# Questions tagged [complexity-theory]

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

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### What are the thermodynamic limits of Shor's algorithm

The asymptotic time complexity of Grover's algorithm is the square root of the time of a brute force algorithm. However, according to Perlner and Liu, the thermodynamic behavior (theoretical minimum ...
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### Could finding Golomb rulers be in $\mathrm{BQP}$?

The problem of factoring large numbers may be in the so-called "intermediate" regime. These are problems that are in $\mathrm{NP}$, but are neither likely to be easy enough to be in $\mathrm{P}$ nor ...
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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   $\vert y \rangle$  ...
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### Do we already know some problems, that would be hard to solve for quantum computers, and use them in cryptography? [duplicate]

I was wondering, whether there are any problems that we already know are difficult to solve for a quantum computer, and that we could potentially use in cryptography, just as we do now with e.g. the ...
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### How to find a marked item out of $K < N$ marked items when K is unknown?

I'm wondering about time complexities of variants of the searching problem in Grover's Algorithm. I know that using G.A. the time complexity required to find a market item reduces to $O(\sqrt N)$. ...
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### Speed up in Bernstein-Vazirani algorithm and Gottesman-Knill theorem

The Bernstein-Vazirani problem: Let $f$ be a function from bit strings of length $n$ to a single bit, $$f: \{ 0, 1\}^n \to \{0, 1\}$$ thus all input bit strings $x \in \{0,1\}^n$. There exists a ...
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### How powerful would quantum computers be if we had direct access to the full state vector?

The state vector is exponential in size, so we can manipulate an exponential quantity of information with a linear quantity of gates. However, this doesn't give us general exponential speedup because ...
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### On the probability of preparing of a uniform superposition by performing a controlled-multiplication and post-selecting $0$

I take as a starting point Watrous's celebrated paper defining the Quantum Merlin-Arthur (QMA) class. He provides a protocol for Arthur to test whether an element $h$ is not in a group $\mathcal{H}$ ...
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### How does number of shots (number of times the computation is repeated) affects time complexity [closed]

I want to know what happens to the time complexity in terms of big O analysis
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### Self reducibility of QCMA problems

Self reducibility is when search version of the problems in a language reduce to decision versions of the same problems. NP-complete problems are self reducible. Are QCMA complete problems self ...
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### Resources to study quantum algorithms and quantum complexity

I have a computer science background, and I'm interested in studying 'quantum algorithms' and anything that is related like 'quantum complexity'. I would like to have all important resources that is ...
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### Quantum algorithms, combinatorial optimization, and approximation bounds

Recently, I saw this article, Classical and Quantum Bounded Depth Approximation Algorithms where the author discusses the limitations of QAOA relative to classical approaches. In particular, they ...
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### Marriott-Watrous style amplification with a quantum input

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$ In MW05 the authors demonstrate so-called "in-place" amplitude amplification for QMA, exhibiting a method for Arthur ...
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### 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 ...
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### Better "In-Place" Amplification of QMA

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$ In MW05 the authors demonstrate so-called "in-place" amplitude amplification for QMA, exhibiting a method for Arthur ...
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### 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 ?...
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### Is quantum complexity basis-invariant?

Quantum computing refers (occasionally implicitly) to a "computational basis". Some texts posit that such a basis may arise from a physically "natural" choice. Both mathematics and physics require ...
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### complexity of classical counting algorithm

Does anyone know the solution of Exercise 6.14 of Nielsen and Cheung: Prove that any classical counting algorithm with a probability at least 3/4 for estimating $M$ correctly to within an ...