Here I want to review some notions of Tensor Network (TN) and a special class of TN called multi-scale entanglement renormalization ansatz (MERA) in order to come up with an outstanding result – computing the geometry of ‘s dual that is .
Tensor Network (TN) [Ref. 1]
Consider an arbitrary separable Hilbert space and its dual . Assume is multi-linear. With respect to a complete basis for and , i.e. and , we can represent as: (omitting the tensor product symbol)
Thus, we can assign to with respect to the basis. Graphically, we can show a tensor with a box, a triangle or an arbitrary symbol, and to show each of its upper (lower) free indices with a free arm on from the symbol going to the upper (lower) half-plane. For example, look at the below figure which shows .
- If two or more tensors are represented on a diagram, we call it a Tensor Network.
- Each tensor in a TN can be connected through their free arm(s) to other tensors. By doing this, we are just tracing over for some .
- Each disconnected components of a TN would be tensor multiplied to each other.
- For a qubit system, the Levi-Civita tensor is .
- A quantum circuit can be though of as sub-class of TN. Consider the state and assign a quantum circuit to it using a finite family of quantum gates, i.e. (see this). It can be seen as a TN since on each gate, a local unitary operator would act and this act can be represented as a tensor contraction in a TN. At the end of the day, you have several tensors with free indices and some contraction on gates. So by this observation, we can assign a complexity to a TN: the complexity of a TN is the number of tensors (gates, inputs, and outputs) in the TN. In this sense, the complexity of the state is the minimum of complexity of TNs that would implement the .
Tensor Renormalization Group (TRG) [Ref. 2]
Consider a lattice that has several sites each labeled by . We want to compute the partition function of this system. Assume the classical Hamiltonian is local in the sense that consists of interactions between neighborhoods. If it is the case, write where labels different “sites” in the lattice and we would like to assume each has the dimension in the sense that it has possibilities. Our goal is to compute over all possibilities. Let me write . Thus, . Since we may write and can only depend on boundary variables since all inner degrees of freedom are traced out, we can associate to site an operator which there are some traces behind these ‘s that would end in an expression for the partition function. Now assume . In the Ref. 2 it is discussed the can be viewed as an operator from a dimensional vector space to itself with an error like . Thus, we may associate to a tensor where each varies from to . In fact, we have just selected the first eigenstates of the density matrix in the site . To summarize, the partition function is nothing but a fully contracted TN.
So assume we have a fully contracted TN in a specific lattice, honeycomb lattice. By the symmetry the tensor on each site is a symmetric tensor. It is argued in Ref. 2 that we can find a tensor using the singular value decomposition of the tensor in order to:
It is equivalent to change the geometry of the TN by this transformation:
After doing this, we can rearrange the TN to coarse-grain the TN in to another TN with smaller number of tensors, however, with higher ranges of indices. In fact, this the part that we need to be careful about assigning a tensor to the operator where the range of all of its indices are constant. Indeed, this method works properly for gapped Hamiltonians. Iterating this procedure leads us with a renormalization group that changes all local operators and tensors and it is possible to compute the partition function. More details are in Ref. 2.
Tensor Network Renormalization (TNR) [Ref. 3]
We want to introduce a coarse-graining transformation on a lattice . Consider we have came up with a fully contracted TN and each tensor is on a complex dimensional Hilbert space namely . What we want to do is computing . Consider a simple 2D square lattice with the Ising model Hamiltonian. In this case . Look at the below figure:
We introduce an operator called a disentangler, i.e. such that . This would not changed the expression .
Next we introduce an isometry operator like such that . However, is a projection. So our coarse-graining transformation is not exact. This transformation is show in the below figure.
In fact, the disentangler operator removes short-range correlations. Several advantages of this method over TRG is discussed in Ref. 3.
MERA [Ref. 4]
Consider a dimensional lattice that has sites. Let a dimensional Hilbert space assigned to each site. Assume the Hilbert space of is . Let an arbitrary state and the ground state. Consider a quantum circuit associated to the state with local unitary operators in a finite family of quantum gates. Using the second example in the section Tensor Network (TN), we can see this quantum circuit as a TN with additional characteristics. Let specify the steps in this quantum circuit. Using the idea of TNR (previous section) this TN can be viewed as a coarse-graining transformation of the lattice . So, can be both viewed as time and scale indicator. Here is a cartoon for .
