Abstract
Qubit regularization is a procedure to regularize the infinite dimensional local Hilbert space of bosonic fields to a finite dimensional one, which is a crucial step when trying to simulate lattice quantum field theories on a quantum computer. When the qubit-regularized lattice quantum fields preserve important symmetries of the original theory, qubit regularization naturally enforces certain algebraic structures on these quantum fields. We introduce the concept of qubit embedding algebras (QEAs) to characterize this algebraic structure associated with a qubit regularization scheme. We show a systematic procedure to derive QEAs for the O(N) lattice spin models and the SU(N) lattice gauge theories. While some of the QEAs we find were discovered earlier in the context of the D-theory approach, our method shows that QEAs are far richer. A more complete understanding of the QEAs could be helpful in recovering the fixed points of the desired quantum field theories.
Highlights
A quantum critical point in the lattice theory provides a way to remove the lattice artifacts and define the continuum quantum field theory. This is due to the fact that the long distance physics of the model in the vicinity of the correct quantum critical point are described by the renormalization group (RG) fixed point that describes the quantum field theory
We develop a systematic way to derive the qubit embedding algebras (QEAs) starting from the Hilbert space of the traditional theory, which can be viewed as a direct sum of many irreducible representations of the symmetry group [55–58]
Using examples from spin models and gauge theories, in this work, we showed that qubit regularizations are characterized by an algebraic structure referred to as the qubit emdedding algebra (QEA)
Summary
Recent development in quantum technologies is making the dream of quantum computing into a realistic and exciting possibility [1]. In bosonic quantum field theories, there is yet another type of infinity, i.e., the infinite dimensional Hilbert space at every local lattice site. All lattice quantum spin models that have been studied in condensed matter physics over the years can be considered as examples of qubit regularized models due to their finite dimensional local Hilbert space. We define each qubit regularization scheme as a projection operator PQ, which projects the original infinite dimensional space to some finite set of irreps Q This projection naturally preserves important algebra related to the symmetries of the original theory. D-theory suggests that one can recover many traditional lattice theories with infinite dimensional Hilbert spaces within acceptable errors, by using simple qubit regularized models.
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