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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3 votes
1 answer
62 views

Optimal dependency of HHL (or any QLSP) algorithm on condition number $\kappa$

This is conserning the optimal dependency on condition number for Quantum linear system problem (QLSP). For solving QLSP, the HHL (algorithm) paper mentions any polylog($\kappa$) quantum algorihm ...
5 votes
0 answers
74 views

What is the total number of qubits required for the Harrow-Hassidim-Lloyd algorithm?

I am fairly new to Quantum Computing and I know a bit of Linear Algebra. I am currently working on the HHL algorithm, I'm having confusion as to how many qubits are actually required in the circuit ...
9 votes
2 answers
382 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 ...
30 votes
3 answers
3k 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 ...
3 votes
2 answers
758 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?
1 vote
1 answer
177 views

No module named 'qiskit.algorithms.linear_solvers'

I am trying to execute HHL algorithm. I referred to the code from https://quantumpedia.uk/quantum-algorithm-4-hhl-algorithm-for-linear-systems-of-equations-40619daec41d. I tried implementing all the ...
4 votes
1 answer
100 views

HHL for the pseudoinverse in Qiskit

The HHL algorithm can be modified to compute the pseudoinverse of $A$ as is shown here and here. Is there anyway to do this in Qiskit without coding the entire thing manually?
2 votes
2 answers
49 views

How to estimate the negative amplitude of multiple qubits?

The probability of measurement is the square of amplitude. After measurement, how to guess the original amplitude of state?? For example, in linear problem, we would like to know the exact solution, ...
1 vote
0 answers
31 views

Can I get valid solution with HHL algorithm even if the QPE is not completely correct?

Can I get the valid solution of linear problem using HHL algorithm even if the QPE is not completely correct? My example is $A=diag(0.5, 0.2, 0.3, 0.6)$, so the solution is $[0.5, 0.2, 0.3, 0.6]$ ...
2 votes
2 answers
400 views

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

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

Role of qubit registers in HHL circuit

I'm trying to figure out how the HHL algorithm works from reading Dervovic's paper and IBM's tutorial. Dervovic shows the following HHL circuit in Fig 5: whereas IBM uses: I want to clarify the IBM ...
4 votes
1 answer
90 views

Can we efficiently run HHL on a computational basis state without violating the BBBV theorem?

Recall that the linear systems algorithm of Harrow, Hassidim, and Lloyd purports to find a quantum state $|x\rangle$ proportional to $\vec x$, where $\vec x$ satisfies $A\vec x=\vec b$. In addition ...
3 votes
0 answers
60 views

How to achieve the controlled rotation in the HHL algorithm

I'm trying to implement the HHL algorithm generally for any 2 x 2 hermitian matrix, but I'm having trouble with the implementation of the controlled rotation of the ancilla qubit. I've read very many ...
4 votes
2 answers
871 views

Why was the HHL algorithm removed from the qiskit library?

I am a beginner in quantum computing and I was meddling with HHL algorithm for my class project. Turns out the linear solver algorithms (mainly HHL algorithm) are removed from qiskit library (see ...
4 votes
0 answers
47 views

We take the reciprocal $\lambda^{-1}$ of eigenvalues in HHL - but what's stopping us from raising them to a positive exponent $\lambda^m$?

The HHL algorithm generally can be thought of as diagonalizing our matrix $A$ with the quantum phase estimation algorithm, and applying a specific function $f(\lambda)=\lambda^{-1}$ to the eigenvalues ...
3 votes
1 answer
150 views

Determining the number of qubits to represent the eigenvalues in HHL algorithm?

I am trying to understand how well the HHL algorithm would scale. Therefore my first inquiry is how the number of qubits scale with the size of the problem ie the size of the Matrix A in the linear ...
6 votes
2 answers
861 views

How to draw Qiskit's HHL algorithm as a circuit?

Qiskit Aqua has a module that implements the HHL algorithm: https://qiskit.org/documentation/stable/0.32/stubs/qiskit.aqua.algorithms.HHL.html How does one draw the implemented circuit? *** EDIT *** I'...
1 vote
1 answer
89 views

How to read the result of this HHL algorithm circuit output?

