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arXiv:2011.09746 · Quantum

Quantum XYZ Product Codes

Anthony Leverrier, Simon Apers, Christophe Vuillot

Published Jul 14, 2022• 12 citations
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Abstract

We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3) . While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

DOI
10.22331/q-2022-07-14-766

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We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3) . While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.
Three-dimensional (3D) quantum XYZ product can construct a class of non-CSS quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of the 3D XYZ product of three repetition codes. Second, we show that the 3D XYZ product can be generalized to four dimension and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, to which we refer as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that, the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-ZZZ-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.
The three-dimensional (3D) quantum XYZ product can construct a class of non-Calderbank-Shor-Stearne (non-CSS) quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of the 3D XYZ product of three repetition codes. Second, we show that the 3D XYZ product can be generalized to four dimensions and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, which we refer to as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-ZZZ-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.
Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.
The hypergraph product creates a quantum stabilizer code from two input classical linear codes; a paradigmatic example being the surface code as a hypergraph product of two classical repetition codes. Many properties of the hypergraph product code can be inherited from those of the classical codes such as the code dimension, minimum distance and certain fault-tolerant gadgets. We investigate ways to reduce the number of physical qubits in hypergraph product codes while maintaining some of their useful properties for fault tolerance. We show that the code dimension, canonical logical basis, and minimum distances of the hypergraph product code are preserved through this reduction. We also provide syndrome-measurement schedules that preserve the effective distance as well as examples of reduced hypergraph product codes with parameter improvements such as [ ⁣[610,64,6] ⁣]→[ ⁣[441,64,6] ⁣][\![610,64,6]\!] \rightarrow [\![441,64,6]\!][[610,64,6]]→[[441,64,6]] and [ ⁣[1225,49,11] ⁣]→[ ⁣[931,49,11] ⁣][\![1225,49,11]\!] \rightarrow [\![931,49,11]\!][[1225,49,11]]→[[931,49,11]]. In memory simulations with circuit-level depolarizing noise, the reduced codes can exhibit logical error rates comparable to their unreduced versions at the sampled low physical-error rates, while using fewer physical qubits. Finally, we show how overhead reduction can be compatible with homomorphic measurement gadgets, fold-transversal gates and automorphisms, which extends the savings to logical computation.
Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.
The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.