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.

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13
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1answer
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Quantum phase estimation and HHL algorithm - knowledge of eigenvalues required?

The quantum phase estimation algorithm (QPE) computes an approximation of the eigenvalue associated to a given eigenvector of a quantum gate $U$. Formally, let $\left|\psi\right>$ be an ...
8
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1answer
919 views

Practical implementation of Hamiltonian Evolution

Following from this question, I tried to look at the cited article in order to simulate and solve that same problem... without success. Mainly, I still fail to understand how the authors managed to ...
7
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1answer
799 views

Efficiently performing controlled rotations in HHL

This question builds off of this question. In the HHL algorithm, how do you efficiently do the $\tilde{\lambda}_k$-controlled rotations on the ancilla qubit? It seems to me that since you don't know ...
8
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1answer
564 views

Quantum algorithm for linear systems of equations (HHL09): Step 1 - Number of qubits needed

This is a continuation of Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm Questions (contd.): Part 2: I'm not exactly ...
12
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2answers
2k views

Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm

I have been trying to get my head around the famous(?) paper Quantum algorithm for linear systems of equations (Harrow, Hassidim & Lloyd, 2009) (more popularly known as the HHL09 algorithm paper) ...
9
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2answers
749 views

Quantum algorithm for linear systems of equations (HHL09): Step 2 - What is $|\Psi_0\rangle$?

This is a sequel to Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm and Quantum algorithm for linear systems of ...
5
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1answer
249 views

Showing that Matrix Inversion is BQP-complete - HHL Algorithm

I am trying to understand an argument that Matrix Inversion is BQP-complete for certain conditions on the matrix. This is explained here on page 39 (this paper is a primer to the HHL algorithm and ...
7
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1answer
468 views

SWAP gate(s) in the $R(\lambda^{-1})$ step of the HHL circuit for $4\times 4$ systems

Context: On the 5th page of the paper Quantum circuit design for solving linear systems of equations (Cao et al, 2012) there's this circuit: Schematic: A brief schematic of what's actually ...
4
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1answer
677 views

HHL algorithm -- controlled-by-eigenvalues rotations

All the references in this question refer to Quantum algorithm for solving linear systems of equations (Harrow, Hassidim & Lloyd, 2009). The question I have is about the step where they apply ...
23
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2answers
2k views

What could be the possible future applications for HHL algorithm?

Note on the vocabulary: the word "hamiltonian" is used in this question to speak about hermitian matrices. The HHL algorithm seems to be an active subject of research in the field of quantum ...
9
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1answer
733 views

HHL algorithm -- why isn't the required knowledge on eigenspectrum a major drawback?

This question is a continuation of Quantum phase estimation and HHL algorithm - knowledge on eigenvalues required?. In the question linked above, I asked about the necessity for HHL to have ...
5
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0answers
424 views

Error: Simulation of "Quantum algorithm for linear systems of equations" for $4\times 4$ systems on Quirk (without SWAP) - Global phase

Following @DaftWullie's answer I tried to simulate the circuit given in Fig. 4 of the paper (arXiv pre-print): Quantum circuit design for solving linear systems of equations (Cao et al, 2012), on ...
9
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2answers
2k views

Quantum algorithm for linear systems of equations (HHL09): Step 2 - Preparation of the initial states $|\Psi_0\rangle$ and $|b\rangle$

This is a continuation of Quantum algorithm for linear systems of equations (HHL09): Step 2 - What is $|\Psi_0\rangle$? In the paper: Quantum algorithm for linear systems of equations (Harrow, ...
5
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2answers
196 views

Using the HHL algorithm to compute $A |b \rangle$ instead of $A^{-1} |b \rangle$

In the paper "Compiling basic linear algebra subroutines for quantum computers" here, they discuss (page 2 bottom right) using the HHL algorithm to multiply a vector by a matrix. However, after ...
4
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1answer
220 views

How to effectively compute eigenvalue rotation in HHL

In the HHL algorithm, how do you efficiently do the $\lambda-$controlled rotation on the ancillary qubit ? It seems to me after reading around some answers that this can be done in two steps : First, ...
3
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2answers
512 views

Problem with controlled rotation in HHL

In HHL algorithm, for subroutine involving controlled rotation, after applying $R_y(\theta)$, where $\theta=2\sin^{-1}\left(\frac{C}{\lambda}\right)$ to the ancilla, the state changes to $\sqrt{1-\...
3
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1answer
73 views

HHL and choice of observable for calculating the expectation value thereof

The chapter about solving linear systems in the qiskit textbook describes the last (6th) step of the HHL algorithm as follows Apply an observable $M$ to calculate $F(x):=\langle x |M|x\rangle$. How ...
0
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1answer
121 views

Qiskit HHL algorithm BrokenProcessPool error

I try to run HHL algorithm on qasm_simulator. I took the example from here. I try to solve 8x8 random matrix and adjust some parameters in create_eigs. ...