What feature of a quantum algorithm makes it better than its classical counterpart? Are quantum computers faster than classical ones in all respects?

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    $\begingroup$ Are you asking about "quantum algorithms" or quantum computers in general? If it's the latter, then the short answer is that "speed" of physical quantum computers are heavily implementation dependent (there's also the issue of noise and fault-tolerance). As of yet, none of them are "faster" (in any legitimate sense) than your typical smartphone. However, some quantum algorithms are (in theory) known to be faster than their classical counterparts. The reason for that isn't straightforward (for instance, there's a lengthy debate about whether entanglement causes any speedup at all). $\endgroup$ – Sanchayan Dutta May 20 '19 at 8:36
  • $\begingroup$ @SanchayanDutta. Thanks for the answer. I was asking about quantum computers, in general. By the way, are you getting paid for answering or monitoring questions on SE? $\endgroup$ – Tobias Fritzn May 20 '19 at 8:45
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    $\begingroup$ To summarize Niel's answer, it can be said that speedup of quantum algorithms is specifically due to destructive interference. To quote Aaronson "The goal in quantum computing is to choreograph a computation so that the amplitudes leading to wrong answers cancel each other out, while the amplitudes leading to right answers reinforce.". I'll later try to write a detailed answer to your question, but this is a topic I too find very interesting. Thanks for asking the question. $\endgroup$ – Sanchayan Dutta May 20 '19 at 9:06

"What feature of a quantum algorithm makes it better than its classical counterpart?"

First, a classical algorithm can be thought of a quantum algorithm that makes no use of quantum superpositions. Therefore a quantum algorithm can be at least as good as its classical counterpart. No classical algorithm can be "better" than quantum algorithms can do, because one example of a quantum algorithm is the classical algorithm itself.

What is a feature of quantum algorithms that make them better than classical algorithms? The feature of quantum entanglement and superposition, which allows us to answer the Deutsch-Josza problem with 1 query instead of $2^n$.

"Are quantum computers faster than classical ones in all respects?"

I believe we do not know much about "speed". We know that we can factor the number $n$ with $log^3(n)$ operations instead of $n$ operations, but how fast are those operations going to be? We only know the answer to this question for quantum computers that have < 10000 qubits, and these have not been able to factor numbers larger than about 7 digits. You can see in WolframAlpha that factoring any number with ~10 digits finishes instantly, so even if the IBM quantum computer is faster, it is insignificantly faster. We need millions of physical qubits (100s of logical qubits) to make a real comparison, and what that million-qubit architecture will look like (if it's even possible to make) is something we don't yet know. Maybe to make a device with a million qubits will come at the sacrifice of gate fidelities or gate speeds.

The answer is "no" quantum algorithms are not better in "all respects" because they are harder to implement physically!

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