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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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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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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 ...
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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 $6$ ...
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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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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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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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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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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 ...
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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 + [...] + - ...
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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 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 ...
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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}...
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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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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 ...
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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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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 "...