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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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Hamiltonian Simulation: What's the meaning of t in $\exp(iAt)$?

My main goal is to find eigenvalues of some hamiltonian matrix $A$. When implementing Quantum Phase Estimation, I need to provide my circuit with informations about $A$. From what I have seen so far, ...
Max's user avatar
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Turning Deutsch-Josza into a continuous problem

I am wondering whether anyone has investigated if there is a notion of a continuous oracle. My starting off point is to consider the Deutsch-Josza problem, in which the oracle acts on the state in a ...
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non-stoquastic vs quantum annealer

Why non-stoquastic Hamiltonian is difficult to implement for a quantum annealer? In another way, why stoquastic vs non-stoquastic matters? Thanks.
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What problems in chemistry or materials science could be solved with 100 fault-tolerant qubits?

Background IBM, Infleqtion, QuEra, and other quantum hardware companies have announced roadmaps where they expect to have 100 or more fault-tolerant qubits by the end of the decade. It seems ...
taciteloquence's user avatar
1 vote
1 answer
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Qiskit TimeEvolutionProblem with complex operator

I have been using Qiskit to simulate some oscillators using Hamiltonian simulation. A next step I would like to make is adding dissipation of these oscillators in some way. I think it would be ...
Thick Harry's user avatar
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How to approximate the time-dependent Hamiltonian in quantum adiabatic theory by the non time-dependent Hamiltonian?

Recently, I am learning how to solve the linear equation $A\left | x \right \rangle =\left | b \right \rangle $ using quantum adiabatic theory. In the solving process, people usually need to set the ...
user30173's user avatar
2 votes
2 answers
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Calculating number of CNOT gates in Pauli evolution gate

How to calculate the number of CNOT gates for a Pauli exponentiation for given time? I am performing Trotterization which involves performing Pauli evolution ...
Zee's user avatar
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Particle number expectation value in QuTip

I am learning now to use QuTiP by going through their documentation site. I am trying to understand what does the argument - particle number expectation value in thermal density matrix do? How does it ...
CuriousMind's user avatar
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1 answer
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Qiskit: Evolve TrotterQRTE from Operator

I am trying to implement the method in the following paper: Exponential Quantum Speedup in Simulating Coupled Classical Oscillators using Qiskit. All is good until I call evolve on ...
guest's user avatar
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4 votes
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Are commuting unitary operators related to commuting Hamiltonians?

TL/DR: Can unitary operators: $$U_a=e^{-it(H_{a1}+H_{a2}+\cdots)}$$ and $$U_b=e^{-it(H_{b1}+H_{b2}+\cdots)}$$ commute, even though $[H_{aj},H_{ak}]\ne 0$ and $[H_{bj},H_{bk}]\ne 0$ for all $j,k$? ...
Mark Spinelli's user avatar
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Modeling a chemical reaction when there is a catalyst

I'd like to construct potential energy surfaces (PES) for chemical reactions with and without the presence of a catalyst. Something like this The closest paper that is https://arxiv.org/pdf/2007....
Minh Triet's user avatar
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Complexity of controlled-$U^j$ operations in QPE applied to Hamiltonian simulation

One method to obtain the eigenvalues of a Hamiltonian $H$ is by applying quantum phase estimation to its time-evolution operator $U(t) = e^{-iHt}$. If I want to obtain an eigenvalue to $k$ bits in ...
Solarflare0's user avatar
3 votes
1 answer
270 views

How to simulate low-rank hamiltonian?

I want to implement a unitary $U\,,$ $$U=\text{exp}(-it|u\rangle\langle u|)\,,$$ where $|u\rangle$ is a known state. Are there any methods to do this efficiently?
mingo's user avatar
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1 answer
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Exponentiating a tensor product of operators acting on disjoint qubit registers

Consider a problem of implementing $\operatorname{e}^{i\bigotimes_j O_j}$, where all the $O_j$ terms act on disjoint sets of qubits. Assume that efficient circuits implementing individual $\...
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What is the intuition behind achieving Quantum advantage in simulating non-hermitian dynamics using Quantum computer?

