9 research outputs found
Quantum Algorithms for Testing Hamiltonian Symmetry
Symmetries in a Hamiltonian play an important role in quantum physics because
they correspond directly with conserved quantities of the related system. In
this paper, 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.Comment: 11 pages, 3 figures Comments welcome! Revision comment: New results
adde
Testing symmetry on quantum computers
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.Comment: v3: 51 pages, 41 figures, 31 tables, final version accepted for
publication in Quantum Journa
Linear optical quantum metrology with single photons: Experimental errors, resource counting, and quantum Cramér-Rao bounds
A Menagerie of Symmetry Testing Quantum Algorithms
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