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7 votes
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Can we always simultaneously diagonalize $H_A \otimes \mathbb{1}$ and $\mathbb{1} \otimes H_B$?

TL;DR: You can always achieve simultaneous diagonalization of $H_A\otimes\mathbb{1}$ and $\mathbb{1}\otimes H_B$ even if $[H_A, H_B]\ne 0$. And yes, this does follow from the fact that $[H_A\otimes\...
Adam Zalcman's user avatar
6 votes

How does a quantum system identify hermitian and unitary matrices?

This is a bit like asking how your car identifies whether the vector you're giving it is its new position or its new velocity. There are various things you can do to quantum systems. All can be ...
Craig Gidney's user avatar
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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
4 votes

Can we always simultaneously diagonalize $H_A \otimes \mathbb{1}$ and $\mathbb{1} \otimes H_B$?

I'm assuming these Hamiltonians act on separate subsystems, since that's the only way to guarantee that $[H_A \otimes \mathbb{1}, \mathbb{1} \otimes H_B] = 0$. Since the two Hamiltonians act on ...
Cody Wang's user avatar
  • 1,233
3 votes

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

Your output size is $4^n$ numbers, so you're certainly not going to do better than $\Omega(4^n)$ complexity. At least, not without some sort of sparsity guarantee and some way to compute those big ...
Craig Gidney's user avatar
  • 38.8k
3 votes

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

For example see the one proposed recently in https://arxiv.org/abs/2401.16378. In my tests, I could decompose random $N=11$-qubit Hamiltonian in about $ \sim 150$ minutes.
R.G.J's user avatar
  • 261
3 votes
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Hamiltonian in CNOT gate implementation

The total Hamiltonian $\hat{H}$ is written as the sum of the following three terms: $$ \hat{H}_1 = \frac{\hbar \omega_1}{2} \hat{\sigma}_{z1} \otimes \hat{I} $$ $$ \hat{H}_2 = \hat{I} \otimes \frac{\...
SimoneGasperini's user avatar
3 votes
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How to splice Hamiltonians corresponding to channels $\Phi_1$ and $\Phi_2$ so as to obtain a Hamiltonian corresponding to $\Phi_2\circ\Phi_1$?

TL;DR: In general, this cannot be done exactly, because of the relationship between the eigenvectors of $A$ and $e^A$. That said, the Feynman's clock construction gets pretty close to realizing the ...
Adam Zalcman's user avatar
3 votes
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How does a quantum system identify hermitian and unitary matrices?

Aha, think I might have an idea about your confusion. So most quantum systems you'll learn about in an introduction to quantum computing evolve according to some unitary process, usually denoted $U(t)$...
Chris E's user avatar
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2 votes
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Can a Hamiltonian of a tripartite system map an product state into a product state?

Let's just do the special case of 3 qubits. Without loss of generality (up to unitary transformations, rescalings,...), $H_A$ and $H_B$ might as well be Pauli $Z$ matrices. There are two possibilities ...
DaftWullie's user avatar
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2 votes

Adiabatic computing basics

Just as with the case of algorithms like QAOA and VQE, the Hamiltonians are actually already known (for example, a Hamiltonian encoding a graph problem); in fact, the use case for any of these ...
Cody Wang's user avatar
  • 1,233
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

About the formulation of an Ising Hamiltonian

The square comes from the definition of the Hamilton $H_A$ given in equation (8) of the referenced paper. This term is described by "... is an energy which provides a penalty if the number of ...
Mathias's user avatar
  • 133
2 votes
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How to write the hamiltonian for w state?

I understand from the comments that the question is really to find a Hamiltonian $H$ and a corresponding time $t$ such that $$ e^{-iHt}|000\rangle=|w\rangle $$ where $$ |w\rangle=\frac{1}{\sqrt{3}}(|...
DaftWullie's user avatar
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1 vote

help understanding gate to hamiltonian and representation

There are many ways that you could go about doing this. (You might also want to take a look at what you"know" about exponentiating a matrix, because you've exponentiated $A$ not $iA$.) One ...
DaftWullie's user avatar
  • 59.4k
1 vote

Underlying Hamiltonians and pulse level controls of different commercially available quantum computers

This hamiltonian is a very basic approximation of the underlying device and in general does not accurately reflect the complexity of the device. It is up to each provider to decide whether to return ...
ThomasAlexander's user avatar
1 vote

Phase accumulation for multiple Rabi driving pulses

Good question! Your first equation mixes a couple of abstractions that we typically sweep under the rug, so let's start by taking a step back. When we apply a control pulse to a qubit, it's usually ...
Chris E's user avatar
  • 1,438
1 vote

Is there a general method for calculating expectation values for time-dependent wavefunctions?

I won't give the final answer (close to it), but instead try and point you in the general direction. Initial State: The given initial state is $ | \psi(t=0) \rangle = | 0 \rangle $. Pauli X Matrix: ...
banercat's user avatar
  • 763
1 vote
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How to combine measurement of each term in Hamiltonian $H = aH_1 + bH_2$ to get final results?

Community Wiki It seems like you wish to perform a "Local Hamiltonian Simulation" where your Hamiltonian $H$ is the sum of two separate terms: $$H=aH_1+bH_2.$$ That is, you wish to ...
1 vote

Knapsack problem formulation

It seems I mixed 2 formulations. It solves the problem with algorithm. Regarding randomness it is connected to backend set up on IBM Q.
JackAW's user avatar
  • 29
1 vote

How can I implement a Hamiltonian which is sum of tensored pauli operators on qiskit?

The simplest way to do that is to use HamiltonianGate ...
Egretta.Thula's user avatar
1 vote

Excplicit Description of Hamiltonians?

Your circuit appears a little confusing to me. The Hamiltonian $H$ is a hermitian matrix, and is not a ket (a vector) itself. The output of the evolution is not $|e^{-iHt}\pm\varepsilon\rangle$, but ...
Mark Spinelli's user avatar
1 vote

Meaning of 'O_i' and 'Δ_i' terms in PulseBackend Hamiltonians

After some research and feedback from the IBM slack channel, I figured out the meaning: $\Delta_i$ represents the anharmonicity of the transmon qubit and has units of frequency $O_i$ is the number ...
Sanson Jones's user avatar
1 vote

How to choose values of phi for Hamiltonian simulation with Quantum Singular Value Transform?

Appendix C in "Grand Unification of Quantum Algorithms" mentioned the pyqsp open source package. QSPPACK package contains an implementation of the Remez exchange algorithm. This package is ...
Egretta.Thula's user avatar
1 vote
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Understanding Hamiltonian's of the Single Qubit Gates and Toffoli gate

It is true that every unitary can be written as $U=e^{i H t}$ where $H$ is a Hermitian matrix. Also, if $H$ is a 2-local Hamiltonian acting on $n$ qubits, then $U=e^{i H t}$ is a $2^n \times 2^n$ ...
MonteNero's user avatar
  • 2,684

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