# 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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### 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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### What is an example of how a Hamiltonian can be decomposed in terms of Pauli matrices?

I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices. I would prefer an option to do this in larger than 2 dimensions, ...
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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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### 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 ...
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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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### 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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### Standard to select base hamiltonaian for Adiabatic quantum computing

I'm learning about connection between QUBO and The Ising Model. It says Take the base Hamiltonian of an adiabatic process as $\sum_i \big(\frac{1-\sigma_i^x}{2}\big)$ to implement Hamiltonian for ...
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### What are examples of Hamiltonian simulation problems that are 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 ...
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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 ...
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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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### 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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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 $\... 2answers 128 views ### What is the correct sign in the unitary evolution operator of a beam splitter? I'm a bit confused about which is the correct sign in the unitary evolution operator of a beam splitter. In paper Digital quantum simulation of linear and nonlinear optical elements author uses the ... 2answers 120 views ### Fermionic occupation operator and nearest neighbor Fermionic hopping interaction as a qubit operator How to express Fermionic occupation operator$(\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 \...
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 ...