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

Egretta.Thula · Accepted Answer

Two things to note:

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().

If the eigenvalue is $e^{2pi itheta}$, then the register contents will be $2^ntheta$

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

Anand Prabhu · Answer

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.

Finding Eigen Values from Quantum Phase Estimation - Using qiskit

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