Dual-isometric Projected Entangled Pair States
Xie-Hang Yu J. Ignacio Cirac Pavel Kos Georgios Styliaris





These authors contributed equally.

These authors contributed equally.

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany


Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany


Efficient characterization of higher dimensional many-body physical states presents significant challenges. In this paper, we propose a new class of Project Entangled Pair State (PEPS) that incorporates two isometric conditions. This new class facilitates the efficient calculation of general local observables and certain two-point correlation functions, which have been previously shown to be intractable for general PEPS, or PEPS with only a single isometric constraint. Despite incorporating two isometric conditions, our class preserves the rich physical structure while enhancing the analytical capabilities. It features a large set of tunable parameters, with only a subleading correction compared to that of general PEPS. Furthermore, we analytically demonstrate that this class can encode universal quantum computations and can represent a transition from topological to trivial order.

Dual-isometric Projected Entangled Pair States
    Georgios Styliaris
    April 29, 2024
==============================================




Introduction.—Efficiently representing strongly correlated many-body quantum states and extracting their relevant physical properties remains a pivotal challenge in dimensions higher than one. Tensor network methods offer an approach to address this problem. Following the successful application of 1D matrix product states (MPS) <cit.>,




Ref. <cit.> introduced their higher-dimensional generalization, projected entangled pair states (PEPS). These states satisfy an entanglement area law <cit.> and are considered a robust ansatz for representing ground states of gapped local Hamiltonians <cit.>.

PEPS also play an important role in quantum dynamics <cit.>, statistical mechanics <cit.>, the classification of phases <cit.>, and quantum machine learning <cit.>.

Despite their advantages, finite PEPS still suffer from the rapid growth of the computational resources required for the calculation of physical quantities, such as expectation values of local observables,

limiting the size of the system that can be investigated. In practice, even approximate algorithms on finite PEPS are costly and the errors are often hard to control <cit.>. This limitation also holds for translationally-invariant systems <cit.>. It has been shown that the exact contraction of a general PEPS is #P-hard <cit.>, even for typical instances <cit.>, which serves as a fundamental limitation.


One way to address this challenge is to consider some subclass of PEPS. One such class is the isometric PEPS (iso-PEPS) <cit.>, which extends the canonical form of 1D MPS to higher dimensions. Their isometric nature allows for the preparation of all iso-PEPS through sequential unitary circuits <cit.>. This sequential generation defines a time axis in the iso-PEPS and thus facilitates the backward contraction along this time direction <cit.>. 
Despite being typically short-range correlated <cit.>, the iso-PEPS family can capture states with complex correlation, such as topological models and the associated phase transitions, which admit an exact representation within this class <cit.>.



However, the computation of the local observables for iso-PEPS is not always efficient. For example, an observable in the bulk of the tensor network may correspond to a later time expectation value in the associated sequential-circuit picture, which one expects to be intractable classically.
Indeed, the computation of a local observable in 2D iso-PEPS is shown to be -complete <cit.>, which, subject to standard complexity theory assumptions, indicates that it is hard to simulate classically. 
Although in principle, some classical algorithms, like the Moses move <cit.>, can be used to approximate the result, the errors may remain uncontrolled <cit.>.
This restricts the practical utility of the iso-PEPS and motivates the search for new classes of contractible PEPS.



In this letter, we introduce dual-isometric PEPS (DI-PEPS), a new subclass of iso-PEPS that enhances the computational tractability of tensor networks in higher dimensions, especially  for computing local observables and certain two-point correlation functions. The class is defined by imposing an additional isometric condition onto iso-PEPS, which reduces the calculation of the above quantities in the 2D tensor network to a 1D tensor network. The latter is manageable and can be efficiently computed. More broadly, the dual-isometric condition is analogous to dual-unitary gates <cit.>, a class of exactly solvable quantum circuits, and can be seen as a natural extension of these ideas to PEPS.
Furthermore, our work reveals that the DI-PEPS preserves the rich physical structure of iso-PEPS despite the additional constraint. For this, we show that DI-PEPS can encode universal quantum computations after post-selection. As a result, their output probability cannot be efficiently sampled on a classical computer. 
Additionally, by counting the number of free parameters of
DI-PEPS, we show that this number is only reduced at subleading order as compared to conventional PEPS.
Lastly, we develop a class of DI-PEPS which exhibits topological order and has a nontrivial transition to decoupled 1D chains. These findings underscore the DI-PEPS as a potent framework for investigating quantum many-body physics, offering new avenues for understanding and manipulating complex quantum systems.









PEPS and the folded picture.—We consider PEPS specified by a rank-5 tensor T_lbrt^p for each vertex (x,y) of a 2D square lattice. The index p represents the physical degree of freedom
with dimension d, and l,b,r,t (left, bottom, right, top) label the virtual degrees of
freedom, each χ-dimensional [Fig. (<ref>a)]. Here  x∈{1,⋯,N} and y∈{1,⋯,M}.
The wave function is obtained by locally contracting all the virtual
degrees of freedom [Fig. (<ref>b)]. At the boundary, the virtual space is isometrically mapped to the physical space; we explain this choice later. Thus, there are in total (N+2)×(M+2)-4 physical sites.



Our setting and results become more transparent via vectorizing; in graphical notation, this corresponds to “folding” [Fig. (<ref>c)]. Given a fixed computational basis, an observable is vectorized to ⟨O| via ∑_ij O_ij|i⟩⟨j|↦∑_ij O_ij⟨i|⟨j|. States are similarly mapped to a bipartite vector such that

    ⟨ψ|Ô|ψ|=⟩⟨O|(|ψ^*|⊗⟩|ψ⟩).

Graphically, T^* is folded in front of T, thereby forming T^*⊗ T = [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.6]
00
=
[baseline=([yshift=-0.4ex] current bounding box.center), scale=0.5]
0.350.3500.

Iso-PEPS.—Here we briefly review iso-PEPS <cit.>, which is a subclass of PEPS satisfying the isometric condition 

    ∑_r,t,pT_l_1b_1rt^pT_l_2b_2rt^*p=δ_l_1,l_2δ_b_1,b_2,


shown also graphically in Fig. (<ref>d). This class can be understood as the 2D analog of the canonical form in MPS. Eq. (<ref>) directly implies that iso-PEPS can be contracted starting from the top-right direction, until an observable is met <cit.>.


Physically, every iso-PEPS tensor can be enlarged to a unitary

    U_lb|0⟩^prt=T_lbrt^p

with one of its inputs initialized to |0⟩. 


This unitary gate also sequentially generates the corresponding state <cit.>
in depth 𝒪(N+M). From this point of view, one can imagine that calculating expectation values of local observables in iso-PEPS is as hard as quantum computation; indeed, this task was recently shown to be -complete <cit.>. Note that the isometric property implies that only a local observable  near the bottom/left boundary of the PEPS can be efficiently calculated <cit.>, which corresponds to an operator along the light cone in a sequential quantum circuit. The latter only involves contribution from the boundary gates of a sequential quantum circuit as all of the bulk gates cancel with their hermitian conjugate <cit.>.


Dual-isometric PEPS.—In this letter, we introduce
a new subclass of iso-PEPS called dual-isometric PEPS (DI-PEPS). This is done by requiring another (dual) isometric condition

    ∑_l,t,pT_lb_1r_1t^pT_lb_2r_2t^*p=δ_r_1,r_2δ_b_1,b_2,


as shown in Fig. (<ref>e).

This is in analogy to dual-unitary circuits, where each gate can be interpreted as a valid evolution along a second “time” direction; DI-PEPS admit an additional sequential preparation direction.




The key point is that the dual condition, together with Eq. (<ref>), allows an efficient calculation of expectation values of local observables.
This becomes transparent in the folded picture, where this expectation value is expressed as

    ⟨ψ|O|ψ|=⟩[baseline=(current  bounding  box.center), scale=0.55]
    ı in 0,1,2ȷ in 0,1,2ı+0.5*ȷȷı+0.5*ȷȷ+0.5-1+ı*0.5ı3+ı*0.5ıı-0.5-1ı+1.53-1+ı*0.5ı+0.53+ı*0.5ı+0.5ı-0.5-0.5ı+1.53.5-1.5-12.5-10.534.53ı in 0,1,2,3,40.65+ı3.65-1.85+ı-1.35-2+ı*0.5-0.85+ı3+0.5*ı-0.85+ı-1.5-0.52.5-0.50.53.54.53.522.6.


Starting from the top-left corner, we can use Eq. (<ref>) [Fig. (<ref>e)] to simplify
the configuration as [baseline=(current  bounding  box.center), scale=0.4]
0010-0.5-10.5-100.510.5-0.50.150.150.651.150.65-0.5-0.50.5-0.5-1-0.85
=
[baseline=(current  bounding  box.center), scale=0.4]
10-0.5-10.5-110.51.150.65-0.5-0.50.5-0.5-0.25-0.50.50-1-0.85.
This procedure can be iterated, as now the same equation can be used in the next row and column. Following this simplification procedure, one finds that the dual-isometric condition Eq. (<ref>) simplifies all of the diagram on the left side of the observable while the isometric condition Eq. (<ref>)
simplifies everything on the right side. As a consequence,

    ⟨ψ|O|ψ|=⟩[baseline=(current  bounding  box.center), scale=0.55]
    -0.5-1ı in 0,1,2ı*0.5ıı in -1,0,1,2ı*0.5ı+0.5-0.5+ı*0.5ıı*0.5+0.5ı1.252.5-0.85-1.3512.6.

