Unified Framework for Quantum Code Embedding

TL;DR

The paper develops a unified homological framework for embedding quantum CSS codes by extending input codes through height-$n$ cones, guaranteeing an isomorphic logical subspace in the output. By representing CSS codes as chain complexes over $\mathbb{F}_2$ and employing mapping-cone constructions, it generalizes code concatenation and encompasses topological and LDPC embeddings, including boundary considerations. A Cleaning Lemma provides a mechanism to bound and relate code distances between the original and embedded codes, tying distance growth to isoperimetric properties. The framework recovers and unifies prior approaches (layered embeddings, square-complex subdivisions, and triangulation/thickening strategies) and informs practical pathways for weight reduction and fault-tolerant measurement in Euclidean-space realizations of quantum codes, with implications for saturating BPT bounds in dimensions $D\ge 3$ and for boundary-enabled topological codes.

Abstract

Given a Calderbank-Shor-Steane (CSS) code, it is sometimes necessary to modify the code by adding an arbitrary number of physical qubits and parity checks. Motivations may include concatenating codes, embedding low-density parity check (LDPC) codes into finite-dimensional Euclidean space, or reducing the weights of parity checks. During this embedding, it is essential that the modified code possesses an isomorphic set of logical qubits as the original code. However, despite numerous explicit constructions, the conditions of when such a property holds true is not known in general. Therefore, using the language of homological algebra, we provide a unified framework that guarantees a natural isomorphism between the output and input codes. In particular, we explicitly show how previous works fit into our framework.

Figures (8)

• Figure 1: Toric Code on Planar Graphs. Edges of the lattice are denote by dashed lines. Each edge hosts a qubit, while each vertex/plaquette hosts an $X$-/$Z$- type Pauli operator acting on adjacent qubits, denoted by a black/dashed dot, respectively. (a) depicts the toric code (with alternating smooth and rough boundaries) on a finite square lattice where qubits are denoted by grey dots. (b) depicts the toric code on the honeycomb lattice with boundary conditions corresponding to (a), where qubits are denoted by grey and blue dots. (c) depicts the toric code on the triangular lattice with boundary conditions corresponding to (a), where qubits are denoted by grey and red dots. Note that certain red dots and neighboring dashed dots are boxed together to emphasize the construction in Eq. \ref{['eq:tri-lvl-2']} in Theorem \ref{['thm:toric-triangular']}.
• Figure 2: Barycentric Subdivision of 2-simplex. (a) depict the induced complex of a 2-simplex, whether the dashed, grey, black dots denote the 2-cell, 1-cells, 0-cells, respectively. The dots also represent the barycenters of the plaquettes, edges and vertices, respectively. (b) depicts the 2-cells, 1-cells, 0-cells of the barycentric subdivision of (a) via the dashed, grey, black dots, respectively. The red and grey circle denote the levels $C^{2},C^{1}$, respectively. (c) depicts a simplicial complex induced by two simplices.
• Figure 3: Toric Codes with Boundaries. Each edge hosts a qubit, while each vertex (plaquette) hosts an $X$-, ($Z$-) type Pauli operator acting on adjacent qubits, denoted by a black (dashed) dot, respectively. (a) depicts the toric code with alternating smooth and rough boundaries on a finite square lattice, and the qubits are denoted by grey dots. $Z$ ($X$) Pauli operators acting on the qubits contained in the red (blue) rectangles denotes an example of nontrivial $Z$- ($X$-) type logical operator, respectively. (b), (c) depict the toric code with rough, smooth boundaries on a finite square lattice, where qubits are denoted by red, blue dots, respectively. (b), (c) do not host nontrivial logical operators.
• Figure 4: Gluing of Toric Codes. (a) tabulates the modification of toric codes parity checks when two planes intersect, i.e., a $Z$- (red) or $X$- (blue) type plane interacting with a qubit plane (grey), or a $Z$- (red) and $X$- (blue) type plane interacting via a string defect (green). For example, in row two, the $Z$-type plaquette operator of the $Z$-type plane (red) along the intersection (dashed red) is modified to also act on the corresponding edge (grey) of the qubit plane, while the $Z$-type plaquette operators of the qubit plane (grey) is not modified. (b) tabulates the modification of toric code parity checks when three planes intersect.
• Figure 5: Logical operators near Intersections. (a) tabulates the behavior of $Z$- and $X$-type logical operators which visit the intersections of toric planes. The dashed lines are to indicate that the $X$-type logical operators are paths on the dual lattice. The first row, for example, indicates that a $Z$-type logical operator in the $Z$-type plane (red) passing through the string defect (green) must also exit into the $X$-type plane (blue). (b) shows an example of the embedding procedure for the CSS code with generators $XXX$ and $ZIZ$. The squiggly lines denote a possible $Z$-type logical operator $\ell_1^1$ (induced by logical $ZZI$) based on the rules in Fig. \ref{['fig:layer-logical-rules']}, where the colors indicate which plane the path live in. The two detached red $yz$-planes denote possible $\ell_2^2$ to use to clean the logical operator $\ell_1$ so that its projection in $C^2_1$ is zero.

Theorems & Definitions (76)

• Theorem 1.1: Height-2 Cone
• proof
• Remark 1
• Remark 2
• Theorem 1.2: Height-$n$ Cone
• proof
• Remark 3
• Lemma 1.3: Cleaning
• proof
• Remark 4