As it is apparent from the TNR method, at each step there are two types of transformations: disentaglers and isometry operations. The former removes the short-range correlations and the latter combines pairs of nearest neighborhood wires into a single wire. Let the total step in the quantum circuit to be and introduce . Hence, at each step, two operations is done. Both are depicted in the figure. Now let define a MERA. A MERA is a representation of a state using the quantum circuit associated to this state from the ground state . One of the important feature of a MERA is that it has a causal cone with a bound width. Another important characteristic of a MERA is that by the stated feature, local operators remain local.
Just as a reminder, we define the complexity of a MERA as number of tensors in the MERA in compatible with our definition of a complexity of a state with the minimum number of nodes in its quantum circuits.
AdS/CFT correspondence [Ref. 5 and Ref. 6]
Why the spacetime associated to CFT is AdS?
In above, the complexity of a MERA defined. However, the complexity of a state is the complexity of an optimal MERA that implements the state. Thus, in order to minimize this complexity, we have to find an optimal MERA for the state. However, one should ask whether the optimal TN is a MERA; it is an ansatz. In Ref. 5 it is conjectured that changing the geometry and external gauge fields would express this optimization. Namely, to each state assign a functional where optimizing that would yield the optimized MERA for the state .
The idea is to discretize Euclidean path integrals and changing them into a TN. Consider we have a state and we start with a flat geometry where point are indicated by such that and is the lattice cut-off:
After “optimizing” the path integral (which will be discussed.) we will come up with a state . The basic idea is that we want where is a constant. We assign to each state a functional that minimizing its value results in the optimization. We called this functional . In there is no coupling that runs with the RG flow so a candidate for such a optimization is the background metric. Thus, we get where is the background metric.
Now we focus on . We know every metric in 2D is conformally (diffeomorphism+Weyl) flat, i.e. there exist a such that . So the only function that would possibly indicate such a optimization is . It is known changing the background metric with a Weyl transformation would lead to change the measure of the path integral by:
It is called the Liouville action. is the Ricci scalar of the original space and in our case zero. It is conjectured in Ref. 5 . But the surprising fact is that we can convince ourselves the relation is true by the supporting arguments using developed TN machinery.
We have seen the number of tensor in a MERA associated to the state is a quantity of complexity of that MERA. There are two types of tensors in the MERA: first the original tensors in the MERA, second the isometry tensors that combine two wires into a single wire. Let’s compute the number of tensor in a MERA. The local density of number of initial tensors is that is apparent from the below figure.
The second components are isometric tensors. These tensors are in the non-continuous part of the diagram, i.e. green ones in the above figure. So they are related to the jump of . The lowest order rotationally symmetric term that is even in (because thinking in reverse time should not change the number of isometric layers.) is: (I have to say I don’t understand this point!)
That is equal to:
So now we have to sum a positive multiplication of these two integral and we conclude where is a constant. So indeed we are trying to minimizing the complexity of the state using its MERA correspondences. Up to now, we have conjectured in order to minimize the complexity of the state, we have conjectured we have to minimize the Liouville action. Doing this leads us to the following equation:
where . We have the boundary condition . One solution is . Thus, the optimized geometry is:
that is a hyperbolic space on . Moreover, if we compact the dimension, we can conclude indeed the optimized value for is where is the . It is interesting to see the lattice cut-off is somehow a cut-off in quantum complexity.
It is an outstanding result! This paper concluded the geometry in the bulk associated to the vacuum is such that its time slices (we have to argue the additional dimension in the MERA is indeed time and it is discussed in Ref. 6.) have the hyperbolic metric in agreement with the geometry of time slices of . For other states in the the same conclusion is discussed in Ref. 5.
- Quantum Tensor Networks in a Nutshell, Jacob Biamonte and Ville Bergholm.
- Tensor renormalization group approach to 2D classical lattice models, Michael Levin and Cody P. Nave.
- Tensor Network Renormalization, G. Evenbly and G. Vidal.
- A class of quantum many-body states that can be efficiently simulated, G. Vidal.
- Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, and Kento Watanabe.
- Einstein’s Equations from Varying Complexity, Bartlomiej Czech.