This is just a 1D tensor network, which
can be efficiently contracted in 𝒪(M). This result also holds for local operators with support larger than 1. In that case,
Eq. (<ref>) includes more than one column but with constant (i.e., system-size independent) width.



Our next point of interest is two-point correlation functions, which play a key role in characterizing many-body properties.

They can be graphically expressed as

    ⟨ψ|O_1O_2|ψ|=⟩[baseline=(current  bounding  box.center), scale=0.55]
    ı in 0,1,2ȷ in 0,1,2ı+0.5*ȷȷı+0.5*ȷȷ+0.5-1+ı*0.5ı3+ı*0.5ıı-0.5-1ı+1.53-1+ı*0.5ı+0.53+ı*0.5ı+0.5ı-0.5-0.5ı+1.53.5-1.5-12.5-10.534.53ı in 0,1,2,3,40.65+ı3.65-1.85+ı-1.35-2+ı*0.5-0.85+ı3+0.5*ı-0.85+ı-1.5-0.52.5-0.50.53.54.53.532.600.6
    .

With the isometric and dual-isometric conditions, one can simplify the diagram from the top-left
and top-right corners similarly as before. The final result is 

    ⟨ψ|O_1O_2|ψ|=⟩[baseline=(current  bounding  box.center), scale=0.55]
    ı in 0,1,2ı0ı+0.512.5+ı*0.5ı+1ı in 0,1,2ı0.5ı+0.51.52.5+ı*0.5ı+1.5ı-0.35-0.352.5+ı*0.5ı-0.50010.751.51.751.52.5233433.753.50.51.63.53.6ı in -0.25,2.25[thick,dashed](ı,-0.75)–(ı,-1.25);
    [x=1,y=-1]|x_1-x_2|[->](0.15,-1)–(-0.17,-1);
    [->](1.85,-1)–(2.17,-1);
    ı in -0.25,1.25[thick, dashed] (3.5,ı)–(4.5,ı);
    [x=4,y=0.5]min{y_1,y_2}[->](4,0.67)–(4,1.17);
    [->](4,0.3)–(4,-0.17);
    .

Here we assume that the two operators
are located at (x_1,y_1) and (x_2,y_2), 
respectively. 
The reduced 2D part of the tensor network is of size t_1× t_2,
with t_1=|x_2-x_1| and t_2=min{y_1,y_2}. 
If one of t_1 or t_2 is small enough, this contraction can be efficiently
calculated using the linear transfer matrix within moderate resources.
Specifically, if either t_1 or t_2 is constant, the correlator still reduces to a
1D tensor network with a possibly increased, but constant, bond dimension. 

We investigate a generalization of the isometric conditions Eqs. (<ref>) and (<ref>) that preserves the solvability of 1- and 2-point correlations in <cit.>.






























Not only in our case, but also in general, the calculation of local or multi-point local expectation values in 2D PEPS can be interpreted
as a circuit of 1+1D completely positive maps acting over the virtual space; in general, however, these are not trace-preserving. 
This is established by interpreting tensor-network diagrams, such as the one in Eq. (<ref>), along a 45^∘ counterclockwise rotated direction, and defining Kraus operators (indexed by p) as E_tr,lb^p=T_lbrt^p.
Pictorially, this can be expressed as 

    [baseline=(current  bounding  box.center), scale=0.55]
    00[x=-0.67,y=0]l[x=0.67,y=0]r[x=-0.33,y=-0.67]b[x=0.33,y=0.7]t00.5
    =
    [baseline=(current  bounding  box.center), scale=0.55]
    00[x=-0.65,y=-0.65]l[x=0.65,y=-0.65]b[x=-0.65,y=0.65]t[x=0.65,y=0.65]r.

where the right side depicts the completely positive map in the folded picture.

For the case of iso-PEPS, this map is in addition trace-preserving, i.e., it corresponds to a valid quantum channel on the folded space. As such, it can be interpreted as a physical evolution.
This evolution is in principle hard to compute, corresponding to the hardness of contracting PEPS in general.
From this point of view, the solvability of DI-PEPS can be connected with the simplifying properties of two-unital space channels from Ref. <cit.>. 

The boundary condition of Fig. (<ref>b) also obtains a physical interpretation at this stage. Viewing the PEPS as a quantum circuit, the boundary conditions correspond to initially preparing N+M EPR-pairs, and inputting half of each pair to the circuit.



Here we also explore an interesting connection between the DI-PEPS and the solvable circuits of quantum channels proposed in <cit.>. We establish the correspondence of a PEPS and a Kraus operator as

    E_rt,lb^p=T_lbrt^p,

namely, the physical index is equated to the Kraus rank and the left, bottom indices are interpreted as input while the right, top indices as output. This effectively models the dynamics of an 1+1D quantum circuit channel in virtual space, following a 45^∘ rotation. From this correspondence, we immediately recognize that the DI-PEPS is just the two-way unital quantum channel of <cit.>. It is also possible to define new classes of PEPS with more isometric conditions corresponding to three-way or four-way unital channels. We leave the property of these states to the future study.



Examples of DI-PEPS.—As mentioned in Eq. (<ref>), DI-PEPS can be prepared with a sequential quantum circuit with the additional condition
∑_ptlU_lb_1|0⟩^pr_1t
(U_lb_2|0⟩^pr_2t)^*=δ_b_1,b_2δ_r_1,r_2, pictorially expressed as
[baseline=(current bounding box.center), scale=0.50]
[very thick] (-0.5,  +0.5) – (+0.75,-0.5);
[very thick] (-0.5,-0.5) – (+0.75,+0.5);
[very thick] (0.125,0.5) – (0.125,-0.5);
[ thick, fill=mygreen, rounded corners=2pt] (-0.25,0.25) rectangle (0.5,-0.25);
[thick] (0.25,+0.15) – (+0.4,+0.15) – (+0.4,0);
[x=1.1,y=-0.55]|0⟩;
-0.50.5-0.5-0.50.750.5
=
[baseline=(current bounding box.center), scale=0.50]
[very thick](0,0.25)–(0,0.75);
[very thick](0,-0.25)–(0,-0.75);
00.250-0.25.
Some classes of unitary gates satisfying the above are:

Permutation-phase gates:
U=P_231D composed with arbitrary single site gates, where P_231 is the shift-permutation P_231=
[baseline=(current bounding box.center), scale=0.50]
[very thick](-0.5,-0.5)–(0,0.5);
[very thick](0,-0.5)–(0.5,0.5);
[very thick](0.5,-0.5)–(-0.5,0.5); 
and D is an arbitrary diagonal gate in the computational basis. This family generalizes 
dual-unitary gates, for which P_21 is just the SWAP <cit.>, to a tripartite setting.



3-qubit gates (d=χ=2):
We take a non-exhaustive ansatz 
U=∏_αexp(iQ_ασ^α_2σ^α_3)∏_βexp(iJ_βσ^β_1σ^β_2),
where α, β run over 1,2,3 and σ_m^α is the α
Pauli matrix acting on the m-th qubit.
We find that Q_1=0, Q_2=π/4, J_3=π/4 or Q_2=π/4, cos2J_1cos2Q_3=cos2J_2cos2Q_3=0
and also  simultaneous exchange of Q_1 and Q_2 as well as J_1  and J_2 all  satisfy the condition.
Moreover, additional arbitrary single site unitary on all five legs but the ancillary one associated with |0⟩ are allowed.


Controlled-dual unitaries:

U_lba^prt=∑_i|i⟩_p⟨i|_aV_tr,lb^i, composed with arbitrary single site gates,
where V^i is a dual-unitary gate <cit.>, i.e., it is unitary and also satisfies ∑_tlV_tr_1,lb_1^iV_tr_2,lb_2^*i=δ_r_1,r_2δ_b_1,b_2.



Next, we discuss another interesting example, which is not constructed from unitary gates, but from the “plumbing tensor” <cit.> which is a special PEPS with D=χ^4.

The local tensor T_lbrt^p is determined by another rank-4 tensor W as 

    T_lbrt^p=T_lbrt^p_1p_2p_3p_4=δ_l,p_1δ_b,p_2δ_r,p_3δ_t,p_4W_lbrt,

where p_i = 1,…,χ.
Graphically,
[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.8]
00
=
[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.55]
[very thick](-0.45,-0.9)–(0.45,0.9);
[very thick](-0.9,0)–(0.9,0);
[ thick, fill=myblue, rounded corners=2pt] (-0.4,+0.4) rectangle (+0.4,-0.4);
[x=0,y=0]W[very thick](0.65,0)–(0.65,0.4);
[very thick](-0.65,0)–(-0.65,0.4);
[very thick](0.325,0.65)–(0.325,1.05);
[very thick](-0.4,-0.8)–(-0.4,-0.4); 
with 

[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.5]
[very thick](-0.5,0)–(0.5,0);
[very thick](0,0)–(0,0.5);
 denoting the delta tensor which is non-vanishing only if all three legs take the same value. The elements of the W matrix are restricted by Eqs. (<ref>) and (<ref>) (see <cit.> for its specific form). Now, if we place the W tensor at the vertices of a square lattice, the physical degrees of freedom lay on the edges[In fact, each edge hosts two spins but they are
just a copy of each other and thus equivalent. One can merge these
two spins into a single one by the transformation ∑_i=1^χ|i⟩⟨ii|.]. Notably, the simplest case χ=2 includes the toric code tensor <cit.> as an example of DI-PEPS. We will return to the topological properties of this class later.







