Optimal Locality and Parameter Tradeoffs for Subsystem Codes

TL;DR

The paper derives fundamental locality–parameter tradeoffs for quantum subsystem and commuting projector codes in arbitrary dimensions. It generalizes and tightens prior 2D results by proving lower bounds on both the number and length of long-range interactions required in any D-dimensional embedding, with bounds scaling as $M^* = c_0 \max(k,d)$ and $\ell^* = c_0 \max\left(\frac{d}{n^{(D-1)/D}},\left(\frac{kd^{1/(D-1)}}{n}\right)^{(D-1)/D}\right)$ (and analogous expressions for commuting projector codes). The authors introduce a suite of geometric and information-theoretic tools, including holographic-like area arguments, tiling and packing lemmas, and a carefully engineered expansion process across dimensions to derive the bounds without relying on the union lemma for subsystem codes. They also provide explicit constructions combining qLDPC, wire, and subdivided codes to show these bounds are tight in both interaction count and length, thereby establishing the optimality of locality constraints for both subsystem and commuting projector codes in any dimension. The results illuminate how locality constraints extend beyond stabilizer codes and quantify the cost of nonlocal gauge generators in high-dimensional quantum error correction.

Abstract

We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any $D$-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters $[[n,k,d]]$ into $\mathbb{R}^D$ must have at least $M^*$ interactions of length at least $\ell^*$, where \[ M^* = Ω(\max(k,d)), \quad\text{and}\quad \ell^* = Ω\bigg(\max\bigg(\frac{d}{n^\frac{D-1}{D}}, \bigg(\frac{kd^\frac{1}{D-1}}{n}\bigg)^\frac{D-1}{D}\bigg)\bigg). \] We also give tradeoffs between the locality and parameters of commuting projector codes in $D$-dimensions, generalizing a result of Dai and Li. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length.

Figures (6)

• Figure 1: The (asymptotically) optimal interaction count and length for subsystem codes in 2D: A $[[n,k,d]]$ subsystem code need at least $\Omega(M^*)$ interactions of length $\Omega(\ell^*)$, where $M^*$ is plotted on the left and $\ell^*$ is plotted on the right. Above, we plot the contours of $k$ vs. $d$ tradeoffs for various values of the Interaction Count or Interaction Length. Everywhere, big-$O$ is suppressed for clarity.
• Figure 2: Schematic diagram illustrating the optimality of our lower bounds for all $n,k,d$: A point $(M,L)$ represents that there is a code with $O(M)$ interactions of length $\omega(L)$. Blue shaded region is achievable, red lined region is unachievable. Our lower bound shows that $(M,\ell)$ with $M\le o(M^*)$ and $\ell\le o\left(\ell^*\right)$ is impossible, where $M^*$ and $\ell^*$ are the optimal interaction count and length, respectively, given by Theorem \ref{['thm:main']}. There is a construction (good qLDPC code) with $O(M^*)$ interactions of any length, and another construction (concatenated local code, Theorem \ref{['thm:constr-1']}) with zero interactions of length $\omega\left(\ell^*\right)$.
• Figure 3: Tiling Lemma: for fixed sets of points $X$ and $Y$ and a random width-$w$ grid tiling, we expect a $O(\ell^2/w^2)$ fraction of $X$ to be within a $O(\ell)$ of a grid codimension-2 face, and a $O(\ell/w)$ fraction of $Y$ to be within $O(\ell)$ of a codimension-1 face.
• Figure 4: Sketch of the expansion process in 2 dimensions. The blue region is $V[a_1]$ and the pink region is $[a_1,\textsf{Next}(a_1+\ell)]\times [0,a_2]$. The crosshatched region (labeled $F$ in the proof of Theorem \ref{['thm:main-d']}) contains the boundary of $V[a_1,a_2]$.
• Figure 5: The covering of the outer boundary $H^+\setminus H$ (magenta) used in the proof of Lemma \ref{['lem:correctable-cube']} for $D=2$. The boundary is covered by four rectangles, each of width $\ell$ and length $w+2\ell$. The rectangles overlap at the corners. The generalization to higher dimensions involve a rectangular slab for each face of $H$ (blue).

Theorems & Definitions (38)

• Theorem 1.2: Main Result for Subsystem Codes
• Theorem 1.3: Generalization of dai2024locality to $D$-dimensions
• Theorem 1.4: Main Result -- Part 1
• Theorem 1.5: Main Result -- Part 2
• Theorem 1.6: Generalization of dai2024locality when $d \leq \sqrt{kn}$
• Definition 2.1: Interactions
• Definition 2.2: Interaction counter
• Definition 2.3: Correctable set
• Definition 2.4: Dressed-Cleanable
• Lemma 2.5: bravyi2009nohaah2012logicalpastawski2015fault