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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Is a unital completely positive map taking this form trace-increasing?

I just came across this problem, in which we consider a CP map $$ \Phi(\rho)=\sum_i K_i\rho K_i^\dagger. $$ Now, let us transform $\Phi$ into a unital map, by composing it with a Kraus rank-1 map $\...
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Unitary operator times projection operator

In this paper, the authors claim that for $C$ a unitary operator and $P$ a projection operator, if $CP \propto P$, then the constant of proportionality must be one. I don't see why this must be the ...
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Does quantum logic describe quantum logic gates?

From what I've gathered, there are multiple quantum computational logics. But, I've been having difficulty figuring out whether they subsume quantum logic or how much overlap there may be. I even ...
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Doubts regarding Local Hamiltonian Problem

I have background in CS but not in Quantum Computing/Quantum Complexity Theory. I am trying to understand the Local Hamiltonian Problem (the formal definition as below): Local Hamiltonians or Q5SAT: ...
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Can you in general make the maximally entangled state in polynomial time?

In a Hilbert space $\mathcal{H}$ with basis $|i\rangle _{i=1,..,n}$ I am interested in producing the maximally entangled state $\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|i\rangle$ in polynomial time. ...
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How to prove one-to-one correspondence in vector space?

I am given a vector subspace $S$ over the space $\{0, 1\}^n$, and a vector space $S^{\perp}$ which contains the set of vectors in $\{0, 1\}^n$ which are perpendicular to all the vectors in $S$. I have ...
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Quantum gate for inner product with given vector

Let $\mathcal{H}$ be a space of $n$-qudits over $\mathbb{F}_d$, i.e. $(\mathbb{C}^d)^{\otimes n}$. And I have a given vector $a=(a_1,\dots,a_n)\in\mathbb{F}_d^n$. I wish to construct an inner product ...
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What is the relation between the classical and the quantum mechanical approach in calculating the Value at Risk?

What is the relation between the quantities in classical Value at Risk (VaR) calculation and the calculation via quantum computers? To specify this question, I would like to briefly explain my ...
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Integrating an ODE with matrices

I've been watching David Deutsch's Lectures on Quantum Computation here https://www.youtube.com/playlist?list=PLqdVnC7OWuEcfKRZXsrooK_EPzwmWSi-N In Lecture 2, he gives these equations to show how X,Y ...
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Unitary transformation on vector is equal to the transpose on its orthogonal vector?

While working on my Quantum course, I have observed that U|0> = UT|1> for a Unitary matrix U. This is solely based on observations and calculating the matrices. I wish to try and prove this ...
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Derivation of Factoring to Order Finding Algorithm

Suppose $N=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_m^{\alpha_m}$ is the prime factorization of an odd composite positive integer. Let $x$ is chosen uniformly at random from $\mathbb{Z}_N^*$, and $r$ be ...
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Explicit examples of oracle in Dihedral Hidden Subgroup Problem

In the general hidden subgroup problem we are given a generating set of a group $G$ (not necessarily abelian), we are given access to a function $f:G\to\mathbb{C}$, such that there is an unknown ...
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Password alternatives in the cryptography

All the modern informational systems are based on a password concept. The password idea is the cornerstone of contemporary cryptography as well. Assume that quantum computers can and will succeed in ...
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Can we construct an ORTHOGONAL ($trace(A^\dagger B) = 0$) basis for Hermitian matrices made of PSD (positive semi-definite) Hermitian matrices?

The space of $N \times N$ Hermitian matrices can be construed as a real vector space with dimension $N^2$. One can define an inner product on this real vector space by $\langle A, B \rangle := trace(A^...
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Generalization of Quantum information?

I've been working a lot with classical information theory results for my undergraduate thesis (thank you Shannon). At the same time, I am taking a course in quantum computing where we discuss briefly ...
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prove that $E(U^m_{\Delta t},e^{-2miH\Delta t})\leq m\alpha \Delta t^3$ where $U_{\Delta t}=e^{-2iH\Delta t}+O(\Delta t^3)$

Let $H=\sum_{k=1}^LH_k$ and define $U_{\Delta t}=[e^{-iH_1\Delta t}e^{-iH_2\Delta t}\cdots e^{-iH_L\Delta t}][e^{-iH_L\Delta t}e^{-iH_{L-1}\Delta t}\cdots e^{-iH_1\Delta t}]=e^{-2iH\Delta t}+O(\Delta ...
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Prove $e^{(A+B)\Delta t}=e^{A\Delta t}e^{B\Delta t}e^{-1/2[A,B]\Delta t^2}+O(\Delta t^3)$

Prove that $$e^{(A+B)\Delta t}=e^{A\Delta t}e^{B\Delta t}e^{-1/2[A,B]\Delta t^2}+O(\Delta t^3)$$ The Baker-Campbell-Hausdorff Formula is $$e^{tX}e^{tY}=e^{t(X+Y)+\frac{t^2}{2}[X,Y]+\frac{t^3}{3!}[[X,[...
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Increasing amplitude of the $|0\rangle$-state in a superposition

Assume you have quantum state $|\varphi\rangle\otimes|\psi\rangle$, where $|\psi\rangle$ is an $n$-qubit state $$ |\psi\rangle = \alpha|0\rangle + \sum_{i=1}^{2^n-1}\beta_i |i\rangle\,, $$ and $\alpha$...
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What is the distance between the 2 ×2 identity matrix and the phase-gate