Parameter counting and computational complexity.—Here we argue about the richness of DI-PEPS, despite them having analytically accessible correlation functions.
As with every PEPS, the representation of the state in terms of tensors is nonunique.

We take into account this so-called gauge freedom by introducing a corresponding canonical form for the family of DI-PEPS that satisfies the generalized solvability conditions (see <cit.>). The resulting number of free real parameters of a normal[A PEPS tensor is called normal if it becomes injective after blocking. Generic PEPS are normal <cit.>.]
DI-PEPS (formed by repeating the same tensor) is 2(d-1)χ^4.
Compared to the number of free real parameters 2dχ^4-4χ^2+2
of a normal PEPS <cit.>, we see that DI-PEPS covers a large subclass of normal PEPS.


We further consider the computational complexity of the DI-PEPS. Although one- and two-point correlators can be efficiently computed, we show that sampling of a DI-PEPS cannot be efficiently simulated in a classical computer. To this end, we demonstrate that the DI-PEPS can encode universal quantum computations with post-selection. Thus, according to Ref. <cit.>, the output probability distribution cannot be sampled to a multiplicative precision by a classical computer efficiently, unless the polynomial hierarchy collapses to its third level, i.e., =.
 
While the proof is available in <cit.>, here we sketch the main points.
The idea is to encode dual-unitary quantum circuits in DI-PEPS and

then utilize known results for their complexity <cit.>. Based on the previous discussion, we can interpret the PEPS as a 1+1D quantum circuit in the virtual (bond) space, while at the last layer the boundary bond outputs the computation result to physical space. We choose the DI-PEPS constructed from the controlled-dual-unitary gates (example above), where the physical index acts as the control. For each configuration of the measurement result, the evolution in the virtual space is then governed by a certain dual-unitary brick-wall quantum circuit. Importantly, the resulting dual-unitary circuit can simulate universal quantum computation after post-selection, which corresponds to measuring the physical state at the last layer of the DI-PEPS. The result then follows since, if a quantum state with post-selection can simulate universal quantum computation, the probability distribution cannot be sampled in polynomial time by a classical computer to a multiplicative precision unless = <cit.>.












DI-PEPS and topological states.—In this section, we consider in detail a χ = 2 subfamily of the DI-PEPS constructed from the plumbing tensor of Eq. (<ref>), with a special
focus on its topological properties.

To this end, we
impose a ℤ_2 symmetry on the W tensor 


    [baseline=([yshift=-0.65ex] current bounding box.center),scale=1.0]
    [very thick](-0.5,0)–(0.5,0);
    [very thick](-0.25,-0.5)–(0.25,0.5);
    [thick, fill=myblue, rounded corners=2pt] (-0.2,0.2) rectangle (0.2,-0.2);
    [x=0,y=0]W
    =
    [baseline=([yshift=-0.65ex] current bounding box.center), scale=1.0]
    [very thick](-0.8,0)–(0.8,0);
    [very thick](-0.4,-0.8)–(0.4,0.8);
    [thick, fill=myblue, rounded corners=2pt] (-0.2,0.2) rectangle (0.2,-0.2);
    [thick, fill=myblue] (-0.5,0) circle (0.2);
    [thick, fill=myblue] (0.5,0) circle (0.2);
    [thick, fill=myblue] (-0.25,-0.5) circle (0.2);
    [thick, fill=myblue] (0.25,0.5) circle (0.2);
    [x=-0.5,y=0]σ^3[x=0.5,y=0]σ^3[x=-0.25,y=-0.5]σ^3[x=0.25,y=0.5]σ^3[x=0,y=0]W
    .

This symmetry restricts its form to 

    W_lb,rt=[   √(α)               √(1-α);          √(β) √(1-β);        √(1-α)   √(α); √(1-β)                 √(β) ],

with α,β∈[0,1], where we omit the complex phases of each element since they are irrelevant
to the topological degeneracy.

Our aim is to probe the topological order of the above subfamily. Each PEPS is associated with a local parent Hamiltonian <cit.>. A PEPS is said to be in the topological order if the parent Hamiltonian has topological degeneracy of its ground states, which can be characterized by the transfer operator 𝕋 <cit.>. Parallel to Ref. <cit.>, here we put the PEPS on a cylinder. We cut out one loop around the cylinder and contract it with its complex conjugate over the physical degrees to form the transfer operator in the doubled virtual space,

    𝕋
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    0.6cm[dotted, thick](0,0) ellipse (0.3cm and 1cm);
    [very thick](80:0.3cm and 1cm) arc (80:280:0.3cm and 1cm);
    [very thick](120:0.3cm and 1cm)–(120:0.5cm and 1.35cm);
    [thick, fill=white] (120:0.5cm and 1.35cm) circle (0.08cm); 
    [very thick](120:0.3cm and 1cm)–(120:0.5cm and 1.35cm);
    [thick, fill=white] (120:0.5cm and 1.35cm) circle (0.08cm);
    [very thick](170:0.3cm and 1cm)–(170:0.6cm and 1.35cm);
    [thick, fill=white] (170:0.6cm and 1.35cm) circle (0.08cm);
    [very thick](220:0.3cm and 1cm)–(220:0.6cm and 1.4cm);
    [thick, fill=white] (220:0.6cm and 1.4cm) circle (0.08cm);
     (120:0.3cm and 1cm) ++(-/2,0) – ++(,0);
     (170:0.3cm and 1cm) ++(-/2,0) – ++(,0);
     (220:0.3cm and 1cm) ++(-/2,0) – ++(,0);
    [thick,fill=mygreen] (120:0.3cm and 1cm) circle (0.15);
    [thick,fill=mygreen] (170:0.3cm and 1cm) circle (0.15);
    [thick,fill=mygreen] (220:0.3cm and 1cm) circle (0.15);
     .



According to the ℤ_2 symmetry, the transfer operator can be block diagonalized
into 𝕋_p,ϕ^p',ϕ', where p,p' denote the parity of
the ket/bra state, respectively; ϕ,ϕ' = 0,π denote whether a
flux is threading the cylinder, i.e., if an additional σ^3 is present in the ket/bra virtual space.

Due to the delta tensor in Eq. (<ref>),
we immediately see that p=p' and the transformation ϕ(ϕ')→ϕ(ϕ')+π
does not change the transfer operator. Thus, we only need to distinguish four transfer operators 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ,𝕋_eϕ^eϕ+π,𝕋_oϕ^oϕ+π. 

The key point is that the leading eigenvalues λ_pϕ^pϕ' and the corresponding degeneracy
in each block determine the topological properties of the tensor. This is since they encode, in the thermodynamic limit, the inner product of the states corresponding to different choices of p,ϕ. Note that the latter may abruptly become linearly (in)dependent when tuning the parameters α, β, signaling a transition.

If α=β=1/2, the resulting state is the toric code <cit.>, which has nontrivial topological order. At
this point, |λ_eϕ^eϕ|=|λ_oϕ^oϕ|=1,|λ_eϕ^eϕ+π|=|λ_oϕ^oϕ+ϕ|=0
and the blocks with the largest eigenvalues, 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ are non-degenerate. The lines α=1 or β=1,
and the point α=β=0 are in the trivial phase, since they correspond
to decoupled 1D chains, each in a GHZ state. Those points have exponentially many degeneracies in the leading eigenvalues of the transfer operators 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ in the vertical dimension M and we refer to them as the GHZ-points. 





In the Supplemental Material <cit.>, we moreover analytically show that except for the GHZ-points mentioned above, it always holds that |λ_eϕ^eϕ|=|λ_oϕ^oϕ|>|λ_eϕ^eϕ+π|=|λ_oϕ^oϕ+ϕ| and the blocks with the largest eigenvalues, 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ are non-degenerate. We achieve that by mapping it to a frustration-free Hamiltonian. We also calculate the full spectrum of the transfer operators along the line α=β or α=1-β in <cit.>.
Thus we can conclude that the DI-PEPS of Eq. (<ref>) exhibit the same topological phase as the toric code, except from the GHZ-points. At those points we have a crossing from a topological phase which has degeneracy 4 to a decoupled phase with exponentially many degeneracies. 
This explicitly demonstrates that DI-PEPS contain topological phases. 


Another important property of a PEPS is the locality of the parent Hamiltonian. In principle, the parent Hamiltonian can be obtained by blocking <cit.>, which however may lead to terms with larger support.

For example, if d=χ=2, generally it is sufficient to block 4×5 tensors and the resulting parent Hamiltonian is locally supported on these 20 sites. Numerically, this may be possible to improve by decomposing the resulting parent Hamiltonian into a sum of terms with smaller support, but it is very costly <cit.>.  
Our examples from Eq. (<ref>) with non-vanishing α,β admit an analytical form of the parent Hamiltonians which are at most 8-local. In particular, if α=1-β, this can be improved to a 4-local Hamiltonian. We illustrate the latter here while the general case follows a similar argument <cit.>. The state with α=1-β can be obtained by acting with a single-body operator 𝒰_v=e^sσ^3 on each vertical edge of the toric code, with s=1/4lnα/1-α. The toric code has the well-known parent Hamiltonian H_TC=∑ h_TC with each h_TC 4-local <cit.>. The parent Hamiltonian of our states H is thus connected to that of toric code as H=∑ h with

    h=∏_v𝒰_v^-1† h_TC𝒰_v^-1.

