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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Implementation of the Phase Estimation algorithm
I've been working on implementing quantum phase estimation in Qiskit for a $2^n \times 2^n$ Hamiltonian as part of my bachelor project, I'm using Trotterization as my Hamiltonian simulation of choice ...
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Are inconsistent results between different VQE runs justified?
I made this post a while ago, where I learned I could use qiskit's VQE to calculate (or approximate) the Transverse Field Ising Hamiltonian and other similar Hamiltonians. After working with my code ...
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What do the numbers in the Ising sampleset mean?
I am trying to create a portfolio optimization with the DWave Quantum Computer. I wrote some code trying to somehow reconstruct the following Ising model paper:
Ai ...
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From QUBO problem to quantum circuit [duplicate]
Can you describe how to convert some QUBO problem $$f(x_1, ..., x_n) = \sum\limits_{1 \le i < j \le n} \alpha_{i,j} \, x_i \, x_j$$ into its equivalent quantum circuits? Is it preferrable to start ...
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Prove that any Hermitian Matrix is a real linear combination of Pauli operators [duplicate]
This is an important result in Quantum Computing because it means that the Hamiltonian of a Quantum System can be encoded as a sequence of real numbers and their corresponding Pauli Operator.
How do ...
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Calculating the ground states of an Ising Hamiltonian on a real quantum computer
I have followed this tutorial and based on it, I've written the following function in qiskit, which can explicitly calculate the ground states of a transverse-field Ising Hamiltonian.
...
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How to construct a Hamiltonian for an ensemble of atoms interacting with each other?
How to construct a Hamiltonian for an ensemble of atoms interacting with each other?
For example if the one atom hamiltonian can be written as:
$$\hat{H}=\left(\begin{matrix}0&\Omega_p(t)&0\\\...
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Why is the time ordering omitted in the trotterised version of the time-dependent evolution operator?
The unitary evolution of a time-dependent hamiltonian is given by the time-ordered matrix exponential
$$\begin{aligned}
U(t)&=\mathcal T\exp\left[-i\int_0^tH(\tau)d\tau\right]\\
&=I-i\int_0^td\...
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Finding the norm of a Hamiltonian
I am experimenting with https://journals.aps.org/prx/pdf/10.1103/PhysRevX.8.041015 and in equation 36 I find that they use the norm of the Hamiltonian.
Is there a clean way to compute it, or an upper ...
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How to get eigenvectors of Hamiltonian in OpenFermion
In OpenFermion you can create a Hamiltonian in terms of creation and annihilation pretty easily:
...
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How do we determine the direction of a time evolution $e^{-itH}$ on the Bloch sphere?
I have a question about the direction of time evolution on a Bloch sphere: suppose I'm performing a unitary time-evolution $\exp(-iHt/\hbar)$ for a single qubit, then on the Bloch sphere it ...
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Has any analogue quantum simulator showed quantum advantage yet?
Quantum advantage/supremacy was achieved by Google using a quantum computer and more recently by Pan Jianwei's group using photons. So I was wondering, has any analog quantum simulator showed quantum ...
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Why are diagonal Hamiltonians considered classical?
I've been following UT QML course (http://localhost:8888/tree/UNI/PHD/UT-QML) and during their lecture on the Ising hamiltonian, they point out that the hamiltonian of an Ising model without a ...
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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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1answer
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VQE - How to get from expectation value to eigenvalue?
In VQE, for a single-qubit Hamiltonian, I can use a standard ansatz to make a state $\psi$ and use two products to compute the expectation value $\langle\psi|{\cal H}|\psi\rangle$. As I vary the ...
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How to construct the two qubit gate generated by the Hamiltonian $H= X\otimes X + Y \otimes Y + Z \otimes Z $?
I know that the two qubit gate generated by $H=X\otimes X$ is $\exp\{-\text{i}\theta X\otimes X\}=\cos{\theta} \mathbb1 \otimes \mathbb1 - \text{i} \sin{\theta} X \otimes X$, where $X$ is the $\...
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Fermionic commutation relation using Jordan-Wigner transformation
How to show in detailed steps that Fermionic annihilation and creation operators under Jordan-Wigner transformation satisfy the Fermionic commutation relation
$\{\hat{a}_i,\hat{a}_j\}= \{\hat{a}_i^\...
