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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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Controlled unitary from the HHL algorithm – practical implementation using Qiskit

I have a question about the implementation of the controlled unitary $e^{iAt}$, from the paper Demonstration of a Quantum Circuit Design Methodology for Multiple Regression, using the Qiskit framework....
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What are applications of HHL's “simple example” to determine similar stable states of different quantum processes?

The HHL paper "Quantum algorithm for linear systems of equations" [link] initially describes a "simple example" to illustrate the potential power of their algorithm. They state: Consider a ...
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Quantum algorithm for linear system of equations (HHL) - Final Step: How can I find my vector of solution $|x\rangle$?

I'm working on solving a linear system with the quantum algorithm HHL. I don't understand how I can recover my vector $|x\rangle$ of real solution of the system starting from the states measured with ...
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Implementing gate with two parameters using Qiskit in Python

I am trying to implement the HHL algorithm (for solving $Ax=b$). I am assuming $A$ to be unitary and Hermitian so that I can find the Hamiltonian simulation for it easily. For any $A$ to be Hermitian ...
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How to find parameters for circuit decomposition of Hamiltonian simulation of any matrix $A$?

In version 2 of the paper Quantum Circuit Design for Solving Linear Systems of Equations by Cao et al., they have given circuit decomposition for $e^{iA\frac{2\pi}{16}}$, given a particular $A_{4\...
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Why is there no $N$ in the time complexity of the QLSP algorithm by Childs et al.?

The paper Quantum linear systems algorithms: a primer by Dervovic et al has this table on page 3: I'm not sure why there's no $N$ in the time complexity of the algorithm by Childs et al. i.e. $\...
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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 ...
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How to speed up the matrix multiplication steps in multi-linear regression?

Context and Motivation: As discussed here, in multilinear regression, we can express the linear system as $AX = b$. This leads to $A^TA \hat{X} = A^T b$. From here, the estimated value of $X$ is ...
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What does $||A|| = 1$ mean in the definition of QLSP?

In their 2017 paper, Childs et al. gave the definition of QLSP beginning with : "Let $A$ be an $N\times N$ Hermitian matrix with known condition numbers $\kappa$, $||A|| = 1$ and at most $d$ non-zero ...
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What exacty is “matrix sparsity” $s$?

In many sources (like on Page 30 here), I found that the complexity of the original Harrow Hassidim Lloyd is stated as $\mathcal{O}(\log (N) s^2 \kappa^2/\epsilon)$ where $s$ is said to be the "matrix ...
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Solving linear systems represented by NxN matrices with N not power of 2

As far as I have seen, when it comes to solving linear systems of equations it is assumed to have a matrix with a number of rows and columns equal to a power of two, but what if it is not the case? ...
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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 ...
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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 ...
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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 ...
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Why does the quantum linear inversion algorithm allow to implement $e^{-ig(\rho)}$ efficiently using multiple copies of $\rho$?

In (Lloyd et al. 2013), the authors write (beginning of page 3) that the quantum matrix inversion techniques presented by some of the same authors in (Harrow et al. 2008) allow to efficiently ...
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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 ...
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HHL algorithm — problem with the outcome of postselection

See edit at the end of the question All the references in this question refer to Quantum algorithm for solving linear systems of equations (Harrow, Hassidim & Lloyd, 2009). HHL algorithm ...
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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 ...
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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 ...
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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 ...
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How exactly is the stated composite state of the two registers being produced using the $R_{zz}$ controlled rotations?

This is a sequel to How are two different registers being used as "control"? I found the following quantum circuit given in Fig 5 (page 6) of the same paper i.e. Quantum Circuit Design for ...
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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, ...
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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 ...
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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 ...
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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) ...