Given any two classical codes with parameters $[{n}_{1},k,{d}_{1}]$ and $[{n}_{2},k,{d}_{2}]$, 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 $D=min({d}_{1},{d}_{2})$. 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.
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≤2n1n2,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 provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsystem quantum codes local in 2-dimensions. The second family consists of the constant rate “generalized Shor” codes of Bacon and Cassicino [1], which we re-brand as subsystem hypergraph product (SHP) codes. We show that any hypergraph product code can be obtained by entangling the gauge qubits of two SHP codes. To evaluate the performance of these codes, we simulate both small and large examples. For circuit noise, a [21], [4], [3] BBS code and a [49], [16], [3] SHP code have pseudthresholds of 2 × 10-3and 8 × 10-4, respectively. Simulations for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates 1 × 10-6and 4 × 10-4, respectively.
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.
In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes Lk and Lk+, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters [[6k2,k2,d]]. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies d=3 for k=3 and d=4 for k≥4, establishing a new framework for structured quantum LDPC code design and optimization.
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 performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pX and the phase error probability pZ, and determine how the probability of a logical error depends on pX and pZ. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX≫1) and the physical error rate pZ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.
Abstract—Quantum information processing is a rapidly mounting field that promises to accelerate the speed up of computations. The field utilizes the novel fundamental rules of quantum mechanics such as accelerations. Quantum states carrying quantum information are tempted to noise and decoherence, that’s why the field of quantum error control comes. In this paper, we investigate various aspects of the general theory of quantum error control- subsystem codes. Particularly, we first establish two generic methods to derive subsystem codes from classical codes that are defined over finite fields Fq and F q 2. Second, we derive cyclic subsystem codes and using our two methods, we derive all classes of subsystem codes. Consequently, we construct two famous families of cyclic subsystem BCH and RS codes. Cyclic subsystem RS codes are turned out to be Optimal and MDS codes saturating the singleton bound with equality. Third, we demonstrate several methods of subsystem code constructions by extending, shortening and combining given subsystem codes. Finally, we present tables of upper and lower bounds on subsystem codes parameters 1. I.