Inner products of two iPEPS states

[zxm403089989]

I noticed that PEPSKit.jl uses expectation_value to compute expectation values for two iPEPSs, which involves reduced_densitymatrix to calculate the density matrix. However, the library doesn’t seem to provide an interface for computing ⟨bra|ket⟩. If bra and ket are two different many-body quantum states and their environments may also differ, is it possible to compute their inner product?

[pbrehmer]

This is indeed possible but not yet well documented. Internally, when you call leading_boundary to contract a PEPS, what you really contract is an InfiniteSquareNetwork. Essentially, that's a type which stores a square lattice of tensors which you can contract using algorithms such as CTMRG and boundary MPS. In order to contract an inner product $\langle \text{bra}| \text{ket} \rangle$, you need to construct such an InfiniteSquareNetwork where the PEPS double layer is comprised of two different PEPS. Here's a MWE:

using TensorKit, PEPSKit

ket = InfinitePEPS(ℂ^2, ℂ^2)
bra = InfinitePEPS(ℂ^2, ℂ^2)
network = InfiniteSquareNetwork(ket, bra)

env₀ = CTMRGEnv(ket, ℂ^10)
env, = leading_boundary(env₀, network; tol = 1.0e-6, alg = :sequential);

Note that, since these are two random PEPS, CTMRG might not converge properly in many cases.

[zxm403089989]

Thanks for response. Is it possible that bra and ket correspond to 2 different environment tensors? Could we contract them?

[pbrehmer]

In standard CTMRG/boundary MPS variants, the bra and ket PEPS tensors connect to the same environment tensors. However, there have always been ideas floating around in the community of splitting up the environment into bra and ket environment tensors. One such way of doing that has been dubbed the split CTMRG approach, which was only published this year (this is not yet implemented in PEPSKit).

[Yue-Zhengyuan]

Is the current implementation able to get rid of the effect of normalizations?

[lkdvos]

The normalization of the environments would be fixed since by convention we have the tensors at norm 1, so in order to compute fidelities you still have to compute the norms of the individual states and divide that out too, with separate environments for each