Questions tagged [hamiltonian-simulation]
Hamiltonian simulation is a class of algorithms that, given a Hermitian matrix A, output a quantum circuit implementing an approximation to the unitary exp[iAt].
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Ground state energy estimation - VQE vs. Ising vs. Trotter–Suzuki
Disclaimer: I am a software engineer who is curious about quantum computing. Although I understand some basic concepts, theory and math behind it, I am by no means experienced in this domain.
I am ...
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Number of Qubits Required for Simulation of Caffeine and Penicillin Molecules
I recently read this report from BCG, which stated:
For scientists trying to design a compound that will attach itself to,
and modify, a target disease pathway, the critical first step is to
...
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Estimating errors in Hamiltonian Simulation paper
I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read
$$ ||PA |0\rangle |\psi \rangle - |0\rangle ...
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2answers
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Which representation describes the composite Hilbert space?
Very often in the standard textbooks on quantum mechanics, one finds that the joint Hilbert space of two systems is given by the tensor product of the individual Hilbert spaces. That is, if $H_1$ and ...
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Encoding Binary Data into Quantum Basis
I am working on implementing a paper on QNNs. I have successfully been able to resize a MNIST digit to be able to meet the size of quantum circuit. But I am not clear of how to convert the resized ...
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Ansatz Techniques to Multi-Body Physics Problems
I have been reading this paper: https://arxiv.org/abs/1906.01563v1.
I am wondering: is it possible to use the idea behind quantum circuits to build classical Hamiltonians represented in the same way?...
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Quantum Optimization via Quantum Label Classification in Quantum Circuits
I have been reading Farhi and Neven's paper on quantum neural networks on quantum circuits. I also found an example - albeit not ideal as pointed out by a couple of users - thank you - in here.
...
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2answers
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GridQubit in Cirq vs LineQubit
It's perhaps a very silly question, but in what ways is it advantageous to use GridQubit vs LineQubits to develop quantum circuits? Specially to develop ansatz ? Are GridQubits Cirq's way of ...
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votes
1answer
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How to implement NM Algorithm for Variational Quantum Eigensolver?
First of all: thanks for reading again. I appreciate the feedback I have gotten from this community the past weeks as I started to feel ready to ask questions about quantum computing topics.
I am ...
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2answers
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Hamiltonian simulation in quantum computation
What is the goal of Hamiltonian simulation?
Is it going to simulate a quantum system on a classical computer or quantum computer or none of them?
What is the relationship between a quantum algorithm ...
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votes
1answer
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Simulating a 3-local Hamiltonian Term
This may be a fairly basic question, but in Nielsen & Chuang, the following circuit is given for simulating $\exp\left(-i\Delta t Z_1 \otimes Z_2 \otimes Z_3\right)$:
which uses an ancilla qubit ...
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1answer
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Clarification of a procedure to compute the product of the exponential of two matrices
In trying to understand a method outlined here (page 3, subroutine 1). Consider
$$R_3 =
\begin{bmatrix} 0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} .$$
Let $A$ be a ...
3
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1answer
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Implement a Hamiltonian in O(n) - exercise question
I have the following exercise to solve:
Consider the Boolean function $f(x_1 . . . x_n) = x_1 \oplus \dots \oplus x_n$
where $x_1 \dots x_n$ is an nbit string and $\oplus$ denotes addition mod $2$...
2
votes
1answer
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How to pick number of simulation qubits for finding eigenvalue of fermionic Hamiltonian?
I am having some trouble understanding how the number of simulation qubits are chosen when finding the eigenvalue of a fermionic Hamiltonian.
For the phase-estimation algorithm, is the number of ...
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Circuit construction for Hamiltonian simulation
I would like to know how to design a quantum circuit that given a Hermitian matrix $\hat{H}$ and time $t$, maps $|\psi\rangle$ to $e^{i\hat{H}t} |\psi\rangle$.
Thank you for your answer.
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Simulate hamiltonian evolution
I'm trying to figure out how to simulate the evolution of qubits under the interaction of Hamiltonians with terms written as a tensor product of Pauli matrices in a quantum computer. I have found the ...
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Numerical approximation to eigenstates and their differentials
I am working in Adiabatic Quantum Computing and I have a $6\times6$ Hamiltonian. I have only the symbolic expression for its eigenstates which have complicated expressions in solutions of degree $6$ ...
5
votes
1answer
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Problem with the mathematical formulation of “qubitization”
In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.
The beginning of their abstract is
Given a Hermitian operator $\hat{H} = \langle G\vert \hat{U} \...
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1answer
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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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Is there a way to express the general 4X4 Hamiltonian in some block diagonal form of 2X2 matrices that I can solve, knowing the exact solution of 2X2?
Is there a way to express the general $4 \times 4$ Hamiltonian in some block diagonal form of $2 \times 2$ matrices that I can solve, knowing the exact solution of $2\times 2$?
This is necessary for ...
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votes
1answer
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Introductory resources for learning about quantum Hamiltonians
I am seeking introductory resources which will enable me to answer these questions (textbooks, lecture series, etc.):
Given a simple quantum system, how do I derive its Hamiltonian?
Given a ...
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How do I construct a Density Matrix corresponding to a Hamiltonian?
I have a Hamiltonian and I want to know the corresponding density matrix. The matrix I'm interested in is the one in this question.
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1answer
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Why is this Hamiltonian matrix diagonal?
I've only recently started using density matrices in my work but I am confused with the following code that I have whether I am getting the right matrix:
...
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3answers
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Example of Hamiltonian Simulation solving interesting problem?
Hamiltionian Simulation (= simulation of quantum mechanical systems) is claimed to be one of the most promising applications of a quantum computer in the future.
One of the earliest – and most ...
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votes
2answers
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Is there a Hamiltonian simulation technique implemented somewhere?
I was wondering if there was some code available for Hamiltonian simulation for sparse matrix. And also if they exist, they correspond to a divide and conquer approach or a Quantum walk approach?
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Advantage of simulating sparse Hamiltonians
In @DaftWullie's answer to this question he showed how to represent in terms of quantum gates the matrix used as example in this article. However, I believe it to be unlikely to have such well ...
10
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Hamiltonian simulation with complex coefficients
As part of a variational algorithm, I would like to construct a quantum circuit (ideally with pyQuil) that simulates a Hamiltonian of the form:
$H = 0.3 \cdot Z_3Z_4 + 0.12\cdot Z_1Z_3 + [...] +
- ...
8
votes
1answer
628 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 ...
4
votes
1answer
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How are two different registers being used as “control”?
On page 2 of the paper Quantum Circuit Design for Solving Linear Systems of Equations (Cao et al.,2012) there's this circuit:
It further says:
After the inverse Fourier transform is executed on ...
9
votes
1answer
602 views
How to implement a matrix exponential in a quantum circuit?
Maybe it is a naive question, but I cannot figure out how to actually exponentiate a matrix in a quantum circuit.
Assuming to have a generic square matrix A, if I want to obtain its exponential, $e^{A}...
9
votes
2answers
368 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 ...
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votes
1answer
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Hamiltonian simulation is BQP-complete
Many papers assert that Hamiltonian simulation is BQP-complete
(e.g.,
Hamiltonian simulation with nearly optimal dependence on all parameters and Hamiltonian Simulation by Qubitization).
It is easy ...
14
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1answer
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Obtaining gate $e^{-i\Delta t Z}$ from elementary gates
I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't ...
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1answer
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How are quantum gates realised, in terms of the dynamic?
When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.
In some sense, these are rather mysterious objects, in that they perform "...
