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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Solving linear system $Ax=b$ with exponential speed-up via binary optimization?

The main disadvantage of HHL algorithm for solving $A|x\rangle = |b\rangle$ is that exponential speed-up is reached only in case we are interested in value $\langle x|M|x\rangle$, where $M$ is a ...
5
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40 views

What is the largest linear system of equations solved using HHL experimentally?

Can someone tell what is the largest system of equations solved using HHL algorithm experimentally? I know the $8$ x $8$ system has been solved experimentally. Has someone gone beyond this to solve $...
3
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1answer
51 views

HHL algorithm for linear systems with a real matrix and a real right side

HHL algorithm can be used for solving linear system $A|x\rangle=|b\rangle$. If we put $|b\rangle$ (to be precise its normalized version) into the algorithm and measuring ancilla to be $|1\rangle$ we ...
3
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34 views

What are the correct eigenvalues to use in controlled rotation in HHL?

I'm studying the HHL algorithm from the qiskit textbook, but I don't understand what $\lambda$ we have to use. If my matrix $A$ has eigenvalues that can be written in a binary representation, then, ...
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1answer
126 views

In the HHL alghoritm, how can I transform my hermitian matrix into a unitary operator?

I'm studying the HHL algorithm and I'm trying to do an his implementation but, there are some points that I don't understand how I can transform my hermitian Matrix into its unitary operator? In ...
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39 views

Question on qiskit.circuit.library.PiecewiseChebyshev Qiskit circuit library

This question follows the post https://quantumcomputing.stackexchange.com/posts/15070/edit. I am working on an implementation of HHL algorithm, to do so, I need to map $|\lambda\rangle\mapsto |\arcsin(...
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1answer
84 views

I don't understand unitary of ${e^{iAt}}$ from HHL algorithm

I tried to implement the following circuit in the image below but with the red circled gates replaced with a unitary controlled ${e^{iAt/2}}$ and controlled ${e^{iAt/4}}$ The image came from this ...
2
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1answer
65 views

Function in Qiskit to get the quantum circuit

Is there any way to view the quantum circuit of pre implemented quantum algorithm in Qiskit? In the Qiskit textbook, there is an example given for HHL algorithm. Is there any function in Qiskit which ...
3
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2answers
78 views

Source code for a Qiskit algorithm: HHL Algorithm

Is it possible to view the source code of one of the Qiskit's algorithm? Actually, I am trying to find how Qiskit implements the HHL algorithm. I want to see the source code for this algorithm.
3
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1answer
145 views

Quantum Circuit for $e^{iAt}$ Hamiltonian Simulation in HHL algorithm

In HHL algorithm, there is a step in Quantum Phase Estimation where we have to apply powers of $e^{iAt}$ to the register (see pic). I am not able to understand how to find the quantum circuit ...
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53 views

In HHL algorithm, how to find the angle of rotation of ancilla qubits?

In HHL algorithm, after the quantum phase estimation, what is the angle of rotation i.e. what is the value of $\theta$ in R$_y$($\theta$) that is used in rotating ancilla qubit? How to find this angle ...
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Quantum implementation of arcsin

I am looking to implement a quantum version of the arcsinus function. Such a problem is motivated by the HHL algorithm where $x\mapsto 1/x$ and $\arcsin$ can be used to get $1/x$ from the ...
3
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1answer
85 views

HHL algorithm, How to implement exp(iAt) gates?

From this paper Quantum Circuit Design for Solving Linear Systems of Equations, in figure 4 The paper shows what inside operator $e^{-iAt}$ but didn't shows how to connect the control qubit (register ...
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52 views

Why does solving linear systems of equations with HHL return imaginary numbers? [closed]

The textbook shows an example system of linear equations $Ax=b$ with $$ A=\begin{pmatrix} -\frac{1}{3} & 1 \\ 1 & -\frac{1}{3} \end{pmatrix} \,\,\,\, b=\begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$ ...
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1answer
84 views

Efficient QFT-based QPEA complexity

The HHL algorithm lies on an implementation of the Quantum Phase Estimation algorithm. One popular implementation is based on the Quantum Fourier Transform which can be divided in three steps. Let $U$ ...
4
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1answer
67 views

HHL algorithm, How can I get result from register $|b\rangle$?

