Preparing code states via seed-entangler-enriched sequential quantum circuits: Application to tetradigit topological error-correcting codes

Demonstrating how long-range entangled states are born from product states has gained much attention, which is not only important for quantum technology but also provides an unconventional tool in characterizing and classifying exotic phases of matter. In this paper, we introduce a unified and efficient framework of quantum circuits (i.e., a series of local unitary transformations), termed the seed-entangler-enriched sequential quantum circuit (SEESQC) to construct long-range entangled states (i.e., code states) in the code space of topological error-correcting codes. Specifically, we apply SEESQC to construct code states of tetradigit models---a broad class of long-range entangled stabilizer codes indexed by a four-digit parameter. These models are not rare but encompass toric codes across arbitrary dimensions and subsume the X-cube fracton code as special cases. Featuring a hierarchical structure of generalized entanglement renormalization group, many tetradigit models host spatially extended excitations (e.g., loops, membranes, and exotic nonmanifold objects) with constrained mobility and deformability, and exhibit system-size-dependent ground-state degeneracies that scale exponentially with a polynomial in linear sizes. In this work, we begin with graphical and algebraic demonstration of quantum circuits for the computational basis states, before generalizing to broader cases. Central to this framework is a key ingredient termed the seed-entangler acting on a small number of qubits termed seeds, enabling a systematic scheme to achieve arbitrary code states. Remarkably, the number of available seeds equals the number of logical qubits for the constructed examples, which leaves plenty of room for future investigation in theoretical physics, mathematics and quantum information science. Beyond the critical limitation of prior state-engineering methodologies, which required entirely distinct, model-specific circuit designs for each class of topological order, this framework transcends spatial dimensions, bridges liquid and nonliquid states, and unifies gapped phases governed by distinct entanglement renormalization group schemes. With experimental feasibility via synthetic dimensions in modern quantum simulators, the SEESQC framework offers a pathway toward engineering topological phases, and manipulating logical qubits.