Error Correction

Abstract: 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 $K=\Theta(N)$, and polynomial distance $D=\Omega(N^{1/3})$. 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 $D=\Omega(\sqrt{N})$ exceeding the $N^{1/3}$ 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 $\Theta(\sqrt{N})$ magic states in parallel in a single round.

Abstract: We introduce transversal dimension jump, a code-switching protocol for lifted product (LP) quantum low-density parity-check (qLDPC) codes across different chain-complex dimensions, enabling universal fault-tolerant quantum computation with low overhead. The construction leverages the product structure of LP codes to implement one-way transversal CNOTs between a 3D code and its 2D component codes, enabling teleportation-based switching with geometrically nonlocal gates. Combined with constant-depth CCZ gates in 3D LP codes and low-overhead transversal Clifford gates in 2D LP codes, this yields universal, high-rate quantum logical computation with high thresholds and low space-time costs. Beyond asymptotic schemes, we identify explicit 3D–2D LP code pairs supporting cup-product CCZ gates, including bivariate tricycle–bicycle families such as the [[81, 3, 5]]–[[54, 2, 6]] pair, where the 3D tricycle codes admit depth-2 CCZ, weight-6 stabilizers, and pseudo-thresholds >~0.4%. As a byproduct, we show that the 3D codes enable highly efficient magic-state preparation: a single round of stabilizer measurements followed by depth-2 CCZ and postselection produces states with error <10^{-9} and success probability ~35%. Our results establish a native integration of qLDPC codes with complementary transversal gates—covering nearly all practically relevant families known so far—and open a broad design space for scalable, low-overhead universal quantum computation.

Abstract: Quantum low-density parity-check (qLDPC) codes are a class of quantum error-correction (QEC) codes with low-weight parity-checks that each require only a few two-qubit gates to implement. In recent years, qLDPC codes have gained popularity, as concrete code constructions have been found that outperform the surface code, and correspondingly performant practical decoders have been built. An outstanding challenge, however, remains that their Tanner graphs are not 2D-local thereby necessitating entangling gates to operate on distant qubits on a 2D device. Trapped-ion and neutral-atom qubits possess the ability to move qubits around when necessary – i.e. “flying qubits” – obviating the need for long-range gates. Here we report on an explicit layout that leverages flying qubits, that is very low-overhead for many families of cyclic codes (including the most promising qLDPC instances found to-date). Crucially, our layout eschews more complicated qubit permutations, and instead favours the cyclic shift a simple re-ordering of qubits along a loop that can be realized in depth 1 even on current generation devices. This contrasts significantly with layouts on fixed qubits that depend on a large number of long-range (and more error-prone) hardware couplers for long-distance gates. We also report on two competitive new sets of cyclic qLDPC codes that we constructed. The first is a set of Bivariate-Bicycle (BB) codes with lower weight parity-checks and higher minimum distance while maintaining the same length and encoding rate as comparable BB codes in. Second, we also constructed new Hypergraph Product (HGP) codes, that significantly outperform previously state-of-the-art HGP instances that were optimized by machine-learning methods. Both sets of new codes are efficiently implementable with our cyclic layout with syndrome circuits of fixed depth, made up of alternating layers of parallel gates and only a very small number of cyclic shifts. Combining competitive new qLDPC codes alongside a simple layout implementable on existing hardware, our work suggest a concrete and practical path towards a fault-tolerant quantum computer.

Abstract: Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation we introduce phantom codes---quantum error-correcting codes that realize entangling gates between all logical qubits in a codeblock purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all 2.71 x 10^{10} inequivalent CSS codes up to n=14 and identify additional instances up to n=21 via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed--Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe one–to–two–order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.