Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2211.02756 · IEEE Transactions on Information Theory

Quantum Weight Enumerators and Tensor Networks

ChunJun Cao, Brad Lackey

Published Dec 7, 2023• 11 citations
arXivHTMLPDF

Abstract

We examine the use of weight enumerators for analyzing tensor network constructions, and specifically the quantum lego framework recently introduced. We extend the notion of quantum weight enumerators to so-called tensor enumerators, and prove that the trace operation on tensor networks is compatible with a trace operation on tensor enumerators. This allows us to compute quantum weight enumerators of larger codes, such as the ones constructed through tensor network methods, more efficiently. We also provide a general framework for quantum MacWilliams identities that includes tensor enumerators as a special case.

DOI
10.1109/tit.2023.3340503

Related papers

Quantum Lego Expansion Pack: Enumerators from Tensor Networks

ChunJun Cao, Michael J. Gullans, Brad Lackey +1 more

We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024

arXiv:2308.05152•PRX Quantum•Jul 22, 2024•18 citations

Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks

ChunJun Cao, Brad Lackey

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation.More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion.Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network.The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as nonstabilizer codes over qubits and qudits.For a contractible tensor network, the sequence of contractions also constructs a decoding or encoding circuit.To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct nontrivial codes.These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a three-dimensional stabilizer code, and other stabilizer codes with interesting properties.Surprisingly, we find that the surface code is equivalent to the two-dimensional Bacon-Shor code after "dualizing" its tensor-network encoding map.

arXiv:2109.08158•PRX Quantum•May 11, 2022•36 citations

Three MacWilliams type identities and quantum error-correcting $ G $ -codes

Chuangxun Cheng, Xiaoguang Shang

We prove three MacWilliams type identities for irreducible projective representations of finite groups. As an application, we deduce the MacWilliams identities of weight enumerators, double weight enumerators and complete weight enumerators for quantum error-correcting $ G $-codes and obtain the Singleton bounds.

Advances in Mathematics of Communications•Dec 27, 2024•0 citations

On the complete weight distributions of quantum error-correcting codes

Chao‐Hai Du, Zhi Ma, Maosheng Xiong

In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

Chinese Physics B•Dec 16, 2022•2 citations

Quantum Error Correction via Tensor Networks

Jincheng Zhang

Quantum error correction (QEC) is crucial for the reliable implementation of quantum computers, as qubits are extremely sensitive to environmental noise. Traditional approaches to QEC often suffer from computational complexity, particularly when scaling to larger qubit systems. This work explores the application of tensor networks as a novel method for representing and manipulating quantum error correction codes. We demonstrate that tensor networks provide a compact and efficient framework for encoding and decoding quantum information, potentially mitigating the scalability issues inherent in conventional QEC schemes. Specifically, we leverage the ability of tensor networks to represent the entanglement structure of codes, allowing for efficient simulation and decoding of noisy quantum computations. This approach offers a promising avenue for developing scalable QEC protocols. The core claim is that tensor networks offer a compact and efficient representation for quantum error correction codes. The core mechanism involves utilizing tensor networks to represent the entanglement structure of quantum error correction codes, enabling efficient simulation and decoding of noisy quantum computations.

•Aug 28, 2026•0 citations

Categorical Tensor Network States

Jacob Biamonte, Stephen R. Clark, Dieter Jaksch

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state |ψ〉, we present a new and general method to factor |ψ〉 into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.

arXiv:1012.0531•AIP Advances•Dec 1, 2011•35 citations
We examine the use of weight enumerators for analyzing tensor network constructions, and specifically the quantum lego framework recently introduced. We extend the notion of quantum weight enumerators to so-called tensor enumerators, and prove that the trace operation on tensor networks is compatible with a trace operation on tensor enumerators. This allows us to compute quantum weight enumerators of larger codes, such as the ones constructed through tensor network methods, more efficiently. We also provide a general framework for quantum MacWilliams identities that includes tensor enumerators as a special case.
We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024
We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation.More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion.Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network.The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as nonstabilizer codes over qubits and qudits.For a contractible tensor network, the sequence of contractions also constructs a decoding or encoding circuit.To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct nontrivial codes.These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a three-dimensional stabilizer code, and other stabilizer codes with interesting properties.Surprisingly, we find that the surface code is equivalent to the two-dimensional Bacon-Shor code after "dualizing" its tensor-network encoding map.
In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.
Quantum error correction (QEC) is crucial for the reliable implementation of quantum computers, as qubits are extremely sensitive to environmental noise. Traditional approaches to QEC often suffer from computational complexity, particularly when scaling to larger qubit systems. This work explores the application of tensor networks as a novel method for representing and manipulating quantum error correction codes. We demonstrate that tensor networks provide a compact and efficient framework for encoding and decoding quantum information, potentially mitigating the scalability issues inherent in conventional QEC schemes. Specifically, we leverage the ability of tensor networks to represent the entanglement structure of codes, allowing for efficient simulation and decoding of noisy quantum computations. This approach offers a promising avenue for developing scalable QEC protocols. The core claim is that tensor networks offer a compact and efficient representation for quantum error correction codes. The core mechanism involves utilizing tensor networks to represent the entanglement structure of quantum error correction codes, enabling efficient simulation and decoding of noisy quantum computations.
We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state |ψ〉, we present a new and general method to factor |ψ〉 into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.