Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes

TL;DR

This work presents SEESQC, a unified, depth-efficient quantum-circuit framework for preparing long-range entangled code states in tetradigit TD stabilizer codes, encompassing Toric Codes in multiple dimensions and fracton-like models such as the X-cube. By introducing seed_entangler steps and a two-stage growth process (U_g and U_c), the authors map seed states to the code space, enabling preparation of both computational bases and arbitrary code states for a broad class of TD models. The paper develops explicit circuit constructions for key TD examples, analyzes stabilizer redundancies, and generalizes the approach to arbitrary [d-1,d,d+1,D] TD models, highlighting a surprising one-to-one correspondence between the number of seeds and logical qubits. This framework advances systematic quantum state engineering in topological and fracton phases, with implications for quantum simulation, error correction, and explorations of ERG structures across dimensions. The results offer a path toward scalable, dimension-agnostic state preparation and open questions about seed locality, universality, and experimental realization on programmable quantum platforms.

Abstract

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 \emph{Seed-Entangler-Enriched Sequential Quantum Circuit} (SEESQC) to construct long-range entangled states (i.e., code states) in code space of topological error-correcting codes. Specifically, we apply SEESQC to construct code states of Tetra-Digit 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 Tetra-Digit models host spatially extended excitations (e.g., loops, membranes, and exotic non-manifold 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 computational basis states, before generalizing to broader cases. Central to this framework is a key ingredient termed the \emph{seed-entangler} acting on a small number of qubits termed \textit{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, ...

Figures (13)

• Figure 1: Schematic illustration of Seed-Entangler-Enriched Sequential Quantum Circuits (SEESQC). All spins, including seeds and other spins, are initiated in product state $|00\cdots0\rangle$. (a)-(c) Unified linear-depth local unitary (LU) circuit $U_c$ for preparing computational basis states ($|0\rangle^{\otimes \log_2 \text{GSD}}_{\text{logic}}$) of logical qubits encoded by stabilizer code topological models: (a) Toric Code $[0,1,2,2]$, (b) X-cube $[0,1,2,3]$, (c) 3D Toric Code $[1,2,3,3]$. This circuit architecture can be extended to a large class of stabilizer code topological models called tetradigit (TD) models, which is reviewed in Sec. \ref{['section_TD_models']}. (d) State preparation protocol: Initial product state undergoes seed entanglement, followed by $U_g$ (a series of CNOT gates) and $U_c$, mapping $|\psi\rangle_{\text{seeds}} \otimes |0\rangle_{\text{non-seeds}}$ to the code state $|\psi\rangle_{\text{logic}}$. Seeds (spins located at the center of each orange wavy line) are: edge-based in (a,b), plaquette-based in (c). When $|\psi\rangle_{\text{seeds}}$ is a Greenberger-Horne-Zeilinger (GHZ) state, $|\psi\rangle_{\text{logic}}$ is a logical GHZ state. For $[d-1,d,d+1,D]$ models (including all shown examples), the number of seeds equals the number of logical qubits.
• Figure 2: Leaves in the $[0,1,2,3]$ TD model (3-dimensional X-cube fracton phase) under open boundary conditions. The black dot denotes a $0$-dimensional node $\gamma_0$ at $(1,2,1)$. Three orthogonal leaves (2-dimensional planes) containing $\gamma_0$ are highlighted: $\langle \hat{x}_1, \hat{x}_2 \rangle$ (red, $x_3=1$ plane), $\langle \hat{x}_1, \hat{x}_3 \rangle$ (green, $x_2=2$ plane), and $\langle \hat{x}_2, \hat{x}_3 \rangle$ (blue, $x_1=1$ plane). Each leaf hosts $B$ terms formed by products of Pauli $Z$ operators on spins (qubits) within the respective plane. The axes $x_1$, $x_2$, $x_3$ are marked in blue. The blue dot stands for the original point $(0,0,0)$.
• Figure 3: Illustration of $A$ terms and $B$ terms in the 2-dimensional Toric Code. The $A$ term associated with plaquette $p$ is $A_p = \prod_{e \subset p} X_e = X_1 X_2 X_3 X_4$. The $B$ term associated with vertex $v$ is $B_v = \prod_{e \hbox{o}rigin=c]{180}{$⊂$} v} Z_e = Z_2 Z_3 Z_5 Z_6$.
• Figure 4: Illustration of $A$ terms and $B$ terms in the X-cube model. The $A$ term associated with cube $c$ is $A_c = X_1 X_2 \cdots X_{12}$. The three different $B$ terms attached to vertex $v$ are $B_{v,xy} = Z_{10}Z_{11}Z_{15}Z_{14}$, $B_{v,xz} = Z_7Z_{11}Z_{13}Z_{14}$, and $B_{v,yz} = Z_7Z_{10}Z_{13}Z_{15}$.
• Figure 5: Illustration of $A$ term and $B$ term in the $[0,1,2,4]$ model. The spins are on edges, and the numbers in this figure are placed at the center of each edge, representing the spin on it. (a) The $A$ term associated with 4-cube $\gamma_4$ is $A_{\gamma_4}=X_1X_2\cdots X_{32}$, supported by all the edges of $\gamma_4$. The dashed cube aids visualizing $x_4$-direction edges: the centers of edges spanning in $x_4$-direction coincide with the centers of vertices of the dashed cube. (b) Each $B$ term attached to vertex $v$ is supported by 4 edges on a cross containing $v$, e.g., $B_{\gamma_0,x_2x_3}=Z_1Z_2Z_3Z_4$, $B_{\gamma_0,x_3x_4}=Z_1Z_2Z_7Z_8$. There are 6 distinct $B$ terms attached to the vertex $v$ due to the combinatorial factor $C_{D-d_n}^{d_l-d_n}=C_{4-0}^{2-0}=6$, which are $B_{\gamma_0,x_1x_2},B_{\gamma_0,x_1x_3}, B_{\gamma_0,x_1x_4},B_{\gamma_0,x_2x_3},B_{\gamma_0,x_2x_4},B_{\gamma_0,x_3x_4}$.