Scaling of symmetry-restricted quantum circuits

Abstract

The intrinsic symmetries of physical systems have been employed to reduce the number of degrees of freedom of systems, thereby simplifying computations. In this work, we investigate the properties of $\mathcal{M}SU(2^N)$, $\mathcal{M}$-invariant subspaces of the special unitary Lie group $SU(2^N)$ acting on $N$ qubits, for some $\mathcal{M}\subseteq M_{2^N}(\mathbb{C})$. We demonstrate that for certain choices of $\mathcal{M}$, the subset $\mathcal{M}SU(2^N)$ inherits many topological and group properties from $SU(2^N)$. We then present a combinatorial method for computing the dimension of such subspaces when $\mathcal{M}$ is a representation of a permutation group acting on qubits $(GSU(2^N))$, or a Hamiltonian $(H^{(N)}SU(2^N))$. The Kronecker product of $\mathfrak{su}(2)$ matrices is employed to construct the Lie algebras associated with different permutation-invariant groups $GSU(2^N)$. Numerical results on the number of dimensions support the the developed theory.

37 pages, 6 figures, 1 table. Submitted to IOP Physica Scripta

References in corpus (21)

• Quantum Machine Learning
• Variational Quantum Algorithms
• Barren plateaus in quantum neural network training landscapes
• Measurement-based quantum computation with cluster states
• Circuit-centric quantum classifiers
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Quantum control theory and applications: A survey
• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
• Bloch vectors for qudits
• Effect of barren plateaus on gradient-free optimization
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Exploiting symmetry in variational quantum machine learning
• Group-Invariant Quantum Machine Learning
• Entanglement and permutational symmetry
• Theory for Equivariant Quantum Neural Networks
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• Entanglement and symmetry in permutation symmetric states
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension
• Thermodynamics of Permutation-Invariant Quantum Many-Body Systems: A Group-Theoretical Framework
• Permutation-invariant quantum circuits

Cited by in corpus (1)

• Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras