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### Is there any mathematical technique to find the exact solution of a QUBO problem?

The short answer: QUBO problems are known to be NP-hard. So, there should be no analytical solution. The long answer: You seem confused about solving quadratic problems, eigenvalues and the meaning of ...
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### Is any multi-qubit unitary operation a rotation about a specific unit vector?

No. This works for a single qubit because a single-qubit unitary can be written as $$U=e^{i\theta}|\psi\rangle\langle\psi|+e^{i\phi}|\phi\rangle\langle\phi|$$ where $\langle\phi|\psi\rangle=0$. ...
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• 7,443

### Can a unitary $U$ transform $H = aX + bY + cZ$ to $U^\dagger H U = -aX + bY + cZ$?

Short answer: if $\xi\in\mathbb R$ is chosen such that $e^{2i\xi}=\frac{-a+ib}{a+ib}$, then $U={\rm diag}(e^{i\xi},e^{-i\xi})$ is the unitary you are looking for. First of all, there is an abstract ...
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### Trace distance between thermal states

Since the multimode Fock states $|\mathbf{n}\rangle=|n_1\rangle\otimes |n_2\rangle\otimes\cdots\otimes |n_M\rangle$ form an orthonormal basis, one can treat each state by their probability ...
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### Heisenberg picture for a quantum measurement where the outcome is recorded

Note that a "post-measurement" $M(\rho)$ cannot be a channel, because the function sending $\rho$ to any of the states left over after a measurement is not linear. And if $M$ is not linear, ...
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### Understanding how to calculate partial trace

The key here is that the tensor product satisfies $(A\otimes B)(C\otimes D)=AC\otimes BD$. Applied to the terms of $\rho$ in (6.27) this means  |ab\rangle\langle cd|\overset{\text{Def.}}=(|a\rangle\...
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### What exactly is the computational or standard basis?

Yes, all of quantum mechanics is essentially linear algebra, so you can indeed take any set of basis vectors or basis states that you want. For the purposes of making proofs in quantum information, ...
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### Computing expectation of a SparsePauliOp using StateVector

You can use Statevector.expectation_value() method as follows: ...
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1 vote
Accepted

$\underline{x} \in \mathbb{F}_{2}^{4}$ as you say, but $x_j \in \mathbb{F}_{2}$ is a scalar. $\underline{g_j}$ is a row of $G_{C/C^{\perp}}$, thus has length 6. Therefore, $\sum_{j=1}^{4}x_{j}\... • 138 1 vote ### What exactly is the computational or standard basis? You are absolutely right in your statement that there is a freedom in the choice of basis, for instance when looking at an expectation value of an observable$O$on a state$\lvert \psi \rangle$, you ... 1 vote ### We take the reciprocal$\lambda^{-1}$of eigenvalues in HHL - but what's stopping us from raising them to a positive exponent$\lambda^m\$?

Self-Answer, making CW Thanks to @incud for pointing out a 2017 paper of Cade and Montanaro - pdf, arXiv. In that paper they do much of exactly what I propose - namely, they calculate, in ...

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