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].
43
questions
1
vote
1answer
29 views
Strange Behaviour of MeSolve, McSolve
I'm using Qutip to plot some basic two level dynamics using hamiltonians with a temporal envelope defined as the sum of two error functions, designed to make it more representative of experimental ...
1
vote
0answers
56 views
Openfermion to qiskit
Is there a direct way to go from an object generated in openfermion to objects usable in Qiskit. I can't find anything about any plugin.
It's not too hard to translate into pyQuil and then to Qiskit ...
3
votes
1answer
147 views
Proof that most Hamiltonian evolutions are not efficiently approximable by quantum circuits
How to rigorously prove that finite Hamiltonians (for $n$-qubit systems), in general, are not efficiently$\dagger$ simulable (in the Hamiltonian simulation sense) using $\mathrm{poly}(n)$ number of ...
1
vote
3answers
88 views
Compiling the Pauli-Z operator to the Rz operator for Hamiltonian simulation
I saw a tutorial on this long ago, but lost it. I know that the Pauli-Z operator compiles to Rz, but how? Here are the steps I remember:
First, we have to solve for $U(t)$ in the Schrodinger equation ...
9
votes
1answer
94 views
Where does precisely the dificulty in exponentiating a Hamiltonian $H$ in the quantum simulation problem lay?
I've read in the Nielsen's, Chuang's "Quantum Computation and Quantum Information":
Classical simulation begins with the realization that in solving a simple differential equation such as $dy/dt = ...
2
votes
2answers
119 views
Nielsen and Chuang, Exercise 6.12: How to simulate the specific Hamiltonian in the search algorithm by the Oracle gates?
In Chapter 6 of "Quantum Computation and Quantum Information" Textbook by Nielsen and Chuang, Exercise 6.12:
Exercise 6.12: (Alternative Hamiltonian for quantum search) Suppose:
$$H=|x\rangle\...
3
votes
0answers
84 views
What is “Lindblad Superoperator” in Stochastic Master Equation
I was reading a paper titled "Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations" and was confused about a stochastic master ...
1
vote
0answers
40 views
Exponentiating Hermitian Matrix for use in QPE/HHL
I am currently trying to understand HHL by implementing a very inefficient Qiskit simulation script that performs HHL on an arbitrary hermitian matrix $A$ and a vector $b$.
Because I am currently ...
3
votes
1answer
136 views
How is the precision of a quantum simulation algorithm actually proved?
The problem of quantum simulation can be formulated as follows:
Given a Hamiltonian $H$ ( $2^n \times 2^n$ hermitian matrix acting on $n$ qubits), a time $t$ and maximum simulation error $\epsilon$,...
9
votes
1answer
160 views
Ground state energy estimation - VQE vs. Ising vs. Trotter–Suzuki
Disclaimer: I am a software engineer who is curious about quantum computing. Although I understand some basic concepts, theory and math behind it, I am by no means experienced in this domain.
I am ...
5
votes
1answer
257 views
Number of Qubits Required for Simulation of Caffeine and Penicillin Molecules
I recently read this report from BCG, which stated:
For scientists trying to design a compound that will attach itself to,
and modify, a target disease pathway, the critical first step is to
...
3
votes
0answers
30 views
Estimating errors in Hamiltonian Simulation paper
I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read
$$ ||PA |0\rangle |\psi \rangle - |0\rangle ...
3
votes
2answers
84 views
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 ...
1
vote
0answers
38 views
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 ...
1
vote
0answers
16 views
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?...
3
votes
0answers
108 views
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.
...
1
vote
2answers
70 views
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 ...
4
votes
1answer
152 views
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 ...
0
votes
2answers
90 views
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 ...
6
votes
1answer
55 views
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 ...
5
votes
1answer
45 views
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
votes
1answer
260 views
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
34 views
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 ...
5
votes
0answers
148 views
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.
11
votes
3answers
498 views
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 ...
3
votes
0answers
40 views
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
278 views
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} \...
5
votes
1answer
133 views
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\...
2
votes
2answers
260 views
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 ...
4
votes
1answer
154 views
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 ...
2
votes
2answers
1k views
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.
1
vote
1answer
356 views
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:
...
6
votes
3answers
345 views
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 ...
7
votes
2answers
272 views
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?
10
votes
1answer
240 views
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 ...
11
votes
2answers
225 views
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
679 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
106 views
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
684 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
432 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 ...
14
votes
1answer
456 views
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
votes
1answer
489 views
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 ...
11
votes
1answer
383 views
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 "...
