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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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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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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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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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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Is there a term for a linear transformation that maps a standard basis element to a scalar multiplication of a standard basis element?

I'm studying quantum computing in purely algebraic sense, and I've come up with a question: Is it possible to decide whether the output of a quantum logic gate is always non-random, when fed non-...
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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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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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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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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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Definition of indistinguishable states in classical statistics and separability

The problem of indistinguishable particles motivated new statistics such as Bose-Einstein or Fermi-Dirac that were later formalized by von Neumann as Quantum Statistics. In quantum statistics, states ...
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Testing a solution of a vector-valued differential equation

I'm working through a book on Quantum Computing. The section is regarding the Time Evolution Postulate of quantum mechanics, and it has sort of thrown me a curveball. Given time-independent ...
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Entries of a unitary matrix

In the solution a a problem in quantum computation I saw this line: $$U_{ij}=\langle\psi_i|\left(\sum_k|\phi_k\rangle\!\langle\psi_k|\right) |\psi_j\rangle.$$ Where $U_{ij}$ are the entries of a ...
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Prove that the size of generating set of a group is at most log(|G|)

I am studying Nielsen & Chuang's book. In the appendix, they prove a little lemma that if a set $\langle g_1, g_2,...,g_l \rangle$ generates a group $G$, then the size of this set would at most $\...
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Quantum mechanics. Potential barrier of magnetic field

I am having trouble to think how to solve the following problem: The plane $x=0$ separates two parts of space: when $x>0$ there is homogeneous magnetic field, which induction vector $B_x$=$B_y$=0, ...
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Help understanding proof that unitary matrices are length preserving

I'm working through the article Quantum Computing for the Very Curious and am stuck on one aspect of the proof it gives for unitary matrices being length preserving. I've included an annotated ...
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Show that surrounding a $CNOT$ with Hadamard gates switches the role of the control-bit and target-bit of $CNOT$

So I wish to show that surrounding a CNOT with Hadamard gates switches the role of the control-bit and target-bit of CNOT. Explicitly I want to show ($H\otimes H$)CNOT($H\otimes H$) is the 2-qubit ...
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Parity of a boolean function

I was doing a course on Quantum Computing. There was a question in problem set. Which is given below (Question No.5) Here the parity function is defined as(As per the instructor): $$y_i = 1-2x_i$$ $$...
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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. ...
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bound for the time needed to find the remainder

Let M be an n-bit integer, and let $a < M$. Give a bound of the form O(s(n)) for the time needed to find the remainder when M is divided into $a^2$. -My attempt: $ M \rightarrow$ n-bit integer $...
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Interest of superdense coding

When you enter the field of quantum communications, you run at some point into the concept of superdense coding. Basically it is a way to encode classical bits on the qubits of a quantum channel, ...
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Expression of the Hadamard transform on $n$ qbits

In "Quantum Computation and Quantum Information" p.75, the reader must show that the Hadamard transform on $n$ qbits, $H^{\otimes n}$, may be written as: \begin{equation}\label{key} H^{\otimes n} = \...
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Can I consider Quantum Logic as Three-valued Logic?

As I understood, Quantum Logic is the logic system in which the truth values can be either (True), (False) or (both True and False). Can I consider this system as one of the Three-valued Logic ...
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Quantum algorithm's reflection clarification

5.5 Reflections Given any unit vector a, we can create the unitary operator Ref a, which reflects any other unit vector b around a. Geometrically, this is done by dropping a line from the tip of b ...
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How to prove a depolarizing channel is entanglement breaking if and only if it is PPT channel?

Let $\Delta _{\lambda}(X)=\lambda X+(1-\lambda)\frac{tr(X)}{n}$ for all $X \in M_n(\mathbb{C})$. Then $\Delta_{\lambda}$ is entanglement breaking if and only if it is PPT(Positive Partial Transpose)...
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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 $X, Y \succeq 0$ ...
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Understanding shared randomness

I'm trying to get some understanding about quantum games from a lecture notes here. Briefly, there are two players Alice and Bob who play cooperatively against a referee. The referee sends the ...
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Is the product of two unitary Householder matrices necessarily of finite order?

Let $\mathbf A$ and $\mathbf B$ be two unitary Householder matrices. They do not necessarily commute. Is $\mathbf{AB}$ necessarily of finite order (i.e. $\exists$ $n\in \Bbb N$ s.t. $(\mathbf{AB})^n = ...
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Shor's Code-Knill Laflamme Theorem.

I'm new to Quantum Error Correction, and I have a question on Shor's Code. If we have a protected subspace, $V \subset \mathbf{C}^2\otimes \cdots \otimes \mathbf{C}^2$ $V=\operatorname{span}\{|0_{...
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What conditions must be met to make f(x) = a^x mod N periodic but not constant?

