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.

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24 views

Help in “commutator expansion” containing creation and annihilation operators

I am trying to expand the commutator $\left[ {{H_0},S} \right]$ to get the ${H_1}$ according to the equation: $${H_1} = {H_I} + \left[ {{H_0},S} \right],$$ where, $${H_I} = {g_k}\left[ {\left( {{b_k} +...
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46 views

Eigenvalues of $H=aX+bY+cZ+dI$

Suppose I have the hamiltonian $H=aX+bY+cZ+dI$, where $a,b,c,d$ are some real constants, and $X,Y,Z,I$ are Pauli matrices. I'm trying to figure out the range of possible energy eigenvalues. If I limit ...
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How do I find the combined Hilbert space for this Hamiltonian?

I knew that the operators in the folwing Hamiltonian act in different Hilbert spaces, so I cannot just multiply them. $$\eqalign{ & H = g\left[ {\left( {a\sigma _1^ + + {a^\dagger }\sigma _1^ - ...
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Can someone explain period finding in either the Discrete or Quantum Fourier Transform (Shor's algorithm)

I am learning about Shor's algorithm, a way to find factors of large numbers using a quantum computer. One of the main steps of this algorithm relies of the Quantum Fourier Transform (QFT) - a quantum ...
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How to multiply the data distribution by the dirac delta function to get the original function?

Given data set $\{x_i, f(x_i)\}$, and $i\in[1, n]$, I try to get the original function for the $f(x)$. However, I can only access the $f(x_i)$ by using the $f(x_i)$ to multipy with dirac delta ...
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Is {Z, CZ} an universal quantum gate set for diagonal gates with eigenvalues +1 and -1?

Consider diagonal quantum gates with eigenvalues $\pm 1$, i.e. all diagonal elements are either $+1$ or $-1$. Can these gates always be decomposed into a finite number of Z and controlled-Z gates? My ...
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22 views

Help in understanding the symmetrical quantum system.

Help in understanding symmetrical quantum system. In some articles I read this sentence: When the tripartite quantum system is symmetrical, i. e., the state of the whole system is invariant under the ...
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What's connection between eigenvalue/eigenvectors and outer products?

What's the connection between eigenvalues and vectors and outer products? For example, if an operator A has eigenvalues $\lambda_1$ and $\lambda_2$ corresponding to eigenvectors $|{\psi}_1\rangle$ and ...
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35 views

Approximate matrix inverse by Fourier approach

Given a hermititan matrix $A$ with the possibility to generate $e^{-iAt}$ for $t\geq 0$ how would I proceed to approxiamte: $A^{-1}\approx\sum_j\alpha_je^{-iAt_j}$ $\quad$ with $\alpha_j\in\mathbb{C}$...
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Question on notation from Quantum Computing book

I've started reading Nielsen and Chuang's book on Quantum Computing, and didn't get far. In their first chapter on "Nomenclature and notation", I saw the following expression: $$ \langle \...
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Bounding Trace Distance with Quantum Fidelity

Let $\rho_{1},\rho_{2}$ be density matrices on finite-dimensional Hilbert spaces over the complex numbers. I need to show $1 - F(\rho_1,\rho_2) \le \frac{1}{2}Tr |\rho_1 - \rho_2 | \le \sqrt{1 - F(\...
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Notation for n-qubits.

I have been asked to, given a n-qubit state and measuring in the computational basis on the first qubit, to calculate the probability of obtaining the outcome 1 and writing the state of the system ...
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30 views

Proving properties' equivalences on unitary matrices

I have been asked to given a matrix A, on the complex numbers, proof the following conditions are equivalent: A is a unitary matrix. ∥A|x⟩∥ = ∥|x⟩∥ for every |x⟩ ∈ C . A transforms orthonormal basis ...
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Decomposition of Quantum Doubles

I'm trying to understand Kitaev's quantum double by following Shawn X. Cui's notes (Topological Quantum Computation). Let $L(s_0,s_1)$ denote the subspace of excited states at sites $s_0,s_1$ and that ...
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41 views

What is the meaning of the words here?

I'm not sure what the meaning of the question is, and I need some help understanding what I'm even trying to do. The unitary operator in question is in $U(4)$ The question is: "Decompose the ...
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44 views

Writing Probability distribution in terms of a trace over a density matrix

I have been given and expression for a probability distribution precisely, $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ and I have been asked to show that the above expression ...
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Can't understand how the authors got this equation?

