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For questions regarding complexity analysis of quantum algorithms and comparisons with complexities of classical algorithms

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Quantum Algorithm for God's Number

God's number is the worst case of God's algorithm which is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles ...
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6answers
230 views

Good metaphors for n-level quantum systems

It seems that a coin flip game is a decent metaphor for a 2-level system. Until 1 of the 2 players picks heads or tails, even if the coin has already been flipped, the win/loss wave form has not yet ...
6
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1answer
112 views

Query regarding BQP belonging to PP

I found the following proof of BQP belonging to PP (the original document is here). There is a part of the proof that I have trouble understanding. First, the structure is given below. We try to ...
6
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1answer
64 views

What classical public key cryptography protocols exist for which hacking is QMA complete or QMA hard?

Such a public key cryptosystem would be "quantum safe" in the sense that quantum computers cannot efficiently solve QMA hard problems.
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2answers
2k views

How many operations can a quantum computer perform per second?

I want to know what time complexity is considered efficient/inefficient for quantum computers. For this, I need to know how many operations a quantum computer can perform per second. Can anyone tell ...
8
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0answers
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Quantum Walk: Why the need of adding “tail” nodes to the root?

As stated in the question, I have found in several papers (e.g. 1, 2) that in order to perform a quantum walk on a given tree it is necessary to add some nodes to the root $r$, say $r^{'}$ and $r^{"}$....
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2answers
554 views

What is the actual power of Quantum Phase Estimation?

I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator $U$ and an eigenvector $|u\rangle$ with related eigenvalue $\text{exp}(2\pi i \phi)$, the ...
9
votes
4answers
257 views

Grover's Algorithm and its relation to complexity classes?

I am getting confused about Grover's algorithm and it's connection to complexity classes. The Grover's algorithm finds and element $k$ in a database of $N=2^n$ (such that $f(k)=1$) of elements with $$...
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0answers
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Is PostBQP experimentally relevant? [duplicate]

Far from my expertise, but sheer curiosity. I've read that PostBQP ("a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with ...
14
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1answer
151 views

Hamiltonian simulation is BQP-complete

Many papers assert that Hamiltonian simulation is BQP-complete (e.g., Hamiltonian simulation with nearly optimal dependence on all parameters and Hamiltonian Simulation by Qubitization). It is easy ...
13
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1answer
116 views

Are there results from quantum algorithms or complexity that lead to advances on the P vs NP problem?

On the surface, quantum algorithms have little to do with classical computing and P vs NP in particular: Solving problems from NP with quantum computers tells us nothing about the relations of these ...
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1answer
54 views

Upper Bounds for QMA Quantum Merlin Arthur, and QMA(k)

QMA(Quantum Merlin Arthur), is the quantum analog of NP, and QMA(k) is the class with $k$ verifiers. These are important classes when studying Quantum Complexity theory. QMA(k) is QMA with $k$ ...
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2answers
592 views

Is BQP only about time? Is this meaningful?

The complexity class BQP (bounded-error quantum polynomial time) seems to be defined only considering the time factor. Is this always meaningful? Do algorithms exist where computational time scales ...
9
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2answers
157 views

Good introductory material on quantum computational complexity classes

I wish to learn more about computational complexity classes in the context of quantum computing. The medium is not so important; it could be a book, online lecture notes or the like. What matters ...
6
votes
1answer
83 views

Quantum Algorithm SAT structure

"Quantum magic won't be enough" (Bennett et al. 1997) If you throw away the problem structure, and just consider the space of $2^n$ possible solutions, then even a quantum computer needs about $\...
4
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0answers
117 views

Consequences of SAT ∈ BQP

"Quantum magic won't be enough" (Bennett et al. 1997) If you throw away the problem structure, and just consider the space of $2^n$ possible solutions, then even a quantum computer needs about $\...
6
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0answers
79 views

Empirical Algorithmics for Near-Term Quantum Computing

In Empirical Algorithmics, researchers aim to understand the performance of algorithms through analyzing their empirical performance. This is quite common in machine learning and optimization. Right ...
7
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1answer
59 views

Can the analysis or design of quantum algorithms benefit from parameterised algorithmics?

In the last decades, the field of parameterised algorithms, with fixed parameter tractibility (FPT) as its main tool has been provided new methods to analyse old algorithms and design techniques for ...
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1answer
116 views

Is there any general statement about what kinds of problems can be approximated more efficiently using a quantum computer?

As the name already suggests, this question is a follow-up of this other. I was delighted with the quality of the answers, but I felt it would be immensely interesting if insights regarding ...
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3answers
1k views

Is there any general statement about what kinds of problems can be solved more efficiently using a quantum computer?

Is there a general statement about what kinds of problems can be solved more efficiently using quantum computers (quantum gate model only)? Do the problems for which an algorithm is known today have a ...
5
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1answer
108 views

How to compare a quantum algorithm with its classical version? [closed]

The Quantum Algorithm Zoo includes a host of algorithms for which Quantum Computing offers speedups (exponential, polynomial, etc). However, those speedups are based on asymptotic computational ...
18
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3answers
426 views

Is there a layman's explanation for why Grover's algorithm works?

This blogpost by Scott Aaronson is a very useful and simple explanation of Shor's algorithm. I'm wondering if there is such an explanation for the second most famous quantum algorithm: Grover's ...
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2answers
412 views

Why is a quantum computer in some ways more powerful than a nondeterministic Turing machine?

The standard popular-news account of quantum computing is that a quantum computer (QC) would work by splitting into exponentially many noninteracting parallel copies of itself in different universes ...
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2answers
286 views

How is the oracle in Grover's search algorithm implemented?

Grover's search algorithm provides a provable polynomial speed-up for unsorted database search. The algorithm is usually expressed by the following quantum circuit: In most representations, a crucial ...
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1answer
164 views

Are there any encryption suites which can be cracked by classical computers but not quantum computers?

Are there any encryption suites that can be cracked by usual computers or super computers, but not quantum computers? If that's possible, what assumptions will it depend on? (Factorizing big numbers, ...
24
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4answers
356 views

Are there problems in which quantum computers are known to provide an exponential advantage?

It is generally believed and claimed that quantum computers can outperform classical devices in at least some tasks. One of the most commonly cited examples of a problem in which quantum computers ...
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2answers
132 views

What is postselection in quantum computing?

A quantum computer can efficiently solve problems lying in the complexity class BQP. I have seen a claim the one can (potentially, because we don't know whether BQP is a proper subset or equal to PP) ...
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4answers
717 views

Is it possible for an encryption method to exist which is impossible to crack, even using quantum computers?

Quantum computers are known to be able to crack in polynomial time a broad range of cryptographic algorithms which were previously thought to be solvable only by resources increasing exponentially ...