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:2305.14560 · Civil War Book Review

A Menagerie of Symmetry Testing Quantum Algorithms

Margarite L. LaBorde

Published Mar 7, 2023• 0 citations
arXivHTMLPDF

Abstract

In Chapter 1, we establish the mathematical background used throughout this thesis. We review concepts from group and representation theory. We further establish fundamental concepts from quantum information. This will allow us to then define the different notions of symmetry necessary in the following chapters. In Chapter 2, we investigate Hamiltonian symmetries. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. Furthermore, we show that this algorithm is that this algorithm is DQC1-Complete. Finally, we execute one of our symmetry-testing algorithms on existing quantum computers for simple examples. In Chapter 3, we discuss tests of symmetry for quantum states. For the case of testing Bose symmetry of a state, there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones. In Chapter 4, we begin by showing that the analytical form of the acceptance probability of such a test is given by the cycle index polynomial of the symmetric group ���� . We derive a family of quantum separability tests, each of which is generated by a finite group; for all such algorithms, we show that the acceptance probability is determined by the cycle index polynomial of the group. Finally, we produce and analyze explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with ��(��2) and ��(�� log(��)) controlled-SWAP gates, respectively, where �� is the number of copies of the state In Chapter 5, we include additional results not previously published; in particular, we give a test for symmetry of a quantum state using density matrix exponentiation, a further result of Hamiltonian symmetry measurements when using Abelian groups, and an alternate Hamiltonian symmetry test construction for a block-encoded Hamiltonian.

DOI
10.31390/gradschool_dissertations.6134

Related papers

Testing symmetry on quantum computers

Margarite L. LaBorde, Soorya Rethinasamy, Mark M. Wilde

Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.

arXiv:2105.12758•Quantum•Sep 25, 2023•18 citations

Quantum Algorithms for Testing Hamiltonian Symmetry

Margarite L. LaBorde, Mark M. Wilde

Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.

arXiv:2203.10017•Physical Review Letters•Oct 14, 2022•13 citations

Entanglement and Asymmetry in Quantum Field Theory

Michele Fossati

This thesis investigates the use of concepts from quantum information theory to analyze and quantify symmetry breaking in extended quantum systems, particularly focusing on the key measure known as Entanglement Asymmetry. The core of the work introduces and applies Entanglement Asymmetry. This quantity is defined based on the difference in information content between a quantum state and a version of that state where the symmetry has been enforced (or "symmetrized"). It represents an alternative approach to traditional, often problematic, local order parameters for measuring symmetry breaking: asymmetry is zero only if the state perfectly preserves the symmetry. The major contributions of the thesis are structured as follows: Symmetry Breaking in Bulk Critical Systems: The research analyzes Entanglement Asymmetry in two-dimensional Conformal Field Theories (CFTs) where the symmetry is explicitly broken. This work derives the asymptotic behavior of the asymmetry as a function of the subsystem size, extending earlier finite-size results to the critical continuum limit. Boundary Effects and Quenches: The analysis is extended to cases where the symmetry holds in the bulk of a CFT but is broken at the edge by a boundary condition. The thesis provides formulas describing this boundary-induced asymmetry, including power-law corrections whose exponents are linked to the properties of boundary-changing operators. It also explores the time evolution of asymmetry in quantum quench protocols. Generalization of Asymmetry to Modern Symmetries: The concept of Entanglement Asymmetry is generalized to include finite generalized symmetries, such as higher-form and non-invertible symmetries, which are a major topic in modern QFT. This involves establishing a generalized procedure for "symmetrizing" a state without relying on traditional group mathematics. Entanglement in Non-Hermitian Systems: The final chapter shifts focus to explore other entanglement properties, specifically symmetry-resolved entanglement and the entanglement Hamiltonian, within non-Hermitian models and their corresponding non-unitary CFTs.

•Dec 18, 2025•0 citations

Quantum Algorithms for Realizing Symmetric, Asymmetric, and Antisymmetric Projectors

Margarite L. LaBorde, Soorya Rethinasamy, Mark M. Wilde

In quantum computing, knowing the symmetries a given system or state obeys or disobeys is often useful. For example, Hamiltonian symmetries may limit allowed state transitions or simplify learning parameters in machine learning applications, and certain asymmetric quantum states are known to be resourceful in various applications. Symmetry testing algorithms provide a means to identify and quantify these properties with respect to a representation of a group. In this paper, we present a collection of quantum algorithms that realize projections onto the symmetric subspace, as well as the asymmetric subspace, of quantum systems. We describe how this can be modified to realize an antisymmetric projection as well, and we show how projectors can be combined in a systematic way to effectively measure various projections in a single quantum circuit. Using these constructions, we demonstrate applications such as testing for Werner-state symmetry and estimating Schmidt ranks of bipartite states, supported by experimental data from IBM Quantum systems. This work underscores the pivotal role of symmetry in simplifying quantum calculations and advancing quantum information tasks.

