How should we regularize this Hilbert space systematically? It would be nice if the group manifold could be discretized by a discrete subgroup, but this does not seem to work except for U(1) theory, where ZN gives an efficient discretization.24
A physically elegant, but practically very hard, approach is to truncate in the loop basis. We start with the ‘strong coupling limit,’ where ˆHB is dropped. The ground state in this limit is given by ˆEµ,~αn|0i = 0. By acting on this state with Wilson loop operators WˆC, obtained by multiplying the link variables along a closed contour C and taking their trace, for various contours, an over-complete basis of the gauge-invariant Hilbert space is obtained. The magnetic term is regarded as the smallest Wilson loop, i.e., the plaquette.
When the loops do not intersect with each other or with themselves, the electric term is proportional to the sum of the lengths of the loops. When the loops intersect, the electric term joins or splits them. By identifying the Wilson loop with the string, this gives an alternative picture to the unitary link variables. A natural cutoff can be introduced by restricting the total length of a string, however, there is no known way to write down an orthonormal basis for the physical states systematically.
For quantum computation, probably the most natural option is to use the Peter-Weyl theorem, which gives the “Fourier expansion” on group manifolds:
Peter-Weyl theorem. For a compact groupG, an orthonormal basis ofL2(G)is given by the matrix coefficients of the unitary, finite-dimensional irreducible representations (irreps) of G, ρ(R)ij , where R runs through all irreps, and i, j = 1,2,· · ·,dimR.
In particular, f ∈ L2(G) can be written as f(g) = PR
PdimR
i,j=1c(R)ij ρ(R)ij (g) for g ∈ G. There are two canonical orthonormal bases on L2(G); in bra-ket notation, {|gi} is the
‘coordinate’ basis of group elements g, and {|R, iji}is the ‘momentum’ basis provided by the Peter-Weyl theorem. We refer to hg|R, iji=ρ(R)ij (g) as a ‘Fourier mode’ on the group G. Accordingly, the Hilbert space can be expressed as
H=⊗µ,~nHµ,~n=⊗µ,~n⊕R⊕dimRi,j=1 |R, ijiµ,~n. (B.19) Note that i and j in |R, ijiµ,~n are transformed by gauge transformations at sites ~n and
~
n+ ˆµ. A natural cutoff is introduced by restricting to a subset of the representations which comprise the Hilbert space [78].
When acting on |R, iji, the electric term ˆHE is proportional to the quadratic Casimir operator,
HˆE|R, iji ∝χ2(R)|R, iji. (B.20)
24See ref. [77] for attempts to use a large discrete subgroup of SU(3).
JHEP09(2021)034
The action of the magnetic term ˆHB is more complicated:
Uˆpq|R, iji=Z
G
dg X
R0,i0,j0
|R0, i0j0ihR0, i0j0|Uˆpq|gihg|R, iji
= X
R0,i0,j0
|R0, i0j0i Z
G
dgρ(k)pq (g)ρ(R)ij (g)ρ(Ri0j00)(g)∗
= X
R0,i0,j0
CR0i0j0;Rij;k,pq|R0, i0j0i, (B.21) so that
hR0, i0j0|Uˆpq|R, iji=CR0i0j0;Rij;k,pq, (B.22) whereCR0i0j0;Rij;k,pq≡ hR0, i0j0| ·(|R, iji ⊗ |k, pqi) is the generalized version of the Clebsch-Gordan coefficient and ‘k’ means that ρ(k)pq (g) is a k ×k matrix, i.e. the fundamental representation of U(k).
As a natural way to regularize the Hilbert space, we can introduce a cutoff for the dimension of the representations. This can be interpreted as the momentum cutoff on the group manifold.
Preparation of ground state. Let us define ˆH(s) as
Hˆ(s) = ˆHE+sHˆB, (B.23) where 0≤s≤1. Ats= 0, ˆH(s= 0) = ˆHE is the ‘strong coupling limit’; the ground state is given by a vanishing electric field, i.e. the trivial representation, on every link. Starting from this trivial limit, we can apply the adiabatic state preparation method to prepare the ground state of ˆHE+ ˆHB. As with the orbifold construction, it is important to look at the gap as a function of sin the extended Hilbert space containing the gauge-non-singlet modes to fully understand the complexity of the adiabatic state preparation procedure.
C Regularization in the coordinate basis
In this appendix, we introduce a regularization in the coordinate basis [4–6]. Let {|xi}be the coordinate basis for a particle in flat space, which satisfies
ˆ
x|xi=x|xi. (C.1)
The simplest way to regularize it is to introduce the cutoff to the value of x as
−R≤x≤R, (C.2)
and introduce Λ lattice points,
xn=−R+nδx, δx= 2R
Λ−1, n= 0,1,· · · ,Λ−1 (C.3)
JHEP09(2021)034
The regularization parameters Λ, δx and R should be sent to infinity, zero and infinity, respectively. Roughly speaking, δx and R correspond to µ and ω in the regularization scheme introduced in section 3.1. By using|ni to denote |xni, we can write
ˆ x=
Λ−1
X
n=0
xn|ni hn|. (C.4)
By using the binary decomposition as in section3.1, we can rewrite it to a sum of the Pauli strings.
The momentum operator ˆp appears in the Hamiltonian only in the form of ˆp2; a convenient way of regularizing it is
ˆ p2= 1
δ2X
Λ−1
X
n=0
{2|ni hn| − |n+ 1i hn| − |ni hn+ 1|}. (C.5) This form can be understood as follows. Above, we introduced the cutoff for the value of x, but we could use the periodic boundary condition |Λi=|0i as well, assuming that the states close to the cutoff do not give non-negligible contributions. In this case, the ‘shift operator’ ˆS≡Pn|n+ 1i hn|is identified witheiδXpˆ, and hence, ˆp= Sˆ1/2iδ−XSˆ−1/2, up to the corrections of orderδX. From this, ˆp2= 2 ˆI−S−ˆδ2Sˆ−1
X follows. This is the same as (C.5) up to the boundary condition. Again, it is straightforward to write the right-hand side of (C.5) as a sum of Pauli strings, upon which efficient simulation algorithms may be applied.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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