Classically Optimized
Variational Quantum Eigensolver
for Topological Ordered Systems

Link to the Extended Abstract (PDF)

I would like to give a presentation entitled Classically Optimized Variational Quantum Eigensolver for Topological Ordered Systems.

This presentation will be given by Keita Osaki at the Graduate School of Engineering Science, Osaka University.

This research was done in collaboration with Kosuke Mitarai and Keisuke Fujii.

First of all, I would like to show you the outline of this presentation.

This work is an improvement of a classical-quantum hybrid algorithm called VQE. We will first introduce the regular VQE and then we will discuss the error problem, which is a weakness of this algorithm.

We propose a new way to overcome the error problem by replacing the quantum part of the algorithm partly with a classical simulation.

We use this method as a proof of concept for detecting the SPT phase in a one-dimensional transverse field cluster model.

Here we tried two ways: looking at the known order parameters and using unsupervised machine learning called clustering without using existing knowledge.

The Variational Quantum Eigensolver, or VQE for short, is an algorithm that works with the help of classical computing to find the parameters such that the quantum circuit minimizes the energy.

First, it runs the quantum circuit with some parameter θ. The result is the expected value of the Hamiltonian and the gradient of it for θ.

This information is passed to the classical computer. The classical computer calculates a new parameter likely to give lower energy.

This new parameter is again fed to the quantum computer, which again evaluates the energy.

By repeating this process, we can find the parameter that gives the lowest energy.

VQE is expected to be an effective use of NISQ computers.

However, VQE has a weakness.

To calculate the expected value in a quantum computer, sampling is required.

This is where statistical and measurement errors occur.

This may not be much of a concern for finding the energy itself, but when it comes to finding the gradient to determine the next parameter, small errors can have a significant impact on computational efficiency and success rate.

The method we propose is to replace this process of energy minimization with classical computer simulations.

In this way, we do not suffer from the statistical and measurement errors from the quantum computation.

Of course, this is not possible in all cases, since classical simulations generally require computation time and memory that is the exponent for the size of the system. However, if the Hamiltonian is local and the quantum circuit is of constant depth, then classical simulations can be performed efficiently.

Why would a classical simulation be efficient under such conditions?

For example, suppose that a term in the Hamiltonian is determined from measurements of only this adjacent three qubits, as shown in the figure.

Then, in this variational quantum circuit, this term will only be affected by the part of the system shown in the figure.

This does not require any computational effort or memory for the exponent of the size of the system, since we can find the expected value of the entire Hamiltonian by simulating such a part of the system separately.

Maybe you are thinking: "What does this have to do with quantum computers? Isn't it just a classic simulatable problem?"

However, there is a continuation of our proposed method, in which the quantum computer plays an indispensable role.

After the optimization stage described above, there is a stage to use the results.

VQE is not just an algorithm for finding the ground energy, but one that can perform various operations on the ground state obtained.

Our method uses a quantum computer for this evaluating step as with normal VQE.

This allows us to obtain expected values of non-local observables or inner products between states.

They cannot be efficiently obtained by classical simulations.

As proof of concept, we have performed numerical experiments on a 1D transverse field cluster model.

The 1D transverse field cluster model is an N-body system with such a Hamiltonian.

The Hamiltonian is local, and the ground state of the system can be obtained efficiently by classical simulations using the present method.

It is known that the ground state of this system undergoes a symmetry protected topological phase transition, or SPT phase transition, depending on the value of J.

To detect this SPT phase transition, we tried two different methods.

The first is to look at the values of the known order parameter. Due to the topological nature of this phase transition, the order parameters are non-local.

The second is to look at the inner products between the ground states to see their similarity. This is a quantity that can be studied even when the nature of the system is not well studied and there is no known phase transition.

To achieve this goal, we built a variational quantum circuit that looks like this.

We took five rotating gates as one unit and assembled them into a brick-like structure like a two-qubit gate. The number of layers of units is defined as the depth.

The simulation was performed using the quantum computing simulator Qulacs.

In the parameter optimization process, we used the BFGS gradient descent method.

Here are the results for 16 qubits and a depth of 4.

We compare the results for the proposed method with those for the exact solution calculated by the Lanczos method.

On the left are the ground state energies for each J value, and we can see that the VQE method is also able to obtain energy states close to the exact solution.

On the right are the values of the order parameters for each J value; in the case of VQE, as in the case of the exact solution, the values change in such a way that a phase transition can be observed.

Such a comparison shows that the proposed method can be used effectively.

Next, let's talk about clustering.

Since VQE allows us to create ground states actually, there is room to use a lot of information about the states.

We tried to detect the phase transition based on the prediction that "the absolute value of the inner product will be large between states in the same phase".

We were able to infer the existence of the phase transition from the values of inner products shown in the next slide, but we also applied a machine learning technique called clustering to show that a machine would be able to do so as well.

Clustering is a type of unsupervised machine learning method that allows us to divide the set into several groups using the similarity between each element of the set.

In this study, we used the absolute value of the inner product as this similarity.

This is the result of the absolute values of the inner products between the states.

The absolute value of the inner product between the ground state in magnetic field J₁ and the ground state in magnetic field J₂ is shown in the table.

The dark blue area is where the inner product is large.

For example, the top row shows the change in the similarity of the ground state for a change in J₁ with J₂ fixed at 2.

This can be interpreted as a large change in the inner product due to a phase transition occurring along the way, as shown in the top right figure.

This speculation is confirmed by the results of the clustering.

Here is a table showing for each J which group its ground state belongs to.

The group to which the state belongs changes after J=1, and we can detect mechanically what seems to be a phase transition.

This is a combination of the above two results.

I will summarize the above and talk about future developments.

We have shown that the proposed method can be used to detect phase transitions in one-dimensional cluster models as proof of concept.

This is useful when we want to see some non-local quantities of the ground state of a system with local Hamiltonians.

An example of such a system would be one with topological invariants.

For some large systems, such problems are difficult to solve with classical computation and can be solved with the proposed method using quantum computation.

In future work, we are considering targeting the perturbed Kitaev Toric code.