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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How to use QubitConverter and ParityMapper to do chemistry simulation?
After viewing the question: Hamiltonian & QubitMappingType cannot be imported from qiskit.chemistry.core,
I also want to implement the same Jupyter notebook and browsed the response below.
However,...
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Solving Hamiltonian eigenvalue problem
I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. ...
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The relationship between max-cut and max independent set
I know that Max Independent set problems are not equivalent to Mac-Cut problems. The cost function of Max-Cut can be described by the classical Ising model
$
H = \frac{1}{2} \sum_{i,j\in E}(1 - Z_i ...
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Converting sigma model Hamiltonian into a quantum circuit
I would like to ask some specific questions about a paper I'm reading. In 1903.06577, authors are playing with the sigma model hamiltonian, and they are showing how they constructed the time evolution ...
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The phase of the eigenvalue for the LCU method and randomized product formula
If we implement the linear combination of unitary(LCU), $\tilde{U}=\sum_i p_i U_i$, onto its eigenstate, $|\psi \rangle$, we can express its eigenvalue as $re^{i\theta}$, where $\sum_ip_i=1$, $0\leq ...
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How to define the phase obtained from the quantum phase estimation when using the randomized product formula as the Hamiltonian simulation?
Several randomized Hamiltonian simulation methods are developed in recent years. For example, the qDRIFT method is developed by Earl Campbell https://arxiv.org/pdf/1811.08017.pdf, or the randomized ...
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From Hamiltonian to the Pulse Sequence
I know my Hamiltonian:
I know which unitary I want to have. Suppose that I want to have this matrix from Hamiltonian:
How can I find the pulse sequence to have this matrix from the Hamiltonian?
What ...
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Convert Hamiltonian to Ising Formulation or QUBO
I have a tridiagonal Hamiltonian matrix that I need to convert to QUBO or Ising format to use D-Wave's quantum annealing solvers. For a generic tridiagonal:
\begin{pmatrix}
a_1 & b_1 \\
c_1 & ...
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Confirming locality of a Hamiltonian through decomposition
I was trying to understand Trotterization. The given Hamiltonian is decomposed into a sum of $k$-local Hamiltonians which can be exponentiated in $O(1)$ gate complexity. After which the Trotter ...
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Is it essential to apply Quantum Singular Value transformation twice for Hamiltonian simulation?
I have been reading the paper A Grand Unification of Quantum Algorithms and I need clarification on the Hamiltonian simulation algorithm provided in the paper on page 23. . In procedure part point 2 ...
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How to do rotations along arbitrary multi-qubit basis
I was trying to implement Trotterization for a $k$-local Hamiltonian simulation using qiskit. For this, say I want to apply $e^{\lambda \sigma^1_z \otimes \sigma^2_z \otimes \sigma^3_z}$ (this being ...
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simulating quantum system on quantum simulator vs classical computer
Suppose I want to simulate a quantum system. Is it true that simulating this on quantum simulator exponential faster than classical computer for arbitrary quantum system and why? If so, does this mean ...
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Can I combine controlled unitaries in IPE?
Suppose I have two controlled unitary ($U_3$) gates which implement controlled $e^{-iHt_1}$ and controlled $e^{-iHt_2}$ (they share the same control qubits), where $H$ is the same Hamiltonian. My ...
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Question about performing VQE with an embedded Hamiltonian
Say I have a physical Hamiltonian $\mathcal{H}$ which is $D$-dimensional and I encode it into a larger matrix $M$ which is $\Delta$-dimensional. In the cases I care about $\Delta$ is strictly greater ...
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How to convert Pennylane decompose_hamiltonian to Cirq PauliString?
I have a matrix that I would like to decompose into a Pauli String. Pennylane's
qml.utils.decompose_hamiltonian does this and returns a list of coefficients and a list of operations representing the ...
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Measurement of a Hamiltonian with Junk States
TL;DR: If I encode a Hermitian matrix in a non-Hermitian matrix, does that matter for VQE? Is the entire $H$ required to be Hermitian or just the parts corresponding to physical states?
Say I have a ...
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How to determine appropriate time evolution for phase estimation algorithm?
In phase estimation algorithms, we have $U|\psi\rangle = e^{2\pi i\theta}|\psi\rangle$, where $|\psi\rangle$ is an eigenvector and $ e^{2\pi i\theta}$ is the corresponding eigenvalue. Since $U$ ...
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What operators preserve antisymmetry?
I would like to understand better what kind of operators maintain antisymmetry as explained in Quantum simulation of chemistry with sublinear scaling in basis size:
Evolution under the Hamiltonian ...
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Did we entangle the qubits in phase estimation algorithm?
This is a follow-up question to my earlier post about the phase estimation algorithms. From the Qiskit tutorials of QPE and IPE, the qubit $q_1$ represents the physical system on which $U$ operates ...
