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:1812.08171 · Physical review. B./Physical review. B

Restricted Boltzmann machines and matrix product states of one-dimensional translationally invariant stabilizer codes

Yunqin Zheng, Huan He, Nicolas Regnault, Bogdan Andrei Bernevig

Published Apr 15, 2019• 21 citations
arXivHTMLPDF

Abstract

We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. We finally conjecture that these features hold true for all the 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, as long as their MPS matrices are of rank 1.

DOI
10.1103/physrevb.99.155129

Related papers

On Translation-Invariant Matrix Product States and advances in MPS representations of the $W$-state

Klimov, Petr, Richik Sengupta, Jacob Biamonte

This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the $n$-party $W$-state and construct a TI MPS representation of bond dimension $\left \lfloor \dfrac{n}{2} \right \rfloor +1$ for it. We further study properties of this class and show that we can can always achieve a bond dimension of $n$ for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension $d(ψ)$ for a given state $ψ$. In particular we introduce a deterministic algorithm for the search of $d(ψ)$ for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the $n$-party $W$-state for small $n$.

arXiv:2306.16456•Jun 28, 2023•1 citation

Efficiency of neural-network state representations of one-dimensional quantum spin systems

Ruizhi Pan, Charles W. Clark

Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis of their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. Published by the American Physical Society 2024

arXiv:2302.00173•Physical Review Research•May 20, 2024•1 citation

An Efficient Algorithmic Way to Construct Boltzmann Machine Representations for Arbitrary Stabilizer Code

Yuanhang Zhang, Zhian Jia, Yu-Chun Wu +1 more

Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states-a class of highly entangled quantum states that are central to quantum error correction. Given a set of stabilizer generators, we develop an efficient algorithm to determine both the RBM architecture and the exact values of its parameters. Our findings provide new insights into the expressive power of RBMs, highlighting their capability to encode highly entangled states, and may serve as a useful tool for the classical simulation of quantum error-correcting codes.

arXiv:1809.08631•Entropy•Jun 13, 2025•9 citations

Efficient machine-learning representations of a surface code with boundaries, defects, domain walls, and twists

Zhian Jia, Yuanhang Zhang, Yu-Chun Wu +3 more

Machine-learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the most important representations, the restricted Boltzmann machine (RBM), in the stabilizer formalism. A general method to construct RBM representations for stabilizer code states is given, and exact RBM representations for several types of stabilizer groups with the number of hidden neurons equal to or less than the number of visible neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze a surface code with boundaries, defects, domain walls, and twists in full detail and find that almost all the models can be efficiently represented via the RBM ansatz: the RBM parameters of the perfect case, boundary case, and defect case are constructed analytically using the method we provide in the stabilizer formalism, and the domain wall and twist case is studied numerically. In addition, the case for Kitaev's $D({\mathbb{Z}}_{d})$ model, which is a generalized model of the surface code, is also investigated.

arXiv:1802.03738•Physical Review A•Jan 4, 2019•31 citations

Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model

Mohammadreza Noormandipour, Youran Sun, Babak Haghighat

Abstract In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. 3 × 3 with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of 0.09 % . Moreover, the wave-functions we find have 99.89 % overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.

arXiv:2003.07280•Machine Learning Science and Technology•Nov 26, 2021•5 citations

Matrix product states with long-range localizable entanglement

Thorsten B. Wahl, David Pérez-Garcı́a, J. I. Cirac

We derive a criterion to determine when a translationally invariant matrix product state (MPS) has long-range localizable entanglement, where that quantity remains finite in the thermodynamic limit. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3.

arXiv:1206.4254•Physical Review A•Dec 14, 2012•9 citations
We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. We finally conjecture that these features hold true for all the 1D stabilizer codes having a nondegenerate ground state with periodic boundary condition, as long as their MPS matrices are of rank 1.
On Translation-Invariant Matrix Product States and advances in MPS representations of the WWW-state
This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the nnn-party WWW-state and construct a TI MPS representation of bond dimension ⌊n2⌋+1\left \lfloor \dfrac{n}{2} \right \rfloor +1⌊2n​⌋+1 for it. We further study properties of this class and show that we can can always achieve a bond dimension of nnn for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension d(ψ)d(ψ)d(ψ) for a given state ψψψ. In particular we introduce a deterministic algorithm for the search of d(ψ)d(ψ)d(ψ) for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the nnn-party WWW-state for small nnn.
Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis of their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. Published by the American Physical Society 2024
Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states-a class of highly entangled quantum states that are central to quantum error correction. Given a set of stabilizer generators, we develop an efficient algorithm to determine both the RBM architecture and the exact values of its parameters. Our findings provide new insights into the expressive power of RBMs, highlighting their capability to encode highly entangled states, and may serve as a useful tool for the classical simulation of quantum error-correcting codes.
Machine-learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the most important representations, the restricted Boltzmann machine (RBM), in the stabilizer formalism. A general method to construct RBM representations for stabilizer code states is given, and exact RBM representations for several types of stabilizer groups with the number of hidden neurons equal to or less than the number of visible neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze a surface code with boundaries, defects, domain walls, and twists in full detail and find that almost all the models can be efficiently represented via the RBM ansatz: the RBM parameters of the perfect case, boundary case, and defect case are constructed analytically using the method we provide in the stabilizer formalism, and the domain wall and twist case is studied numerically. In addition, the case for Kitaev's D(Zd)D({\mathbb{Z}}_{d})D(Zd​) model, which is a generalized model of the surface code, is also investigated.
Abstract In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. 3 × 3 with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of 0.09 % . Moreover, the wave-functions we find have 99.89 % overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.