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11 votes
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Ground state energy estimation - VQE vs. Ising vs. Trotter–Suzuki

In each of the examples you mentioned, the task breaks very roughly down into two steps: finding a Hamiltonian that describes the problem in terms of qubits, and finding the ground state energy of ...
Chris Granade's user avatar
5 votes
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Jordan-Wigner Transform and Trotterization: which goes first?

Short answer: When you do the Jordan-Wigner transformation, you essentially insert a linear combination of tensor products of Pauli matrices for each fermionic creation and annihilation operator. As ...
cheetah's user avatar
  • 433
5 votes
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Is it possible to implement any random Hamiltonian using quantum circuit

I think you are asking whether a quantum computer can efficiently simulate the evolution of any Hamiltonian, as long as the Hamiltonian is represented by a hermitian matrix. There are some details ...
Mark Spinelli's user avatar
3 votes
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Calculation of Trotter-Suzuki error bound

There are two aspects of your question that I will address separately: the computational and analytical. I believe the underlying issue is the same, but I will nonetheless address them separately. ...
Jacob's user avatar
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3 votes

What is the difference between Trotter, Lie-Trotter and Trotter-Suzuki approximations?

I believe they all refer to the same methods, which is to expand the unitary evolution into small steps. The most basic one is the following: $$\left[\exp\left(-\frac{it}{n}H_1\right) \exp\left(-\frac{...
sailx's user avatar
  • 355
2 votes

Investigating the scaling of the error of a Trotter-Suzuki-approximation

You can use the Baker–Campbell–Hausdorff formula that states that for $e^Ae^B = e^C$ (assuming $e^A,e^B \approx I$) $C$ is given by: $$C = A + B + \frac12[A, B] + \frac1{12}[A, [A, B]] + \frac1{12}[B, ...
Eelvex's user avatar
  • 261
2 votes
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Confirming locality of a Hamiltonian through decomposition

As long as you tweak your condition to require strict inequalities, $$ \exists \sigma_{\gamma_1\dots\gamma_n} \text{ s.t } c_{\gamma_1\dots\gamma_n} > 0 \tag{1} \\ |\{i|\gamma_i > 0\}| > k $$...
forky40's user avatar
  • 7,183
2 votes

How to perfrom a time-dependent Hamiltonian simultation using the Trotter-Suzuki formula?

There's really not any difference. Imagine I'm trying to simulate a Hamiltonian $$ H(t)=f(t)H_1+g(t)H_2 $$ from time $t=0$ to $T$. I'm going to break this down into $N$ little time steps $\delta=T/N$. ...
DaftWullie's user avatar
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2 votes
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How to implement Hamiltonian $0.01Z$?

I found the answer. The R_x and R_z gates do exactly that. Decomposing the Pauli Trotter Evolution funktion helped answer that.
Fabrice Schöneberger's user avatar
2 votes

How to implement Hamiltonian $0.01Z$?

If I got it well, to implement what you want in Qiskit you can do the following: ...
SimoneGasperini's user avatar
2 votes

Time evolution of Hamiltonian

I found out, that one can do that using the $Z$-rotations instead of $X$-rotations because the $Z$-rotation is already diagonal. that makes it way easier to see what $e^{\sigma^{z}}$ should be. So one ...
Ruebli's user avatar
  • 85
1 vote

Qiskit TimeEvolutionProblem with complex operator

You can use quantum imaginary time evolution (QITE) techniques to simulate $e^{Ht}$. Qiskit algorithms package contains ImaginaryTimeEvolver interface with two ...
Egretta.Thula's user avatar
1 vote

Qiskit: Evolve TrotterQRTE from Operator

You can use SparsePauliOp.from_operator() method to construct a SparsePauliOp from an ...
Egretta.Thula's user avatar
1 vote
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Applying Suzuki-Trotter approximation to a Quantum Circuit in Qiskit

The method SuzukiTrotter.synthesize() takes an instance of PauliEvolutionGate as a parameter. The issue in your code is that you ...
Egretta.Thula's user avatar
1 vote

The approximation of the time evolution operator U, using Trotter formula, can't hold anymore taking a time step of $\Delta t = \pi$

As you allude to, Nielsen and Chuang say on pp. 259-260, Now initially we imagined that $\Delta t$ was small, since we were considering the case of quantum simulation, but Equation (6.28) shows that ...
Banach space fan's user avatar
1 vote

Exponentiating a multi-controlled NOT gate for trotterization

MCXGate exists in Qiskit's circuit library. So, you can get the operator as follows: ...
Egretta.Thula's user avatar
1 vote
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Using QDRIFT on qiskit

An easy way to use QDrift in Qiskit is to pass it as a synthesis method to PauliEvolutionGate ...
Egretta.Thula's user avatar
1 vote

Investigating the scaling of the error of a Trotter-Suzuki-approximation

For this particular calculation, you can keep your results exact for quite a long time. To see this, start with the exact thing $$ H_0=e^{i\pi/2(X+Z)/\sqrt{2}}=i\frac{X+Z}{\sqrt{2}}. $$ Now for the ...
DaftWullie's user avatar
  • 59.4k
1 vote
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How to do rotations along arbitrary multi-qubit basis

One way to do that is by using PauliEvolutionGate: ...
Egretta.Thula's user avatar
1 vote

What is the usefulness of the Suzuki-Trotter formula?

Your question is not so much about the usefulness of the Trotter-Suzuki formula per se, but is rather about the usefulness of Hamiltonian simulation in general (and eigenvalue sampling in particular), ...
Mark Spinelli's user avatar

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