I have computed the final states of a 1-qubit HHL circuit using initial conditions $$ A|x\rangle = |b\rangle$$ $$\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix} |x\rangle = \begin{bmatrix} \...
2 votes
2 answers
121 views

What is the algebraic result of the matrix exponential operation $e^{i A}|b\rangle$?

The circuit for the HHL algorithm looks as follows: I am uncertain what is the algebraic operation of the matrix exponential $e^{i A}$ on $|b\rangle$? If $$|b\rangle = b_0|0\rangle + b_1|1\rangle$$ ...
7 votes
0 answers
306 views

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 ...
4 votes
1 answer
73 views

How is $\vec{b}$ loaded in terms of computational basis if we don't know the eigenvectors $\vec{u}_i$ of A?

In HHL, vector $\vec{b}$ is assumed to be decomposed in the eigenbasis {$u_i$}$_{i=1}^n$ of a Hermitian matrix $A$. However, as we do not calculate explicitly the eigenvectors $u_i$ in course of the ...
1 vote
1 answer
71 views

How to construct the matrix in the equation $Ax=b$ in the HHL algorithm?

I am trying to apply the HHL algorithm to a fluid mechanics problem, but I can't figure how to construct the matrix for the equation $ Ax=b $. Specifically, I am trying to reproduce this paper in code ...
1 vote
0 answers
97 views

HHL phase estimation step

I have got an HHL circuit that looks as follows: In the phase estimation part we are trying to find the eigenvalues of the matrix A. But what is the role of the piece of circuit hightlighted below? ...
3 votes
1 answer
105 views

HHL Algorithm: How to compute the signs of the solution vector

Let us assume we have used the HHL algorithm to approximately prepare $|x\rangle$, where $|x\rangle$ is a normalized quantum state corresponding to $\vec{x}$, the solution of a system of linear ...
0 votes
0 answers
81 views

HHL Algorithm for Classical Syndrome Decoding

Classical Syndrome decoding involves finding the e vector for the given syndrome s and Parity Check matrix H i.e., H.e' = s. This while implemented trying HHL Quantum Algorithm gives error after ...
2 votes
1 answer
123 views

Is a diagonal matrix with one non-zero element a measurable observable?

The HHL algorithm prepares the output state $|x\rangle$. However, we cannot efficiently measure the state directly to get its components. Instead, we can construct an operator $M$ to find $\langle x|M|...
0 votes
1 answer
128 views

hhl algorithm output

I'm trying to implement the HHL algorithm for the matrix and vecor as follows: $ A = \begin{bmatrix} 11 & 5 & -1 & -1 \\ 5 & 11 & 1 & 1 \\ -1 & 1 & 11 & -5 \\ -1 &...
3 votes
2 answers
506 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 $\tilde{O}\left(\log (N) s^{2} t_0\right)$. ...
1 vote
1 answer
103 views

Confusing notation in Block-Encoding

I'm reading the lecture notes from Ronald de Wolf and got confused at the part where it introduces Block encoding, specifically by this notation at page 76: More generally we can define an $a$-qubit ...
2 votes
0 answers
50 views

Simulation of algorithms with QFT on a classical computer

In paper The Quantum Fourier Transform Has Small Entanglement the authors showed that strong entanglement of qubits caused by QFT comes mainly from ordering the qubits. QFT itself prepares only weak ...
1 vote
0 answers
57 views

What is the implementation of AQE on classic?

I want run the HHL, but if the matrices are big, I need to use HHL hybrid, but I was not able to found an implementation or pseudo algorithm of AQE or HHL hybrid. Any solution?
4 votes
1 answer
207 views

HHL algorithm: how running the process $N$ times gives the vector components of the solution?

In the HHL paper the authors have mentioned that if the user wants to get all the components of vector $\vec{x}$, one needs to run the process at least $N$ times, where $N$ is the dimension of the ...
2 votes
0 answers
84 views

What useful information can be efficiently extracted from solutions provided by the HHL-algorithm?