There have been several works on simulating ODE for classical systems like here and here. They are quantum techniques to solve the ODE related to classical systems. A generic methodology is: To solve ...
Manish Kumar's user avatar
2 votes
1 answer
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Do some Hamiltonian simulations require an irreversible process?

I just stumbled upon this research paper https://arxiv.org/abs/2309.16596. They claim to have found a problem which is easy to solve quantumly but hard classically: to find local minima of 2D ...
Matteo's user avatar
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Entanglement generation for commuting Hamiltonian

Consider an $n$ qubit Hamiltonian $H$ given by \begin{equation} H = \sum_{i=1}^{m} H_i, \end{equation} where each $H_i$ is a $k$-local term and it holds that \begin{equation} e^{H} = e^{H_m} \cdot e^{...
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Time-derivatives of Observables over Hamiltonian evolution

I am reading about algorithms to simulate Hamiltonian evolution by means of quantum computers, e.g. a transverse field Ising model. As far as I see one is interested in getting expectation values of ...
Josu Etxezarreta Martinez's user avatar
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What work has been done on hamiltonian simulation of operators that are integrals instead of sums?

I'm looking at a problem where I want to do hamiltonian simulation of an operator integral. That is, I want to implement the unitary $$\mathrm U = \exp[-i \mathrm H t]$$ where $\mathrm H$ is of the ...
Dyon J Don Kiwi van Vreumingen's user avatar
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Spin Hamiltonian to Quantum Circuits and are there any group theory associated with the quantum circuits?

Can we think of Quantum Circuits as another representation to describe the dynamics of a system other than its Hamiltonian? How can we go from the spin Hamiltonian version (for eg: SSH Model ...
CuriousMind's user avatar
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Deriving circuit templates for Hamiltonian simulation

Background I've been reading the paper entitled Some improvements to product formula circuits for Hamiltonian simulation. The authors propose three improvements motivated by phase estimation type ...
Callum's user avatar
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2 votes
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QPE algorithm to the Hubbard Model

I'm trying to perform the Eigenvalue Estimation algorithm to the Hubbard model with two sites, one spin-up fermion and one spin-down fermion, with Qiskit. Given the Hamiltonian: $\hat{H}=u\sum_{i=1}^N\...
Quantum Lele's user avatar
3 votes
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30 views

Why is "reducing Hamiltonian energy" also optimizing a Quantum Machine Learning model?

From what I observed, most hybrid qml architectures surround the ideas of Hamiltonian states, and it seems like our goal to optimize a circuit is to keep energy states as low as possible. But why is ...
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How to define an energy function, for predicting protein structure in quantum computing? [closed]

Let q denote a particular configuration of the protein in a grid, written in the form where $x_i$ and $y_i$ are the $x$ and $y$ coordinate of the $i$th amino acid. For this configuration, how to ...
Pretty's user avatar
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2 votes
1 answer
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How is geometric frustration different from (not) being frustration free?

In the context of Ising models, some Hamiltonians can be described as geometrically frustrated - such as, I think, the antiferromagnetic kagome lattice, as well as a one-dimensional anisotropic, next-...
Mark Spinelli's user avatar
3 votes
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81 views

What properties of a local Hamiltonian are basis-(in)dependent?

Some properties of a Hamiltonian are unique to its spectrum and are basis-independent. For example, I think whether the Hamiltonian's gap remains constant as $n$ goes to infinity, or whether the ...
Mark Spinelli's user avatar
2 votes
1 answer
151 views

If all terms of a local Hamiltonian commute, how hard is it to learn the ground state (energy)?

