A Menagerie of Symmetry Testing Quantum Algorithms

• URL: http://arxiv.org/abs/2305.14560v1
• Date: Tue, 23 May 2023 22:55:02 GMT
• Title: A Menagerie of Symmetry Testing Quantum Algorithms
• Authors: Margarite L. LaBorde
• Abstract summary: We show how to generate a notion of symmetry from a discrete, finite group and how this generalizes to a continuous group. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This thesis aims to establish notions of symmetry for quantum states and channels as well as describe algorithms to test for these properties on quantum computers. Ideally, the work will serve as a self-contained overview of the subject. We begin by establishing the necessary mathematical background. We show how to generate a notion of symmetry from a discrete, finite group and how this generalizes to a continuous group. We then use these notions to investigate Hamiltonian symmetries. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group and show that this algorithm is DQC1-Complete. We next discuss tests of symmetry for quantum states. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested and establish various generalizations of the resource theory of asymmetry. In the next chapter, we show that the analytical form of the acceptance probability of such a test is given by the cycle index polynomial of the symmetric group $S_k$. 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 $O(k^2)$ and $O(k\log(k))$ controlled-SWAP gates, respectively, where $k$ is the number of copies of the state. Finally, we include additional results not previously published; specifically, 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.

Related papers

• Quantum Algorithms for Realizing Symmetric, Asymmetric, and Antisymmetric Projectors [3.481985817302898]
  Knowing the symmetries of a given system or state obeys or disobeys is often useful in quantum computing. We present a collection of quantum algorithms that realize projections onto the symmetric subspace. We show how projectors can be combined in a systematic way to effectively measure various projections in a single quantum circuit.
  arXiv  Detail & Related papers  (2024-07-24T18:00:07Z)
• Revealing symmetries in quantum computing for many-body systems [0.0]
  We deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form for evaluation on quantum computers. We prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations.
  arXiv  Detail & Related papers  (2024-07-03T18:53:09Z)
• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Quantum Computational Complexity and Symmetry [3.9134031118910264]
  Testing the symmetries of quantum states and channels provides a way to assess their usefulness. We prove that various symmetry-testing problems are complete for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP. We also prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and QAM.
  arXiv  Detail & Related papers  (2023-09-18T18:48:44Z)
• A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules [62.997667081978825]
  We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions. The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix. We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
  arXiv  Detail & Related papers  (2023-04-12T18:10:47Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• Cycle Index Polynomials and Generalized Quantum Separability Tests [3.481985817302898]
  The mixedness of one share of a pure bipartite state determines whether the overall state is a separable, unentangled state. 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 of the group.
  arXiv  Detail & Related papers  (2022-08-31T02:13:16Z)
• Symmetric Pruning in Quantum Neural Networks [111.438286016951]
  Quantum neural networks (QNNs) exert the power of modern quantum machines. QNNs with handcraft symmetric ansatzes generally experience better trainability than those with asymmetric ansatzes. We propose the effective quantum neural tangent kernel (EQNTK) to quantify the convergence of QNNs towards the global optima.
  arXiv  Detail & Related papers  (2022-08-30T08:17:55Z)
• Quantum Algorithms for Testing Hamiltonian Symmetry [4.62316736194615]
  We show that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance 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  Detail & Related papers  (2022-03-18T15:30:50Z)
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution [52.77024349608834]
  Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. We present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians.
  arXiv  Detail & Related papers  (2021-08-06T19:38:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.