Here v belongs to the vertical edges which overlap with h_TC. The above transformation does not change the locality of h which remains 4.





Discussion and outlook.—
In this letter, we proposed a new class of PEPS called DI-PEPS, with two isometric conditions. This PEPS allows for efficient computational of local and certain two-point correlation functions. Furthermore, we have shown that this class exhibits interesting physical properties. It has a large number of free parameters, encodes universal quantum computation after post-selection and can represent interesting transitions from topological to trivial order.


Let us now discuss possible generalizations.  First of all, we can impose further isometric conditions, from bottom-left and (or) bottom-right.

In this case, higher-point correlation functions are also tractable. One can figure out that for a PEPS with n-isometric conditions, all the (n-1)-point correlation functions are solvable. For example, if we consider a 4-isometric PEPS, all the three-point correlation functions can be reduced to a 1D tensor network. 

Although in our study we focus on the 2D square lattice, the generalization to higher dimensions or different lattices is also interesting. We may consider the 3D simple cubic lattice, or the triangle lattice where one can define up to six isometric conditions.
Another further direction is to go beyond correlation functions and investigate other quantities, such as the entanglement entropy.



Lastly, the equivalence between PEPS and 1+1D quantum dynamical completely-positive maps may guide us to construct new classes of solvable local quantum evolutions for any solvable PEPS. 
As an example, we look at Sequentially Generated States (SGS) <cit.>, which satisfy

    [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.550]
    0000.50.50
    =
    [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.550]
    0000.50.5000.5-0.50[very thick](-1.3,0)–(-0.8,0);
    -0.80 .

We can use our proposed duality to define a new kind of solvable quantum map with the property

    [baseline=(current  bounding  box.center), scale=0.550]
    000.5-0.5
    =
    [baseline=(current  bounding  box.center), scale=0.550]
    000.5-0.5-0.50.5[very thick](-1.3,1.3)–(-0.8,0.8);
    -0.80.8.

This class allows the efficient calculation of local and two-point expectation values, as it can decouple different layers along the diagonal line. A somehow related unitary example was recently proposed in <cit.>. Importantly, this class is solvable even for non-trace preserving dynamics, like for
measurement-induced phase transitions. 

Acknowledgements.—The authors thank Giacomo Giudice, Yu-Jie Liu, Frank Pollmann, and Rahul Trivedi for fruitful discussions.
The research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus. The authors acknowledge funding from the projects FermiQP of the Bildungsministerium für Bildung und Forschung (BMBF).
P. K. acknowledges financial support from the Alexander von
Humboldt Foundation.






















§ APPENDIX




 §.§ Computational complexity

In this subsection, we investigate the computational complexity of DI-PEPS. We will demonstrate that, with post-selection, DI-PEPS can simulate universal quantum computations. Thus the output probability distribution cannot be sampled to a multiplicative precision by a classical computer efficiently unless the Polynomial Hierarchy collapses to its third level, i.e., = <cit.>. We prove this by encoding dual-unitary quantum circuits into the DI-PEPS and referring to known results on their complexity <cit.>.

We consider the following DI-PEPS with the same boundary condition as in Fig. (<ref>b) of the main text

    [baseline=(current  bounding  box.center), scale=0.5]
    002.557.5550ı in 0,1ı+100.5+ı*0.5ı+16.5-ı57-ı*0.54-ı1.5164241.5330405.51623.554.552.5122ı in 0,1,2,32.5+ı3ı in 0,1,23+ı43+ı2ı in 0,13.5+ı1.

Here we use four different kinds of local tensors, denoted by their colors. From the explicit construction shown below, one can verify that all of them satisfy the condition of DI-PEPS. We choose physical dimension d=16 and virtual dimension χ=2. However, locally the tensor can have an effective physical dimension[I.e., the dimension of the image of the tensor, when viewed as a map from virtual to physical space.] lower than 16, denoted as d_e, as only a subspace of the larger physical Hilbert space is relevant.

Orange tensor: It is a special case of the class “controlled-dual unitaries” proposed

in the main text, but with d_e=1, i.e., 

    T_lbrt^p=δ_p,1V_tr,lb.
 Here V is dual unitary <cit.>. This tensor can be interpreted as a 1+1D quantum circuit composed of dual-unitary gates acting over the virtual space from left-bottom to right-top. This tensor is the main ingredient for providing the computational complexity of DI-PEPS.


Light-green tensor: This tensor with d_e=16 is composed of a bipartite physical space,

each with dimension 4, such that 

    T_lbrt^p_1p_2=⟨lt|ϕ^p_1|⟨%s|%s⟩⟩rb|ϕ^p_2.




Here |ϕ^p_1(2)⟩ with p_1(2)∈{1,2,3,4} are the four Bell basis states
|ϕ^1⟩=|00⟩+|11⟩/√(2), |ϕ^2⟩=|00⟩-|11⟩/√(2), |ϕ^3⟩=|01⟩+|10⟩/√(2), |ϕ^4⟩=|01⟩-|10⟩/√(2). The state |ϕ^1⟩ is also called an EPR pair. The light-green tensor serves as the boundary condition of the dual-unitary circuit. For those on the left-top, we post-select the measurement outcome to p_2=1 and on the right-bottom, we post-select the output to p_1=1 such that the boundary condition for the dual-unitary circuit is

    [baseline=(current  bounding  box.center), scale=0.5]
    [very thick](0,2)–(-1,1);
    [very thick](0,0)–(-1,1);
    [very thick](0,0)–(-1,-1);
    [very thick](-1,-1)–(-0.5,-1.5);
    [very thick](0,2)–(-1,3);
    [very thick](-1,3)–(-0.5,3.5);
    001102.

where the blue rectangles are some dual-unitary gates, following the same notation as in <cit.>.
Thus, the dynamic in the virtual space is driven by a dual-unitary quantum circuit. 


Dark-blue tensor: This tensor with d_e=4 is defined as

    T_lbrt^p=U^p_lb⟨rt|ϕ^1|.⟩

Here U is an isometry from the left-bottom virtual spaces to physical space, and |ϕ^1⟩ corresponds to an EPR pair of the right-top virtual space of the dark-blue tensor. The dark-blue tensors together prepare a chain of EPR pairs that serve as the initial state for the dual-unitary circuit dynamics in the virtual space. The dark-blue tensors also effectively decouple the left-bottom space from the right-top space.



Grey tensor: This tensor with d_e=1 is defined as 

    T_lbrt^p=δ_p,1δ_l,rδ_b,t.

This grey tensor is just the SWAP gate in virtual bonds as can be seen in Eq. (<ref>). Pictorially this can be viewed as 

    T^p_lbrt
    =
    [baseline=(current  bounding  box.center), scale=1.0]
    [thick](-0.5,0)–(-0.1,0)–(-0.1,0.1)–(0.1,0.1)–(0.1,0)–(0.5,0);
    [thick](-0.25,-0.5)–(0.25,0.5);
    [x=-0.65,y=0]l[x=0.65,y=0]r[x=-0.33,y=-0.65]b[x=0.33,y=0.65]u[x=0,y=0.65]p[thick, fill=black](-0.1,0.4) circle (2pt);
    
    .

The information of state is thus swapped to the boundary which is finally read out. 


Therefore, with the above choices, the resulting DI-PEPS can be interpreted as a 1+1D dual-unitary quantum circuit (orange PEPS) over the virtual space with a chain of EPR pairs as the initial state (dark-blue PEPS). The output of this circuit is transmitted to the top-right boundary, which is further transmitted to the physical space and “read out” by the last layer of the PEPS. From Ref. <cit.>, a dual-unitary quantum circuit with initial state a chain of EPR pairs can prepare the cluster state, which serves as the well-known measurement-based universal quantum computation <cit.>. Thus, DI-PEPS under post-selection can simulate universal quantum computation. According to Ref. <cit.>, quantum states which can encode postselected quantum computation are unlikely to be classically samplable to multiplicative precision, unless =.



 §.§ Dimension of the DI-PEPS manifold, generalized families, and canonical form


In this subsection, we explore some mathematical properties of DI-PEPS and their generalization. We discuss the dimension of the underlying manifold and a canonical form. The number of free parameters here always refers to the real degrees of freedom.



  §.§.§ Dimension of the DI-PEPS manifold


We first consider the injective PEPS, which is a generic case for d≥χ^4. Recall that a PEPS tensor is called injective if the corresponding map from virtual to physical space is injective <cit.>. In order to calculate the dimension of the manifold, we recall that a folded PEPS, after contracting the physical degrees of freedom, can be viewed as a completely positive map, shown in Eq. (<ref>). The number of free parameters of

completely positive maps corresponding to DI-PEPS is then equal to the one of the two-unital channels in Ref. <cit.>. The latter can be obtained by perturbing around the totally depolarizing channel <cit.>. For completeness, we repeat this analysis here.


