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.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.

arXiv:2411.03302Nov 5, 20240 citations

Constant-rate quantum codes with low-weight stabilizers and full logical Clifford actions via transversal and fold-transversal gates

Takaya Matsuura, Yohji Chin, Shohei Kiryu +1 more

Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.

quant-pharXiv:2609.37699Sep 30, 20260 citations

Fast computational primitives for La-cross qLDPC codes

Andrea Di Fini, Tom Hartweg, Laura Pecorari +2 more

Quantum LDPC codes promise to reduce the overhead of fault-tolerant quantum computation, but their practical use requires efficient logical gates, measurements, and access to non-Clifford resources. Here, we develop a complete set of computational primitives for La-cross hypergraph product codes. For a subclass of these codes, we show that seed automorphisms together with fold-transversal gates generate the full logical Clifford group up to Pauli corrections. We further prove that every rank one logical operator admits $D$ pairwise disjoint physical representatives, enabling parallel gauging and a space--time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalisation of surface code computation, retaining analogous computational primitives while providing a complete Clifford+ $T$ instruction set across multiple encoded qubits.Quantum LDPC codes promise to reduce the overhead of fault-tolerant quantum computation, but their practical use requires efficient logical gates, measurements, and access to non-Clifford resources. Here, we develop a complete set of computational primitives for La-cross hypergraph product codes. For a subclass of these codes, we show that seed automorphisms together with fold-transversal gates generate the full logical Clifford group up to Pauli corrections. We further prove that every rank one logical operator admits $D$ pairwise disjoint physical representatives, enabling parallel gauging and a space--time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalisation of surface code computation, retaining analogous computational primitives while providing a complete Clifford+$T$ instruction set across multiple encoded qubits.

quant-pharXiv:2610.12685Oct 12, 20260 citations

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Ali Lavasani, Guanyu Zhu, Maissam Barkeshli

A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/log⁡d) . For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/log⁡d) . For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.

arXiv:1901.11029QuantumAug 26, 201935 citations

Logical Entangling with Phantom Codes in Hypergraph Products

Keming He, Ziao Tang, Zetong Li +2 more

Logical entangling gates are a major source of physical spacetime overhead in fault-tolerant quantum computation. Phantom codes reduce this cost by implementing every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates. Whether this mechanism can coexist with the low-weight stabilizer structure of qLDPC codes is a central question for low-overhead fault-tolerant architectures. We give a deterministic answer within binary CSS hypergraph product (HGP) codes. Up to natural equivalences, the simplex-repetition family is the unique HGP family satisfying the phantom condition. We then evaluate this family under circuit-level noise in logical GHZ-state preparation and Trotterized many-body quantum simulation. The codes retain low-weight stabilizer checks and yield concrete advantages over rotated surface-code baselines in both benchmarks. Reconfigurable neutral-atom arrays offer a natural setting for this approach, supporting nonlocal qLDPC operations while enabling in-block logical CNOTs without additional physical operations. Together, these results make precise how permutation-based logical entangling constrains code design within the HGP framework, demonstrate the circuit-level benefits of the unique family, and guide the search for phantom qLDPC families with better asymptotic parameters for low-overhead fault tolerance on neutral-atom hardware.Logical entangling gates are a major source of physical spacetime overhead in fault-tolerant quantum computation. Phantom codes reduce this cost by implementing every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates. Whether this mechanism can coexist with the low-weight stabilizer structure of qLDPC codes is a central question for low-overhead fault-tolerant architectures. We give a deterministic answer within binary CSS hypergraph product (HGP) codes. Up to natural equivalences, the simplex-repetition family is the unique HGP family satisfying the phantom condition. We then evaluate this family under circuit-level noise in logical GHZ-state preparation and Trotterized many-body quantum simulation. The codes retain low-weight stabilizer checks and yield concrete advantages over rotated surface-code baselines in both benchmarks. Reconfigurable neutral-atom arrays offer a natural setting for this approach, supporting nonlocal qLDPC operations while enabling in-block logical CNOTs without additional physical operations. Together, these results make precise how permutation-based logical entangling constrains code design within the HGP framework, demonstrate the circuit-level benefits of the unique family, and guide the search for phantom qLDPC families with better asymptotic parameters for low-overhead fault tolerance on neutral-atom hardware.

quant-pharXiv:2607.12948Jul 15, 20260 citations

Logical gates on Floquet codes via folds and twists

Alexandra E. Moylett, Bhargavi Jonnadula

Floquet codes have recently emerged as a new family of error-correcting codes, and have drawn significant interest across both theoretical and practical quantum computing. A central open question has been how to implement logical operations on these codes. In this work, we show how two techniques from static quantum error-correcting codes can also be implemented on Floquet codes. First, we present a way of implementing fold-transversal operations on Floquet codes in order to yield logical Hadamard and S gates. And second, we present a way of implementing logical CNOT gates on Floquet codes via Dehn twists. We discuss the requirements for these techniques, and show that they are applicable to a wide family of Floquet codes defined on colour code lattices. Through numerical benchmarking of the logical operations on the CCS Floquet code, we establish a logical-gate threshold of 0.25-0.35% and verify sub-threshold exponential error suppression. Our results show that these logical operations are robust, featuring a performance that is close to the baseline set by a quantum memory benchmark. Finally, we explain in detail how to implement logical gates on Floquet codes by operating on the embedded codes.Floquet codes have recently emerged as a new family of error-correcting codes, and have drawn significant interest across both theoretical and practical quantum computing. A central open question has been how to implement logical operations on these codes. In this work, we show how two techniques from static quantum error-correcting codes can also be implemented on Floquet codes. First, we present a way of implementing fold-transversal operations on Floquet codes in order to yield logical Hadamard and S gates. And second, we present a way of implementing logical CNOT gates on Floquet codes via Dehn twists. We discuss the requirements for these techniques, and show that they are applicable to a wide family of Floquet codes defined on colour code lattices. Through numerical benchmarking of the logical operations on the CCS Floquet code, we establish a logical-gate threshold of 0.25-0.35% and verify sub-threshold exponential error suppression. Our results show that these logical operations are robust, featuring a performance that is close to the baseline set by a quantum memory benchmark. Finally, we explain in detail how to implement logical gates on Floquet codes by operating on the embedded codes.

quant-pharXiv:2512.17999Dec 23, 20250 citations

\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.

D

pairwise disjoint physical representatives, enabling parallel gauging and a space--time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalisation of surface code computation, retaining analogous computational primitives while providing a complete Clifford+

T

instruction set across multiple encoded qubits.