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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Representation of rotation operators $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ about arbitrary axis for $3$ qubits
I was wondering in how to interpret and represent the operator $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ for a 3 qubit system in a circuit using qiskit.
I was thinking I could just perform an ...
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Why does one need a non-commuting Hamiltonian for an algorithm to exhibit “quantumness”
In two places so far, I've heard statements of the sort "... and we need the Hamiltonian to be non-commuting. If not, the algorithm is classical, and we get no benefit from using a quantum computer."
...
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Product of block-encoded matrices
I am trying to understand just the first step of the proof fo Lemma 53 of this paper, with scarce success.
Before starting, let me state this definition:
Definition: Block encoding of operator A.
Let $...
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What is the usefulness of the Suzuki-Trotter formula?
I can't seem to wrap my head around the suzuki-trotter formula. I have seen This answer but I am still confused of the applicability of the formula. Let me explain:
As I understand it Trotterization ...
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Resource recommendation on quantum simulations
I would like to know more about quantum simulations, so as to start on a few standard physical models (maybe particle in a box, harmonic oscillator, etc.) and then build up on more complex things. But ...
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What are the biggest obstacles currently preventing us from solving real world problems defined in terms of quantum simulation?
Quantum simulation (also referred to as Hamiltonian simulation) is defined as follows:
In the Hamiltonian simulation problem, given a Hamiltonian $H$ ($2^n \times 2^n$ hermitian matrix acting on $n$...
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Creating Hamiltonian Simulation Operator in Q#
I am trying to create a unitary operator $U = \sum^{T - 1}_{k=0}$ $|k\rangle$ $\langle k |$ $ \otimes$ $e^{i A k}$ in Q#, where A is a Hermitian matrix. For the beginning, I just want A to be a ...
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How to build a circuit for simulation of a simple Hamiltonian?
Consider very simple Hamiltonian $\mathcal{H} = Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$. It has eigenvalues 1 and -1 with coresponding eigenstates $|0\rangle$ and $|1\rangle$, ...
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Generic matrix exponential in Q#
I am trying to find a way to implement a unitary transformation in Q# that implements e^(iA) where A is a square matrix. However, I only found ways to do this in Q# if A can be represented as a ...
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Example of Hamiltonian decomposition into Pauli matrices [closed]
I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices. Please show me how this is written in Python.
What I have tried ...
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Problem with building quantum circuit for Hamiltonian operation
In the book, Nielsen and Chuang, there is a section on quantum simulation of the quantum search algorithm. Hamiltonion operator is defined as follows-
$$
H = |x\rangle\langle x| + |\psi\rangle\langle\...
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Question Regarding Simulating Hamiltonian With Quantum Circuit
There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/...
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How to convert QUBO problem to Ising Hamiltonian?
According to paper Ising formulations of many NP problems an unconstrained quadratic programming problem
$$
f(x_1, x_2,\dots, x_n) = \sum_{i}^N h_ix_i + \sum_{i < j} J_ix_ix_j
$$
can be expressed ...
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How to convert a qubit hamiltonian to QUBO and vice versa?
This is my hamiltonian. Solving this by hand, Numpy Python package and VQE algorithm gives the minimum energy eigenvalue -2. If we want to find the minimum energy of this hamiltonian with Quantum ...
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What Are The Most Promising Real-World Applications For Quantum Machine Learning
I know this has been asked before in different ways, however, I am interested in something with a degree of clarity and focus not found in other questions.
What I am looking to get is a list of the ...
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1answer
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Two commuting Hamiltonians
Let's say I have 2 commuting Hamiltonians that are not degenerate, I know it means that they a have a common energy basis, yet does it mean that they also have the same ground state? Or is there any ...
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Ansatz state for finding the lowest eigenvalue of a $2^n\times 2^n$ real matrix using VQE
I would like to find the lowest eigenvalue of a $2^n\times 2^n$ real matrix $H$ using the VQE procedure. The measurement part is simple — I just expand $H$ in a sum of all possible $n$-qubit Pauli ...
