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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Can I use the Lie product formula to simulate the Hamiltonian of an adjacency matrix by using the QPE to take Nth roots of permutation matrices?

I have gotten some great help recently on Hamiltonian simulation, and am interested in using Hamiltonian simulation to explore (classical) random walks on large graphs, but I'm running up against ...
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Wick rotation of the Schrödinger equation

Studying the following paper: https://www.nature.com/articles/s41534-019-0187-2.pdf Trying to figure out how $ E_T$ shows up from (1) and (2). Any suggestion or guidance would be appreciated. We ...
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Is the square-root-of-SWAP for a pair of 4-dimensional qudits isomorphic to two square-root-of-SWAPS for two pairs of qubits?

This may be a very naïve question indicative of a lot of confusion, but I am trying to understand more about Hamiltonian simulation. I'm starting to intuit that the $n^{th}$-root-of-SWAP acting on a ...
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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 ...
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Do VQE and QAOA use the same Hamiltonian?

In this paper, it talks about the 2-local Hamiltonian in the form: $H = \sum_{(u,v)\in E} H_{uv} + \sum_{k \in v} H_k $ It also says the Ising model, Heisenberg model, XY model and QAOA are in the ...
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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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Are these two circuits equivalent in performing controlled time-evolution?

I want to perform the controlled time-evolution of some 2 or 3-qubit Hamiltonian. Say we have this example: $$ H= Z_0\otimes Z_1 + Z_1\otimes Z_2 $$ The circuit performing the time-evolution ...
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QAOA for Binary Optimization

I have reduced an optimization problem to a Binary Integer Linear Programming model as follows: $$\sum_{j=1}^{2^n-1} f(C_j)x_j \rightarrow \max$$ $$\text{subject to} \sum_{j=1}^{2^n-1} S_{i,j}x_j=1\,\,...
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Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product

Given a fermionic Hamiltonian in a matrix form, we can write it as a sum over Kronecker products of Pauli matrices using the Hilbert-Schmidt inner product. However if the same Hamiltonian is given in ...
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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 type of tasks it is possible to solve on a quantum simulator?

In this article, the author claimed that researches from Harvard and MIT created 256 qubits quantum simulator. However, we are not talking about piece of software on a classical computer but actual ...
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How does Qiskit's VQE encode binary digits into a solution from a given Hamiltonian?

So far I have been working with the VQE on different Hamiltonians that happened to have degeneracies, that is there were always at least two different global minima, because the Hamiltonians I was ...
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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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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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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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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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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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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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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 ...