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Questions tagged [quantum-state]

In quantum physics, quantum state refers to the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. A mixture of quantum states is again a quantum state. (Wikipedia)

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What is the meaning of the state $|1\rangle-|1\rangle$?

I don't understand it. But suppose you have a state of two qubits in superposition and it looks like $|10\rangle-|11\rangle$. Now you have in your circuits two gates one is a controlled-NOT from the ...
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Quantum machine learning after Ewin Tang

Recently, a series of research papers have been released (this, this and this, also this) that provide classical algorithms with the same runtime as quantum machine learning algorithms for the same ...
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Concurrence for a two qubit state

The concurrence for a state $\rho$ as defined here is \begin{equation} C(\rho) = {\rm max}\{0, \lambda_1-\lambda_2-\lambda_3-\lambda_4\}. \end{equation} Where $\lambda_i$ are the eigenvalues of matrix ...
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A two qubit state in a special form

How can a pure two-qubit state $|\psi\rangle = a |00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$, be written in the following form \begin{equation} |\psi_{\alpha}\rangle = \sqrt{\alpha}|01\...
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Is it possible to create a superposition in IBMQ QISkit which has probability amplitudes $|a|\neq |b|$?

For example, we can create a single qubit state with a polar angle of $\pi/2$ with the Hadamard gate. But, can we create a state such as this, $$\Psi = \cos(\pi/8)|0\rangle + \sin(\pi/8)|1\rangle$$ ...
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Why does $|P_U − P_V |$ equal $\langle \psi |U^{\dagger} M U|\psi\rangle −\langle \psi |V^{\dagger} M V |\psi\rangle$?

In QC and QI by Chuang and Nielsen, they state that the $P_U$ of operation $U$ acting on $\psi$ can be reached by $\langle \psi |U^{\dagger} M U |\psi\rangle$. Where $P_U$ (or $P_V$) is the ...
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If $|\psi\rangle, U|\psi\rangle$ are known, how many pairs of such qubits are required to find the operator $U$?

Assume that we know a quantum state and the result of applying an unknown unitary $U$ on it. For example, if the quantum states are pure qubits, we know $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ ...
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Do the probability amplitudes describe the probability of projective measurement?

I don't really know the difference between projective measurement and regular measurement or POVM. Until now, wherever I read about the subject I saw that the amplitude describes the probability of ...
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Why does $x\sqrt{1-x^2}$ enhance the ability to approximate analytical functions in quantum circuit learning?

In this paper Quantum Circuit Learning they say that the ability of a quantum circuit to approximate a function can be enhanced by terms like $x\sqrt{1-x^2}$ ($x\in[-1,1])$. Given inputs $\{x,f(x)\}$, ...
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Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$

I'm curious about how to form arbitrary-sized uniform superpositions, i.e., $$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$ for $N$ that is not a power of 2. If this is possible, then one can ...
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Better Way Of Separating Two CQ-States

I have this cq-state: $$\frac{1}{2} \times (|0\rangle \langle0|_A \otimes \rho^0_E + |1\rangle \langle1|_A \otimes \rho^1_E)$$ Where Alice (A) is classical and an adversary Eve (E) has some ...
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Rabi oscillations with different energy differences

Please check this Rabi oscillations image1: Now, let's say I want to create a superposition of E1 and E2, so I started an EM field tuned to a frequency related to E2-E1. But, let's say there is ...
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Hamiltonian of the valence electron of Yb+ ion

Let's say I have chosen the valence electron of Yb+ as my qubit. I want to consider it's hyperfine structure as the two-level energy states. Are my following assumptions correct? If the magnetic ...
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Can I call $\{|0\rangle, |1\rangle\}$ the “standard computational basis” as it is done in linear algebra?

$\newcommand{\ket}[1]{|#1\rangle}$Is there some standard computational basis defined in quantum computing? Can I just call $\{\ket{0}, \ket{1}\}$ the standard computational basis?
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How to read Dirac notation (without algebra)?