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Problem with commutation of $e^{-iH_1t}$ and $e^{-iH_2t}$, where $H_1$ commutes with $H_2$
I'm given with a Hamiltonian, $H=H_1+H_2$, where $H_1=\sigma_x\otimes\sigma_z$ and $H_2=\sigma_y\otimes\sigma_y$, and want to built a circuit which will implement $e^{-iHt},t=\pi/6$. We see that as $\...
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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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Questions about the Hamiltonian of a decay
In paper Simulating quantum systems on a quantum computer the author mention in section 3, simulating a decay to obtain the ground state, and give the Hamiltonian for that:
$$H=H_1+H_2 + H_{1I}\otimes ...
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VQE Cirq example
Is my understanding correct that in this example the Hamiltonian measurement is not performed through measuring individual Pauli operators because all its terms are mutually commuting? So, for each ...
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1answer
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Modifying Qiskit Hamiltonian
I'm fairly new to the Qiskit API. I was wondering if I could get some assistance with trying to implement our technique of projecting out a new Hamiltonian from the ground-state Hamiltonian. The ...
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What is the correct sign in the unitary evolution operator of a beam splitter?
I'm a bit confused about which is the correct sign in the unitary evolution operator of a beam splitter.
In paper Digital quantum simulation of linear and nonlinear optical elements author uses the ...
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Constructing a time evolution operator $e^{it H}$ for $H^2=I$
Consider a Hamiltonian $H = \sigma_x \otimes \sigma_z$
Construct the time evolution operator $U(t) = \mathrm{e}^{-\frac{iHt}{\frac{h}{2\pi}}}$ [Hint:Write down the expansion of $\mathrm{e}^x$ and use ...
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Primer for Quantum Circuits and Optimization
I am interested in studying physical systems and trying to build circuits to simulate them. Now, of course, all the systems I could try to work with at simple, toy, systems - and that's fine.
...
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2answers
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Circuit of a very trivial thing
I am writing to double check that if have a hamiltonian of the form $H = I_1 \otimes I_2$, when I seek to find the unitary, $e^{-i\gamma I_1 \otimes I_2}$, there really is no need to convert this into ...
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XX, YY, ZZ circuit representations?
Is there a good primer or set of lectures\examples that show entirely how to take a given matrix and developing a circuit that represents it. I am trying to implement a program to find the lowest ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 = ...
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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:
$$H=|x\rangle\...
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What is “Lindblad Superoperator” in Stochastic Master Equation
I was reading a paper titled "Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations" and was confused about a stochastic master ...
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Exponentiating Hermitian Matrix for use in QPE/HHL
I am currently trying to understand HHL by implementing a very inefficient Qiskit simulation script that performs HHL on an arbitrary hermitian matrix $A$ and a vector $b$.
Because I am currently ...
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1answer
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How is the precision of a quantum simulation algorithm actually proved?
The problem of quantum simulation can be formulated as follows:
Given a Hamiltonian $H$ ( $2^n \times 2^n$ hermitian matrix acting on $n$ qubits), a time $t$ and maximum simulation error $\epsilon$,...
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Ground state energy estimation - VQE vs. Ising vs. Trotter–Suzuki
Disclaimer: I am a software engineer who is curious about quantum computing. Although I understand some basic concepts, theory and math behind it, I am by no means experienced in this domain.
I am ...
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Number of Qubits Required for Simulation of Caffeine and Penicillin Molecules
I recently read this report from BCG, which stated:
For scientists trying to design a compound that will attach itself to,
and modify, a target disease pathway, the critical first step is to
...
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Estimating errors in Hamiltonian Simulation paper
I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read
$$ ||PA |0\rangle |\psi \rangle - |0\rangle ...
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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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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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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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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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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 ...
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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 $2$...
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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 ...
