Locality vs Quantum Codes

TL;DR

This paper proves optimal tradeoffs between the locality and parameters of quantum error-correcting codes and exhibits quantum codes that show, in strong ways, that their interaction length ℓ* and interaction count M* are asymptotically optimal for all n,k,d.

Abstract

This paper proves optimal tradeoffs between the locality and parameters of quantum error-correcting codes. Quantum codes give a promising avenue towards quantum fault tolerance, but the practical constraint of locality limits their quality. The seminal Bravyi-Poulin-Terhal (BPT) bound says that a $[[n,k,d]]$ quantum stabilizer code with 2D-locality must satisfy $kd^2\le O(n)$. We answer the natural question: for better code parameters, how much "non-locality" is needed? In particular, (i) how long must the long-range interactions be, and (ii) how many long-range interactions must there be? We give a complete answer to both questions for all $n,k,d$: above the BPT bound, any 2D-embedding must have at least $Ω(\#^*)$ interactions of length $Ω(\ell^*)$, where $\#^*= \max(k,d)$ and $\ell^*=\max\big(\frac{d}{\sqrt{n}}, \big( \frac{kd^2}{n} \big)^{1/4} \big)$. Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length $\ell^*$ and interaction count $\#^*$ are asymptotically optimal for all $n,k,d$. Our results generalize or improve all prior works on this question, including the BPT bound and the results of Baspin and Krishna. One takeaway of our work is that, for any desired distance $d$ and dimension $k$, the number of long-range interactions is asymptotically minimized by a good qLDPC code of length $Θ(\max(k,d))$. Following Baspin and Krishna, we also apply our results to the codes implemented in the stacked architecture and obtain better bounds. In particular, we rule out any implementation of hypergraph product codes in the stacked architecture.

Figures (11)

• Figure 1: The (asymptotically) optimal interaction count and length for stabilizer codes: A $[[n,k,d]]$ stabilizer code need at least $\Omega(\#)$ interactions of length $\Omega(\ell)$, where $\#$ is plotted on the left and $\ell$ is plotted on the right. Above, we plot the counters of $k$ vs. $d$ tradeoffs for various values of the Interaction Count or Interaction Length. Everywhere, big-$O$ is suppressed for clarity. The bounds in the purple regions were shown (with lost log-factors) in baspin2022quantifying.
• Figure 2: Schematic diagram illustrating the optimality of our lower bounds for all $n,k,d$: A point $(\#,\ell)$ represents that there is a code with $O(\#)$ interactions of length $\omega(\ell)$. Blue shaded region is achievable, red lined region is unachievable. Our lower bound shows that $(\#,\ell)$ with $\# \le o(\#^*)$ and $\ell\le o\left(\ell^*\right)$ is impossible, where $\#^*$ and $\ell^*$ are the optimal interaction count and length, respectively, given by Theorem \ref{['thm:main']}. There is a construction (good code) with $O(\#^*)$ interactions of any length, and another construction (concatenated construction, Theorem \ref{['thm:construct']}) with zero interactions of length $\omega\left(\ell^*\right)$.
• Figure 3: Achievable qLDPC parameters for the stacked architecture as defined in baspin2022quantifying Big-$O$ is suppressed for clarity. Red regions are not achievable. Blue region (below BPT bound) is achievable by surface code, and we are not aware of any better constructions. Our bounds imply $d\le O(n^{2/3})$, $kd^4\le O(n^3)$, and $k^3d^2\le O(n^3)$. Baspin and Krishna baspin2022quantifying showed $d\le O(n^{2/3}\log^{2/3} n)$ and $k^3d^4\le O(n^5\log^4n)$.
• Figure 4: Illustration of the growing-the-square process. The yellow internal square $V$ is the small correctable set with side length $w$, the orange external square $T$ is the larger set grown from it with side length $w + 2\ell$. The labeled edges are (a) an interaction between two qubits in $V$, (b) a short interaction between a qubit in $V$ and a qubit in $T$, and (c) a long interaction between a qubit in $V$ and a qubit outside of $T$.
• Figure 5: The partition into sets $A, B, C$ when there are (1) no long interactions and (2) qubits are arranged in a lattice. $A$ is the union of the blue squares, $B$ is the union of the red rectangles, and $C$ is the union of the yellow squares. A long interaction is drawn connecting two previously disconnected blue squares.

Theorems & Definitions (32)

• Theorem 1.2: Main result
• Theorem 1.3: Interaction Count is Optimal, panteleev2021asymptoticallyleverrier2022quantum
• Theorem 1.4: Interaction Length is Optimal
• Corollary 1.5
• proof
• Theorem 1.6: Main result, part 1
• Theorem 1.7: Main result, part 2
• Proposition 2.1
• Definition 3.1: Interactions
• Remark 3.2: Connectivity Graph