Non-Clifford and Parallelizable Fault-Tolerant Logical Gates on Constant and Almost-Constant Rate Homological Quantum Low-Density Parity-Check Codes via Higher Symmetries
Guanyu Zhu, Shehryar Sikander, Elia Portnoy, Andrew W. Cross, Benjamin J. Brown
We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical gates on any triplet of logical qubits with their logical- X membranes having a Z 2 triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the Z 2 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate k / n = O ( 1 / log ( n ) ) and O ( log ( n ) ) distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with O ( 1 / log 1 2 ( n ) ) rate and O ( log 1 2 ( n ) ) distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- X measurements.
We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical gates on any triplet of logical qubits with their logical- X membranes having a Z 2 triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the Z 2 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate k / n = O ( 1 / log ( n ) ) and O ( log ( n ) ) distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with O ( 1 / log 1 2 ( n ) ) rate and O ( log 1 2 ( n ) ) distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- X measurements.
A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the N1/3 distance barrier
We develop a topological theory for fault-tolerant quantum computation in quantum low-density parity-check (qLDPC) codes. We show that there exist hidden simplicial or CW complex structures encoding the topological data for all qLDPC and CSS codes obtained from product construction by generalizing the Freedman-Hastings code-to-manifold mapping. This is achieved by building manifolds from the Tanner graphs of the skeleton classical or quantum codes, which further form a product manifold and an associated thickened product code defined on its triangulation. One can further deformation retract the manifold back to a CW complex which supports a non-topological code with minimal overhead suitable for near-term implementation. Both types of codes admit cohomology operations including cup product which can induce non-Clifford gates. When applying this mapping to a 3D hypergraph product code obtained from the product of 3 copies of good classical expander codes, we obtain non-Clifford logical CCZ gates via constant depth circuits on a code with constant stabilizer weight w=O(1), constant rate K=Θ(N), and polynomial distance D=Ω(N1/3). When applied to logical CCZ on 3D homological product codes consisting of the product of a pair of good quantum and classical LDPC codes, we can further improve the distance to D=Ω(N) exceeding the N1/3 distance barrier implied by the Bravyi-König bound for conventional topological codes with the aid of non-Euclidean geometries. Our work suggests that it is feasible to apply native logical non-Clifford gates on qLDPC codes or directly inject high-fidelity magic states as resources ('magic state fountain') without the distillation process. For the homological product construction, the fountain can inject Θ(N) magic states in parallel in a single round.
A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical non-Clifford gates, one needs to resort to 3D product-code constructions. In this work, we extend Kitaev's framework of non-Abelian topological codes on manifolds to non-Abelian qLDPC codes (realized as Clifford-stabilizer codes) and the corresponding combinatorial topological quantum field theories (TQFT) defined on Poincaré CW complexes and certain types of general chain complexes. We also construct the spacetime path integrals as topological invariants on these complexes. Remarkably, we show that native non-Clifford logical gates can be realized using constant-rate 2D hypergraph-product codes and their Clifford-stabilizer variants. This is achieved by a spacetime path integral effectively implementing the addressable gauging measurement of a new type of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford gates and become higher-form symmetries when lifted to higher-dimensional CW complexes or manifolds. Building on this structure, we apply the gauging protocol to the magic state fountain scheme for parallel preparation of O(n) disjoint CZ magic states with code distance of O(n), using a total number of n qubits.
Transversal non-Clifford gates on qLDPC codes breaking the N distance barrier and quantum-inspired geometry with Z2 systolic freedom
Historically, a Nlog1/2(N) distance barrier for quantum low-density parity-check (LDPC) codes with N qubits persisted for nearly two decades, until the recent discovery of the fibre-bundle code. An open question is whether such a distance barrier can be broken while preserving the ability to perform transversal non-Clifford gates. In this direction, another long-standing distance barrier of N1/3 for LDPC stabilizer codes -- present since the discovery of the 3D color code -- was only recently overcome by a construction achieving an Ω(N) distance (arXiv:2501.19375). The present work further breaks the N distance barrier by taking a homological product of three good qLDPC codes, combined with the Freedman-Hastings code-to-manifold mapping and the triple cup product to implement transversal CCZ gates. The resulting code achieves an Ω(N2/3) distance (a linear X-distance of Θ(N)) and a dimension of Θ(N2/3), which enables fault-tolerant preparation of Θ(N1/3) independent logical CCZ magic states in a single shot, without distillation (`magic state fountain'). This new quantum code also inspires the discovery of a family of exotic 3q-dimensional manifolds M, which exhibit both a power-law Z2-(q, 2q)-systolic freedom and Θ(vol(M)) triple intersection points of 2q-dimensional submanifolds.
We establish a systematic mathematical framework for constructing and computing (co)homological invariants that induce logical gates on quantum low-density parity-check (qLDPC) codes. By synthesizing tools from graph theory, algebraic topology, and number theory, we study constructions ranging from hypergraph product (HGP) codes to sheaf codes. Using commutative algebra, we generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a longstanding challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, {we establish a nearly linear family of independent cup product cochains on almost-good qLDPC codes. Under a separate nonvanishing pairing condition, they are expected to yield parallel logical multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes, we construct interlaced families through odd-degree lifts that preserve compatible invariant pairings, allowing nontrivial, constant-depth logical gates throughout an infinite family to be certified on a constant-size HGP seed. Treating local code constructions and expansion bounds as external inputs, our framework identifies the compatibility conditions for combining large dimension and distance, local testability, and constant-depth non-Clifford gates, thereby provides a mathematical foundation for separate constructions that realize these properties simultaneously.
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1Z gate, for every constant r≥3. In particular, we obtain length-n quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/r supporting transversal Cr−1Z gates on a close-to-linear number k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n) logical Cr−1Z gates, though with stabilizer weight and physical circuit depth nε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits.
To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
We systematically construct and classify fault-tolerant logical gates implemented by constant-depth circuits for quantum codes using cohomology operations and symmetry. These logical gates are obtained from unitary operators given by symmetry-protected topological responses, which correspond to generators of group cohomology and can be expressed explicitly on the lattice using cohomology operations including cup product, Steenrod squares and new combinations of higher cup products called higher Pontryagin powers. Our study covers most types of the cohomology operations in the literature. This hence gives rise to logical Cn−1Z gates in n copies of quantum codes via the n-fold cup product in the usual color code paradigm, as well as several new classes of diagonal and non-diagonal logical gates in increasing Clifford hierarchies beyond the color code paradigm, including the logical Rk and multi-controlled CmRk gates for codes defined in projective spaces. Implementing these gates could make it more efficient to compile specific types of quantum algorithms such as Shor's algorithm. We further extend the construction to quantum codes with boundaries, which generalizes the folding approach in color codes. We also present a formalism for addressable and parallelizable logical gates in LDPC codes via higher-form symmetries. We further construct logical Clifford gates in expander-based codes including the asymptotically good LDPC codes and hypergraph-product codes. As a byproduct, we find new topological responses of finite higher-form symmetries using higher Pontryagin powers.