The result of the HHL-algorithm is the (amplitude-encoded) solution vector $|x\rangle$. I was wondering, which useful information could be extracted from this solution without loosing the algorithms ...
1 vote
0 answers
144 views

Amplitude Amplification applied to HHL Algorithm

I’m trying to understand and implement the amplitude amplification algorithm described in the HHL paper. I’m using the cirq implementation of the HHL algorithm as my starting point. I have a couple of ...
1 vote
0 answers
104 views

Changing the eigenvalues used in HHL algorithm

For an HHL algorithm implemented exactly as depicted in Figure 2 of this paper by Dutta et. al (https://arxiv.org/abs/1811.01726), how do I go about changing the eigenvalues that they use? Obviously, ...
0 votes
0 answers
181 views

Implementation of conditional Rotation on ancilla qubit in HHL

In HHL algorithm, after we have encoded the eigenvalues onto the clock register qubits, we apply conditional rotation on ancilla qubit such that it changes from $|0\rangle$ to: To achieve this ...
2 votes
1 answer
133 views

I think I’m having endianness problems replicating the results of this paper?

So I am trying to implement the HHL algorithm depicted in this paper titled 'Quantum Circuit Design for Solving Linear Systems of Equations' by Cao et. al (https://arxiv.org/abs/1110.2232v2), but if I ...
8 votes
2 answers
237 views

What exactly 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 ...
3 votes
1 answer
179 views

Is it possible to compute a matrix inverse for an arbitrary matrix within a subroutine?

Suppose I want solve a lower diagonal linear system of equations given in block form by $ \left( {\begin{array}{cccc} I & 0 & \cdots & 0 &0\\ M & I & \cdots & 0 &...
2 votes
1 answer
210 views

HHL for non power of 2 matrix

I'm trying to see how HHL does on a series of matrices coming from a linear problem I'm interested in. These matrices are always square, real, and symmteric. They are not, however, very often exactly ...
4 votes
1 answer
174 views

Representation of a vector in the HHL algorithm

Reading about the HHL algorithm, which is used to solve the equation $Ax=b$, on Wikipedia, they say to represent $b$ as $|b\rangle=\sum_{i \mathop =1}^N b_i|i\rangle.$ I'm assuming $b$ is a vector ...
2 votes
0 answers
75 views

How to extract solution from HHL statevector on qskit?

I want to extract the HHL solution from the StateVector. I am doing this on a $4\times4$ matrix. For a $2$ by $2$ the solution is available in the StateVector $8$, $9$. However for a $4$ by $4$ matrix ...
1 vote
2 answers
224 views

How to extract a solution to HHL from the full statevector?

I am having a difficult time trying to extract an HHL matrix inversion solution from the full system statevector. I have a 32x32 size matrix A and a 32x1 vector b, and I ran HHL on 13 qubits. HHL ...
8 votes
1 answer
226 views

Why is HHL the popular choice to solve QLSP and not the Childs et al. (2017) algorithm?

The Childs, Kthari, and Rolando (2017) (CKS) 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 $...
1 vote
0 answers
49 views

Plots of the result for the qiskit HHL tutorial

Could anyone make explicit the code which gives the two plots at the end of the HHL tutorial. They show the results of the jobs on a real IBM quantum device.
4 votes
0 answers
124 views

Is the exponential speedup and output $\langle x|M|x\rangle$ in contradiction in HHL algorithm?

Isn't the exponential speedup and the output $\langle x|M|x\rangle$ in contradiction in HHL algorithm? How can we print the solution vector $|x\rangle$ without losing the exponential speedup?
3 votes
1 answer
132 views

Which observable $M$ provides the Absolute Average of a statevector?

My question should be fairly simple, though I did not find an answer to it here or anywhere else. I have been working on an algorithm which, similarly to the HHL algorithm, provides a state $|x\rangle$...
5 votes
1 answer
459 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 ...
4 votes
1 answer
300 views

HHL - the result is correct for one matrix but wrong for another one

I tried to run HHL algorithm in new version of Qiskit (0.29). Firstly I tried to solve a diagonal system 4x4 with matrix [[1,0,0,0],[0,2,0,0],[0,0,3,0],[0,0,0,4]] ...
0 votes
1 answer
203 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. ...