8 votes
Accepted

If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?

Hint: Instead of using the BCH formula in the form usually presented, for example at the top of this Wikipedia page, use this consequence of Hadamard's Lemma: $$\tag{1} e^{iHt}\hat{a}e^{-iHt} = \hat{a}...
  • 12.5k
7 votes
Accepted

What are the preferred numerical methods to simulate the evolution of a state through a time-dependent Hamiltonian?

It depends on the Hamiltonian. There are three particular questions whose answers might influence your choice of strategy: Does the Hamiltonian have any particular structure or symmetry? How quickly ...
  • 50.4k
7 votes

How are quantum gates realised, in terms of the dynamic?

Generally speaking, a realization of a quantum gate involves coherent manipulation of a two-level system (but this is nothing new to you, maybe). For example, you can use two long-lived electronic ...
6 votes
Accepted

How to prove that a naive quantum random walk is non-unitary

I'm going to define $|n\rangle$ to be "the walker is at site $n$". Now imagine the walk as specified: $$ |n\rangle\rightarrow (|n-1\rangle+|n+1\rangle)/\sqrt{2}. $$ You can put some phases ...
  • 50.4k
6 votes

If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?

Use the differential form of the time evolution, $$dO/dt=i[H, O]\ .$$
5 votes
Accepted

Why does joint ground state not change under action of beam splitting unitary operator?

Calculate $$ \begin{align} \hat{U}|00\rangle &= \exp\left(-igt(\hat{a}^\dagger_2\hat{a}_1+\hat{a}^\dagger_1\hat{a}_2)\right)|00\rangle \\ &= \sum_{k=0}^\infty \frac{(-igt)^k}{k!}(\hat{a}^\...
4 votes

How to construct solution based on the Schrödinger equation and split it into gates?

Recall that a time-dependent unitary operator (technically a one-parameter group) $U$ can be generated by a Hermitian operator $H$(the generator) $$ U(t) = \exp(-itH) $$ This in turn gives us the ...
  • 800
4 votes

Why does joint ground state not change under action of beam splitting unitary operator?

There's more than one way, and I'll suggest two of them here: Expand $\hat{U}$ using the formula for the Taylor series of an exponential ($e^\hat{A}$) centered around $\hat{A}=\hat{0}$, and then you ...
  • 12.5k
2 votes

Why does joint ground state not change under action of beam splitting unitary operator?

Let $|\psi\rangle$ be an eigenstate of an operator $A$, $A|\psi\rangle=\lambda|\psi\rangle$. Then $$e^A |\psi\rangle = \sum_{k=0}^\infty \frac{A^k}{k!}|\psi\rangle = \sum_{k=0}^\infty \frac{\lambda^k}{...
  • 20.8k
2 votes

If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?

Note that $$[(a^\dagger)^n,a] = -n(a^{\dagger})^{n-1}, \qquad [(a^\dagger)^n a^m,a] = -n (a^\dagger)^{n-1}a^m, \qquad [a^n,a]=0.$$ Consider an arbitrary function of the mode operators, that we assume ...
  • 20.8k
2 votes

Quantum gates for more than two basis states

If you want the best possible evolution for your qubit space, then you simply don't define what your target is on the rest of the space - so long as you always start in the two-dimensional subspace/...
  • 50.4k
1 vote

How to comparing a quantum channel with a unitary?

I found the following two works solving this problem. They selected several reference states and took the HS product between the final states evolved by the target unitary and the quantum channel. ...
  • 155
1 vote

Only assuming the universe evolves according to a positive trace-preserving map, is there a proof that all subsystem evolution must be CPTP?

There are two possible answers. Let's say the universe evolves from $t=0$ to $t_f$ then the unitary evolution $U$ from $0$ to $t_f$ induces a CP evolution on the subsystem. To see this, note that ...
1 vote

What is the physical meaning of the Hamiltonian $H = \alpha ( |01 \rangle \langle10| + | 10 \rangle \langle 01| )$?

I'm not sure for this specific problem, and more broadly Hamiltonians are typically in the "eye of the beholder." For example, for quantum chemistry problems, Hamiltonians are really clean mappings ...
  • 1,551
1 vote

Quantify the probability in guessing the Hamiltonian?

This is my attempt. Let's say the list looks like: $\lambda_1$, $\lambda_2$, $\lambda_1$, $\lambda_3$, $\dots$, $\lambda_n$ where $\lambda_i$ are numbers. The variables (unknowns) are the ...

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