From the paper A survey on HHL algorithm: From theory to application in quantum machine learning , I use qasm code from here. I try to follow the example in page 7. with Ax = b and the answer x ...
3
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1answer
65 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 ...
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1answer
57 views

Query on the HHL Algorithm in the Qiskit Aqua Library [closed]

This is a question regarding the Qiskit Aqua package. I have been studying HHL algorithm from QISKIT textbook and after understanding the math I finally got to the code part, I am having a hard time ...
8
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2answers
91 views

How to recover the normalization constant of the HHL solution

HHL solves the linear equation $Ax=b$ by the quantum state $|x\rangle=A^{-1} |b\rangle$. However, the quantum state $|x\rangle$ is normalized and thus diffs a normalization constant from the solution ...
3
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1answer
115 views

Gate-level implementation of Eigenvalue-Inversion in HHL

I am trying to understand how does the gate-level implementation of eigenvalue-inversion step in the HHL algorithm works. I am following this reference, where it is stated (Lemma 4) that this can be ...
4
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1answer
158 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, ...
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Why is HHL the popular choice to solve QLSP and not the Childs et al. (2017) algorithm?

The Childs et al. (2017) algorithm can solve the quantum linear systems problem (QLSP) in $\operatorname{poly}(\log N, \log(1/\epsilon))$ time while the HHL algorithm solves it in $\operatorname{poly}(...
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HHL algorithm Qiskit version

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1answer
116 views

Given a matrix, how do I proceed with the quantum phase estimation algorithm and choose $\theta$?

Given a matrix, say $\begin{bmatrix} 1.5 & 0.5\\ 0.5& 1.5 \end{bmatrix}$, with eigenvalues $1$ and $2$, how do I proceed with the quantum phase estimation algorithm? In particular, how do I ...
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1answer
100 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. ...
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37 views

In the HHL algorithm, how large should the eigenvalue estimation register be?

In the HHL algorithm, we need to estimate the eigenvalues of a matrix $A$. The result is stored in a register with $l$ qubits. Let $T = 2^l$. $T$ cannot be too small, otherwise the result would have a ...
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72 views

HHL example solution on Qiskit

In the example on the Qiskit page (https://qiskit.org/textbook/ch-applications/hhl_tutorial.html#A.-Some-mathematical-background) section 3 for the HHL algorithm the author is trying to solve the ...
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1answer
72 views

How to increase matrix size and getting high fidelity for HHL algorithm in Qiskit?

I have followed an example that Qiskit provides here. I tried to increase the matrix size to 16x16 and change num_ancillae and ...
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0answers
87 views

Implementing the HHL algorithm with negative eigenvalues (Cirq)

How do I implement the HHL with negative eigenvalues? This paper (https://arxiv.org/abs/1803.01486) says that: what if $\lambda<0$? This problem actually does not hard to solve, since we can ...
2
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1answer
183 views

Error when running HHL algorithm in Qiskit

I tried to solve a simple system of two simultaneous linear equations in with HHL algorithm in Qiskit. In particular the system is $Ax=b$, where $$ A = \begin{pmatrix} 1.5 & 0.5 \\ 0.5 & 1.5 \...
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1answer
130 views

How to optimize my HHL algorithm on qiskit?

I am following this HHL tutorial to solve the $Ax=b$ problem and have been using the general (inefficient) approach with the BasicAer simulator that they describe in section 4a. I would now like to ...
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1answer
51 views

How can I use the quantum volume to design the scale of my experiment?

I am designing an experiment which involves solving a linear system of equations of the form $Ax=b$. To do this, I am using the HHL algorithm on the IBMQ system. My experiment is scalable such that ...
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1answer
83 views

Is it required to input full matrix when using Qiskit HHL algorithm for sparse matrices?

I am working with a very sparse matrix and it seems inefficient to load the full matrix as input into the Qiskit HHL algorithm. Is it possible to input only the non-zero elements, instead ? I am ...
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0answers
57 views

Help Understanding Line in Source Code

In Qiskit's implementation of the HHL algorithm, they have the following line at the end of their code that allows them to compute the global phase we need to add on to the solution. ...
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Is HHL still BQP-complete when the matrix entries are only in {0,1}?