arXiv:2407.17563•Jul 24, 2024•1 citation

Efficient Quantum Algorithms for Testing Symmetries of Open Quantum Systems

Rahul Bandyopadhyay, Alex H. Rubin, Marina Radulaski +1 more

Symmetry is an important and unifying notion in many areas of physics. In quantum mechanics, it is possible to eliminate degrees of freedom from a system by leveraging symmetry to identify the possible physical transitions. This allows us to simplify calculations and characterize potentially complicated dynamics of the system with relative ease. Previous works have focused on devising quantum algorithms to ascertain symmetries by means of fidelity-based symmetry measures. In our present work, we develop alternative symmetry testing quantum algorithms that are efficiently implementable on quantum computers. Our approach estimates asymmetry measures based on the Hilbert–Schmidt distance, which is significantly easier, in a computational sense, than using fidelity as a metric. The method is derived to measure symmetries of states, channels, Lindbladians, and measurements. We apply this method to a number of scenarios involving open quantum systems, including the amplitude damping channel and a spin chain, and we test for symmetries within and outside the finite symmetry group of the Hamiltonian and Lindblad operators.

arXiv:2309.02515•Open Systems & Information Dynamics•Sep 1, 2023•4 citations

Designing tight frames for quantum computing

Luis Quezada

The aim of this thesis is to explore the implementation of a special kind of quantum measurements, so called harmonic tight frames. To achieve this goal, representation theory is addressed, emphasizing its construction from irreducible representations and the relations they satisfy. These tools simplify the study due to the existence of symmetries, leading to the concept of group frames, defined as orbits under the unitary action of a group and characterized by their symmetry groups. Among the various groups, the simplest are the abelian ones, giving rise to harmonic frames, which are analyzed through the classification theorem of abelian groups and the characters of their irreducible representations. In the quantum mechanics context, the frame elements can be interpreted as pure states of a system. Therefore, the necessary and sufficient conditions for separability are explored, as separable states are easier to implement due to their local nature. Alternatively, when viewing frames as measurements, the concept of POVMs and Naimark's theorem are studied as key tools for designing measurements in quantum computers. Using this knowledge, a characterization of the harmonic frames is given from the abelian group structure theorem, which allows obtaining a necessary and sufficient condition for their separability. The conditions of maximum entanglement of these states for bipartite systems are also studied, obtaining a necessary condition on the dimensions of the subsystems. Finally, a quantum circuit is obtained that allows implementing the harmonic frames associated to cyclic groups as POVMs using a Fourier matrix and a permutation matrix. We conclude by giving simple examples where the results are applied to quantum computers and proposing future avenues of research that could serve to improve the circuit design and extend it to the case of all harmonic frames.

