D Toric Code NP Breuckmann, K Duivenvoorden, D Michels, BM Terhal QIC 17 (3 and 4), 0181-0208 , 2016 2016 Citations: 94
Scalable neural network decoders for higher dimensional quantum codes NP Breuckmann, X Ni Quantum 2, 68 , 2018 2018 Citations: 87
Fault-tolerant connection of error-corrected qubits with noisy links J Ramette, J Sinclair, NP Breuckmann, V Vuletić npj Quantum Information 10 (1), 58 , 2024 2024 Citations: 69
Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes O Higgott, NP Breuckmann PRX Quantum 5 (4), 040327 , 2024 2024 Citations: 64
Space-time circuit-to-Hamiltonian construction and its applications NP Breuckmann, BM Terhal Journal of Physics A: Mathematical and Theoretical 47 (19), 195304 , 2014 2014 Citations: 59
Single-shot decoding of linear rate LDPC quantum codes with high performance NP Breuckmann, V Londe IEEE Transactions on Information Theory 68 (1), 272 - 286 , 2021 2021 Citations: 55
Quantum phase transitions of interacting bosons on hyperbolic lattices X Zhu, J Guo, NP Breuckmann, H Guo, S Feng Journal of Physics: Condensed Matter 33 (33), 335602 , 2021 2021 Citations: 42
Quantum pin codes C Vuillot, NP Breuckmann IEEE Transactions on Information Theory 68 (9), 5955-5974 , 2022 2022 Citations: 40
Renormalization group decoder for a four-dimensional toric code K Duivenvoorden, NP Breuckmann, BM Terhal IEEE Transactions on Information Theory 65 (4), 2545-2562 , 2018 2018 Citations: 40
Critical properties of the Ising model in hyperbolic space NP Breuckmann, B Placke, A Roy Phys. Rev. E 101 (2), 022124 , 2020 2020 Citations: 39
The small stellated dodecahedron code and friends J Conrad, C Chamberland, NP Breuckmann, BM Terhal Philosophical Transactions of the Royal Society A: Mathematical, Physical … , 2018 2018 Citations: 33
GRANT DETAILS
Co-applicant Making noisy quantum processors practical'' iUK-NSERC Canada-UK Quantum Technologies [GBP 300,000 in funds] (2021)
UCLQ Post-Doctoral Research Fellowship in Quantum Technologies [GBP 235,640 in funds] (awarded 2017, deferred for one year)
HPC project of 1.3 million core-hours at the RWTH Compute Cluster to simulate the performance of quantum fault-tolerance schemes (2016)
Travel grant to visit the conference QStart at Hebrew University Jerusalem [EUR 1,000 in funds] (2013)
RESEARCH OUTPUTS (PATENTS, SOFTWARE, PUBLICATIONS, PRODUCTS)
U.S. Patent Application No. 17/444 943, August 2021
