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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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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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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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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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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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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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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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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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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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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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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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 ...
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How to solve QUBO problems in Q#?

Short version: I'm trying to solve a traveling salesman problem very similar to the traveling Santa example here: http://quantumalgorithmzoo.org/traveling_santa/, which is also included in the samples ...
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How do I efficiently implement a POVM using a fixed universal gate set and the ability to measure in the standard basis?

Let's say I am given a Hamiltonian \begin{equation} H = \sum_{i = 1}^{m} H_{i}, \end{equation} where $H$ acts on $n$-qubits, and each $H_{i}$ acts non-trivially on at most $k$ qubits. The eigenvalues ...
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Why VQE gives an incorrect ground state energy for a chain of 4 hydrogen atoms?

I am a bit hesitant to ask this very specific question, as I feel other people need not benefit from it. But since I have struggled for a while, and I think I should get some help. So I am using VQE ...
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How to realize Su-Schrieffer-Heeger model in Qiskit

This is a very specific question, which I try to implement a simple dimerized tight-binding Hamiltonian on qiskit. The model is one dimensional, and defined as $$ H = \sum_{\langle i,j\rangle} t_{ij} ...
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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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Simulate a random quantum state time evolution in Qiskit Aqua

I am trying to evolve a quantum state through a PauliTrotterEvolution in aqua and I'm trying to do so by initializing a random state, by using ...
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XX and YY and ZZ Hamiltonians in vqe

I'm trying to implement a vqe in cirq and I have sort of a brain knot. I have a 4 qubit chain with periodic boundary condition. So in fact a 2x2 qubit grid. Now 2 of them each are coupled. How do I ...
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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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Changing the Basis

I am attempting to use a VQE algorithm to find the ground state of a deuterium nucleus by applying a constructed hamiltonian to an ansatz state with one parameter created by a circuit. While I am ...
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Is there a tool to get the quantum circuit corresponding to a sparse matrix?

If I know a sparse matrix, is there any tool that allows me to get the corresponding quantum circuit directly? If not what should I do? For example,I want to try hamilton simulation and I have 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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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

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 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 ...