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arXiv:2409.15203 · ACM Symposium on Theory of Computing (STOC)

Locality vs Quantum Codes

Samuel Dai, Ray Li

Published Jun 15, 2025• 1 citation
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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.

DOI
10.1145/3717823.3718113

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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]][[n,k,d]][[n,k,d]] quantum stabilizer code with 2D-locality must satisfy kd2≤O(n)kd^2\le O(n)kd2≤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,dn,k,dn,k,d: above the BPT bound, any 2D-embedding must have at least Ω(#∗)Ω(\#^*)Ω(#∗) interactions of length Ω(ℓ∗)Ω(\ell^*)Ω(ℓ∗), where #∗=max⁡(k,d)\#^*= \max(k,d)#∗=max(k,d) and ℓ∗=max⁡(dn,(kd2n)1/4)\ell^*=\max\big(\frac{d}{\sqrt{n}}, \big( \frac{kd^2}{n} \big)^{1/4} \big)ℓ∗=max(n​d​,(nkd2​)1/4). Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length ℓ∗\ell^*ℓ∗ and interaction count #∗\#^*#∗ are asymptotically optimal for all n,k,dn,k,dn,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 ddd and dimension kkk, the number of long-range interactions is asymptotically minimized by a good qLDPC code of length Θ(max⁡(k,d))Θ(\max(k,d))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.
Abstract For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi and Terhal (2009 New J. Phys. 11 043029) (BT) and Bravyi et al (2010 Phys. Rev. Lett. 104 050503) (BPT) established that geometric locality constrains code properties—for instance [ [ n , k , d ] ] quantum codes defined by local checks on the D-dimensional lattice must obey k d 2 / ( D − 1 ) ⩽ O ( n ) . Baspin and Krishna (2022 Quantum 6 711) studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.
Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension 3−ε3-ε3−ε and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.
Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi et al. [Phys. Rev. Lett. 104, 050503 (2010)] have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension kkk and distance ddd. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters kkk and ddd. In particular, in 2D we show that a quantum LDPC code with distance $d\ensuremath{\propto}{n}^{1/2+ϵ}$ requires $\mathrm{\ensuremath{\Omega}}({n}^{1/2+ϵ})$ interactions of length $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}({n}^{ϵ})$. Further, a code satisfying $k\ensuremath{\propto}n$ with distance $d\ensuremath{\propto}{n}^{\ensuremath{\alpha}}$ requires $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}(n)$ interactions of length $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}({n}^{\ensuremath{\alpha}/2})$. As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.
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 R^D must have at least M^* interactions of length at least 𝓁^*, where M^* = Ω(max(k,d)), and 𝓁^* = Ω(max(d/(n^((D-1)/D)), ((kd^(1/(D-1))/n))^((D-1)/D))). We also give tradeoffs between the locality and parameters of commuting projector codes in D-dimensions, generalizing a result of Dai and Li [Dai and Li, 2025]. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length.
Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.
This thesis concerns itself with the role locality plays in the performance of quantum codes. We first address the difficulty of producing codestates from relatively good codes, specifically we investigate the depth of geometrically local encoding circuits. Previous results struggled to produce bounds allowing for general, non-local classical communications in the circuit. We show that, owed to the entanglement hidden in codestates, the depth of encoding circuits is lower bounded, even when allowing for boundless classical computation. We then move on to address a converse question: given a particular code, what can we say about its geometrical structure? We show that implementing stabiliser codes requires satisfying some connectivity requirements that depend on k and d. We quantify these requirements through the notion of graph expansion. Finally, in the last chapter, we ask the question of the achievability of these requirements. The celebrated BPT bound states upper bound on the parameters that can be achieved for codes local in three dimensions. We provide the first construction that matches this bound.