I have no idea how to read Dirac notation. $$\left|↑↑ \right\rangle \tag{1}$$ $$\left|↑↓\right\rangle+\left|↓↑\right\rangle \tag{2}$$ $$\left|↑↓\right\rangle-\left|↓↑\right\rangle \tag{3}$$ $$\left|↓...
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How does a $2 \pi$ pulse in Cirac Zoller give a -1 sign to the state?

I understand the first step in the Cirac-Zoller controlled-phase gate; about how to move the state from the electronic state to the vibrational mode state. However, I am unable to understand how a $2\...
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Can “experimental data from a quantum computer” be used to test separability probability conjectures?

An article entitled "Experimental data from a quantum computer verifies the generalize Pauli exclusion principle" by Scott E. Smart, David I. Schuster, and David A. Mazziotti has just appeared In the ...
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Ion trap qubit Hamiltonian calculation manipulation

What is the Hamiltonian of a single ion in an ion trap? How to derive that formula? Also, how exactly can we create superposition by sending laser beams? I am interested in knowing the exact solutions ...
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Optimal strategy to a quantum state game

Consider the following game: I flip a fair coin, and depending on the outcome (either heads/tails), I'll give you one of the following states: $$|0\rangle \text{ or } \cos(x)|0\rangle + \sin(x)|1\...
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Measurement of a qubit and storage of the information on a bit

Suppose we have the quantum circuit below with a quantum register of 2 qubits and a classical register of 2 bits. The Hadamard gates and CNOT gate are not important for the question. When we measure a ...
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Quantum proof for the group non-membership problem

Group non-membership problem: Input: Group elements $g_1,..., g_k$ and $h$ of $G$. Yes: $h \not\in \langle g_1, ..., g_k\rangle$ No: $h\in \langle g_1, ..., g_k\rangle$ Notation: $\...
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How to translate matrix back into Dirac notation?

In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $|\Phi^{+}\rangle$ state is: $ M \...
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Why do we have to uncompute rather than simply set registers to zero?

In implementing a quantum subroutine it is important to uncompute temporary registers after use, to ensure the output state of the subroutine is not entangled with them (which would affect its ...
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What is the notational convention for the single-qubit states $\lvert0\rangle\pm i\lvert1\rangle$?

The features of major search engines (looking at you, google!) prevent me from searching for an answer to this question, so I will poll the community here. Is the state $|{\pm} i\rangle$ generally ...
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What is the difference between a relative phase and a global phase? In particular, what is a phase?

I know that $re^{i\theta}$ = $x + iy$ for some imaginary number $z$ by Euler's formula. How do you calculate relative and global phase?
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Evaluate the given quantum circuit

The input state on the qubit is $|1\rangle\otimes(2|0\rangle+|1\rangle)$. I am assuming this means that the first qubit has the state $|1\rangle$, and the second qubit has the state $2|0\rangle+|1\...
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Representing a real valued vector with qubits

I have some understanding that with qubits, I can represent a $2^n$ size vector using only $n$ qubits. However, I'm having trouble putting this together in a way that can make a useful circuit. Say I ...
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Phase shifter acting on double rail states

In Nielsen and Chuang, it is stated that the photonic phase shift gate acts on the single photon states as $P|0\rangle \ = \ |0\rangle$ and $P|1\rangle \ = \ e^{i\Delta}|1\rangle$, where $\Delta \ = \ ...
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What are some examples of measuring qubits in different basis states?

What are some examples of measuring qubits in different basis states (it's known that preferable choices are the computation basis states $|0\rangle$, $|1\rangle$ and the Bell basis states $|+\...
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What is the post measurement state given an input and the outcome measurement?

Given a two-qubit state of equal superposition, what is the post-measurement state (should be the same number of qubits that have changed as a result of the measurement) on one of the two qubits and ...
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Default input states of qubits to quantum circuits

It is conventional to assume that the state input to the circuit is a computational basis state, usually the state consisting of all $|0\rangle$s. This rule is broken frequently in the literature ...
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How is postselection used in quantum tomography?