Suppose we have a $k$-local Hamiltonian with each of $m$ terms acting on $k$ of $n$ qudits of constant dimension $d$: $$H=H_1+H_2+\cdots+H_m.$$ If at least some of the terms don't commute, e.g., if $[...
Mark Spinelli's user avatar
4 votes
0 answers
82 views

How useful is it to know the ground state energy of an arbitrary $k$-local Hamiltonian, if Nature herself can never find such energy?

We know that the $2$-local Hamiltonian problem is (promise) QMA-complete, which under the reasonable assumption that BQP$\subsetneq$QMA implies that no fast quantum algorithm exists to determine the ...
Mark Spinelli's user avatar
3 votes
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75 views

The no fast forwarding theorem and exponential advantage for many body Hamiltonians

When simulating Hamiltonians in first quantization there are $\eta$ particles occupying a grid of $N$ grid points. In the first quantization, you directly discretize the differential operators onto a ...
Cuhrazatee's user avatar
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1 answer
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time evolution of Hamiltonian and observables

I am given the following Hamiltonian on 4 qubits. $$ H = - J_x (X_0 X_1 + X_2 X_3) - J_z (Z_0 Z_2 + Z_1 Z_3) - h\sum_{j=0}^3 X_j + Z_j $$ I have already implemented the time evolution of this ...
Ruebli's user avatar
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0 answers
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How is implemented the hamiltonian simulation of Hermitian operator multiplied by projection

The article "Quantum Topological Data Analysis with Linear Depth and Exponential Speedup" (Ubaru et al) discusses the implementation of the Hamiltonian $\Delta_\Gamma$, named the ...
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How to model the transtition part of dispersive frequency shift in dispersive readout?

The phenomenon of interest is the resonance frequency shift and the larger noise in the measurements inside the intermediate-powered region as shown below (from Fig. 3.3 of Characterisation of ...
Ziyuan's user avatar
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2 votes
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Efficient gate executing the time evolution of a Hamiltonian using Runge-Kutta method

You can find a minimal working example below. In particular, I want to replace the scipy.linalg.expm() matrix exponential by a Runge Kutta time evolution method as ...
ANDREAS kruckenhauser's user avatar
2 votes
0 answers
47 views

Do Aharanov and Ta-Shma treat the entries of a sparse Hamiltonian as edges of a graph?

Background and history The mid-90's to early 2000's work on Hamiltonian simulation saw some pretty rapid advances. Within two years of Shor's algorithm, Lloyd outlined how Trotterization can lead to ...
Mark Spinelli's user avatar
0 votes
1 answer
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How to find explicit gate decomposition of a circuit implementing a unitary using HamiltonianGate()?

I'm new to Qiskit. I am trying to construct a gate from HamiltonianGate(), available on Qiskit. The Hamiltonian in question is: $$H = - \pi\delta(Z_1 - Z_2) + 2\pi J ~ \mathbf{I}_1 \cdot \...
Pratham Hullamballi's user avatar
2 votes
0 answers
67 views

Weird behavior when simulating vacuum Rabi oscillation on QuTip

I've been playing around with vacuum Rabi oscillation on QuTip and found an odd behavior. My Hamiltonian is as follows: $$ H=\omega_n n^\dagger n + \omega_c c^\dagger c -6K(n+n^\dagger)^4 - g(n^\...
wannaqc's user avatar
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1 vote
2 answers
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Notation: Hamiltonian Simulation of Pauli Gates

Let $\sigma^j_x$ describe the following unitary over $n$ qubits: on the $j$-th qubit, it acts as the Pauli $x$ operator; instead, on any other qubit, it acts as the identity. A paper states now that \...
user20374's user avatar
3 votes
0 answers
36 views