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Why does joint ground state not change under action of beam splitting unitary operator?
How can one show that $\hat{U}|00\rangle=|00\rangle$ where $\hat{U}=e^{-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)}$ and $|00\rangle$ is the unique joint zero eigenstate of the ...
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If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?
How can one show that $\hat{U}^\dagger\hat{a}\hat{U}$ (with $\hat{U} =e^{-i\hat{H}t}$) involves only linear orders of the ladder operator, when $H$ is the general quadratic Hamiltonian $(\hat{H} = \...
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2answers
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Exponentiating Pauli matrices using trapped ion native gates (single-qubit rotations + XX, YY, ZZ)
I'm wondering what are the known/good/standard ways of exponentiating Pauli terms (i.e. constructing circuits, which implement $\exp(i\alpha XIIZYI...)$) using gates supported by trapped ion quantum ...
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Fermionic occupation operator and nearest neighbor Fermionic hopping interaction as a qubit operator
How to express Fermionic occupation operator $(\hat{a}_j^\dagger\hat{a}_j)$ and nearest neighbor Fermionic hopping interaction ($H_h= J\sum_{i=1}\hat{a}_i^\dagger \hat{a}_{i+1}+\hat{a}_{i+1}^\dagger \...
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1answer
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Distant quantum gates between uncoupled qubits
Is there any formalism to perform quantum gates between two qubits (let's say in a superconducting quantum network) to perform a quantum gate between two qubits which are not directly coupled? I want ...
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1answer
135 views
What's the state-of-the-art to calculate $|Ab\rangle$, given a matrix $|A\rangle$ and a vector $|b\rangle$ in QRAM encoding
Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e.
$$|A\rangle= \frac{1}{\|A\|_F}\sum_{i,j=0}^{n-1}{a_{ij}}|i,j\rangle$$
and a vector $b\in \mathbb{R}^...
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Matrix multiplication through Block Encodings
For a project, I want to simulate a matrix multiplication on a simulated quantum circuit. Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e.
$$|A\...
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1answer
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What does the identity operator represent when computing $\langle\varphi|I\otimes Z|\varphi\rangle$?
Consider a single qubit state $|\varphi\rangle$ and a hamiltonian $H = Z$. Evaluating $\langle \varphi | H | \varphi \rangle$ corresponds to a measurement of $|\varphi\rangle$ in the computational ...
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From QUBO matrix to Ising model in Qiskit
Given a general QUBO matrix $Q$ for a quadratic minimization problem, is there a Qiskit way to obtain the Pauli gate list or the Ising model for it? A related question is Qiskit: Taking a QUBO matrix ...
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Trotter error for bosons in various encodings
Mapping second-quantized bosonic modes onto qubits can be done using various encodings. Each of those have their pro et contra. Fewer qubits — more gates, and vice versa. Encoding an $N$-level bosonic ...
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How can I convert exponentials of pauli matrices to circuits of this form in Qiskit?
For example the following circuit is for $e^{-i(Z\otimes Z\otimes Z)\Delta t} $
I know this can even be done without the ancilla qubit, having the CNOTs control the last qubit and applying an RZ on ...
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Is there a systematic way how to generate the Hamiltonian from a given circuit?
If I have a designed circuit to solve a particular problem. Is there a systematic way how to generate the Hamiltonian from it?
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Is $\gamma \in [0,2 \pi]$ or $\gamma \in [0,\pi]$ in $CU1(2\gamma)_{(i,j)} $?
When wanting to find the groundstate of this Hamiltonian with QAOA:
\begin{equation}
H_{C} =\sum_{i }^{n}(1 - Z_{i})/2 + \sum_{\{i,j\}\in \overline{E} } - 2(1 - Z_{i})(1 - Z_{j})/4
\end{equation}
...
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1answer
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Applications of Quantum Computing to Economics
I have recently been interested in the field of 'Econophysics' which as I understand it is the practice of basically applying results of physics in areas such as non-linear dynamics and stochastic ...
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Why QAOA with $p \rightarrow \infty $ gives the optimal solution?
In the QAOA paper, it is shown that the optimal value of the p-ansatz $M_p$ converges to $\max_z C(z)$ as $p \rightarrow \infty$ on page 10. The proof is to relate to QAOA by considering the time-...