We denote the (conveniently normalized) vectorized identity operator as

    |
    [baseline=(0,0.3), scale=0.75]
    [very thick] (0.0,0.5) – (0.5,0.5);
    0.50.5⟩=
    ∑_i|ii⟩   on physical space
    1/√(χ)∑_i|ii⟩   on virtual space,

where χ is the dimension of the virtual space. With this notation, the decomposition of a contracted PEPS over the doubled space can be schematically expressed as


    [baseline=(0,-0.1), scale=0.75]
        0000.5
    =a_∘∘∘∘[baseline=(0,-0.1), scale=0.75]
    [very thick] (0.25,0)–(0.75,0);
    [very thick] (-0.25,0)–(-0.75,0);
    [very thick] (0,0.25)–(0,0.75);
    [very thick] (0,-0.25)–(0,-0.75);
    0.250-0.25000.250-0.25
    +a_∘∘∘ -[baseline=(0,-0.1), scale=0.75]
    [very thick] (0.25,0)–(0.75,0);
    [very thick] (-0.25,0)–(-0.75,0);
    [very thick] (0,0.25)–(0,0.75);
    [very thick] (0,-0.25)–(0,-0.75);
    0.250-0.2500-0.25
    +⋯+a_∘∘–[baseline=(0,-0.1), scale=0.75]
    [very thick] (-0.25,0)–(-0.75,0);
    [very thick] (0,-0.25)–(0,-0.75);
    [very thick] (0,0.75)–(0,0)–(0.75,0);
    -0.2500-0.25
    +⋯+a_—-[baseline=(0,-0.1), scale=0.75]
    [very thick] (0.75,0)–(-0.75,0);
    [very thick] (0,0.75)–(0,-0.75);
    .


Here [baseline=(0,0.3), scale=0.75]
[very thick] (0.0,0.5) – (0.5,0.5);
denotes an element in the space orthogonal to |
[baseline=(0,0.3), scale=0.75]
[very thick] (0.0,0.5) – (0.5,0.5);
0.50.5⟩. This space is thus spanned by (vectorized) traceless Hermitian matrices which have χ^2-1 free parameters.




























The subscript of a follows the usual order left, bottom, right, top. Therefore the DI-PEPS condition amounts to a_∘∘∘∘=1 and a_-∘∘∘=a_–∘∘=a_∘ -∘∘=a_∘∘ -∘=a_∘–∘=0 with no restriction on the other components. Thus the space of quantum channels corresponding to DI-PEPS forms a convex subset in the space of completely positive and trace-preserving maps (i.e., quantum channels) with dimension  χ^8-2χ^4+χ^2.

Here the additional requirement that the corresponding channel should be a completely positive map does not change the above parameter counting <cit.>.


To obtain the number of free parameters of the original PEPS T_lbrt^p, we need to consider the freedom in the physical index p, which corresponds to the Kraus index in the corresponding representation of a quantum channel. Recall that two Kraus representations are equivalent if and only if they are related by an isometry <cit.>. In our case, we can select an (arbitrary) set of Kraus operator, E_tr,lb^i with i ∈{1,⋯,χ^4}, to represent the channel corresponding to the contracted DI-PEPS. The original PEPS is thus expressed as
T_lbrt^p=∑_iV_piE_tr,lb^i. Here V satisfies V^† V=I_χ^4 with 2dχ^4-χ^8 free parameters. Since here we assume the tensor to be injective, E_tr,lb^i are linearly independent and thus different isometries V must lead to different tensors T. The total number of free parameters of injective DI-PEPS is thus

    2dχ^4-χ^8+χ^8-2χ^4+χ^2=2(d-1)χ^4+χ^2.


For non-injective tensors, it is possible that different isometries lead to the same tensor, so the above counting does not apply directly. Nevertheless, we have numerically calculated the dimension of the tangent space of DI-PEPS and found that Eq. (<ref>) holds for generic (non-injective) PEPS for d<χ^4. A notable exception is the permutation-phase gates of the main text [also in Eq. (<ref>)], which have a larger tangent space dimension.

In fact, the following intuitive argument suggests that Eq. (<ref>) serves as a lower bound for the number of free parameters. A general PEPS has 2dχ^4 free parameters. The isometric condition Eq. (<ref>) imposes χ^4 constraints and the dual isometric condition Eq. (<ref>) imposes another χ^4 constraints. However, these constraints are not linearly independent since both Eqs. (<ref>) and (<ref>) require that 
[baseline=(0,-0.05), scale=0.7]
0000.50.50-0.500.250.5
=
[baseline=(0,-0.1), scale=0.7]
[very thick](0,0)–(0,-0.5);
00,
which includes χ^2 equations. Therefore the number of linearly constraint equations is at most 2χ^4-χ^2. The free parameters are thus, at least, 2dχ^4-2χ^4+χ^2.

Note that, so far, we have focused on counting parameters that specify a local tensor and not a physical state. We address this in the following, after first introducing more general isometric conditions.









  §.§.§ Generalized families and canonical form


The constraints of Eqs. (<ref>) and (<ref>) for a PEPS are not the most general ones allowing the efficient calculation of expectation values using our strategy. A more general definition of a solvable PEPS may be written as

    ∑_r,t,r',t',pT_l_1b_1rt^pS_r,r'R_t,t'T_l_2b_2r't'^*p=S_l_1,l_2R_b_1,b_2,


    ∑_l,t,l',t',pT_lb_1r_1t^pB_l,l'R_t,t'T_l'b_2r_2t'^*p=B_r_1,r_2R_b_1,b_2.

Here S,R,B are arbitrary matrices. Graphically, this can be expressed as 

    [baseline=(current  bounding  box.center), scale=0.7]
    000.5000.50.250.5
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick](-0.75,0)–(-0.25,0);
    [very thick](-0.375,-0.75)–(-0.125,-0.25);
    -0.250-0.125-0.25,



    [baseline=(current  bounding  box.center), scale=0.7]
    0000.50.250.5-0.50
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick] (0.25,0)–(0.75,0);
    [very thick] (-0.125,-0.25)–(-0.375,-0.75);
    -0.125-0.250.250.

Here we use [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
00
, 
[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
00
, 
[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
00

to represent the vectorized S,R,B matrices, respectively.

We choose open boundary conditions described by rank-4 tensors (rank-3 at the corners). The requirement for solvability over those boundary tensors will be discussed in the next subsection. Here we only focus on the bulk tensor, which is assumed to be translationally invariant for simplicity, and consider its degrees of freedom.


Although two PEPS tensors may be different, the resulting states may be the same; this is known as gauge freedom <cit.>. We now use this freedom to bring the matrices S,R,B into a canonical form. In the following, we restrict our discussion to normal PEPS <cit.>. For this class, the gauge freedom is well-understood <cit.> (including for open boundary conditions <cit.>).
In particular, two normal tensors represent the same state if and only if they can be connected by a gauge transformation

    [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.8]
    00→[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.8]
    [very thick](-1.0,0)–(1.0,0);
    [very thick](-0.5,-1.0)–(0.5,1.0);
    00[ thick, fill=myblue, rounded corners=2pt] (-0.85,+0.25) rectangle (-0.35,-0.25);
    [x=-0.6,y=0]Q[ thick, fill=myblue, rounded corners=2pt] (0.35,+0.25) rectangle (0.85,-0.25);
    [x=0.6,y=0]Q^-1[thick, fill=myblue, rounded corners=2pt] (0.05,0.85) rectangle (0.55,0.35);
    [x=0.3,y=0.6]J[thick, fill=myblue, rounded corners=2pt] (-0.05,-0.85) rectangle (-0.55,-0.35);
    [x=-0.3,y=-0.6]J^-1,

(see Ref. <cit.>), with Q,J arbitrary invertible matrices. After the gauge transformation, the corresponding eigenvectors S,R,B in Eqs. (<ref>) and (<ref>) will also be changed. In the following, we use this gauge freedom to bring the matrices S,R,B into a canonical form.


A normal PEPS can be blocked into a large rectangle tensor such that it is injective. The blocked region is depicted as

    [baseline=(current  bounding  box.center), scale=0.7]
    1020[x=3,y=0]⋯40[x=2,y=-1]⋮0-21-2[x=2,y=-2]⋯3-2.

The corresponding folded tensor after contracting the physical degrees then defines a completely positive map, when interpreted from top-right to bottom-left. We recognize that

    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick](0,1)–(0,0.5);
    [very thick](1,1)–(1,0.5);
    [very thick](3,1)–(3,0.5);
    011131[x=2,y=0.75]⋯[very thick](3,0)–(3.5,0);
    [very thick](3,-2)–(3.5,-2);
    [x=3.25,y=-1]⋮3.503.5-2

is an eigenvector with the eigenvalue 1. We assume that this map has no eigenvalue larger than 1 for a large enough blocking, which will be justified in the next paragraph.
Since this blocked tensor is injective, we can use a lemma from <cit.> which states that the completely positive map corresponding to an injective tensor is primitive and thus its leading eigenvector must be unique and strictly positive definite. This implies that the matrices S,R are strictly positive definite. Parallel to Ref. <cit.>, we choose J=R^1/2 and Q=S^-1/2 such that for the new tensor T, Eq. (<ref>) reduces to the isometric condition Eq. (<ref>) and Eq. (<ref>) simplifies to a new one with R replaced by the identity matrix and B replaced by B=S^1/2BS^1/2.