I am working through R.de Wolf lecture notes on Quantum Computing and have difficulties figuring out a certain property of matrix norms. Consider the operator norm (induced norm on matrices) : $\|A\| =...
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Prove that $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from $E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|1-\exp(i\beta/2)|$

This is from my reference Page 196, Quantum Computation and Quantum Information by Nielsen and Chuang. How do we prove that for any $\epsilon>0$ there exists an $n$ such that $E(R_n(\alpha),R_n(\...
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How does pigeonhole principle imply that $|\theta_k-\theta_j|\leq 2\pi/N<\delta$ where $\theta_k=(k\theta)\mod 2\pi$

I am trying to understand this part from the book Page 196, Quantum Computation and Quantum Information by Nielsen and Chuang. Here $R_{\hat{n}}(\theta)=\cos(\theta/2)I-i(\hat{n}.\vec{\sigma})\sin(\...
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For bounded operators $A,B$ on a Hilbert space prove that $A^*A \leq cB^*B \Leftrightarrow \lVert A \rVert \leq \sqrt{c}\lVert B \rVert$.

For bounded operators $A,B$ on a Hilbert space $\mathbb{H}$ prove that for any $c \geq 0$, $A^*A \leq cB^*B \Leftrightarrow \lVert A \rVert \leq \sqrt{c}\lVert B \rVert$. This is an exercise from ...
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Prove $|\langle\psi|U^\dagger MU|\psi \rangle-\langle\psi|V^\dagger MV|\psi \rangle|\leq ||(U-V)|\psi\rangle||+||(U-V)|\psi\rangle||$

Prove that $$ |\langle\psi|U^\dagger MU|\psi \rangle-\langle\psi|V^\dagger MV|\psi \rangle|\leq |||\Delta\rangle||+|||\Delta\rangle|| $$ where $|\psi\rangle,|\Delta\rangle=(U-V)|\psi\rangle$ are ...
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Are $\pi$ and $\tan^{-1}\left(2\right)$ rational multiples of each other? [duplicate]

For a proof of quantum universality, I need to show that $\tan^{-1}\left(2\right)$ is not a rational multiple of $\pi$. How do I show this? I feel like showing algebraic independence over the ...
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How many matrices do you need to generate a dense subset of $U(n)$?

In quantum computing, people talk about universal gates, which are a finite collection of matrices $\{U_1,\dots,U_k\}$ that generate a dense subset of $U(n)$. The wiki gives various examples of ...
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To what extent will quantum computing cause disruption in mathematics?

Currently I am involved in a research (German public sector) that aims to assess the security implications and the potential of advances in quantum computing. In Germany we have the first Quantum ...
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Prove that $U=e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)$ for non-parallel unit vectors $\hat{m},\hat{n}$

Suppose $\hat{m},\hat{n}$ are non-parallel real unit vectors in 3D. Show that an arbitrary single qubit unitary operator $U$ may be written as $$ U=e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\...
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Can a Turing machine with a quantum oracle emulate a quantum Turing machine without penalty?

Is there a way to build a mostly classical Turing machine with a special quantum oracle that's equivalent in power to a quantum Turing machine? I am vaguely familiar with the concept of an oracle ...
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Can you make a Gaussian-weighted superposition state with polynomially many gates?

Suppose I have N register qubits, so they can represent the range [0, 2^N-1]. They are initialized in the all-zeroes state. I want my final state to approximate $$|\phi \rangle = \frac{1}{2^{2^N-1}} \...
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Understanding the Unitary Operator $U|\psi\rangle\otimes|0\rangle=\sum_m M_m|\psi\rangle\otimes|m\rangle$

The operator U defined as $U|\psi\rangle\otimes|0\rangle=\sum_m M_m|\psi\rangle\otimes|m\rangle$ where $\sum_mM_m^\dagger M_m=I$, preserves the scalar product $(\langle \phi|\otimes\langle 0|U^\...
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Subsystem representation of quantum system as independent state vector

Suppose I have some entangled pure state of some $n$ qubits. Then, I want to look at a subsystem of some $k<n$ qubits. Can I extract a state vector over $k$ qubits disconnected from the other $n-k$,...
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Graph states: two equivalent definitions

I know two definitions of graph states: $|G\rangle = K^{a} |G\rangle = \sigma_x^{a} \otimes \sigma_z^{b} |G\rangle$ for all $a \in V$ vertices and for all $b\in Na$ connected to $a$ through an edge. $...
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Open problem in quantum computing: find eigenvalues of matrix or differential operator

Solving the following problem would help better understand a stabilization scheme for a quantum computer, originally developed at Yale and currently pursued by Amazon. Consider an infinite tridiagonal ...
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Penalty Function for XOR Gate.

I have been reading a paper on Gates for Adiabatic Quantum Computing. The paper consists of penalty functions for different classical gates like AND, OR, XOR, etc. Can someone explain how to get the ...
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How to create a Hermitian Matrix from the a Unitary one describing a LDE system

I am studying a quantum algorithm for solving systems of linear differential equations (LDEs), which can be written in the following form: $$ \frac{d}{dt}(\vec{x})=M\vec{x}+\vec{b} $$ More ...
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understanding Grover's quantum search algorithm

I had a question about Grovers algorithm. Let $f:\{0,1\}^n \rightarrow \{0,1\}$ and a unique $s\in\{0,1\}^n$ that has $f(s) =0$. Start from uniform superpositionover all n-bit strings u. Step 1: Set $...
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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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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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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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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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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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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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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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1 vote
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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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