Unitary designs emerge from the temporal ensemble of two chaotic Hamiltonian evolutions separated by a random Pauli operation, based on the universal Pauli spectrum.
Evenly distributed unitaries: on the structure of unitary designs
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abstract
We clarify the mathematical structure underlying unitary $t$-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any $t$-th order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.
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Numerical study of five symmetry-preserving HVAs for Z2 gauge theory finds overparametrization eliminates local minima and loss decay rate scales linearly with number of parameters.
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Unitary Designs from Two Chaotic Hamiltonians and a Random Pauli Operation
Unitary designs emerge from the temporal ensemble of two chaotic Hamiltonian evolutions separated by a random Pauli operation, based on the universal Pauli spectrum.
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Symmetries and overparametrization properties of Hamiltonian variational ansatzes for the $(1+1)$d $\mathbb{Z}_2$ lattice gauge theory
Numerical study of five symmetry-preserving HVAs for Z2 gauge theory finds overparametrization eliminates local minima and loss decay rate scales linearly with number of parameters.