8 votes

Why are diagonal Hamiltonians considered classical?

Classical Hamiltonians By the spectral theorem, for every Hamiltonian there exists a basis in which it is diagonal. Thus, it is not correct to say that diagonal Hamiltonians are classical since this ...
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6 votes
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Can a quantum computer run classical algorithms?

Quantum computers can run classical computations using exactly the same algorithms, and hence have the same running time in terms of scaling. For example, if you look at shor’s algorithm, a major ...
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6 votes
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Trace distance of two classical-quantum states

Yes, since the trace norm is the sum of the absolute value of the singular values, and the singular values can be found for each of the $a$ blocks independently.
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5 votes

Can error correction for a classical algorithm with bit flips be easier than for a general quantum circuit?

Can implementing error correction in this case be any easier than in the case of a general quantum circuit? Yes, for example you could use a classical error correcting code such as a repetition code. ...
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5 votes
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Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?

That quantity appears to be identical to Holevo information, which turns out to be the upper bound on how much classical information you can transmit using a quantum channel [1]. More generally the ...
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5 votes

Is the set of classical-quantum states convex?

Your mistake is that you assume that $\rho$ and $\sigma$ are classical-quantum in the same classical basis on $X$. However, there is no need to do so -- all which is necessary is that there exists ...
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5 votes

Example of a quantum algorithm better than its classical counterpart which involves only $1$ qubit?

There aren't many examples! The main reason for advantages in quantum computers is the ability to constructively combine amplitudes - if you've only got 1 qubit, there aren't any amplitudes to combine!...
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4 votes
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What does superposition do for quantum probabilistic sampling?

Classical computers are inherently deterministic, so they either generate pseudorandom numbers, or use an external physical process with statistically random noise to generate random numbers. Quantum ...
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4 votes
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Bounding diamond norm distance using probability of error in transmission of classical information

Intuition The expression $\|\mathcal{A} - \mathcal{I}\|_\diamond$ quantifies how close the channel $\mathcal{A}$ is to the identity channel $\mathcal{I}$ which is the channel that preserves quantum ...
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3 votes

Can error correction for a classical algorithm with bit flips be easier than for a general quantum circuit?

Peter Shor has two error correcting methods. One is the bit flip method and the other is the phase shift method. The bit flip method is similar to what you could use in classical computing, and is ...
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3 votes
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Understanding the association rule between classical to quantum data $|x\rangle=\frac{1}{|\vec x|_2}\sum_{i=1}^d x_i|i\rangle$

It may be easier to understand how this works in terms of the computational basis if you choose $d$ to be some power of $2$. So let $d=2^n$ and then we will use $n = \log_2 d$ qubits to represent the ...
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3 votes
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Prove that for a pure tripartite state $\rho_{ABE}$, $H(RB) = H(RE)$

As $\rho_{ABE}$ is pure we have $\rho_{ABE} = |\psi\rangle \langle \psi|$. We'll rewrite the output of the channel $\mathcal{R}$ as $$ \rho_{ABE}' = \sum_j (P_j \otimes I_{BE}) |\psi\rangle \langle \...
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3 votes
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Prove that the conditional entropy of a classical-quantum state is non-negative

We will use the upper bound on the entropy of a mixture (for proof see for example theorem 11.10 on p.518 in Nielsen & Chuang) $$ S\left(\sum_k p_k \rho_k\right) \leq H(p) + \sum_k p_k S(\rho_k)\...
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3 votes
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Relating quantum max-relative entropy to classical maximum entropy

As far as I'm aware there isn't much of a meaningful connection. The corresponding entropy for $D_{\max}$ is the min-entropy (written $H_{\min}$ or $H_{\infty}$). It measures a sort of `worst case' ...
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3 votes
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Translating classical math and code, to quantum math and code

Bra-ket notation is not necessarily tied to "quantum math," it's simply a convenient notation in many circumstances. It may seem intimidating at first, but once you understand the basics (ket = ...
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3 votes

Translating classical math and code, to quantum math and code

At least currently, most of the translations being made are in extraordinarily specialized areas - for example, quantum chemistry / computational chemistry. A lot of the math involves mapping domain ...
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2 votes

How to initialize classical register in Qiskit?

Classical registers are typically used for capturing measurement results, and may also be used for conditionally applying quantum operation. See: https://github.com/Qiskit/openqasm/blob/master/spec/...
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2 votes

Classical and quantum limits to classical copying?

You seem to be mixing two very different concepts here. Quantum cloning is talking about the absolute limits of what is theoretically possible in a perfect world. In this absolute theoretical limit, ...
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2 votes

Better Way Of Separating Two CQ-States

In quantum information theory, the standard way to obtain what it is called reduced density operator from a quantum system composed by several quantum states is to use the so-called partial trace ...
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2 votes

Is "classical information" the same as "Shannon information"?

I don't think there is a canonical "right" answer to this question as there is no universal formulation of the terminology, so let me try and pick apart a few of the things you mention, and ...
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2 votes
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Trace distance of two classical-quantum state with hashing

No, this is not possible. The existence of such a hash function requires the (smooth) min-entropy of the initial state to be large enough but does not depend on its trace distance from a uniform state....
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2 votes

Why are diagonal Hamiltonians considered classical?

To pose a very simple answer to compete with all these complex (but also excellent) answers: the Ising model is a classical Hamiltonian because it is diagonal as it's written and therefore all of its ...
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2 votes

Why are diagonal Hamiltonians considered classical?

While Adam's very detailed answer is probably emaculate, it's a bit long so for people that want a shorter answer, I'll give a much more compact alternative. This is not at all to challenge or try to ...
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2 votes
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Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states

As you say, $$ \mathrm{Tr}[\rho_{XB} \log \rho_{XB}] = -S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_{B}^x \log \rho_B^x]. $$ But if you can prove the above statement, then the exact same derivation gives ...
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1 vote
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How does qiskit's CircuitQNN calculate the gradients of circuits?

Qiskit implements the parameter shift rule and the linear combination of unitaries to calculate the gradients for a QNN. These techniques are described in detail in Section 3 of this paper. If we ...
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1 vote

What is the general form of a classical-quantum state?

Your description has X as a mixed state (a quantum state with classical uncertainty) and not a classical state. For example you can apply quantum gates to X but that shouldn’t be allowed if X was a ...
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  • 473
1 vote

Given averages of powers of position and momentum in quantum mechanics what information can be secured about the wave-function?

The covariance matrix is a function of the expectation values of powers of position and momentum associated to some state in a continuous-variable system. $$ \mathbf{\sigma} = \begin{pmatrix} \langle\...
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1 vote

Counting Achievable Operations

Why is 4! valid? We can imagine the desired operation to implement as a truth table / permutation matrix. Recognize that we may do this because none of the operations actually modify the amplitudes - ...
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  • 1,521
1 vote

The effect of available information on random quantum channels

This question gets right to the heart of what information does a density matrix contain about the state of a qubit. Critically, it is a subjective state of knowledge. So, if I don't know the outcome ...
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