Quantum bootstrap product codes

TL;DR

The paper addresses limitations of traditional quantum product-code constructions, notably code-rate bounds and the challenge of capturing fracton codes. It introduces the quantum bootstrap product (QBP), which bootstraps from a segment of the tensor-product chain complex and solves the bootstrap equation $R_{I^l_t}[\partial^{(p,q,w)}_1] \partial^l_2 = 0$ to complete the full chain complex, yielding fork complexes when multiple independent $\partial^l_2$ exist. The framework unifies high-dimensional hypergraph-product codes and fracton codes such as the X-cube, demonstrates self-correcting codes (e.g., a 4D toric code) from input codes with constant energy barriers, and shows tetra-digit fracton codes can achieve polynomial growth of code dimension $k$ with system size, surpassing HGP rate limits. This approach broadens the quantum product-code toolbox, offering a versatile path to fault-tolerant quantum memories and deeper insights into fracton order through fork complexes.

Abstract

Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.

Figures (6)

• Figure 1: A schematic picture of fork complexes. For clarity, here we draw a fork complex based on the chain complex representing a 3D cubic lattice. The 1-chain group $C_1$ and 0-chain group $C_0$ respectively correspond to 2-cells (plaquettes) and 0-cells (vertices) of the lattice, while the 2-chain group $C_2$ is the direct sum of three copies of the 3-cells (cubes); each copy $C^i_2$ is equipped with a unique boundary operator $\partial^i_2$ mapping to $C_1$.
• Figure 2: Chain complex representation of the 4D toric code. In (a), we demonstrate the boundary operators $\partial_2$ and $\partial_1$ defining $Z$-checks and $X$-checks, respectively. More concretely, $\partial_2$ maps a $Z$-check on a cube to the six qubits on the plaquettes surrounding the cube, and $\partial_1$ maps a qubit on a plaquette to the four $X$-checks on the links surrounding the plaquette. Thus, an $X$-check on a link involves six qubits on plaquettes adjacent to the link. In (b), we demonstrate a loop-like syndrome (colored red) created by Pauli $Z$ errors distributed on a membrane (colored yellow). Such loop-like syndromes underpin the self-correcting property of the 4D toric code.
• Figure 3: Fork complex representation of the X-cube code on the dual lattice. In (a), we illustrate the boundary operators $\partial_1$ and $\partial^1_2$ defining $X$-checks and one type of $Z$-checks, respectively ($\partial^2_2$ can be represented similarly). Specifically, $\partial^1_2$ maps a $Z$-check on a cube to four qubits on the plaquettes that are both adjacent to the cube and perpendicular to the $x$ or $y$ axis. Meanwhile, $\partial_1$ maps a qubit on a plaquette to the four $X$-checks on the vertices surrounding the plaquette. Thus, an $X$-check on a vertex involves twelve qubits on the nearest plaquettes. These checks exactly define the X-cube code on the dual lattice. In (b), we show four point-like syndromes (colored red) created by Pauli $Z$ errors distributed on a membrane (colored yellow). Such syndromes generated at the corners of membranes provide more redundant information for decoding than typical point-like syndromes generated at the endpoints of error strings.
• Figure 4: Comparison of code parameters for a series of QBP codes. All codes are defined on (hyper)cubic lattices of linear size $L$ with dimensions specified by the code; the left and right panels show $L=5$ and $L=15$, respectively. For clarity, we normalize $k$ and $d$ by considering $k/n$ and $d/n$ and plot the data on a logarithmic scale. As observed, while the code dimensions of $[p-2,p-1,p,p]$ codes ($p$-dimensional toric codes) obtained by the HGP of $\mathcal{C}^{1D}$ can only be constants, QBP-generated $[0,1,2,p]$ codes show superior performance in terms of code rates. Here, 4D TC refers to the 4D toric code with only loop-like syndromes, while $[2,3,4,4]$ refers to the 4D toric code with both point- and membrane-like syndromes.
• Figure 5: A comparison between the 3D HGP and the $(3,2,0)$ QBP of input classical codes $\mathcal{C}^i: C^i_1\xrightarrow{\delta^i} C^i_0,\ i=1,2,3$. As shown, although both products are based on the tensor product of input classical codes $\mathcal{C}^{TP}$, the HGP formalism extracts a segment of three terms from $\mathcal{C}^{TP}$ to represent the resulting code, whereas the QBP formalism generates the complete resulting code by solving a bootstrap equation. The multiplicity of solutions leads to a fork structure in the resulting complex.