The renowned quantum algorithm Shor's Algorithm relies on the periodicity of the function $f(x)=a^x mod N$. The a, x, and N are all positive integers. By observation, we know the function is constant ...
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Let H be the Hadamard operator, prove $H^{\otimes n} \left| 0 \right>^{\otimes n} = \sum_{i=0}^{2^n -1} \left| i \right>$

Let H be the Hadamard operator. $$ H = (\left| 0 \right> \left< 0 \right| + \left| 0 \right> \left< 1 \right| + \left| 1 \right> \left< 0 \right| -\left| 1 \right> \left< 1 \...
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How difficult would it be to find valid answers for this hash arrangement?

If $A$ is a 160-bit number, and $X \& Y$ are two SHA-1 hashes, to be generated such that the 320-bit number $X\mathbin\|A$ hashed to $Y$, and the 320-bit number $A \mathbin\| Y$ hashed to $X$? ...
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Is there ever a need to place two Hadamar gates side by side in a quantum circuit?

I'm just starting to learn quantum computing and playing around with my own circuits. I've seen a few examples of circuits where people are placing two Hadamar gates side by side. I've worked out the ...
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How does reversal symmetry demand the opposite pseudo spin in K and -K valley in TMDC?

I am trying to understand the valleytronics in TMDC. I have searched quite a large number of research articles but they just write the valleytronics is due to time-reversal symmetry. Can someone shed ...
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Understanding a discrepancy in tensor multiplication

I have seen in texts about quantum computing people take two vectors and do tensor multiplication. Now, what confuses me is that vectors are (1,0)-tensors. This means that when I multiply two of them, ...
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Pauli-Z gate clarification

In this following document: https://people.cs.umass.edu/~strubell/doc/quantum_tutorial.pdf Page 16 says the following about Pauli Z gate However the right-hand most (ket-bra) side of the equation ...
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Calculate the square root of Euler angles

I am trying to find a nice way to represent the square root of an arbitrary single qubit unitary to implement Lemma 6.1 from this paper: https://arxiv.org/pdf/quant-ph/9503016.pdf Given the Euler ...
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definition of the partial trace and its complete positivity

I know the definition of the partial trace by its Kraus decomposition (which converges in trace norm to some trace class operator) $$ \mathrm{tr}_2(A) = \sum_k ( \mathbb{1} \otimes \langle f_k \rvert) ...
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Understanding the Pauli-Y gate in the Bloch sphere

I'm having some trouble understanding the Bloch representation of qubits in some cases. The canonical representation $\cos(\psi/2) |0\rangle + \sin(\psi/2)e^{i\theta}|1\rangle$ has the first ...
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How to calculate the unitaries satisfying $U_YXU_Y^\dagger=Y$ and $U_ZXU_Z^\dagger=Z$?

These are the Pauli $X$, $Y$ and $Z$ matrices respectively: $$X=\begin{bmatrix}0&1\\1&0\end{bmatrix},\ Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix} \text{ and } Z = \begin{bmatrix}1&0\\...
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Proving a finite Exponential Sum (textbook expression)

enter image description here I saw a mathematical expression in a Quantum Computation Textbook that states $$\sum_{k\in\{0,r,2r,...,N-r\}} \exp\left(\frac{2\pi ikl}{N}\right) = \sqrt{N/r} \text{...
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How to find the value with which I need to divide a unitary matrix such that its first component lies within the range of cosine?

In quantum mechanics, we generally deal with unitary matrices. In the IBM cloud computers and Qiskit, the general $2\times 2$ unitary is defined as $$U(\theta, \phi, \lambda) = \begin{pmatrix} \...
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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 $\...
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Showing that any linear operator can be written as a sum of Hermitian matrices [duplicate]

Let $(V, \mathbb{C})$ be a complex - valued vector space. Let $A$ be any linear operator acting on this vector space. Suppose that $B = \{|v\rangle_{k}\}_{k=1}^{n}$ is a basis set for $(V, \mathbb{C})$...
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Apply the Hadamard transform to different state vectors

I am in the process of understanding a proof. First, the following is said there: $$H\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}=\frac{1}{\sqrt{N}}\begin{pmatrix}1\\1\\1\\1\end{pmatrix}$$ This is ...
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Hadamard transform for two qubit

This question has been fussy me for two hours so I'd appreciate some help. The Hadamard operator on one qubit may be written as $H = \frac{1}{\sqrt{2}}[(|0\rangle + |1\rangle)\langle0| + (|0\...
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79 views

Determine the matrix representation for an operator written as an outer product

Suppose $|v_{i} \rangle$ is an orthonormal basis for an inner product space $V$. What is the matrix representation for the operator $|v_{j}\rangle \langle v_{k}|$, with respect to the $|v_{i}\rangle$ ...
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How to prove Shannon entropy inequality with something that seems to be some sort of taylor expansion

I'm slightly confused about some sort of "proof" (probably not a real proof since it's physics math) I have the formula $f(x) = f(y) + (x-y)f'(y) + \frac{f''(y)}{2\epsilon}, \quad \epsilon \in (x,y)$...
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Deeper understanding of the adjoint of a linear operator

My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality. At this point, I'd like a deeper and heuristic understanding of it. My questions are ...

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