Compiling pulses with time dependent $\sigma_x$ and $\sigma_y$ control

I have a Hamiltonian of the form: $$ H = c_x(t) \sigma_x + c_y(t) \sigma_y$$ and I want to compile a pulse "P" that has both $\sigma_x$ and $\sigma_y$ control, with different time dependent ...
Annie's user avatar
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3 votes
1 answer
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Simulating Sparse Hamiltonians: help understanding query complexity bounds

tl;dr: How can I show that $e^k/k^k$ is less than $\epsilon^2/2$ when $k=\Omega\left(\frac{\log(1/\epsilon)}{\log \log(1/\epsilon)}\right)$, where $k,\epsilon\in \mathbb{R}$ and > 0? Context: Berry ...
muru's user avatar
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What is the closest experimental platform to $H=\sum_{ij} J^X_{ij}X_iX_j+J^Y_{ij}Y_iY_j+J^Z_{ij}Z_iZ_j$?

Consider the Hamiltonian $$H=\sum_{ij} J^X_{ij}X_iX_j+J^Y_{ij}Y_iY_j+J^Z_{ij}Z_iZ_j$$ where $X,Y,Z$ are Pauli spin operators and $J_{ij}^\alpha$ are arbitrary couplings that can be positive and ...
Nichola's user avatar
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4 votes
2 answers
290 views

When is it justified to have an oracle in a quantum algorithm?

I've always been confused as to when a quantum algorithm is allowed to have an oracle and what kind of a function the oracle can have. For instance, I know in Hamiltonian simulation algorithms, you ...
confusion's user avatar
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1 vote
0 answers
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Problems with Exact Diagonalization Implementation in Quantum System Evolution

I've been trying to implement exact diagonalization to study the time evolution of quantum states and subsequently compute local magnetizations. I have written functions for evolving the state and ...
Hakan Akgün's user avatar
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0 answers
26 views

Difference in the Order of Applying Quantum Gates in Qiskit

I've been experimenting with Qiskit and came across a scenario where I'm uncertain about the effects of the order in which I apply quantum gates. Imagine we have the Hamiltonian Which one would be ...
Hakan Akgün's user avatar
1 vote
0 answers
374 views

Applying Trotterization to a Hamiltonian for Time Evolution in Qiskit

I'm currently working on a project where I need to simulate the time evolution of a quantum system using Qiskit. The Hamiltonian of my system is given by: $$H = -J \sum_{j=1}^{N-1} (\sigma_j^x \...
Hakan Akgün's user avatar
5 votes
2 answers
1k views

What does Qubitization mean?

I was listening to an advanced lecture about quantum computing, when the professor introduced a chapter called "Qubitization and the quantum singular value transform", but never really cared ...
Lagrange's user avatar
1 vote
1 answer
188 views

How can I get a time evolution operator for imaginary time?

I want to implement time evolution operator of a hamiltonian in qiskit. I am using circ.hamiltonian(H,time,list(range(N))) to get the circuit. This method does not ...
Cheshta Joshi's user avatar
1 vote
0 answers
22 views

Intuitive explanation on dependence of Hamiltonian simulation on norm?

Suppose I have two Hamiltonians, $H_1$ and $H_2$, that I want to simulate for time $T$. If $\|\|H_1\|\|>\|\|H_2\|\|$, why is it more costly to simulate $H_1$ compared to $H_2$? Is there an ...
confusion's user avatar
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2 votes
1 answer
83 views

How many eigenstates are accessible in polynomial time?

A result of Hamiltonian complexity theory by Poulin et al. shows that only a small fraction of the volume of Hilbert space can be reached in polynomial time for any physical system or quantum computer....
Dr. T. Q. Bit's user avatar
1 vote
0 answers
11 views

From what distribution is QuTip's rand_herm sampled from?

I am trying to figure out how QuTip samples Hamiltonians using its rand_herm function. It seems to use SciPy's sparse.rand ...
Silly Goose's user avatar
1 vote
1 answer
73 views

How can I simulate the following 2×2 Hamiltonian $e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}$?

How can I simulate the following 2×2 Hamiltonian $$ e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}}|\Psi\rangle$$ ie. how to rewrite that matrix exponential in terms of other, well-used ...
James's user avatar
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