# Questions tagged [quantum-computation]

Quantum computing is a model of computation that uses quantum bits instead of classical $0/1$ bits. This allows for the superposition of classically allowable states. Relevant topics include quantum algorithms (e.g. Shor's factoring algorithm), quantum information theory, quantum entanglement, and quantum annealing.

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### Estimating ground state energy of $n$-qubit $2$-local Hamiltonian $H$ with known coefficients

Suppose we have an $n$-qubit $2$-local Hamiltonian $H$ with known coefficients. The eigenvalues of $H$ lie in $[0,1)$ and can all be written exactly with $[2 \log_2n]$ bits of precision. You would ...
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### Classification of unitary modular tensor categories (UMTCs)

Context/background: I'm approaching this topic from the perspective of anyonic systems. In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that i) The ...
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### Quantum P vs NP equivalent problem

If $P = NP$, does it follow that $BQP = NP^{BQP}$? I came up with this question when I was thinking about how $P = NP$ can be described as "does every decision problem where a proof for YES can be ...
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### Unique words in dihedral groups

Suppose $x$ is a word over the alphabet $\{0,1\}$. Let $a$, $b$ be elements of the group Dih$_k$ for some $k$. Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the ...
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### Fixed point of quantum operations

A quantum operation is defined as $$\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}$$ where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
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### Constructing the oracle for Grover's algorithm

For a final project in my class, I decided to try to simulate a quantum computer and implement Grover's algorithm. I followed this excellently written blog post by Craig Gidney, and was successful in ...
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### Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
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### Bounding the von Neumann entropy of a density matrix with the Hilbert-Schmidt norm

Question Suppose I have a $D$-dimensional density matrix $\rho_0$ $\rho_0^\dagger = \rho_0 \quad, \quad \mathrm{Tr} \rho_0 = 1 \quad, \quad \rho_0 > 0,$ with a known spectrum $\{\lambda_i^0\}$ and ...
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### Are all quantum cellular automata invertible & representable?

A little background: As far as I know there is no standard definition of a quantum cellular automaton in the literature. Different authors use different definitions. Here I propose my own definition (...
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### A non-associative three-valued logic

There are three elements: x, y, z and a relation C:         x C y,  y C z,  z C x,     x C x,  y C y,  z C z. Let us introduce two binary operations with respect to the C: "the leftmost" (L) ...
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### Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$

The equivalence I describe below is well-known, but I'd like a simple standard reference for it. Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
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### Why isn't Montgomery modular exponentiation considered for use in quantum factoring?

It is well known that modular exponentiation (the main part of an RSA operation) is computationally expensive, and as far as I understand things the technique of Montgomery modular exponentiation is ...
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### Quantum PCP Theorem

Although I think I know the answers to these, I'd just like to collect them all in one place. What is the quantum PCP theorem, what implications does its proof have for simulation of Hamiltonians and ...
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### Grover's Quantum Search Algorithm

I am confused about an extremely basic point concerning Grover's quantum search algorithm; my confusion suggests to me that maybe I've missed the entire point. My understanding of the algorithm is ...
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I have been reading the paper - "Introduction to Quantum Fisher Information". In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows: Let $D \in M_n$ be a ...