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Accepted

• 22.9k

### 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 ...
• 1,223

### 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 ...
Accepted

• 25.4k

### 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 ...
• 25.4k

### 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/...
• 58.8k
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. ...
• 177
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

### How are elementary quantum gates realised?

I'll posit that, much as classical NAND and NOR gates can be generated with NMOS and PMOS transistors arranged in series and/or in parallel, quantum gates such as CCNOT (Toffoli) and CSWAP (Fredkin) ...
• 12.6k
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 ...
• 1,756
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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