Questions tagged [complexity-theory]

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

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36
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2answers
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How is the oracle in Grover's search algorithm implemented?

Grover's search algorithm provides a provable quadratic speed-up for unsorted database search. The algorithm is usually expressed by the following quantum circuit: In most representations, a crucial ...
33
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4answers
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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 ...
31
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4answers
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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 ...
43
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4answers
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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 ...
20
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2answers
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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) ...
4
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1answer
729 views

Simpler implementation of the Toffoli gate on IBM Q for special circumstances

On current quantum hardware, a depth of circuit is constrained because of noise. In some cases, results are totally decoherent and as a result meaningless. This is especially true when Toffoli gates ...
30
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4answers
1k 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 ...
19
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2answers
4k views

What does Google's claim of "Quantum Supremacy" mean for the question of BQP vs BPP vs NP?

Google recently announced that they have achieved "Quantum Supremacy": "that would be practically impossible for a classical machine." Does this mean that they have definitely proved that BQP ≠ BPP ?...
6
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1answer
167 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.
7
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1answer
390 views

Classical algorithm with complexity similar to Shor's discovered: Are there more efficient quantum algorithms than Shor's?

In the article Fast Factoring Integers by SVP Algorithms the author claims that he discovered classical algorithm for factoring integers in polynomial time. The Quantum Report mentioned here that it ...
7
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2answers
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Will quantum computers be able to solve the game of chess?

Will it be possible to use quantum computing to one day solve the game of chess? If so, any estimate as to how many qubits it would require? The game of checkers has already been solved through back ...
5
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1answer
261 views

Showing that Matrix Inversion is BQP-complete - HHL Algorithm

I am trying to understand an argument that Matrix Inversion is BQP-complete for certain conditions on the matrix. This is explained here on page 39 (this paper is a primer to the HHL algorithm and ...
3
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1answer
409 views

Can quantum computers be used to solve P = NP

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. It is one of the seven ...
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Question about the Grover-Sysoev algorithm [duplicate]

We consider  a quantum circuit that takes as input two vectors $\vert x \rangle$  and $\vert y \rangle$. The output of this quantum circuit must contain the reflected vector of   $\vert y \rangle$  ...
12
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1answer
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Why are non-Clifford gates more complex than Clifford gates?

There are two groups of quantum gates - Clifford gates and non-Clifford gates. Representatives of Clifford gates are Pauli matrices $I$, $X$, $Y$ and $Z$, Hadamard gate $H$, $S$ gate and $CNOT$ gate. ...
17
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1answer
756 views

What are examples of Hamiltonian simulation problems that are 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 ...
10
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2answers
475 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 ...
12
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4answers
1k 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 $$...
10
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2answers
7k views

What can we learn from 'quantum bogosort'?

Recently, I've read about 'quantum bogosort' on some wiki. The basic idea is, that like bogosort, we just shuffle our array and hope it gets sorted 'by accident' and retry on failure. The ...
4
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2answers
883 views

Do there exist problems known to be computationally intractable for quantum computer, but tractable for classical computer?

Or alternatively phrased, is it believed that the complexity class P is a complete subset of BQP? Consider the following diagram à la MIT OpenCourseWare, which seems to explicitly state as much.
9
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2answers
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Why is quantum Fourier transform required in Shor's algorithm?

I’m currently studying the Shor’s algorithm and am confused about the matter of complexity. From what I have read, the Shor’s algorithm reduces the factorization problem to the order-finding problem ...
6
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1answer
305 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 ...
5
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6answers
430 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 ...
8
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0answers
212 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 $\...
7
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1answer
206 views

Does the GLOA have any advantage over the Solovay-Kitaev algorithm?

The Solvay Kitaev algorithm was discovered long before the Group Leaders Optimization algorithm and it has some nice theoretical properties. As far as I understand, both have exactly the same goals: ...
7
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1answer
383 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 ...
4
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2answers
196 views

How exactly is solving the random circuit sampling problem a computation in the Church-Turing thesis sense?

Note: This has been cross-posted to CS Theory SE. If we assume $\mathsf{BQP} \neq \mathsf{BPP}$, then we can say with reasonable certainty that Google's random sampling experiment falsifies the ...
3
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2answers
97 views

How to implement exponentiation of a gate without breaking complexity?

In the application of QFT for quantum phase estimation (QPE) of a unitary $\mathbf{U}$, one has to perform successive controlled operations using powers of $\mathbf{U}$. In order not to break the ...
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1answer
30 views

How precise are BQP measurements?

Let's say I am given a Hamiltonian $H$, whose ground state is efficiently preparable. We know that $||H|| \leq 1$. Let that ground state be $|\psi_{0}\rangle$, with energy $E_{0}$. We also know that ...