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 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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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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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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3answers
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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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1answer
492 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 "...
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1answer
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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}...
11
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1answer
297 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 = ...
11
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1answer
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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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752 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?
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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 ...
9
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1answer
658 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 ...
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921 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 ...
8
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1answer
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Can we simulate the Hamiltonian for the Rubik's Cube with "nth-root of SWAP" gates?

I'm interested in, but confused about, local Hamiltonian simulation. I don't yet have enough intuition regarding even the approach set forth by Lloyd in 1997. I think Lloyd's recipe is to repeatedly ...
7
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3answers
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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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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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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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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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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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Impact of ordering Hamiltonian terms for Trotterization

In Trotterization, the typical Hamiltonian considered is: $$ H = \sum_{p, q} h_{pq} a^{\dagger}_p a_q + \sum_{p, q, r, s} a^{\dagger}_p a^{\dagger}_q a_r a_s $$ Which is then converted into a sequence ...
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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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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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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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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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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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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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1answer
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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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Simulating $e^{iX_1\otimes X_2t}$

I'm trying to figure out the quantum circuit to simulate the time-evolution of a 2-qubit Hamiltonian $e^{iX_1\otimes X_2t}$, where $X$ is a Pauli gate. From this answer, the quantum circuit performs $...
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1answer
132 views

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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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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1answer
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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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1answer
181 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\...
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Can we use quantum phase estimation to count how often we can walk from one Rubik's cube position to another?

Consider a state $\vert\psi\rangle$ as below, which is in a superposition of a difference between a Rubik's cube in a solved state and a Rubik's cube in the "superflip" state. Here, with $8$...
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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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1answer
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Proof that any unitary can be written as $U=e^{-iH}$ with $H$ Hamiltonian with bounded norm

I am looking for a proof that any unitary matrix can be written as: $$U = e^{-iH}$$ where $H$ is some Hamiltonian with bounded norm. That is $$||H||_{2} = O(1).$$
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1answer
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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. ...
4
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1answer
201 views

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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1answer
537 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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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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1answer
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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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1answer
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How to exactly implement Trotter-Suzuki formula on quantum computer

Recently, I am studying some topics related to product formula, and I am curious about how to implement such formula on real quantum devices. The $(2k)$-th order product formula can be witten as \...
4
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1answer
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Hamiltonian simulation for 2 qubit gates

I have been reading this paper, and at the end they have given exact decomposition of Hamiltonian simulation step, where they have decomposed a matrix $A$ into pauli matrices and done the operation $e^...
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1answer
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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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2answers
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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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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1answer
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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
129 views

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: ...