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.
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.
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.
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 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,d: above the BPT bound, any 2D-embedding must have at least Ω(#∗) interactions of length Ω(ℓ∗), where #∗=max(k,d) and ℓ∗=max(nd,(nkd2)1/4). Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length ℓ∗ 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.
We study the structure of logical operators in local D-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is d, then any logical operator can be supported on a set of specified geometry containing $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ qubits, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}{d}^{1/(D\ensuremath{-}1)}=O(n)$ and n is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that for any two-dimensional local commuting projector code there is a nontrivial logical ``string'' operator supported on a narrow strip, where the operator is only slightly entangling across any cut through the strip.
Given any two classical codes with parameters [n1,k,d1] and [n2,k,d2], we show how to construct a quantum subsystem code in two dimensions with parameters [[N,K,D]] satisfying $N\ensuremath{\le}2{n}_{1}{n}_{2}, K=k$, and D=min(d1,d2). These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi [Phys. Rev. A 83, 012320 (2011)]. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound KD=O(N) on the code parameters of two-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge fixed to certain hypergraph product codes that do.
We study subsystem codes whose gauge group has local generators in two-dimensional (2D) geometry. It is shown that there exists a family of such codes defined on lattices of size $L\ifmmode\times\else\texttimes\fi{}L$ with the number of logical qubits k and the minimum distance d both proportional to L. The gauge group of these codes involves only two-qubit generators of type XX and ZZ coupling nearest-neighbor qubits (and some auxiliary one-qubit generators). Our proof is not constructive as it relies on a certain version of the Gilbert-Varshamov bound for classical codes. Along the way, we introduce and study properties of generalized Bacon-Shor codes that might be of independent interest. Secondly, we prove that any 2D subsystem [n,k,d] code with spatially local generators obeys upper bounds kd=O(n) and d2=O(n). The analogous upper bound proved recently for 2D stabilizer codes is kd2=O(n). Our results thus demonstrate that subsystem codes can be more powerful than stabilizer codes under the spatial locality constraint.
Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.