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XY Hamiltonian in a 1D Heisenberg Chain
I've been trying to implement the 1D Heisenberg chain (i.e. the XXZ model) on Qiskit but have been having trouble. To recap, the Heisenberg hamiltonian is as follows:
$$H_{XXZ} = \sum^{N}_{i = 1} [J(S^...
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2answers
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Sign problem and stoquastic Hamiltonians
What is the sign problem in quantum simulations and how do stoquastic Hamiltonians solve it? I tried searching for a good reference that explains this but explanations regarding what the sign problem ...
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Cirq.simulate expectation value of a Hamiltonian
I want to simulate the final state of an ansatz in cirq using simulate.
Now I want to calculate the expectation value of a Hamiltonian.
How do I do this? I can only find simulator.run examples in cirq....
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Hamiltonian simulation: how can I incorporate the constant before each term?
I got another follow-up question about Hamiltonian simulation from the previous post: if I perform the controlled time-evolution of the Hamiltonian:
$$
H_{3} = \alpha\ X_1\otimes Y_2 + \beta \ Z_1\...
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1answer
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Gate cancellations in Hamiltonian simulation
I'm a bit confused about in which case the two unitary gates in a quantum circuit could be canceled? I'm reading an example in this paper. In the following diagram, Figure (b) is a simplified circuit ...
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Definitions of $D_y$ gate in Hamiltonian simulation: are they the same?
I'm reading a Hamiltonian simulation example proposed in this paper. From their notation, the operator $D_y$ (sometimes it's called $H_y$) serves the function to diagonalize the Pauli matrix $\sigma_y(...
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1answer
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Simulate Hamiltonians with Pauli operations (controlled time evolution)
I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ...
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Reducing cost of Phase Estimation for Trotterization
Even though Trotterized Hamiltonians have polynomial time scaling directly, the process of quantum phase estimation means that the controlled unitaries $ CU$ scale exponentially with number of ...
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1answer
433 views
How can I simulate Hamiltonians composed of Pauli matrices?
Suppose I want to perform the time-evolution simulation on the following Hamiltonians:
$$
H_{1} = X_1+ Y_2 + Z_1\otimes Z_2
\\
H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2
$$
Where $X,Y,Z$ are Pauli ...
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1answer
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How to get a molecular Hamiltonians in OpenFermion
I want to get a jordan_wigner_hamiltonians of a molecule-ion by using ...
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How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere?
For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
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How close is the history state to the ground state in the Kitaev clock construction?
Consider a standard circuit to Hamiltonian reduction in QMA. For example, refer here (Vazirani's lecture notes) or page 235 of here (survey by Gharibian et al).
The history state of the Kitaev clock ...
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How to implement the Mixer of Quantum Alternating Operator Ansatz for Max-Independent-Set
I am trying to implement the Mixer of the Max-Independent Set from The Quantum Alternating Operator Ansatz.
From this paper: https://arxiv.org/pdf/1709.03489.pdf in Chapter 4.2 page 15 to 17.
For ...
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1answer
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Is there any algorithm that implements wavelet?
Is there any algorithm that implements wavelet (like there is Quantum Fourier Transform)? I've tried looking online, but couldn't find any, I wonder if something like this exists.
Thank you.
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2answers
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How does VQE performs the measurement on a given Hamiltonian?
I'm trying to understand, given a specific Hamiltonian, for example $H = Z\otimes Z+X\otimes Z$, does the VQE algorithm calculates the expectation value of $Z\otimes Z$ first or does it calculates the ...
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1answer
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Intuition behind the construction of an ansatz circuit
I'm learning about the VQE algorithm. When I looked at the declaration in Qiskit I saw you need to pass an ansatz which prepares the state. I looked at some commonly used ansatz functions, e.g. ...
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How precise are BQPSPACE measurements?
This is in a similar spirit to another question I asked here.
Let's say I am given a $k$-local Hamiltonian $H$. We know that $||H|| \leq 1$. Let the ground state be $|\psi_{0}\rangle$, with energy $E_{...
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How precise are BQP measurements?
Let's say I am given a Hamiltonian $H$, whose ground state is efficiently preparable. We know that $||H|| \leq 1$. Let that ground state be $|\psi_{0}\rangle$, with energy $E_{0}$. We also know that ...