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State evolution and phase correction for iterative phase estimation
I'm learning about the iterative phase estimation (IPE) algorithm from the qiskit textbook. Here's a circuit I generated to implement this algorithm on a random single-qubit Hamiltonian. Instead of ...
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Qiskit: PauliTrotterEvolution for Hamiltonian simulation
Context:
I have H (in qiskit.opflow notation).
I want a circuit which does exp(-itH)
Solution attempt:
I think qiskit.opflow.evolutions.PauliTrotterEvolution should do the trick.
Also, I found this ...
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Measurement in the Pauli X basis
A qubit in the state $|1\rangle + |0\rangle$, normalized, is time-dependent, right? So it sweeps around the equator of the Bloch sphere at a frequency proportional to the energy gap according to ...
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Implementing controlled unitary gate in QPE
I'm reading the Qiskit tutorial on quantum phase estimation. In this tutorial, the controlled unitaries on the diagram are denoted as $U^{2^{t-1}}, ..., U^{2^{0}}$:
In the actual quantum circuit, ...
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Paradox on the evolution direction in controlled Hamiltonian simulation for Quantum Phase Estimation
Suppose we want to perform Quantum Phase Estimation over a Linear Combination of Unitaries Hamiltonian. One of the most efficient ways to do so is to use qubitization:
\begin{equation}
Q=(2|0\rangle\...
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Simulating the Ising-like model as a quantum circuit
We are interested in simulating the 1d Ising model Hamiltonian using a Quantum Circuit (QC). A similar question was posted before with no answers. Here we will assume, for simplicity, 3 lattice sites ...
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Why does quantum phase estimation complexity scale with maximum representable energy?
In Quantum simulation of chemistry with sublinear scaling in
basis size Ryan Babbush and other authors from Google Quantum team argue, when talking about performing Quantum Phase Estimation in 1st ...
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Applying QPE on a large matrix on amazon-braket
I'm running a QPE algorithm on the amazon-braket but it can only apply on a 22 or 44 matrix, when I want to expand it into a 5*5 or more, it will come an error. As I know, there is no theoretical ...
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How is Quantum Phase Estimation useful for simulating dynamics of a many-body system?
I am quite aware of the Quantum Fourier Transform (QFT) as well as the very closely related topic of Quantum Phase Estimation (QPE). The latter is usually motivated as follows:
Given a unitary $U$ and ...
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Could the Hamiltonian of a 2x2 Rubik's Cube be simulated with a NISQ device?
Consider the four cells on each of the six faces of the 2x2x2 Rubik's cube (the pocket cube). We can construct and simulate a quarter-turn Hamiltonian as below. $^*$
Let $\langle F_1,U_1,R_1\rangle$ ...
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In the HHL algorithm, does the controlled unitary depend on the Hermitian matrix coefficients?
In HHL algorithm, does the controlled unitary (Hamiltonian simulation part of Quantum phase estimation) depend on Hermitian matrix coefficients and how?
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Eigenvalues and energy levels of 1D Heisenberg model using real Quantum Computers?
The 1D Quantum Heisenberg model is
$$H_\textrm{Heisenberg} = -~J \sum_{\langle i\ j\rangle} \hat{S}_{i} \cdot \hat{S}_{j}$$
where each spin is an operator.
For simple cases, for example, a system with ...
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Why does Hamiltonian simulation seek to find the energy minimum, if eigenvalues of unitaries are always unimodular?
I know I am wrong here and trying to find out where I am making a logical mistake. I'd appreciate it if you can help me untangle.
A. We know that the eigenvalues of Unitaries are all unimodular (...
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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
\...
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How to find minimum time needed for Hamiltonian evolution?
Database search can be looked upon as Hamiltonian evolution, with kinetic and
potential energy operators. Let the evolution follow the Schrodinger equation:
$$i\frac{d}{dt}|\psi⟩= H|ψ⟩$$
with $H = E|s⟩...
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Nyquist–Shannon sampling theorem for Quantum Evolution
In classical digital signal processing one can try to identify the dynamics of a system by sampling its evolution from an initial time $t_0$ to a final time $t_1$. Sampling $N$ times results in a ...
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Question on Aharanov and Ta-Shma (ATS)'s Sparse Hamiltonian Simulation notation
In the equations in section 3.4.2 of Aharonov and Ta-Shma's paper (pdf, arxiv abstract), they define the operator: $$T_1:|k,0\rangle\rightarrow|b_k,m_k,M_k,\tilde{A_k},\tilde{U_k},k\rangle,$$ where $...
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How can extract reduced dynamics of a bipartite system from unitary evolution in quite
Let us assume that I have a bipartite system $A\otimes B$ and an initial product state undergoing some evolution $H^{AB} = H^A+H^B+V^{AB}$, which is time independent. I want to simulate the reduced ...
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Can we use quantum phase estimation to learn anything about the dynamics of puzzles like the Rubik's cube?
Introduction
Consider a state $\vert\psi\rangle$ such 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" ...
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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 ...
