Comparison of global minimum ratios of QUBO and Ising models

Solving linear system

Solving linear system: Ax = b
This folder solve the below matrix and vector with 12 qubits
     1   -1    1
A = -2    3    4 , b = (4, -4, 8)
     1   -4   -1

File description

QA_12qubits.ipynb: Original QUBO and Ising models
QA_12qubits_penalty.ipynb: QUBO and Ising models with constraint terms equal to 0
QA_12qubits_penalty.ipynb: QUBO and Ising models with constraint terms equal to 1000
QA_12qubits_rmvd_penalty.ipynb: QUBO and Ising models with removed constraint terms

Success rate finding global minimum (10,000 anneals)

                Original        Panelty 0        Panelty 1,000    Panelty removed
QUBO                  57         299, 279              67, 278           508, 605
Ising                175         363, 703             176, 114           258, 960

<Sylvester_Theorem Folder>

Summary

• This folder contains sage & python code for implementing QUBO formulations for system of linear equations using Sylvester's law of inertia.
• Joint work with Sun Woo Park
• The same code can be found in the following link: https://github.com/spark483/Quantum_Computing_sub
• Arxiv preprint link: https://arxiv.org/abs/2111.10084

Matrix_congruence_update.ipynb

• This calculator prints out the d-wave quantum annealer command for solving a system of linear equations using the quadratic unconstrained binary optimization formulation.

QA_X-qubits.ipynb

• The following code implements the following QUBO models which utilize X qubits.
  • QUBO formulation utilizing Sylvester's law of inertia & qubit relations (compiles up to 132 qubits)
  • QUBO formulation utilizing only the quadratic relation q^2 = q (compiles up to 64 qubits)
  • (The error message "no embedding found" shows that the D-wave quantum annealer cannot process the second QUBO formulation.)
• QUBO model utilizing Sylvester's Theorem:
  • Empirical maximum number of utilizable qubits: 132 qubits (D-wave 2000Q quantum annealer)
  • Models utilizing more than 136 qubits cannot be compiled (D-wave 2000Q quantum annealer)
• QUBO model utilizing only the quadratic relation:
  • Empirical maximum number of utilizable qubits: 64 qubits (D-wave 2000Q quantum annealer)