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I was wondering if the following is a viable method using machine learning and neural networks to get to the ground states of the toric code and also understand the phase transition in the presence of perturbation that is $H_{p} = H_{TC} + h_{x(z)}\sum_{i}\sigma_{x(z)}^{i}$, where $H_{TC}$ is the toric code hamiltonian given by $H_{TC} = -\sum_{v}A_{v} - \sum_{p}B_{p}$, where $A_{v}, B_{p}$ are the vertex and face operators respectively

Consider toric code with periodic boundary conditions, we know that the ground state is given by $$ \psi_{gs} = \prod_{v}(1+A_{v})\lvert\bf{0}\rangle,$$ given we are in the $\lvert0\rangle, \lvert1\rangle$ basis.

  1. For system sizes upto $N=24$ (on a system with decent memory) we can compute the ground state using the above expression and express it as a superposition of bit strings (as a superposition of 2^24 basis states, given by combination of the 24 bit strings of 0's and 1's, $\lvert000.....0\rangle, \lvert000.....1\rangle, \lvert111.....1\rangle$). We see that, if we represent the ground state as a superposition, not all 2^24 states are present in the superposition. Now we can use the number of spins (here 24) as an input to CNN (may be other architectures, I do not know much on this) to classify the toric code ground states as follows, if a bit string (a particular combination appearing in the basis) is present in the superposition we can label it as 1, and if it not we label it as 0. As we have labelled the entire data set, it should be possible to train a neural network. Given we do the above there are two points that arise

    • If we look at the basis states which appear in the superposition, there are very few appearing in the superposition compared to the entire basis space (for $N=12$, we have 4096 bit strings (basis states) out of which only 36 appear in the ground state, even if we include the other ground states arising out of the application of the non-trivial loop operators, it is around 144 which is still small compared to 4096), there is a cornering of the hilbert space and how can this be captured in the neural network analysis (because it makes the learning completely biased) ?

    • Also, if we can somehow get past the bias mentioned in the above point it means we have a neural network which has learnt about the states, now given we have neural network for $N=24$, how can this be extended to higher lattice sizes. In sense we have trained on a system with inputs $N=24$ and now we want to extend the knowledge of the trained network on lesser input and apply to a higher input, say $N=32$? In the sense the network trained on $N=24$ would predict whether a particular 32 bit string appears in the superposition or not, thereby providing a classification for higher system sizes and thereby getting to the ground states in higher system sizes.

  2. Now consider the toric code in the presence of perturbation, performing a similar analysis of representing the ground state in a superposition of bit-strings (here the ground state is obtained by ED for similar system sizes). The aim here is not to predict the ground state for higher system sizes but to predict the phase transition as the strength parameter is varied. For this the input to the neural network would be a bit string along with the strength parameter $h_{x(z)}$ and the output would be a 11 bit string output (here we see that the ground state is no longer a equal superposition of bit strings, to perform a binary classification, therefore we need more strings in the output to label the input bit string). For example if the probability of a bit-string, at say strength $h_{x}=0.3$ to appear in the ground state is given by 0.23, we assign the output 00100000000, if the probability is 0.i we assign the output which has 1 at lower bound of i and so on so forth. Here we do not have issue of cornering of Hilbert space, or heavy bias as for each strength the same basis state appear in the superposition but with a different probability. Therefore we can train the neural network at high strengths and low strengths of perturbation and then predict for all strengths whether a particular basis state appears or not and thereby from the superposition of these states conclude if it is in the ordered phase or trivial phase.

    • The problem here is I do not see the classification happening using CNN, so it would nice if someone could suggest a better method and if the method itself is viable ?
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  • $\begingroup$ The ground states of toric codes can be generated by a finite length quantum circuit, so definitely it can be represented by a deep neural network. In fact there is a explicit representation of it using RBM as given in the paper 'Efficient Representation of Quantum Many-body States with Deep Neural Networks'. $\endgroup$ – XXDD Nov 3 '17 at 3:26
  • $\begingroup$ Although it's quite popular to connect NN with many-body quantum states, I am still wondering if this is a break-through. Since the number of states is double-exponential w.r.t. the number of qubits. Intuitively NN can only represent a 0-measure subset of all the states and therefore NN is not valid for representing general states except for very special states. $\endgroup$ – XXDD Nov 3 '17 at 3:31
  • $\begingroup$ I guess there has been a RBM representation of the toric code ground state, Exact Machine Learning Topological States but I think RBM's are more on the lines of unsupervised learning. In the above context, I was looking at a supervised learning scenario, which I am not clear if CNN's are the way to go forward, because of the bias in 1 and the construction of the NN in the 2. $\endgroup$ – esornep Nov 3 '17 at 10:11
  • $\begingroup$ I do not think CNN (or CNN like network) can be used to efficiently represent quantum states. If a state can be generated by a limited length quantum circuit, then essentially between the input and the output we need to combine an exponentially large number of items, which can not be achieved efficiently by a CNN like (even not a convolutional NN) network since we need two many hidden neurons. DBN should be the proper network. At least DBN can represent any 'simple' state with a simple rule but for CNN there is no such method till now. $\endgroup$ – XXDD Nov 3 '17 at 11:23
  • $\begingroup$ For me, CNN is a kind of 'classical' network, it use a classical computation to connect the inputs and outputs. But for RBM/DBM, it's in fact can be regarded as a quantum network since each hidden neuron can take different values ($h=0,1$) as in a superpositioned stated. I have no idea how to convert a quantum circuit into a classical CNN. $\endgroup$ – XXDD Nov 3 '17 at 11:42

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