Here we argue that the assumption that the map after large enough blocking has no eigenvalue larger than 1 is sensible. To this end, we require without loss of generality that the state is normalized in the thermodynamic limit. Equation (<ref>) is further blocked horizontally and then vertically to form a corner transfer operator <cit.>, i.e.,
the basic tensor of the corner transfer operator is now replaced by the blocked one of Eq. (<ref>). According to Ref. <cit.>, the norm of the state can be approximated by the power of the leading eigenvalue of the corner transfer operator, which suggests that no eigenvalue larger than 1 is allowed as long as the state is normalized in the thermodynamic limit. For the corner transfer operator constructed above, Eq. (<ref>) with extension to infinity is just the eigenvector with eigenvalue 1, which means that it is the leading eigenvector. Since the corner transfer operator is formed by folding an injective tensor [constructed from Eq. (<ref>)], we immediately obtain that S,R are strictly positive definite <cit.> and everything else follows as before.

Now, the only gauge freedom remaining which does not change Eq. (<ref>) is the unitary gauge, i.e., we can choose J,Q to be both unitary. Since the matrix B is positive definite (following a similar argument), this gauge choice can be used to diagonalize it. For simplicity, we assume that the eigenbasis of B is non-degenerate, which is the generic case. Thus, the canonical form of the solvable PEPS can be expressed as (here we use the symbol T to also represent the tensor after the gauge transformations from above were implemented)

    ∑_r,t,pT_l_1b_1rt^pT_l_2b_2rt^*p=δ_l_1,l_2δ_b_1,b_2,


    ∑_l,t,pT_lb_1r_1t^pΛ_l,lT_lb_2r_2t^*p=Λ_r_1,r_1δ_r_1,r_2δ_b_1,b_2

for some diagonal matrix Λ. Or equivalently in graphical notation

    [baseline=(current  bounding  box.center), scale=0.7]
    000.5000.50.250.5
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick](-0.75,0)–(-0.25,0);
    [very thick](-0.375,-0.75)–(-0.125,-0.25);
    -0.250-0.125-0.25,



    [baseline=(current  bounding  box.center), scale=0.7]
    0000.50.250.5-0.50
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick] (0.25,0)–(0.75,0);
    [very thick] (-0.125,-0.25)–(-0.375,-0.75);
    -0.125-0.250.250,

with the white square now the vectorized form of Λ. At this stage, the remaining gauge corresponds to Q being the diagonal phase matrix in the same eigenbasis as Λ,  
and J being any unitary matrix.

The number of free parameters associated with the physical state described by a uniform solvable normal PEPS [i.e., satisfying Eqs. (<ref>) and (<ref>)] can be estimated 
from the free parameters of the canonical form [Eqs. (<ref>) and (<ref>)] minus the remaining gauge. Eq. (<ref>) or (<ref>) with any fixed diagonal matrix Λ allows a similar decomposition as in Eq. (<ref>).  
We just need to replace the right empty bullet [baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
[very thick](0,0)–(0.5,0);
00 in Eq. (<ref>) with baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
[very thick](0,0)–(0.5,0);
[thick, fill=white](0.08,0.08) rectangle (-0.08,-0.08); (vectorized Λ),
and also the left solid line in Eq. (<ref>) with an element in the orthogonal space of Λ. This corresponds to the biorthogonal decomposition in the horizontal direction where the left and right eigenvectors are different. 
For example, the first term would be
[baseline=([yshift=-0.65ex] current bounding box.center), scale=0.7]
[very thick] (0.25,0)–(0.75,0);
[very thick] (-0.25,0)–(-0.75,0);
[very thick] (0,0.25)–(0,0.75);
[very thick] (0,-0.25)–(0,-0.75);
0.250
-0.250
00.25
0-0.25
.
Thus, for any fixed Λ, a generic PEPS satisfying Eqs. (<ref>), (<ref>) has the same number of free parameters as the one satisfying Eqs. (<ref>), (<ref>), but with Λ contributing additional χ-1 free parameters (here we have subtracted the global scaling Λ→ c Λ with c a constant).


Combining all the things together, we arrive at the total number of free parameters associated with the generic physical state (i.e., the one satisfying all previous assumptions) as

    2(d-1)χ^4+χ^2+χ-1-(χ-1+χ^2-1)-1 
    
    =    2(d-1)χ^4.

Here the term in the second parenthesis represents the gauge freedom of J,Q (whose global phase is irrelevant since it cancels with J^-1,Q^-1). The last -1 comes from the fact that two tensors T differing by a phase represent the same physical state.


For comparison, we also derive the number of free parameters of a physical state associated with a generic (i.e., normal) PEPS. For such a PEPS, the rank-5 tensor T has 2dχ^4 free parameters. Now, because of the gauge freedom, J,Q can be chosen as arbitrary invertible matrices. Moreover, the total scaling factor of J,Q is irrelevant since it cancels with J^-1,Q^-1. Thus, each of J,Q has 2χ^2-2 free parameters. A general PEPS is not normalized, which corresponds to subtracting two additional free parameters from T. The number of free parameters is thus 2dχ^4-4χ^2+2.



  §.§.§ Solvable boundary tensors


Lastly, we consider the solvable boundary tensors associated with Eqs. (<ref>) and (<ref>). We explicitly analyze the left-boundary tensor; the rest is understood similarly. Here, we assume that the left boundary is described by a rank-4 tensor L^p_brt
as 

    [baseline=(current  bounding  box.center), scale=0.7]
    00[x=0.65,y=0]r[x=-0.33,y=-0.65]b[x=0.33,y=0.65]t[x=0,y=0.7]p
    =
    L^p_brt.

The boundary condition is solvable if L satisfies 

    [baseline=(current  bounding  box.center), scale=0.7]
    0000.50.250.5
    =
    [baseline=(current  bounding  box.center), scale=0.7]
    [very thick] (0.25,0)–(0.75,0);
    [very thick] (-0.125,-0.25)–(-0.375,-0.75);
    -0.125-0.250.250,

where the green ball depicts the doubled tensor L^p_brt⊗ L^*p_brt.
One can immediately see that the boundary condition shown in Fig. (<ref>b) of the main text is a special case of Eq. (<ref>) with Λ replaced by the identity and L^p_brt=δ_p,rδ_t,b.



 §.§ Analytic expression of the transfer operator


Here we derive analytic expressions for the transfer operators of Eq. (<ref>).



Due to the presence of the delta tensor in the plumbing tensor Eq. (<ref>), folding results in bipartite spaces, but with each part carrying a copy of the other one in the computational basis. That is, contracting two delta tensors

    [baseline=(0,-0.1), scale=0.7]
    [very thick](-0.5,0.5)–(0.5,0.5);
    [very thick](-0.5,-0.5)–(0.5,-0.5);
    [very thick](0,0.5)–(0,-0.5);
    
    =
    |00⟩⟨00|+|11⟩⟨11|,

results in a rank-2 projector.

Thus, for convenience we only work in the subspace of its image and view |00⟩→|0⟩, |11⟩→|1⟩. Therefore, the green tensor in Eq. (<ref>) (PEPS tensor after contracting the physical degrees) corresponds to a different plumbing tensor,

obtained by squaring W element-wise as


    W_lb,rt=[   α         1-α;       β 1-β;     1-α   α; 1-β           β ].

For the latter use, we reorder W as


    W_rt,lb=   [   α         1-β;       β 1-α;     1-β   α; 1-α           β ]
    
    =    I_H⊗[ α;   β ]_V+σ^1_H⊗[     1-β; 1-α ]_V.

Here the last line suggests that we can interpret the tensor W as a joint operator acting on the product of two 2-dimensional spaces. We denote these spaces as horizontal, indexed with H, and vertical, indexed with V. The former corresponds to the indices l,r of the original W tensor and the latter corresponds to the b,t indices. We emphasize that, in the transfer operator given by Eq. (<ref>), each tensor W at different positions i has its own horizontal space. But all tensors share a common vertical space since they are connected vertically. The transfer operator with ϕ'=ϕ thus can be expressed as 

    𝕋_ϕ^ϕ=Tr_V(∏_i^M W_i)

with M the number of sites in the transfer operator. Without loss of generality, we assume M to be even. Now the transfer operator acts on a product of M horizontal spaces.
Substituting Eq. (<ref>) into Eq. (<ref>) we directly see that 𝕋 only consists of Identity and σ^1 operators. Moreover, due to the trace in Eq. (<ref>), 𝕋 must include an even number of σ^1. After some algebraic calculations, we can obtain the expression



    𝕋_ϕ^ϕ=   I(α^M+β^M)+∑_1≤ i<j≤ Mσ^1_iσ^1_j(1-α)(1-β)(β^j-i-1α^M+i-j-1+α^j-i-1β^M+i-j-1)
    
    +   ∑_1≤ i<j<k<n≤ Mσ^1_iσ^1_jσ^1_kσ^1_n(1-α)^2(1-β)^2(β^j-i+n-k-2α^M+i-n+k-j-2+α^j-i+n-k-2β^M+i-n+k-j-2)+⋯
    
    +2   ∏_i=1^Mσ^1_i(1-α)^M/2(1-β)^M/2


where ⋯ denotes terms with a larger (even) number of σ^1 matrices.
Since Eq. (<ref>) is frustration-free and each factor is non-negative, we immediately see that there are two degenerate leading eigenvectors |ψ_+⟩=⊗_i^M|+⟩_i,|ψ_-⟩=⊗_i^M|-⟩_i. Here |+⟩=|0⟩+|1⟩/√(2),|-⟩=|0⟩-|1⟩/√(2) are the eigenvectors of σ^1. The linear combinations of these two vectors |ψ_+⟩+|ψ_-⟩/√(2), |ψ_+⟩-|ψ_-⟩/√(2) have even and odd parity, corresponding to 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ, respectively. Thus, λ_eϕ^eϕ=λ_oϕ^oϕ always holds.