Let us consider a two-qubit system (A and R) initially entangled as given by: $$ \left| {{\Psi _{AR}}} \right\rangle = \alpha \left| {{0_A}} \right\rangle \otimes \left| {{1_R}} \right\rangle + \...
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How much classical algorithmics and probability theory is necessary for quantum algorithmics?

I would like to learn more about quantum algorithmics in next months/years. I wonder - do I need to be very very good in "classical" algorithmics to be good in quantum algorithmics, or are ...
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Measuring entangled quantum state

Just starting out with QM here. I believe I'm missing something simple but can't get my head around it. Consider the state: $$ \left | \psi \right> = \frac{1}{\sqrt 2}\left|0\right> + \frac{i}{\...
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Question: are quantum computers a type of Turing machine or something else?

I read that in quantum computations you can not examine the internal state of the computation while it is happening. This is something I thought is always possible with Turing machines. Does this mean ...
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Continuous time quantum walk on directed graph

Continuous time classical random walk is described by probability vector $p(t)$ and generator matrix $Q$: \begin{equation*} \frac{dp(t)}{dt}=Qp(t) \end{equation*} Continuous time quantum walk is ...
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Find a POVM to distinguish the two states

Suppose $|\psi \rangle = |1\rangle$ and $| \phi \rangle = | + \rangle = \frac{1}{\sqrt{2}} (|0 \rangle + |1 \rangle )$. Write a POVM that allows for imperfect distinguishability between two states. ...
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74 views

Bipartite Version of Schur's Lemma

Let $V$ be a finite-dimensional vector space, let $\pi$ be a representation of some group $G$ (I'm really interested in $G=SO(n)$) on $V$, and suppose that a linear operator $S$ on $V\otimes V$ ...
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51 views

Quantum Typicality

Let $\rho, \sigma \in \mathcal{D}(\mathcal{A})$ with $\text{supp}(\rho) \subseteq \text{supp}(\sigma)$, and spectral decomposition \begin{align*} \rho = \sum_{x}p_x |\psi_x\rangle\langle\psi_x| ~~\...
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88 views

Prerequisites for Quantum cryptography

Sir i am from mathematics background. I am aware of pure mathematics subjects. My university is offering quantum cryptography in Jan semister. I am an absolute beginner. Can you please list down the ...
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63 views

$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ satisfies Data Processing Inequality

$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{H}_1, \mathcal{H}_2$ be a Hilbert spaces and $\rho, \sigma$ be density matrices on $\mathcal{H}_1$. Define $$D(\rho\parallel\sigma) := \sup_{0 \leq M ...
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211 views

Prove that the tensor product of unitary matrices is also unitary

Show that if $A$ and $B$ are unitary matrices, then $C = A \otimes B$ is unitary.
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25 views

Computing Inner Product of Tensor Product

Let $|w_2 \rangle = | + \rangle |-\rangle$ and $|w_3\rangle = |-\rangle |+\rangle$. Show that $\langle w_3 | w_3 w_2 \rangle = 0$. I get $\langle w_3 | w_3 w_2 \rangle = (\langle - | \langle + | ) (| -...
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Operation of NOT and Z gate on the qubit in the state $|H\rangle = a|0\rangle + b|1\rangle$?

I have this last question for an assignment and I've been stuck on it for hours. Pauli operators for a two-level system(qubit), $$ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \...
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41 views

Maximum number of independent commuting Paulis

Let $P_n$ be the Pauli group on $n$ qubits. Let $H \subseteq P_n$ s.t. $H$ is commuting and consist of independent elements. it is well known that $|H| \leq n$. However I have not been able to find a ...
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Output register in Shor's algorithm

In Shor's original paper he creates two registers, sets the first to uniform superposition for each possible number $a \text{(mod $q$)}$, and then computes $x^a \text{(mod $n$)}$ into the second ...
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Given an element in a sequence space, find elements which are orthogonal

Let $l^2$ be the Hilbert space consisting of all square summable sequences over $\mathbb{C}$ equipped with inner product $<a,b>=\sum_{k\in\mathbb{N}}\bar{a_k}b_k$. Let $a\in l^2$ be fixed and ...
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Are the multiplications in $\langle v | \bar U^{\operatorname{T}} U | v \rangle$ commutative?

I'm learning about operations on qubits, and I came across this statement: Suppose $|w\rangle = U |v\rangle$, and we want $U$ to preserve state norms. Then $\langle w|=\langle v|\bar U^{\operatorname{...
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258 views

How to prove Projection operators are Hermitian/positive?

