Questions tagged [hhl-algorithm]
For questions related to the quantum algorithm for inverting linear systems of equations, developed by Harrow, Hassidim and Lloyd in 2009.
147 questions
2 votes
1 answer
341 views
Resources on Quantum Linear Solvers (e.g., HHL) and Their Application to Differential Equations
I am currently writing my Bachelor’s thesis on Quantum Algorithms for Solving Differential Equations. I have a background in both differential equations (including partial differential equations) and ...
- 141
1 vote
0 answers
60 views
HHL eigenvalue inversion step with larger number of clock qubits that number of eigenvalues
Reading about the HHL algorithm (this paper/arXiv and this paper/arXiv) I came up with the following question: if I have, say, 2 eigenvalues for which I need, say, 5 qubits for the best QPE algorithm, ...
4 votes
1 answer
186 views
HHL with eigenvalues that are not integer multiples
Pretty much all implementations of the HHL algorithms support cases where it is possible to map the eigenvalues not just to integers, but to powers of 2. Power of 2 map nicely to individual bits in an ...
- 1,142
0 votes
1 answer
119 views
HHL hermitian to unitary matrix convertation and output results
I'm reading the paper describing the numerical example HHL. The first question related to the Hermitioan-unitary matrix transformation. We can use numpy.linalg.expm ...
1 vote
1 answer
109 views
Why should the inverse phase estimation in HHL change the ket $|b\rangle$?
In HHL the forward phase estimation allows to set the phases of eigenvalues. It does not change the input ket $|b\rangle$ as far as this is a combination of the eigenvalues of the operator (and ...
1 vote
2 answers
163 views
Derivation in "Step-by-Step... HHL" is unclear
In the excellent paper "Step-by-Step HHL Algorithm Walkthough..., https://arxiv.org/abs/2108.09004, it makes a derivation that is not clear to me at all. In Equation (15) it combines the effect ...
- 1,142
4 votes
0 answers
88 views
What are classroom examples for the HHL algorithm?
I am teaching a course in QC and one of the topics I would like to put a good emphasis on HHL algorithm for solving systems of linear equations. Anyway, besides the theory, I would like to add some ...
- 141
2 votes
1 answer
311 views
HHL eigenvalue inversion and further inverse QPE
Reading the paper (arXiv) about HHL I misunderstood the following points: $$|{x}\rangle = A^{-1} \cdot |b\rangle = \sum^{2^nb-1}_{i=0}\lambda_i^{-1}b_i|u_i\rangle$$ How do we inverse the eigenvalues? ...
0 votes
1 answer
111 views
Modify a quantum state prepared with HHL [closed]
I can prepare a state $|1\rangle|x\rangle|0\rangle$ using HHL. How do I prepare $|1\rangle|x\rangle|0\rangle+|0\rangle|y\rangle|0\rangle$ or simply $|1\rangle|x\rangle|0\rangle+|0\rangle|0\rangle|0\...
2 votes
1 answer
175 views
Why does QPE still produce phase eigenvalues in HHL even though it operates on non-eigenvectors?
I'm trying to understand mathematical intuition of HHL algorithm using original paper (arXiv) For now I stuck at the part of Phase estimation. If I understand correctly, if vector b is the eigenvector ...
2 votes
1 answer
112 views
How to construct the HHL oracle circuit when the entries of $A$ are described by a polynomial?
Among other assumptions, HHL's algorithm assumes that the entries of the coefficient matrix $A$ (where $Ax=b$ is the linear system to be solved) can be realized by means of an oracle circuit $U_A$. ...
- 153
2 votes
1 answer
168 views
Applicability of the error analysis of HHL algorithm assuming spectrum could contain negative and positive eigenvalues
I have a question concerning the applicability of the error analysis of the HHL algorithm , specifically the QPE subroutine. The authors state that they assume all the eigenvalues lie between $[\frac{...
- 82
1 vote
1 answer
178 views
How to calculate an unnormalized solution x in HHL?
I'm going through this paper Step-by-Step HHL Algorithm Walkthrough, and wondering how to calculate an unnormalized solution $|x \rangle$ after measuring its b-register and ancilla qubit. In equation (...
- 1,532
1 vote
1 answer
120 views
HHL: Numerical example without using hidden information
I have the following resource request: I am looking for a numerical example of an implementation of HHL that does not "cheat" by using knowledge that would be hidden in a realistic scenario. ...
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3 votes
1 answer
152 views
What is the intuition behind achieving Quantum advantage in simulating non-hermitian dynamics using Quantum computer?
There have been several works on simulating ODE for classical systems like here and here. They are quantum techniques for solving ODEs related to classical systems. A generic methodology is: To solve ...
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