As long as we are not at the GHZ-points, where α=1 or β=1 or α=β=0, each coefficient in Eq. (<ref>) is strictly positive. Any other state not spanned by |ψ_+⟩ and |ψ_-⟩ must violate some terms in Eq. (<ref>) and thus decrease the leading eigenvalue. 
Therefore, the leading eigenvalue of 𝕋 is at most 2-fold degenerate, each corresponding to one of the blocks 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ separately. 

If ϕ'=ϕ+π, the transfer operator is given by

    𝕋_ϕ^ϕ+π=Tr_V(∏_i^MW_iσ^3_V).

After reasoning as above, the expression is 


    𝕋_ϕ^ϕ+π=   I(α^M-β^M)+∑_1≤ i<j≤ Mσ^1_iσ^1_j(1-α)(1-β)(β^j-i-1α^M+i-j-1-α^j-i-1β^M+i-j-1)
    
    +   ∑_1≤ i<j<k<n≤ Mσ^1_iσ^1_jσ^1_kσ^1_n(1-α)^2(1-β)^2(β^j-i+n-k-2α^M+i-n+k-j-2-α^j-i+n-k-2β^M+i-n+k-j-2)+⋯,


which has a similar form as Eq. (<ref>) but with some coefficients changed from positive to negative. Thus, as long as we are not at the GHZ-points, the leading eigenvalue of Eq. (<ref>) must be lower than that of Eq. (<ref>). Since it still has the symmetry ⊗_i^Mσ^3_i, each eigenvalue is at least twice degenerate, one corresponds to 𝕋_eϕ^eϕ+π and the other to 𝕋_oϕ^oϕ+π. Thus, we proved that as long as we are not at the GHZ-points, |λ_eϕ^eϕ|=|λ_oϕ^oϕ|>|λ_eϕ^eϕ+π|=|λ_oϕ^oϕ+ϕ| and the blocks with the largest eigenvalues 𝕋_eϕ^eϕ,𝕋_oϕ^oϕ are nondegenerate. This implies that those states are in the same topological order as the toric code.

We illustrate the above ideas with two examples. 
The first example is β=1-α. In this case, with a straightforward calculation, Eq. (<ref>) can be rewritten as 

    𝕋_ϕ^ϕ=∏_i^M-1(α I+ (1-α)σ^1_iσ^1_i+1)(α I+(1-α)σ^1_Mσ^1_1).

This transfer matrix has the leading eigenvalue 1 for eigenvectors |ψ_+⟩, |ψ_-⟩ as the only leading eigenvectors. The second leading eigenvalue is (1-2α)^2 with the eigenvectors obtained by replacing one |+⟩ (|-⟩) from |ψ_+⟩ (|ψ_-⟩) with |-⟩ (|+⟩). As α→0, the gap closes as 4α. Similarly, Eq. (<ref>) simplifies to 

    𝕋_ϕ^ϕ+π=∏_i^M-1(α I+ (1-α)σ^1_iσ^1_i+1)(α I-(1-α)σ^1_Mσ^1_1).

The absolute value of its leading eigenvalue is |1-2α| which is always smaller than 1 as long as α≠ 0,1.

The second example is β=α where, after straightforward calculations, Eq. (<ref>) can be rewritten as 

    𝕋_ϕ^ϕ=∏_i^M(α I+(1-α)σ^1_i)+∏_i^M(α I-(1-α)σ^1_i).

In the above equation, the sum of the two terms ensures the symmetry ⊗_i^Mσ^3_i thus only terms with an even number of σ^1 can survive in the final result. This transfer operator has the leading eigenvalue 1+(2α-1)^M. The absolute value of the second one is |2α-1|+|2α-1|^M-1. The corresponding eigenvectors are the same as in the first example. As long as α≠ 0,1, the gap does not close. When α→ 0, the gap closes as 4(M-1)α^2. On the other hand, Eq. (<ref>) vanishes 

    𝕋_ϕ^ϕ+π=0.




 §.§ Parent Hamiltonian


In this subsection, we discuss the parent Hamiltonian of the DI-PEPS constructed from the plumbing tensor Eq. (<ref>) with α,β∈ (0,1).  The special point α=β=1/2  corresponds to the toric code |TC⟩, which has the parent Hamiltonian 

    H=∑_v 𝒜_v+∑_p ℬ_p,

where 𝒜_v=I-∏_i∈ vσ^3_i and ℬ_p=I-∏_i∈ pσ^1_i are four-body projectors constructed from the products of Pauli operators around each vertex v and plaquette p, respectively. Recall that the physical spins are at the edges of the lattice. The state associated with general α and β can be obtained from the toric code by the local transformation up to a normalization factor as

    |ψ⟩=∏_v 𝒰_v|TC⟩,

where 𝒰_v=exp{h_bσ^3_b+h_tσ^3_t+h_btσ^3_bσ^3_t} are two-body operators acting vertically on each vertex. Here, the labels b and t represent the bottom and top edge of a vertex and h_b=1/8ln((α
   -1) α/(β -1) β), h_t=1/8log(α  (β -1)/(α -1)
   β), h_bt=1/8log(αβ/(α -1) (β
   -1)). The state |ψ⟩ is therefore annihilated by 


    𝒜_v=∏_v'𝒰_v'𝒜_v∏_v'𝒰_v'^-1=𝒜_v,


    ℬ_p=∏_v'𝒰_v'ℬ_p∏_v'𝒰_v'^-1,

where in the first equation we use the fact that 𝒜 and 𝒰 only consist of σ^3 and thus commute with each other.
The corresponding parent Hamiltonian can be chosen as

    H=∑_v 𝒜_v^†𝒜_v+∑_p ℬ_p^†ℬ_p.

Note that in Eq. (<ref>), only the terms associated with the vertices which sit on the corner of the plaquette p contribute to ℬ_p, as shown below 

    [baseline=(current  bounding  box.center), scale=1.0]
    00.510.50.051.50.951.5-0.05-0.51.05-0.50.500.51-0.50-0.511.501.51-1-0.5-10.5-11.52-0.520.521.5[very thick, color=red] (0,0) rectangle (1,1);
    [very thick, color=blue] (-0.1,-0.8)–(-0.1,1.0);
    [very thick, color=blue] (1.1,-0.8)–(1.1,1.0);
    [very thick, color=blue] (0.1,1.8)–(0.1,0.0);
    [very thick, color=blue] (0.9,1.8)–(0.9,0.0);
    
    
    
    
     .


Here the red line denotes one plaquete operator ℬ_p, and the blue lines denote the four 𝒰_v which contribute in Eq. (<ref>). The crosses represent the physical spins, which are at the edge of the lattice. From this, we can see that ℬ_p is at most an 8-body operator. Since 𝒜_v=𝒜_v is a 4-body operator, the parent Hamiltonian of the example Eq. (<ref>) is at most 8-local. This can be compared to a general “plumbing” PEPS describing a eight-vertex model, where the parent Hamiltonian is 12-local <cit.>.

For the special points where β=1-α, we found that h_bt=0, thus 𝒰_v is decomposed to a product of two single-site operators. As can be seen in Eq. (<ref>), ℬ_p has the same locality as ℬ_p. Therefore, the parent Hamiltonians for those states are at most four-local.




 §.§ Explicit form of W tensor


Here we report the general expression for a  plumbing tensor to be a DI-PEPS. To this end, the explicit form of W tensor can be obtained by directly substituting Eq. (<ref>) to Eqs. (<ref>) and (<ref>). The parametrization of the resulting W tensor can be written as


    W_lb,rt=[  cosαcosθ_1e^iϕ_1  cosαsinθ_1e^iϕ_2  sinαsinθ_2e^iϕ_3  sinαcosθ_2e^iϕ_4;  cosβsinθ_5e^iϕ_9 cosβcosθ_5e^iϕ_10 sinβcosθ_6e^iϕ_11 sinβsinθ_6e^iϕ_12;  sinαsinθ_3e^iϕ_5  sinαcosθ_3e^iϕ_6  cosαcosθ_4e^iϕ_7  cosαsinθ_4e^iϕ_8; sinβcosθ_7e^iϕ_13 sinβsinθ_7e^iϕ_14 cosβsinθ_8e^iϕ_15 cosβcosθ_8e^iϕ_16 ].


with all the angle variables take values in [0,2π). Note that the toric code corresponds to a special choice of
the parameters θ_i=ϕ_j=0 for all i∈{1,⋯,8},j∈{1,⋯,16}
and α=β=π/4.