I'm using Dirac notation here. Given only that a projection operator is defined by the property $P=P^2$, prove that $P$ is a positive operator on the Hilbert Space, i.e. $ \langle v|P|v\rangle \geq 0 \...
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When is the bipartite state $\rho ^{ab} $ a classical state

According to "Lectures on General Quantum Correlations and their Applications" page (13), for a bipartite state $\rho ^{ab} $, if one performs a local von Neumann measurement $\Pi = \{ \Pi ...
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Understanding the probability of measure in the Deutsch-Jozsa quantum algorithm

I am trying to understand my teacher's mathematical development. Just to give you some context, $\vec{c}$ is a boolean vector and this is the probability of measuring $\vec{c} = (c_1,...c_n)$. When ...
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45 views

quantum tensor equation simplification

I've started learning quantum computing of late and got interested in some secret sharing. In a recent post on the site https://quantumcomputing.stackexchange.com/questions/13195/grover-search-with-...
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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 ...
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When getting the outer product representation w.r.t an input and output basis, what matrix is $\langle w_{j}|A|v_{i}\rangle$ actually in?

If $A:V\to W$, where $|v_{i}\rangle$ is an orthonormal basis for $V$ and $|w_{j}\rangle$ is an orthonormal basis for $W$, then $A=I_{W}AI_{V} = \sum_{ij}|w_{j}\rangle\langle w_{j}|A|v_{i}\rangle\...
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Can logical system rules be transformed to some kind of quantum logical circuits?

I have read some of the basic theory involving quantum logic circuit theory. Now, if I understood correctly, every quantum circuit: Has the same number of input as outputs Is reversible (we can ...
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if $(|a\rangle - |b\rangle)\langle b| + |b\rangle(\langle a| - \langle b|) = 0$ then $|a\rangle = |b\rangle$, where $|a\rangle, |b\rangle$ unit

Essentially the problem above. I've tried approaching it by arguing that $|b\rangle(\langle a| - \langle b|)$ is the adjoint of $(|a\rangle - |b\rangle)\langle b|$, and since $|b\rangle$ nonzero, ...
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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 ...
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A logic better adapted to quantum phenomena?

Our way of mathematical thinking is totally controlled by a simple two-valued logic $(\mathbf{false}, \mathbf{true})$. All deductions are due to this logic and we are unable to think otherwise. But ...
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50 views

Showing if scalar product of vectors is 1 then they are the same quantum state

How do you show that if $|\langle \psi|\phi\rangle| = 1$, then $\phi$ and $\psi$, both of dimension $d$, represent the same quantum state? (Same quantum state iff there exists a $\theta$ s.t. $|\psi\...
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39 views

Group generated by quantum H-gate and T-gate

What is the group generated by Hadamard gate and T-gate? Or, in terms of Algebra: Let $U$ be the multiplicative group of 2-by-2 unitary matrices, and let $P$ be $\{e^{i\phi}I : \phi \in [0,2\pi)\}$. ...
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1answer
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Spotting the matrix representation of the quantum AND operation

https://people.maths.bris.ac.uk/~csxam/teaching/qc2020/lecturenotes.pdf Applying the above construction to AND we get the map $(x1,x2,y) \rightarrow (x1,x2,y⊕(x1∧x2))$ for $x1,x2,y \in \{0,1\}$. The ...
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Unclear theorem about unitary transformations

I am currently working on quantum finite automata, when I came across a paper by Ambainis and Nahimovs. I am having some struggle with the "theorem from linear algebra" stated by them in ...
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1answer
101 views

Probability of error for randomized algorithm solution to the Deutsch-Jozsa problem

The Deutsch-Jozsa problem asks to determine whether a function $f: \{0, \ldots, 2^n - 1 \} \to \{ 0, 1\}$ is constant or balanced (half of the inputs yield 1, the other half yield 0). Suppose we ...
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71 views

Can I find the exponent of a matrix given only a vector and its image?

here is my problem : Given a $GL(m,2)$ matrix $A$, and $x,y$ two non-zero $F_2$ vectors of length $m$ with the premise that $y = x(A^n)$ for some positive integer $n$. The goal is to find n. Is it ...
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87 views

If $m > n$ are coprime, then there (often) exists $p$, $q$ where $mq+1=n^p$. why?

One of the key reductions in Shor's algorithm in quantum computing for finding prime factors of $m$ is that if $n < m$ is coprime with $m$, then there likely exists integers $p$ and $q$ where $mq+...

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