Tensor network methods for invariant theory

Abstract

Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state factorization) when studying polynomial invariants. As our main example, we consider invariants of matrix product states. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Rényi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as matrix product states, contain excellent tools that can be applied in the study of invariants.

Invariant perfect tensors

“…In particular, the perfect tensor network provides an interesting illustration of how the Ryu-Takayanagi formula of holographic entanglement entropy (HEE) 12 emerges from a many body quantum system. It is also proposed by Biamonte et al that a tensor network can be used in the study of polynomial invariants of a quantum state [22].…”

Charged string tensor networks

“…t k [13] |t k := |0 + i k |1 lifts to a unitary operator where the Clifford gate S is recovered for k = 1-thereby establishing (iv). For k = 2 we recover the standard Pauli Z matrix, then HZH = X and SXS 3 = Y .…”

“…[(a) associativity; (b) gate unit laws; (c) symmetry; (e) copy laws; (f) unit scalar given as a blank on the page.] of condensed matter theory and quantum computation-by myself with several colleagues (11)(12)(13). We adapted these tools and discovered efficient tensor network descriptions of finite Abelian lattice gauge theories (14).…”

Multiphoton states related via linear optics

“…A theorem by Hilbert states that, for a compact group G acting in a unitary fashion on a finite dimensional vector space, there exists a finite number of independent invariants (which are polynomial in the coordinates of |ψ ) that are able to distinguish whether two vectors belong to the same orbit of G [21][22][23]. A convenient way to write down the invariants involves using tensor diagrams [24] -they make it explicit why certain polynomials are invariant and allow us to avoid multiple index contractions. Thus, the LU-equivalence problem can be solved completely once the minimal set of independent polynomial invariants is known.…”