The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical CCCZ gate transversally.The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyperdiamond.This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical X and Z and only allows for the implementation of a transversal Clifford gate.This paper further develops the established connection between topological dimension and transversal gates in the Clifford hierarchy, generalizing the known designs for the implementation of transversal CZ and CCZ in two and three dimensions, respectively.
The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical CCCZ gate transversally.The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyperdiamond.This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical X and Z and only allows for the implementation of a transversal Clifford gate.This paper further develops the established connection between topological dimension and transversal gates in the Clifford hierarchy, generalizing the known designs for the implementation of transversal CZ and CCZ in two and three dimensions, respectively.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for n-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of n-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23 gauge theory (equivalent to D4 topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d) rounds via code switching. In 3D, we construct a transversal logical T gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24 gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.
We construct a four-dimensional toric code on the D4 lattice, the densest lattice packing in four dimensions, by placing qubits on the 16L^4 triangles of its Delaunay decomposition into regular 16-cells. Each X-stabilizer acts on the 4 triangles of a tetrahedron, the smallest weight such a check can have, and each Z-stabilizer on the 8 triangles meeting along an edge. We prove that the code is [[16L^4, 6, 2L^2]] for every even L >= 4. A logical operator must cast a shadow covering a two-dimensional cross-section of the torus, and a single triangle shadows an area of at most 1/2, so at least 2L^2 qubits are needed; a close-packed plane of the lattice attains this bound exactly. Because the distance is twice the hypercubic value at equal side length, the code reaches distance d with 4d^2 physical qubits against 6d^2 for the hypercubic four-dimensional toric code. A hook-error analysis shows that the weight-8 checks cost no more at circuit level than the weight-4 ones. Neither sector has point-like excitations, and the code's equal-time slices are the face-centred-cubic lattice. Flat four-space has three regular honeycombs, and this code completes the regular case. All results are verified by published scripts.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S, SHS=X, and CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
The two-dimensional color code is an alternative to the toric code that encodes more logical qubits while maintaining crucial features of the Z2×Z2 toric code in the long wavelength limit. However its short range physics include single qubit Pauli operations that violate either three or six stabilisers as opposed to the toric code where single qubit Pauli operations violate two or four stabilisers. Exploiting this fact we construct several non-CSS versions of the two-dimensional color code falling into two families - those where either three, four or five stabilisers are violated and those which violate exactly four stabilisers for all the three types of single qubit Pauli operations. These models are not equivalent to the original color code by a local unitary transformation. Nevertheless the code spaces of the CSS and non-CSS versions are related by local unitaries which reflects the fact that their long range physics coincide. This also implies that the non-CSS versions host transversal Clifford gates and hence support fault-tolerant computations. As a consequence of the non-CSS structure, the logical operators are of a mixed type which in some cases include all the three Pauli operators making them potentially useful for protection against biased Pauli noise.
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice.
We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group.
For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
In universal fault-tolerant quantum computing, implementing logical non-Clifford gates often demands substantial spacetime resources for many error-correcting codes, including the high-threshold surface code. A critical mission for realizing large-scale quantum computing is to develop simple and resource-efficient implementations of logical non-Clifford gates. We propose a novel way of implementing non-Clifford operations in the standard surface code based on hybrid lattice surgery. First we generalize the standard lattice surgery to hybrid lattice surgery, where operations of rough merge and rough split happen across different topological codes. Then we apply such procedures between Abelian and non-Abelian codes and show that this can provide non-Clifford operations in the standard surface code, in the form of a magic state or a non-Clifford gate teleportation. Complementing this, we provide a continuum topological field theory description of this hybrid lattice surgery utilizing interfaces between (2+1)d topological orders. From these considerations, we can generalize our protocol to non-Clifford gates and magic states at all finite levels of the Clifford hierarchy, as well as gates beyond the hierarchy. We also discuss protocols extending this framework to qutrits.