We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.

arXiv:2208.11567Aug 24, 20220 citations

Symmetries in Quantum Physics

Elsevier eBooksJan 1, 199636 citations

Quantum symmetry vs nonlocal symmetry

David E. Roberson, Simon Schmidt

We introduce the notion of nonlocal symmetry of a graph $G$ , defined as a winning quantum correlation for the $G$ -automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on $C(\text{Qut}(G))$ -- the algebra of functions on the quantum automorphism group of $G$ . This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of $C(\text{Qut}(G))$ . Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, $S_4^+$ , does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group $Γ$ and a permutation $π$ on $|Γ|$ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations $π$ result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when $\mathbb{Z}_2^3$ is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.We introduce the notion of nonlocal symmetry of a graph $G$, defined as a winning quantum correlation for the $G$-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on $C(\text{Qut}(G))$ -- the algebra of functions on the quantum automorphism group of $G$. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of $C(\text{Qut}(G))$. Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, $S_4^+$, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group $Γ$ and a permutation $π$ on $|Γ|$ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations $π$ result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when $\mathbb{Z}_2^3$ is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.

arXiv:2012.13328Dec 24, 20206 citations

Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos

Kohei Kawabata, Anish Kulkarni, Jiachen Li +2 more

We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

arXiv:2212.00605PRX QuantumAug 29, 202373 citations

Symmetries in quantum physics

Michel Le Bellac

Cambridge University Press eBooksMar 23, 20061 citation

We introduce the notion of nonlocal symmetry of a graph

G

, defined as a winning quantum correlation for the

G

-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on

C(\text{Qut}(G))

-- the algebra of functions on the quantum automorphism group of

G

. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of

C(\text{Qut}(G))

. Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points,

S_4^+

, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group

Γ

and a permutation

π

|Γ|

elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations

π

result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when

\mathbb{Z}_2^3

is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.