 §.§ ***

In the (semi)infinite system, one may equivalently consider the corner
transfer operator. In that case, the same result can be obtained. I dont understand what this means.
For a dual-ISO PEPS, the tensor W has a specific form


    W=[  cosαcosθ_1e^iϕ_1  cosαsinθ_1e^iϕ_2  sinαsinθ_2e^iϕ_3  sinαcosθ_2e^iϕ_4;  cosβsinθ_5e^iϕ_9 cosβcosθ_5e^iϕ_10 sinβcosθ_6e^iϕ_11 sinβsinθ_6e^iϕ_12;  sinαsinθ_3e^iϕ_5  sinαcosθ_3e^iϕ_6  cosαcosθ_4e^iϕ_7  cosαsinθ_4e^iϕ_8; sinβcosθ_7e^iϕ_13 sinβsinθ_7e^iϕ_14 cosβsinθ_8e^iϕ_15 cosβcosθ_8e^iϕ_16 ].

Notice that the toric code just corresponds to a special choice of
the parameter where θ_i=ϕ_j=0 for all i∈{1,⋯,8},j∈{1,⋯,16}
and α=β=π/4. We will discuss the property of
this class detailedly in the following chapter.
Canonical Form and Solvable Boundary Conditions—Actually,
Eq. (<ref>) is not the most general form. A naive
observation is that a gauge transformation can not change the physical
property of the state but will violate Eqs. (<ref>) and
(<ref>). We answer this problem by considering a
more general form of dual-ISO-PEPS. Here we focus our discussion on
the normal PEPS. Analogy to Eqs. (<ref>) and (<ref>),
one can consider the conditions 

    ∑_r,u,r',u'T_l_1d_1ru^pS_r,r'R_u,u'T_l_2d_2r'u'^p=S_l_1,l_2R_d_1,d_2,


    ∑_l,u,l',u'T_ld_1r_1u^pB_l,l'R_u,u'T_ld_2r_2u'^p=B_r_1,r_2R_d_1,d_2.

Here S,R,B are arbitrary matrices. Eqs. (<ref>) and
(<ref>) are just the special cases where these three
matrices are Identity. Eqs. (<ref>) and (<ref>)
states that the leading eigenvectors from right-up and left-up are
factorized. With the similar technique we employed before, one can
verify the calculation of the local observable or two-point correlators
under suitable boundary conditions can also be reduced to the same
form as the dual-ISO-PEPS. Eqs. (<ref>) and (<ref>)
are the most general form one can assume for a solvable PEPS. In the
following, we propose a canonical form of the solvable PEPS and show
that each PEPS satisfying Eqs. (<ref>) and (<ref>)
is gauge equivalent to these canonical form. 

For a normal PEPS, following the proof in <cit.>,
one can show that, in the thermodynamic limit, the matrix S,R can
be chosen as positive definite. Thus, we can multiple the PEPS T
by S^-1/2, S^1/2 from the left and right,
while R^-1/2, R^1/2 from the bottom and
up. The new PEPS T then satisfy Eq. (<ref>).
Eq. (<ref>) is then transformed into a new one

    ∑_l,l',uT_ld_1r_1u^pB_l,l'T_ld_2r_2u^p=B_r_1,r_2δ_d_1,d_2.

Here B is also a positive definite matrix which has the
decomposition B=UΛ U^† where U is a unitary
matrix and Λ is diagonal. We can multiply tensor T
by U and U^† from left and right, respectively such
that the new tensor T' satisfies 

    ∑_r,uT'_l_1b_1ru^p(T'_l_2b_2ru^p)^*=δ_l_1l_2δ_b_1b_2


    ∑_l,uT'_lb_1r_1u^pΛ_l,lT'_lb_2r_2u^p=Λ_r_1,r_1δ_r_1,r_2δ_b_1,b_2

We call these two equations the canonical form of the dual-ISO-PEPS.
The above argument shows that each normal solvable PEPS satisfying
Eqs. (<ref>) and (<ref>)
is gauge equivalent to the canonical form.

Using a similar decomposition as in <cit.>,
we can calculate the dimensions of the dual-ISO-PEPS. Here, we recognize
the two PEPS which is gauge equivalently as the same one since they
give the same physical state. We focus on the normal PEPS. Since the
solvable normal PEPS all gauge equivalently to the canonical form,
we directly calculate the dimensions of the canonical form. For a
normal PEPS, the gauge reminiscent for the canonical form is the unitary
transformation in the vertical direction and the diagonal phase matrix
(the same basis as Λ) in the horizontal directions, which
equals to χ-1+χ^2, where -1 comes from the total phase.
Because Eq. (<ref>) is scale invariant for the diagonal
matrix Λ, we can recognize that the degrees of freedoms from
Λ is χ-1. Thus, the total degrees of freedom is 

    2(D-1)χ^4+χ^2+χ-1-(χ-1+χ^2)
    
    =     2(D-1)χ^4.


For Eqs. (<ref>) and (<ref>), the other
boundary conditions can be chosen the same as before except the left
boundary conditions, where we can think that the boundary is given
by a four-rank tensor L_dru^p, i.e., without
the leg pointing to the left. As long as this tensor satisfies the
condition 

    ∑_p,uL_dru^pL_d'r'u^p=δ_d,d'δ_r,r'Λ_r,r,

the local expectation value and correlators can be simplified the
same way as before. One immediately realized that the previous boundary
condition is just a special case of Eq. (<ref>).

Here we emphasize that it is possible to consider a more complicated
boundary condition such as 

    ∑_p,u,u'L_dru^pM_u,u'L_d'r'u'^p=M_d,d'δ_r,r'Λ_r,r.

However, for a normal tensor, we can show that it is equivalent to
Eq. (<ref>) in the thermal dynamic
limit following the logic of <cit.>.


Computational Complexity— In this section, we consider
the computational complexity of the dual-ISO-PEPS. Specifically, we
consider two problems, sampling and calculating local expectation
values. This result is based on the equivalence between the dual-ISO-PEPS
and the 1+1D quantum evolution. We will show that our dual-ISO-PEPS
can simulate a universal quantum computation after carefully choosing
the measurement results, which equivalent to the post-selection of
a quantum circuit. According to the complexity Hierarchy, a circuit
with post-selection can simulate a universal quantum computation implies
the sampling is hard, we conclude that our dual-ISO-PEPS is also sampling
hard. Our setting is a non-uniform PEPS with each tensor satisfies
the dual-ISO property. The boundary condition is the same as before
that we transmit the virtual bound state to the physical degrees of
freedom at the boundary. The virtual space can be thought as a 1+1D
quantum circuit. Since the controlled-Dual-unitary circuit is one
example of the dual-ISO-PEPS, we can choose that the dynamics of the
virtual space is constructed from the dual-unitary circuits. At the
boundary of the 1+1D quantum circuit, we choose a tensor as |0⟩_p⟨ϕ^+|_v+|1⟩_p⟨ϕ^-|_v+|2⟩_p⟨φ^+|_v+|3⟩_p⟨φ^+|_v
where |ϕ^±⟩,|φ^±⟩are the Bell states.
After we measure this tensor, we post-select the results of getting
|0⟩, thus the information can remain in the 1+1D quantum circuit
and become a closed system. According to <cit.>,
this dual-unitary circuits can prepare a cluster state. A cluster
state is able to simulate the universal quantum computation. Thus,
we claim our dual-ISO-PEPS is hardness in sampling.

We can also discuss the computational hardness when calculating expectation
values. From the previous discussion, calculation of one local observable
or the correlator is classically easy. However, since the post-selection
makes the PEPS a universal quantum computation, the calculation
of the conditional local expectation value in worst-case will be BQP-complete.
Alternatively, the post-selection can be thought as measure the projection
operator |0⟩⟨0|. These projection operators lie along the
boundary of the 1+1D quantum circuits and form a 1-dimension submanifold
in the lattice space. We thus conclude that the expectation value
for an 1-dimensional product of local operators in worst case will
be BQP-complete. 


We can put most of the details from here to SM (things about gauge freedom...) to save space, and give only the results here?
can skip this sentence  later
Here we calculate the number of free parameters in this class. 
This can be done by choosing an orthogonal basis, including the Identity operator, of the folded tensor, a decomposition parallel to <cit.>. Not clear even to me, let alon general reader.
However, here we have to take the gauge freedom of a PEPS into account. For a normal PEPS, the fundamental theoremadd citation states that if and only if two PEPS are connected by a gauge transformation
letters in the diagrams too big

    [baseline=(current  bounding  box.center), scale=1.0]
    00→[baseline=(current  bounding  box.center), scale=1.0]
    [very thick](-1.0,0)–(1.0,0);
    [very thick](-0.5,-1.0)–(0.5,1.0);
    00[ thick, fill=myblue, rounded corners=2pt] (-0.85,+0.25) rectangle (-0.35,-0.25);
    [x=-0.6,y=0]Q[ thick, fill=myblue, rounded corners=2pt] (0.35,+0.25) rectangle (0.85,-0.25);
    [x=0.6,y=0]Q^-1[thick, fill=myblue, rounded corners=2pt] (0.05,0.85) rectangle (0.55,0.35);
    [x=0.3,y=0.6]R[thick, fill=myblue, rounded corners=2pt] (-0.05,-0.85) rectangle (-0.55,-0.35);
    [x=-0.3,y=-0.6]R^-1,

they describe the same physical state. Here, R and Q are two arbitrary invertible matrices. In fact, the gauge transformation will modify the (dual-)isometric conditions (<ref>) and (<ref>). In <cit.>, we give a definition of the DI-PEPS which is independent of the gauge used and we also propose a corresponding canonical form. 