quant-pharXiv:2607.27247•Jul 29, 2026•0 citations
In Chapter 1, we establish the mathematical background used throughout this thesis. We review concepts from group and representation theory. We further establish fundamental concepts from quantum information. This will allow us to then define the different notions of symmetry necessary in the following chapters. In Chapter 2, we investigate Hamiltonian symmetries. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. Furthermore, we show that this algorithm is that this algorithm is DQC1-Complete. Finally, we execute one of our symmetry-testing algorithms on existing quantum computers for simple examples. In Chapter 3, we discuss tests of symmetry for quantum states. For the case of testing Bose symmetry of a state, there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones. In Chapter 4, we begin by showing that the analytical form of the acceptance probability of such a test is given by the cycle index polynomial of the symmetric group ���� . We derive a family of quantum separability tests, each of which is generated by a finite group; for all such algorithms, we show that the acceptance probability is determined by the cycle index polynomial of the group. Finally, we produce and analyze explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with ��(��2) and ��(�� log(��)) controlled-SWAP gates, respectively, where �� is the number of copies of the state In Chapter 5, we include additional results not previously published; in particular, we give a test for symmetry of a quantum state using density matrix exponentiation, a further result of Hamiltonian symmetry measurements when using Abelian groups, and an alternate Hamiltonian symmetry test construction for a block-encoded Hamiltonian.
Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.
Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.
This thesis investigates the use of concepts from quantum information theory to analyze and quantify symmetry breaking in extended quantum systems, particularly focusing on the key measure known as Entanglement Asymmetry. The core of the work introduces and applies Entanglement Asymmetry. This quantity is defined based on the difference in information content between a quantum state and a version of that state where the symmetry has been enforced (or "symmetrized"). It represents an alternative approach to traditional, often problematic, local order parameters for measuring symmetry breaking: asymmetry is zero only if the state perfectly preserves the symmetry. The major contributions of the thesis are structured as follows: Symmetry Breaking in Bulk Critical Systems: The research analyzes Entanglement Asymmetry in two-dimensional Conformal Field Theories (CFTs) where the symmetry is explicitly broken. This work derives the asymptotic behavior of the asymmetry as a function of the subsystem size, extending earlier finite-size results to the critical continuum limit. Boundary Effects and Quenches: The analysis is extended to cases where the symmetry holds in the bulk of a CFT but is broken at the edge by a boundary condition. The thesis provides formulas describing this boundary-induced asymmetry, including power-law corrections whose exponents are linked to the properties of boundary-changing operators. It also explores the time evolution of asymmetry in quantum quench protocols. Generalization of Asymmetry to Modern Symmetries: The concept of Entanglement Asymmetry is generalized to include finite generalized symmetries, such as higher-form and non-invertible symmetries, which are a major topic in modern QFT. This involves establishing a generalized procedure for "symmetrizing" a state without relying on traditional group mathematics. Entanglement in Non-Hermitian Systems: The final chapter shifts focus to explore other entanglement properties, specifically symmetry-resolved entanglement and the entanglement Hamiltonian, within non-Hermitian models and their corresponding non-unitary CFTs.
In quantum computing, knowing the symmetries a given system or state obeys or disobeys is often useful. For example, Hamiltonian symmetries may limit allowed state transitions or simplify learning parameters in machine learning applications, and certain asymmetric quantum states are known to be resourceful in various applications. Symmetry testing algorithms provide a means to identify and quantify these properties with respect to a representation of a group. In this paper, we present a collection of quantum algorithms that realize projections onto the symmetric subspace, as well as the asymmetric subspace, of quantum systems. We describe how this can be modified to realize an antisymmetric projection as well, and we show how projectors can be combined in a systematic way to effectively measure various projections in a single quantum circuit. Using these constructions, we demonstrate applications such as testing for Werner-state symmetry and estimating Schmidt ranks of bipartite states, supported by experimental data from IBM Quantum systems. This work underscores the pivotal role of symmetry in simplifying quantum calculations and advancing quantum information tasks.
Symmetry is an important and unifying notion in many areas of physics. In quantum mechanics, it is possible to eliminate degrees of freedom from a system by leveraging symmetry to identify the possible physical transitions. This allows us to simplify calculations and characterize potentially complicated dynamics of the system with relative ease. Previous works have focused on devising quantum algorithms to ascertain symmetries by means of fidelity-based symmetry measures. In our present work, we develop alternative symmetry testing quantum algorithms that are efficiently implementable on quantum computers. Our approach estimates asymmetry measures based on the Hilbert–Schmidt distance, which is significantly easier, in a computational sense, than using fidelity as a metric. The method is derived to measure symmetries of states, channels, Lindbladians, and measurements. We apply this method to a number of scenarios involving open quantum systems, including the amplitude damping channel and a spin chain, and we test for symmetries within and outside the finite symmetry group of the Hamiltonian and Lindblad operators.
The aim of this thesis is to explore the implementation of a special kind of quantum measurements, so called harmonic tight frames. To achieve this goal, representation theory is addressed, emphasizing its construction from irreducible representations and the relations they satisfy. These tools simplify the study due to the existence of symmetries, leading to the concept of group frames, defined as orbits under the unitary action of a group and characterized by their symmetry groups. Among the various groups, the simplest are the abelian ones, giving rise to harmonic frames, which are analyzed through the classification theorem of abelian groups and the characters of their irreducible representations. In the quantum mechanics context, the frame elements can be interpreted as pure states of a system. Therefore, the necessary and sufficient conditions for separability are explored, as separable states are easier to implement due to their local nature. Alternatively, when viewing frames as measurements, the concept of POVMs and Naimark's theorem are studied as key tools for designing measurements in quantum computers. Using this knowledge, a characterization of the harmonic frames is given from the abelian group structure theorem, which allows obtaining a necessary and sufficient condition for their separability. The conditions of maximum entanglement of these states for bipartite systems are also studied, obtaining a necessary condition on the dimensions of the subsystems. Finally, a quantum circuit is obtained that allows implementing the harmonic frames associated to cyclic groups as POVMs using a Fourier matrix and a permutation matrix. We conclude by giving simple examples where the results are applied to quantum computers and proposing future avenues of research that could serve to improve the circuit design and extend it to the case of all harmonic frames.