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Questions tagged [quantum-computation]

Quantum Computation deals with considering computation as fundamentally physical, as well as replacing the classical binary digit (bit) with the quantum binary digit (qubit). While the classical bit is either 0 or 1, the qubit can be in a superposition of these states. Computation systems that use quantum phenomena, such as superposition and entanglement, can solve certain complex problems very quickly.

Related Links

Quantum Computing on Wikipedia

(Textbook) Quantum Computation and Quantum Information: The de-facto standard textbook for learning about quantum computing.

(Video Series) Quantum computing for the determined: Khan-academy-style videos explaining quantum computation, by Michael Nielson (co-author of Quantum Computation and Quantum Information).

370 questions
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Is a bra the adjoint of a ket?

The instructor in my quantum computation course sometimes uses the equivalence $$(\left|a\right>)^\dagger\equiv\left
orome
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Notation in formula for tensor product of Hadamard matrix

I'm having trouble understanding the notation used in a linear algebra exercise (it's exercise 2.33 of Nielsen and Chuang's "Quantum Computation and Quantum Information", page 74). The exercise gives the Hadamard matrix $$H = \frac{1}{\sqrt{2}}…
Ricardo Massaro
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Exponential of Pauli Matrices

Let $\vec{v}$ be any real three-dimensional unit vector and $\theta$ a real number. Prove that $\exp(i\theta \vec{v}\cdot\vec{\sigma}) = \cos(\theta)I + i\sin(\theta)\vec{v}\cdot\vec{\sigma}$, where $\vec{v}\cdot\vec{\sigma} \equiv…
Mathematicing
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$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ representation of $B_3$ braid group

I've been trying to find a representation of the braid group $B_3$ acting on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ but I can't find it anywhere. From what I understand I have to find two $8 \times 8$ matrices $\sigma_i$ satisfying…
GKiu
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Probabilistic classical algorithm on Deutsch-Jozsa problem

Here is the description of Deutsch-Jozsa problem: Let $f: \{0,1\}^n \mapsto \{0,1\}$ be a function promised to be either constant or balanced ('balanced' means that $f$ outputs as many 0's as 1's). I need to show that a probabilistic classical…
Guangliang
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How to apply the Schur-Weyl duality to a three-qubit system?

I am interested in applying Schur-Weyl duality to quantum information theory, specifically "qubits". But I have been stuck for some time on understanding how the Young symmetrizers work in this situation. Let $V$ be a 2 dimensional complex vector…
Simon Burton
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Set of Density Operators with Separable Eigenbasis

Let $\mathbb{H}^A$ and $\mathbb{H}^B$ be finite dimensional Hilbert spaces. Consider the set $S$ of all bipartite density operators $\rho \in D(\mathbb{H}^A \otimes \mathbb{H}^B)$ such that every eigenspace of $\rho$ has an orthonormal basis…
Michael Liu
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How many unique values of $\cos(\frac{a\pi}{N})\cos(\frac{b\pi}{N})$ are there for the positive integers $a,b < N$

The Question How many unique values of $\cos(\frac{a\pi}{N})\cos(\frac{b\pi}{N})$ are there for the positive integers $a,b < N$ for a given $N$? I would like a function $f(N)$ which gives that number of unique values. Upper Bound It can be shown…
Elliot E
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2 answers

How are spin network edges related to anti-symmetric projectors on the Hilbert space of the fundamental rep of SU(2)?

In the paper here https://arxiv.org/pdf/gr-qc/9905020.pdf we see an introduction to Spin-networks of the original Penrose type i.e an undirected open graph whose edges have labels that are irreducible representations of SU(2). In particular given a…
East
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Once a mathematical theorem is proven true like the Halting problem can it ever be disproven?

Just curious about this article I read today in the Google News. I am not a mathematician but enjoy the history of mathematics and the article seems to suggest the Halting problem has been disproven. I always thought once a theorem is proven it…
Sedumjoy
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2 answers

Does the tensor product respect semidefinite ordering in this way?

I'll use $\succeq$ to denote the positive semidefinite ordering: for square matrices $X,Y$, one has $X \succeq Y$ iff $X - Y$ is positive semidefinite. It's a well known fact that if $X, Y \succeq 0$ then $X \otimes Y \succeq 0$. However, if one has…
b2coutts
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Sum of outer product of vectors in a basis

If $\{|u_1\rangle, ..., |u_n\rangle \}$ are an orthonormal basis for $\mathbb{C}_n$, then $$ \sum_{j=1}^{n} |u_j\rangle\langle u_j| = I_n$$ I can see that this is true in the standard computational basis, but I'm having trouble seeing it…
Brian Carmicle
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Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a Hadamard transformation for the coin flip…
sonicboom
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Defining conditional quantum probability

My knowledge of quantum mechanics is very limited, but I will try to ask a purely mathematical question here. If there is a text or resource that explains this, I would definitely appreciate any pointers! I have been unable to find any explanations…
usul
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Prove that set of matrices is dense in $U(2)$

Consider the group of matrices $B$ generated by taking products of the matrices \begin{equation} \rho_1 = \begin{pmatrix}\exp\left(\dfrac{-4\pi i}{5}\right) & 0\\ 0 & \exp\left(\dfrac{3\pi i}{5}\right)\end{pmatrix}\\ \rho_2 =…
PhPanda
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