I'm studying BQP-completeness proofs of a number of interesting problems of Janzing and Wocjan, and Wocjan and Zhang. Janzing and Wocjan show that estimating entries of matrix powers $(A^m)_{ij}$ with ...
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77 views

Comparison of matrix inversion algorithms

Since 2009, many matrix inversion algorithms have appeared. Is there somewhere a table, or recently released overview, comparing the speed of matrix inversion algorithms, like done in this table taken ...
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27 views

Using matrix inversion and summation together. How to compute $(A+B^{-1})^{-1}|x\rangle$

I need to compute $(A+B^{-1})^{-1}|x\rangle$, where A and B is hermitian for simplicity. I read out about matrix inversion algorithms (HHL, QSVE,...), and algorithms of summing up of matrices(LCU, Lie–...
3
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1answer
151 views

Confusion regarding time complexity in the HHL algorithm

In the paper of HHL algorithm (Quantum algorithm for linear systems of equations), the time complexity of simulating $e^{i A t}$ for a hermitian matrix A is $\tilde{O}\left(\log (N) s^{2} t_0\right)$. ...
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66 views

Adding Quantum State Tomography Step to HHL Algorithm (Cirq)

Over the past month, I have been learning about the HHL algorithm and am interested in extending the Cirq implementation to include the quantum state tomography step to extract the amplitudes or ...
4
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1answer
124 views

How do I optimize HHL algorithm in Qiskit?

How do I optimize HHL algorithm in Qiskit? I tried to follow this tutorial on HHL in Qiskit. My project requires solving a very specific type of linear equations $Ax=b$ like the one below. ...
2
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1answer
119 views

The maximum depth possible on quantum computers

I hope you don't mind me having two questions. Firstly, I was running a Qiskit HHL simulation on a 12x12 matrix and a 12x1 vector, leading to a 16x16 matrix after expansion, and it resulted in a ...
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1answer
114 views

Angles of rotation in HHL example

I am trying to implement the 4-qubit example outlined in showed in section 3 of the qiskit tutorial on the HHL algorithm. Does anyone know what are the 2 angles that the ...
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1answer
71 views

In the rotation part of HHL algorithm, how do you decompose $R$ in terms of universal gates?

In this paper, there is a diagram explaining how HHL works, which I attached below: My question is, for the rotation part, how do you write $R$ as a combination of universal set of gates without ...
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1answer
165 views

New Hybrid-HHL algorithm vs VQLS

A team of researchers has realized hybrid quantum algorithm for solving a linear system of equations with exponential speedup that utilizes quantum phase estimation, the algorithm demonstrates quantum ...
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1answer
132 views

Problem with HHL algorithm on Qiskit

I try to run HHL algorithm on quantum computer in Qiskit Notebooks on the site. I take it from here https://qiskit.org/textbook/ch-applications/hhl_tutorial.html but it doesn't work. The error message ...
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2answers
430 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-\...
2
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1answer
202 views

Quantum circuit to get expectation values of Pauli matrices, given state $|\psi\rangle$

I'm trying to solve 2 linear equations with the help of HHL algorithm. I've taken $A=\begin{pmatrix} 1.5 & 0.5\\ 0.5 & 1.5\\ \end{pmatrix}$ and $b=\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$. I've ...
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3answers
668 views

Quantum circuit to implement matrix exponential

I want to build a circuit which will implement $e^{iAt}$, where $ A= \begin{pmatrix} 1.5 & 0.5\\ 0.5 & 1.5\\ \end{pmatrix} $ and $t= \pi/2 $. We see that $A$ can be written as, $A=1.5I+0.5X$. ...
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0answers
83 views

Explanation of HHL Cirq implementation [closed]

Can someone help me in understanding the program on the link given below? I would really appreciate if someone could give a detailed explanation ... https://github.com/quantumlib/Cirq/blob/master/...
3
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1answer
302 views

Implementation of matrix A in HHL algorithm circuit

In HHL algorithm, how do we implement matrix $A$ (where $A|x\rangle = |b\rangle$) in the circuit?