Questions tagged [quantum-computing]

Quantum computation and computational issues related to quantum mechanics

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8
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1answer
298 views

BQP algorithm for two graph bisection problems and its implications on NP $\subseteq$ BQP

I read the paper Ahmed Younes, "A Bounded-error Quantum Polynomial Time Algorithm for Two Graph Bisection Problems", 2015. doi:10.1007/s11128-015-1069-y which is published in Springer's journal ...
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4answers
703 views

Bounding the gap between quantum and deterministic query complexity

Although exponential separations between bounded-error quantum query complexity ($Q(f)$) and deterministic query complexity ($D(f)$) or bounded-error randomized query complexity ($R(f)$) are known, ...
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2answers
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Runtime of Grover's algorithm

What is the time complexity (not query complexity) of Grover's algorithm? It seems clear to me that it is $\Omega(\log(N) \sqrt{N})$ since there are $\Omega(\sqrt{N})$ iterations and each iteration ...
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1answer
636 views

On optimality of Grover algorithm with high success probability

It is well-known that bounded error quantum query complexity of the function $OR(x_1,x_2,\ldots, x_n)$ is $\Theta(\sqrt{n})$. Now the question is what if we want our quantum algorithm to succeed for ...
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1answer
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Dimension of the Fourier transform for $S_5$ [closed]

My question: What is the dimension of the Fourier transform for $S_5$? My effort: The dimensions of the seven irreps of $S_5$ are $1,1,4,4,5,5,6$. According to the notes of Andrew Childs, the ...
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2answers
378 views

Consequences of $NP\subseteq P/poly$ to $BQP$

A post here Consequences of $BQP \subseteq P/poly$? queried on Consequences of $BQP \subseteq P/poly$. It is not known if $NP\subseteq BQP$. In general, what are the consequences of $NP\subseteq P/...
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1answer
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Geometric picture behind quantum expanders

(also asked here, no replies) A $(d,\lambda)$-quantum expander is a distribution $\nu$ over the unitary group $\mathcal{U}(d)$ with the property that: a) $|\mathrm{supp} \ \nu| =d$, b) $\Vert \mathbb{...
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1answer
301 views

Quantum Hardness of Approximating Lattice Problems

A common claim in lattice-based cryptography is that cryptosystems based on the Learning with Errors ($\mathsf{LWE}$) problem are hard to break (for a per-system definition of "break") for quantum ...
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Quantum Hardness of Finding Nash Equilibria

This question is inspired by the recent, beautiful work On the Cryptographic Hardness of Finding a Nash Equilibrium by Bitansky, Paneth, and Rosen. Their main result is that the existence of ...
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56 views

First register in the hidden subgroup representations of Simon's and graph isomorphism problems

The Simon's problem involves a function which takes binary strings as inputs. One seeks to find the period of the function which acts on those inputs. In the standard method, the first register has ...
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What is the relationship between quantum tomography and quantum error correction?

To realize correct quantum computing it seems that both quantum error correction and quantum tomography would be necessary. Is this true? What is the relationship between these two fields?
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Quantum annealing or adiabatic quantum optimization with continuous optimization problems

How do quantum annealing or adiabatic quantum optimization deal with continuous optimization problems such as SDP?
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Evidence that there is some problem in BQP distinct from BPP?

Are there any evidences (1 physics, 2 mathematics AND 3 computer science) that particular problems such as integer factorization, discrete logarithm are in BQP but not in BPP? There do not seem to be ...
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Is Logic Done on Superpositional Bit Values Useful?

Let's say I have a way to represent $N$ bits such that those bits are in a superposition of the $2^N$ possible states those bits can have and that I can do XOR and AND on those superpositional bits to ...
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1answer
673 views

How powerful is exact “quantum” computing if you suspend unitarity?

Short Question. What is the computational power of "quantum" circuits, if we allow non-unitary (but still invertible) gates, and require the output to give the correct answer with certainty? This ...
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472 views

Adiabatic quantum computing with level crossings

Question. In adiabatic evolution, to ensure that the ground state high overlap with the unique ground state of the system (i.e. to achieve arbitrarily small error) using adiabatic theorems, it is ...
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1answer
357 views

How Much Computing Power would be Required to Fully Simulate a Cubic Meter?

