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:2311.17249

On Commutative Penalty Functions in Parent-Hamiltonian Constructions

Jacob Biamonte

Published Nov 28, 2023• 0 citations
arXivHTMLPDF

Abstract

There are several known techniques to construct a Hamiltonian with an expected value that is minimized uniquely by a given quantum state. Common approaches include the parent Hamiltonian construction from matrix product states, building approximate ground state projectors, and, in a common case, developing penalty functions from the generalized Ising model. Here we consider the framework that enables one to engineer exact parent Hamiltonians from commuting polynomials. We derive elementary classification results of quadratic Ising parent Hamiltonians and to generally derive a non-injective parent Hamiltonian construction. We also consider that any $n$-qubit stabilizer state has a commutative parent Hamiltonian with $n+1$ terms and we develop an approach that allows the derivation of parent Hamiltonians by composition of network elements that embed the truth tables of discrete functions into a kernel space. This work presents a unifying framework that captures components of what is known about exact parent Hamiltonians and bridges a few techniques across the domains that are concerned with such constructions.

DOI
10.48550/arxiv.2311.17249

Related papers

Large classes of quantum scarred Hamiltonians from matrix product states

Sanjay Moudgalya, Edward O'Brien, Bogdan Andrei Bernevig +2 more

The authors develop a general formalism to systematically construct parent Hamiltonians with quasiparticle excited states starting from a matrix product state (MPS) wave function. The resulting parent Hamiltonians are generically nonintegrable, and the quasiparticle excited states are thus examples of quantum many-body scars of the Hamiltonian. This method is used to recover well-known quantum scarred models, such as the AKLT spin chain, as well as to construct many new families of models, thus illustrating the close connection between parent Hamiltonians of MPS and quantum scarred models.

arXiv:2002.11725•Physical review. B./Physical review. B•Aug 11, 2020•109 citations

Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States

Ruohan Shen, Yuchen Guo, Shuo Yang

There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.

arXiv:2301.12448•Physical Review Letters•May 30, 2023•13 citations

Parent Hamiltonians of Jastrow wavefunctions

Mathieu Beau

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.

•Nov 1, 2021•9 citations

Parent Hamiltonians of Jastrow wavefunctions

Mathieu Beau, Adolfo del Campo

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.

arXiv:2107.02869•SciPost Physics Core•Nov 18, 2021•0 citations

Parent Hamiltonians for stabilizer quantum many-body scars

Shane Dooley

Quantum many-body scars (QMBS) have attracted considerable interest due to their role in weak ergodicity breaking in many-body systems. We present a general construction that embeds stabilizer states as QMBS of local Hamiltonians. The method relies on a notion of factorizability of Pauli strings on a lattice, which is used to convert stabilizer elements into local, few-body operators that annihilate the stabilizer state. This enables the systematic construction of parent Hamiltonians with zero-energy stabilizer QMBS typically near the middle of the spectrum. The method reproduces several known results in a unified framework, including recent examples of volume-law entangled QMBS, such as the ``rainbow'' QMBS and the entangled antipodal Bell pair state. We also apply the framework to construct examples of stabilizer QMBS with a more complex entanglement structure, such as the cluster state, the toric code state, and a volume-law entangled state we dub the antipodal toric code (ATC) state. Exact diagonalization confirms our results and reveal the stabilizer states as exact eigenstates of their parent Hamiltonian.

quant-pharXiv:2601.10805•Physical Review Letters•Jan 19, 2026•0 citations

Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions

Eli Chertkov, Bryan K. Clark

Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output.However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians.We introduce an alternative computational "inverse method," the eigenstate-to-Hamiltonian construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems.EHC takes as input a wave function jψ T i and produces as output Hamiltonians for which jψ T i is an eigenstate.This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix.EHC is widely applicable to a number of models and, in this work, we consider seven different examples.Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model.EHC gives an alternative inverse approach for studying quantum many-body phenomena.

arXiv:1802.01590•Physical Review X•Jul 27, 2018•94 citations
There are several known techniques to construct a Hamiltonian with an expected value that is minimized uniquely by a given quantum state. Common approaches include the parent Hamiltonian construction from matrix product states, building approximate ground state projectors, and, in a common case, developing penalty functions from the generalized Ising model. Here we consider the framework that enables one to engineer exact parent Hamiltonians from commuting polynomials. We derive elementary classification results of quadratic Ising parent Hamiltonians and to generally derive a non-injective parent Hamiltonian construction. We also consider that any nnn-qubit stabilizer state has a commutative parent Hamiltonian with n+1n+1n+1 terms and we develop an approach that allows the derivation of parent Hamiltonians by composition of network elements that embed the truth tables of discrete functions into a kernel space. This work presents a unifying framework that captures components of what is known about exact parent Hamiltonians and bridges a few techniques across the domains that are concerned with such constructions.
The authors develop a general formalism to systematically construct parent Hamiltonians with quasiparticle excited states starting from a matrix product state (MPS) wave function. The resulting parent Hamiltonians are generically nonintegrable, and the quasiparticle excited states are thus examples of quantum many-body scars of the Hamiltonian. This method is used to recover well-known quantum scarred models, such as the AKLT spin chain, as well as to construct many new families of models, thus illustrating the close connection between parent Hamiltonians of MPS and quantum scarred models.
There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.
We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.
We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.
Quantum many-body scars (QMBS) have attracted considerable interest due to their role in weak ergodicity breaking in many-body systems. We present a general construction that embeds stabilizer states as QMBS of local Hamiltonians. The method relies on a notion of factorizability of Pauli strings on a lattice, which is used to convert stabilizer elements into local, few-body operators that annihilate the stabilizer state. This enables the systematic construction of parent Hamiltonians with zero-energy stabilizer QMBS typically near the middle of the spectrum. The method reproduces several known results in a unified framework, including recent examples of volume-law entangled QMBS, such as the ``rainbow'' QMBS and the entangled antipodal Bell pair state. We also apply the framework to construct examples of stabilizer QMBS with a more complex entanglement structure, such as the cluster state, the toric code state, and a volume-law entangled state we dub the antipodal toric code (ATC) state. Exact diagonalization confirms our results and reveal the stabilizer states as exact eigenstates of their parent Hamiltonian.
Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output.However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians.We introduce an alternative computational "inverse method," the eigenstate-to-Hamiltonian construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems.EHC takes as input a wave function jψ T i and produces as output Hamiltonians for which jψ T i is an eigenstate.This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix.EHC is widely applicable to a number of models and, in this work, we consider seven different examples.Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model.EHC gives an alternative inverse approach for studying quantum many-body phenomena.