24
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What is postselection in quantum computing?
"Postselection" refers to the process of conditioning on the outcome of a measurement on some other qubit. (This is something that you can think of for classical probability distributions and ...
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19
votes
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Why is a quantum computer in some ways more powerful than a nondeterministic Turing machine?
From a pseudo-foundational standpoint, the reason why BQP is a differently powerful (to coin a phrase) class than NP, is that quantum computers can be considered as making use of destructive ...
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18
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What are examples of Hamiltonian simulation problems that are BQP-complete?
There are plenty of different variants, particularly with regards to the conditions on the Hamiltonian. It's a bit of a game, for example, to try and find the simplest possible class of Hamiltonians ...
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16
votes
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What does Google's claim of "Quantum Supremacy" mean for the question of BQP vs BPP vs NP?
Google's paper/results are kind of sideways to questions in computational complexity about the relation between $\mathrm{BPP}$ and $\mathrm{BQP}$ (and even further from questions about whether $\...
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12
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What is postselection in quantum computing?
As the other answer conveyed (and to which I am just trying to provide some clarification), post-selection is about just looking at a subset of possible measurement outcomes. To my mind, this falls ...
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12
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Do there exist problems known to be computationally intractable for quantum computer, but tractable for classical computer?
It is indeed true that $P \subseteq BQP$ and so any problem solvable on a classical computer is solvable on a quantum computer.
Physics intuition
The physics intuition behind $P \subseteq BQP$ is ...
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10
votes
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Why doesn't Deutsch-Jozsa Algorithm show that P ≠ BQP?
I believe there are two issues here. The first isn't anything wrong with your statement, but rather that you could make a far stronger (non-quantum) statement by the same reasoning: $\mathsf{P}\neq \...
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10
votes
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Is BQP only about time? Is this meaningful?
BQP is defined considering circuit size, which is to say the total number of gates. This means that it incorporates:
Number of qubits — because we can ignore any qubits which are not acted on ...
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9
votes
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Jones Polynomial
This answer is more or less a summary of the Aharonov-Jones-Landau paper you linked to, but with everything not directly related to defining the algorithm removed. Hopefully this is useful.
The ...
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7
votes
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Query regarding BQP belonging to PP
Two quick comments before explaining this:
The notes don't actually contain a proof of the claim made about the simulation; the intention was only to give a basic idea of how the simulation works. It ...
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6
votes
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CS conjecture that Quantum Computer cannot solve NP-complete problems, but Boson Samplers do a #P-hard problem. How is it?
Boson sampling samples from a distribution, but does not compute the full distribution. While computing the distribution is linked to computing permanents, which is #P-hard, we would expect that ...
- 5,598
5
votes
Is BQP only about time? Is this meaningful?
Not for memory, at least, as every memory access requires $O(1)$ 'time'.
In the term time complexity, 'time' is a bit misleading, as we actually count the number of elementary operations required to ...
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4
votes
Jones Polynomial
You have mentioned five papers in the question, but one paper that remains unmentioned is the experimental implementation in 2009. Here you will find the actual circuit that was used to evaluate a ...
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4
votes
Do there exist problems known to be computationally intractable for quantum computer, but tractable for classical computer?
I would think that if a problem is tractable on a classical computer then it is tractable on a quantum computer as any classical circuit can be replaced by an equivalent circuit containing only ...
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4
votes
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BQP and PH separation
The Deutsch-Josza problem provides an oracle separation between $\mathsf{EQP}$ (exact quantum-polynomial time) and $\mathsf{P}$, but there's no preclusion against adding randomization to get an ...
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4
votes
Is APPROX-QCIRCUIT-PROB a BQP-complete problem?
In BQP-Complete Problems by Zhang (2012)
Like many [...] "semantic" complexity classes, BQP is not known to contain complete problems.What people usually study for completeness, in such a ...
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4
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Clarification needed for the N&C proof that BQP ⊆ PSPACE
Basic Definitions:
If you don't know the definitions of the basic computational complexity classes well, I strongly recommend going through Watrous' lecture. We won't be using the quantum Turing ...
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3
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CS conjecture that Quantum Computer cannot solve NP-complete problems, but Boson Samplers do a #P-hard problem. How is it?
This is a well-framed question that highlights subtleties about what is known and unknown on the strengths and limitations of quantum computers.
Initially, it is completely consistent with what we ...
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3
votes
Do there exist problems known to be computationally intractable for quantum computer, but tractable for classical computer?
Other great answers address the question in the body of the posting, specifically about the relationship between BPP and BQP.
However the question in the title can be construed a bit more broadly. ...
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3
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Is it known that BQP is not contained within NP?
There is no known relationship of BQP and NP. The wikipedia page is up to date on the relationship of BQP to other classes.
The pdf you linked is not a peer-reviewed publication, and should not be ...
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3
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Is APPROX-QCIRCUIT-PROB a BQP-complete problem?
Just to further @Egretta.Thula's answer, a portion of the Wikipedia article on the APPROX-QCIRCUIT-PROB mentions $\alpha$ and $\beta$ and stated:
Note that the problem does not specify the behavior ...
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3
votes
Accepted
Could finding Golomb rulers be in $\mathrm{BQP}$?
Here's a theorem that gives a nice, elegant (yet not optimal in the ruler sense) algorithm that can run on any computer (classical, quantum, basically any turing complete system):
Theorem : For any $...
- 524
3
votes
Accepted
$\sf BQP$ and general $\mathrm{SU}(2^n)$ gates
I think the issue here is that you've got to be careful with families of circuits. If you're picking a single fixed gate from $SU(2^k)$ for some $k$, then that doesn't necessarily help you with $L$ ...
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3
votes
What does Google's claim of "Quantum Supremacy" mean for the question of BQP vs BPP vs NP?
Paraphrasing some tweets on the matter earlier, the result is rather underwhelming because it plays on a discrepancy between what they mean by quantum supremacy (QS) and what people tend to think QS ...
2
votes
Consequences of SAT ∈ BQP
Boaz Barak has a lovely essay on various hypothetical worlds with quantum computers. In particular, he calls your world where NP$\subseteq$BQP "popscitopia", and provides:
...in ...
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2
votes
Accepted
Polynomial time reductions vs. Quantum Polynomial time reductions
Following O'Donnell's answer from a student's question about uniformity in O'Donnell's lecture on the BQP complexity class, I claim that it does not matter whether the reduction is polynomial-time ...
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2
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How powerful are boundedly many $T$-gates?
I think your hierarchy collapses, or at least would never get beyond $P$, following the top-line results of Bravyi and Gosset.
Bravyi and Gosset's paper gives an algorithm to classically simulate a ...
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2
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Why doesn't Deutsch-Jozsa Algorithm show that P ≠ BQP?
I will try to give an answer from complexity theory's point of view. This question should be asked in cs.stackexchange by the way. The Deutsch-Jozsa problem has an efficient algorithm on quantum ...
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2
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Confusion regarding hardness of BQP
I don't think your hierarchy is all that well-defined as-is; I think you're confusing the width $n$ of the circuit with the length or depth $d$ of the circuit.
For example for a fixed universal gate ...
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2
votes
Accepted
Relation between BQP-Complete and $BQP\setminus PH$
Indeed I think it follows from (1) showing that evaluating the Jones polynomial is (Promise)BQP-complete, and (2) the existence of an oracle separation between BQP and the polynomial hierarchy PH, ...
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