Imagine you want to simulate a cubic meter down to the particle. By following the Standard Model and other basic physical equations, how much computing power would be required to do this, in say, a ...
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1answer
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$BQP$ vs $QMA$?

The central problem of complexity theory is arguably $P$ vs $NP$. However, since Nature is quantum, it would seem more natural to consider the classes $BQP$ (ie decision problems solvable by a ...
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Does simulating chiral gauge theories lie within BQP?

In theoretical physics, there is a branch of quantum field theory dealing with chiral gauge theories. It has been conjectured by Feynman [1] and others that all quantum field theories can be simulated ...
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3answers
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Physical realization of nonlinear operators for quantum computers.

I have read in a paper where nonlinear operators for quantum computers implies the solving of problems in #P time. See http://arxiv.org/pdf/quant-ph/9801041 . What would be the simplest realization of ...
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2answers
600 views

Quantum algorithms based on transforms other than Fourier transforms

In Quantum Computation and Quantum Information by Nielsen and Chuang they say that many of the algorithms based on quantum Fourier transforms rely on the Coset Invariance property of Fourier ...
7
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1answer
4k views

Quantum annealing vs adiabatic quantum computation

I had this impression that quantum annealing is an optimization technique which may or may not produce exact solutions. On the other hand adiabatic quantum computation always gives exact solutions ...
4
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1answer
360 views

How does one determine if a mixed bipartite quantum state is entangled or not?

My question is based on the structure of the NP-hardness proof in section 6 (page 17) of this paper, http://arxiv.org/pdf/quant-ph/0303055v1.pdf Mathematically one can think of being given a positive ...
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Claw finding using quantum walk: superposition for Szegedy's framework

Within Claw Finding Algorithms Using Quantum Walk there is the subroutine $claw_{detect}$ described. As in above paper: Let $J_f(N, l)$ and $J_G(M, m)$ be Johnson graphs. Let $F$ and $G$ be vertices ...
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1answer
964 views

Why does Odlyzko improvement of Shor's Algorithm reduces the number of trials to $O(1)$

In his 1995 paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter W. Shor discusses an improvement on the order-finding part of his ...
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180 views

Two questions on Shor's algorithm

Does Shor's algorithm produce factors of a $n$-bit number and discrete log modulo $n$-bit prime in $O((\log n)^{2+\epsilon})$ bit operations using fast multiplication? I am trying to read from ...
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1answer
284 views

Verifying Shor's quantum error correction code

I know that Shor's 9 bit code can correct phase or bit flip, but I'd like to show that it can correct any type of error on a single qubit. I know that an arbitrary error can be expressed with the ...
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1answer
929 views

If BQP contains NP, does this mean that P=NP?

There is a question raised by Scott Aaronson in one of his papers [1]: "Could we show that if NP ⊆ BQP, then the polynomial hierarchy collapses?". Assuming the answer is yes, and it is also know that ...
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1answer
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Understanding efficient classical simulation of quantum computing

I want to understand the Gottesman-Knill theorem, which basically says that using some subclass of unitary transformations (from the Clifford group) there is no quantum speed-up ie. we can simulate ...
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2answers
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Was the reduction in Shor's algorithm originally discovered by Shor?

This is a "historical question" more than it is a research question, but was the classical reduction to order-finding in Shor's algorithm for factorization initially discovered by Peter Shor, or was ...
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Is it conceivable at all that the standard model of physics can outperform a quantum computer in any sense?

The Standard Model of physics (the mathematical model which predicts the Higg's boson) is, as far as I understand, our most complete model of the universe. That is to say, it is the best description ...
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510 views

Suppose $\mathbf{P} = \mathbf{BQP}$. Then what is randomness? Would it even exist at all?

DISCLAIMERI do apologize in advance if this question turns out to be silly, for some trivial reason that I may be overlooking in this moment. Suppose for a moment that $\mathbf{P} = \mathbf{BQP}$ ...
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1answer
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What is the proper role of verification in quantum sampling, simulation, and extended-Church-Turing (E-C-T) testing?

Since no answer was given, a flag has been set requesting that this question be converted to a community wiki. The comments by Aaron Sterling, Sasho Nikolov, and Vor have been synthesized into the ...
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1answer
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Is it possible to design an efficient approximation algorithm for one NP-complete problem based on Shor's algorithm?

