Quantum simulation close to critical point

[sonaQC]

Description

In project, we extend our existing work to simulate some exotic phenomena close to critical point

Deliverables

extend the code for detect some symmetry, familiar with qiskit and state preparation required

Mentors details

• Mentee 1
  • Required:
  • item 1
  • Nice to have:
  • item 1
• Mentee 2
  • Required:
  • item 1
  • Nice to have:
  • item 1

[BStar14]

Hi, I am interested in this topic. May I ask what problems you are planning to address?

[Qcatty]

Hi, Sona. can you make more details about the project?

[shuraim642]

Hi Sona. I just go through this project title. I already have skills in Qiskit and simulation models, and interested in the topic. Can you please more elaborate on this?

[Siheon-Park]

Hi Sona. As a student who majored in Physics, this is not only trendy but also a personally interesting subject. I would like to join this project. May I know the details or roadmap you pictured in mind?

[ruihao-li]

Hi @GemmaDawson, could you please assign me to this issue?

[ruihao-li]

Checkpoint 1 presentation slides: checkpoint1_slides_rl.pdf

[ruihao-li]

Checkpoint 2:

So far, we have been trying to build and simulate the quantum circuit for computing the partition function of the Ising model (based on A. Krishnan et al., PRA 100, 022125 (2019)) on Qiskit. The basic idea is that the modulus of the partition function can be effectively measured through a quantum circuit involving simple one- and two-qubit rotation gates.

The authors proposed two approaches used to compute the partition function of the two-dimensional (2D) classical Ising model with a longitudinal magnetic field on a square lattice of size $N \times L$ with cylindrical boundary conditions, meaning that it has a periodic boundary of length $N$ and open boundary of length $L$. One way is to implement the circuit directly based on the 2D geometry and such implementation requires $4NL - N$ qubits in total. This includes $NL$ "physical" qubits, $2NL - N$ ancilla qubits for nearest-neighbor couplings, and another $NL$ ancilla qubits for the external field. The other way is to first map the 2D classical Ising model without the longitudinal field to a 1D periodically kicked quantum transverse-field Ising model and then build the circuit based on the 1D model. Such method only requires $2NL$ qubits, but the presence of a longitudinal field in the classical model would need to be taken into account separately.

We implemented both methods in Qiskit and our preliminary results based on the Aer simulator have qualitatively verified the results established in the paper, particularly, the Fisher zeros on the complex-$K$ plane, where $K$ represents the nearest-neighbor coupling, and the Lee-Yang zeros on the complex-$H$ plane, where $H$ is the external magnetic field strength. We include these simulation results below as the visual representation for this checkpoint.

There currently still exist two problems with the simulation. One is the qubit scaling. Simulating a $3\times 3$ square lattice with the first method mentioned above would require 33 qubits, making it difficult to complete in a reasonable amount of time on a laptop. This makes the simulation limited to only small system sizes. The other problem is that as the system size gets larger, the size of the Hilbert space increases exponentially, and therefore, the return probability, which is the quantity we measure from the quantum circuit that is used to compute the partition function, is likely to get exponentially suppressed (to be confirmed). Therefore, we would need many shots to measure the probability accurately. This further hinders the computational efficiency.

Fig.1 - Illustration of Fisher zeros (left) and Lee-Yang zeros (right) of the 2D Ising model on a 2 x 2 square lattice. The resolution is 40 x 40 for both plots. Each point was evaluated based on 5000 repeated measurements of the quantum circuit.

[ruihao-li]

Final Showcase slides: QAMP21_final_slides_RL.pdf