I refer to this paper but reproduce a simplified version of their argument. Apologies if I have misrepresented the argument of the paper! Alice has a classical description of a quantum state $\rho$. ...
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Is the set of classical-quantum states convex?

I read about the classical-quantum states in the textbook by Mark Wilde and there is an exercise that asks to show the set of classical-quantum states is not a convex set. But I have an argument to ...
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Why does state preparation of 'off the shelf' qubits not follow from the Born rule (Mermin)?

In Mermin's Quantum Computer Science, section 1.10 (Measurement gates and state preparation), Mermin writes that: This role of measurement gates in state preparation follows from the Born rule if ...
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Multi-photon states in photonic quantum computing?

Within photonic quantum computing, one of the ways to represent information is the dual-rail representation of single-photon states ($c_0|01\rangle \ + \ c_1|10\rangle$). Is it possible to utilize ...
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Ways in which $\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$ can be expressed as $\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle)$

I want to find out what values $|u\rangle$ and $|v\rangle$ can take if I want to write $$\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$$ as $$\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle).$$ Say $$|u\...
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Proof of no-cloning

I was reading a proof of No-cloning theorem, there are a couple of steps that are not clear to me, but the book does not give explanation for them. So here it is: Theorem: It is impossible to create ...
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Non-layperson explanation of why a qubit is more useful than a bit?

I have a computer science and mathematics degree and am trying to wrap my head around quantum computing and it just doesn't seem to make sense from the very beginning. I think the problem is the ...
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A question about notation for quantum states

I am reading an article on representation of digital images using qubits. But I am not able to understand the notations of the article. Can somebody help me: $|0\rangle$: It means the first basis ...
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FPGA qubit simulation

This question is regarding the simulation of qubits, using FPGAs. My question is: how does using FPGAs to simulate qubits help us understand or give us an insight into how quantum computers could be ...
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How to encode a qubit in standard basis?

I wonder what the steps for encoding a qubit in a certain basis are (you can give your answer in terms of the standard basis for simplicity). Example and a little explanation about steps would be ...
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How to measure in the standard basis?

I saw some related topics, but none of them give step by step instructions on measuring in standard basis (or some other basis). Could you please give such instructions? Giving an example would be ...
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Why is the state of multiple qubits given by their tensor product?

How did we derive that the state we get by $n$ qubits is their tensor product? You can use $n=2$ in the explanation for simplicity.
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Why do $n$ inputs to a ket give a vector of dimension $(2^n,1)$?

NB - notation from Octave. I understand that ket([0]) is the $(1,0)$ in a $(x,y)$ plane. But when I try a ket with three numbers I get a column vector of dimension $(8,1)$. I assume that comes ...
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Circuit to construct a $n$-qubit state which is a superposition of states with only a single qubit being $\lvert1\rangle$ [duplicate]

So the question came up in a book I am working through. Given a circuit with $n$ qubits, construct a state with only $n$ possible measurement results, each of which has only $1$ of $n$ qubits as $1$, ...
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Transformation of a Bell state

I am relatively new and interested in quantum computing. Specifically, I am interested in transforming an equation that I found on Wikipedia. But I did not quite understand the transformation. $ \...
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Could we use varying voltage with programmable gates?

One of the benefits I'm reading about qubits is that they can be in an infinite number of states. I'm aware of Holevo's bound (even though I don't fully understand it). However, it made me think of ...
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Imperfect Quantum Copying

It is known by the no-cloning theorem that constructing a machine that is able to clone an arbitrary quantum state is impossible. However, if the copying is assumed not to be perfect, then universal ...
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Outcome of Hadamard transformation on a complex state

I would like to calculate the state after a transformation using the Hadamard gate on a complex state. I get stuck mid-calculation, most likely due to not being able to dealing with the global state. ...
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What happens when I input two qubits starting at state $|00\rangle$ into a hadamard gate?

I was watching the following video https://www.youtube.com/watch?v=IrbJYsep45E And around 3 minutes in, they do an example computation with two qubits. The qubits started off in the state $|00\rangle$...