Is it possible to design an efficient approximation algorithm for an $\sf{NP\text{-}complete}$ problem based on reductions from Shor's algorithm? Are known any (classical) approximation algorithms ...
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1answer
161 views

Probabilistic and quantum analog of $FP$ and $FNP$?

Is there any analog of the computational classes $FP$ and $FNP$ with probabilistic or quantum Turing machines? If so, what are the relation with other computational classes?
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Coset state of $3$-node graph isomorphism problem

The hidden subgroup representation of a $3$-node graph isomorphism problem is defined over the symmetric group, $G = S_6$. So, any hidden subgroup algorithm that wishes to solve the problem should ...
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2answers
310 views

Is the Presburger arithmetic decision problem known to be outside of BQP or BPP?

Presburger arithmetic is well-known to be decidable but intractable, requiring doubly exponential time even with nondeterminism (Fischer and Rabin, 1974). I am wondering if it is also known whether ...
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1answer
562 views

Is there any problem which is in AWPP but conjectured to be not in BQP?

Is there any known problem which is in $\mathsf{AWPP}$ but conjectured to be not in $\mathsf{BQP}$? What about relative to an oracle? Is there any known problem in $\mathsf{MQ^2}$ which is ...
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1answer
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Travelling sales man with Quantum Computers [closed]

I know that it takes billions of years to solve the travelling sales man when n = 25 (Number of cities). I am wondering how fast can a quantum computer solve the travelling sales man problem (for ...
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1answer
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Applications of HHL's algorithm for solving linear equations

In HHL's algorithm for solving a system of linear equations (HHL = Harrow, Hassidim and Lloyd) the output is a quantum state rather than explicit information. Has anyone been able to apply knowledge ...
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4answers
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If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
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0answers
94 views

Proofs to verify quantum states without revealing their description

Consider the following function $$f_s: k \rightarrow \lvert \psi_k \rangle$$ where $s,k$ are bit strings, and $\lvert \psi_k \rangle$ is a $n$-qubit state. Assume the function is a one-to-one mapping....
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Hardness of finding similar (quantum) states

Consider a quantum state $\lvert \psi \rangle$, we know from the no cloning theorem, that it cannot be perfectly cloned. Also, loosely speaking, that it can be imperfectly cloned s.t. one can produce ...
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1answer
162 views

How well can an arbitrary (unknown) quantum state be imperfectly cloned?

How well can an arbitrary unknown (quantum) state $\rvert \psi \rangle = \alpha\rvert 0 \rangle + \beta \rvert 1 \rangle$, be imperfectly/approximately cloned? Given an unknown state ${\rvert \psi \...
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1answer
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Clarification for argument in proof of search in N^1/3 queries with hidden variables/non-collapsing measurements

Let $N=2^n$. In Aaronson's Quantum Computing and Hidden Variables (1) and the recent follow up by Aaronson, Bouland, Fitzsimons, and Lee The space "just above" BQP (2), we consider models of ...
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1answer
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What is the Quantum Cheshire Cat experiments' import to Quantum Computing?

What is the significance of the Quantum Cheshire Cat to Quantum Computing? To recap, the Quantum Cheshire Cat experiment proved it was possible to separate a neutrons' spin from the neutron. Or, in ...
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3answers
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Will quantum computing pave the way for native, true RNGs?

Obviously, regular computers can't generate random numbers on their own, since they're inherently systematic machines. Would quantum computing be able to run a true RNG without a seed based off user ...
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1answer
150 views

How does one extend local checkability to quantum complexity classes?

How does one extend local checkability to quantum complexity classes like BQP? Or are quantum algortihms inherently holistic?
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1answer
177 views

Efficient generation of permutational invariant quantum states

Starting from $|00\cdots 0\rangle$, can permutational invariant quantum states, i.e. the following one: $$ |\psi_n\rangle = \frac1{n!} \sum \prod_{\pi\in S_n} |\pi(0)\rangle|\pi(1)\rangle\cdots|\pi(n-...
4
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1answer
359 views

The relationship between QCMA and QMA in the Turing and Communication model

First my background about computational complexity is still beginner. Recent paper published by Klauck and Podder [KP14] show that for the first time an exponential gap between computing partial ...

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