Finding Eigen Values from Quantum Phase Estimation - Using qiskit

Question

I am trying to use the quantum phase estimation(EigsQPE) of qiskit to find the eigen values of a matrix. As I am new to quantum computing so I am confused what to measure in the circuit to derive the eigen values of the input matrix.

I know how to identify phase and derive eigen value using phase derived from the most probable bit string for a single eigen value. But deriving multiple eigen values from QPE circuit is confusing. Any help will be much appreciated.

Code : https://github.com/Anand-GitH/HLL-QauntumComputing/blob/main/Qiskit-QPEStandalone.ipynb

Answer 1

Two things to note:

1. EigsQPE needs the eigenvalues to be scaled onto the range (0,1]. You can use evo_time to set the scaling. If you don't pass this value, a scaling value will be set automatically. You can get this value using eigs.get_scaling().

2. If the eigenvalue is $e^{2\pi i\theta}$, then the register contents will be $2^n\theta$

That means if the register contains the value $x$, and evo_time equals $s$, then your eigenvalue will be $2\pi x/(2^ns)$

Answer 2

Thanks a lot with your two points I was able to proceed further.

https://github.com/Anand-GitH/HLL-QauntumComputing/blob/main/Qiskit-QPEStandalone.ipynb

I was able to find the eigen values of the two matrices with greater precision- A=[[1,-1/3],[-1/3,1]] Expected Eigen Values: [1.33333333 0.66666667] Counts: {'11111111': 0.499999999999994, '11111110': 0.499999999999994} Actual Eigen Value from Circuit: 1.3333333333333333 0.6614173228346456

A=[[1,-1/2],[-1/2,1]] Expected Eigen Values: [1.5 0.5] Counts: {'111': 0.500000000000001, '110': 0.500000000000001} Actual Eigen Value from Circuit: 1.5 0.5

However, when I try other matrix: A = [[1.5, 0.5], [0.5, 1.5]] Expected Eigen Values: [2. 1.] Counts: {'011': 0.213388347648319, '010': 0.213388347648319} Actual Eigen Value from Circuit: 2.0 0.6666666666666666

I do see there is so much difference in second eigen value. If I use the same matrix in HHL to solve linear equation and match results with paper(https://www.sciencedirect.com/science/article/pii/S037596012030462X?via%3Dihub) it works absolutely fine. https://github.com/Anand-GitH/HLL-QauntumComputing/blob/main/Qiskit-HHL.ipynb

I doubt now I am doing something wrong in reading out eigen values from QPE. Any suggestions on that part. I am just trying to read out all the eigen values of any matrix using qiskit EigsQPE.