Questions tagged [quantum-state]

Quantum systems can mathematically be described by their 'quantum state'. When the system is closed/isolated, the state is 'pure' and can be written as a sum (i.e. 'superposition') of basis vectors. When the system is a subsystem of an open system, the state is instead usually 'mixed' and cannot be written as a pure state, so has to be written as a density matrix. Consider using the density-matrix tag when relevant

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1answer
74 views

What happens if a Pauli $X$ gate is applied to part of a Bell state?

I have started to learn about the mathematics behind ebits and I have a question. Assume $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \...
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3answers
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Saving statevector on more than one location in a quantum circuit in Qiskit

So, I'm fairly new to Qiskit, and I've been playing around and following the tutorials from the Qiskit textbook. However, there is one thing I fail to understand/implement: for a quantum circuit with ...
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45 views

Derivation for the result of performing the Hadamard transform on $|0\rangle^{\otimes n}$ being $2^{-n/2}\sum_x|x\rangle$

It's said that the result of performing the Hadamard transform on n qubits initially in the all |0> state is $$ \frac{1}{\sqrt{2^n}}\sum_x|x\rangle $$ where the sum is over all possible values of x....
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How to prepare a quantum state of the form $\frac1{2^{n/2}}\sum_{x \in \{0, 1\}^{n}} |x\rangle |y_x\rangle$ with $y_x$ random variables?

Let's say I am given an efficiently samplable probability distribution $D$, over $n$ bit strings. I want to efficiently prepare the following state \begin{equation} |\psi\rangle = \frac{1}{\sqrt{2^{n}}...
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Confusion regarding the tensor product usage in book

I have recently started with quantum computing, and I've found great book about it - Learn Quantum Computing with IBM Quantum Experience, which explains a lot of things in quite a simple language. ...
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1answer
76 views

$\mu$ matrix construction for quantum state tomography

In the paper Maximum Likelihood, Minimum Effort, given an orthonormal Hermitian operator basis $\{\sigma_i\}_{i=1}^{d^2}$ of $d \times d$ matrices and a set of measured values $m_{ij}$ corresponding ...
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191 views

What is the role of entanglement in quantum-computational speed-up?

The way I see it, there are three main quantum properties utilized in quantum computing - superposition, quantum interference, and quantum entanglement. I'm looking to understand which one is ...
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What is a term for a basis state along with its corresponding complex amplitude?

For some arbitrary state $|\psi\rangle = c_0|x_0\rangle + c_1|x_1\rangle + c_2|x_2\rangle ... + c_{2^n}|x_{2^n}\rangle$, where each of $|x_i\rangle$ is a basis state, and each of $c_i$ is the ...
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Copying quantum state [closed]

I'm confused about the last complete sentence in the following paragraphs. If ab=0, that means either a or b equals to 0. As a result, doesn't $|\psi\rangle|\psi\rangle$ equal to either $b^2|11\rangle$...
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Why does $a ⊕ (a ⊕ b) = b$ and $(a ⊕ b) ⊕ b = a$? [closed]

For the following circuit that swap two qubits The sequence of gates is said to have the following sequence of effects on a computational basis state |a, b> where all additions are done modulo 2. ...
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The use of modulo 2 in state representation after CNOT

The following circuit with a CNOT gate has the following effect on a computational basis state $|a, b\rangle$, where all additions are done modulo 2. Why is the state of the second qubit changed to $...
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How to describe a known quantum state using classical information?

In Nielsen and Chuang, it's said that to describe a known quantum state precisely takes an infinite amount of classical information since $|\psi\rangle$ takes values in a continuous space (from the ...
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Unitary Transformations for States with Same Entanglement [duplicate]

$\newcommand{\Ket}[1]{\left|#1\right>}$ I know this has been asked before in another context (How to construct local unitary transformations mapping a pure state to another with the same ...
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How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

I create a quantum state $| \psi \rangle = \frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ for a randomly chosen $x_0,x_1$ of 50 bits. I give this quantum state $|\psi \rangle$ to you and you return ...
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Can quantum circuits/operations have truth tables?

In the caption for the following figure, the word "truth table" is put inside a quotation. I am wondering if this means that the truth table the caption refers to isn't exactly a real truth ...
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1answer
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Why can every Bell state be written as $|\beta_{xy}\rangle=\frac1{\sqrt2}(|0,y\rangle + (-1)^x|1,\bar y\rangle)$?

In Nielsen and Chuang, there's the following paragraph: The mnemonic notation $|\beta_{00}\rangle, |\beta_{01}\rangle, |\beta_{10}\rangle, |\beta_{11}\rangle$ may be understood via the equations $$ |\...
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How to construct local unitary transformations mapping a pure state to another with the same entanglement?

$\newcommand{\Ket}[1]{\left|#1\right>}$In Nielsen's seminal paper on entanglement transformations (https://arxiv.org/abs/quant-ph/9811053), he gives a converse proof for the entanglement ...
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For any two quantum states does there exist a gate that takes you from one to the other

For any two states $|\phi\rangle$ and $|\psi\rangle$ Does there exist a gate $U$ such that $U|\phi\rangle = |\psi\rangle$ ? I suppose that we know for a vector space $V$ then $\forall \quad a, b \quad ...
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How to derive $|0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$?

When learning measurement basis, my teacher told us $|0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$ and said that we can derive it ourselves. Along this, he also mentioned $|+\rangle=\frac{1}{\...
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2answers
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Does QRNG generates True Random numbers?

I read an article that claims, QRNG can produce a true random number. So I wonder, how could they prove that this is a true random numbers generator? In fact, imagine I look at my memory state and ...
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1answer
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How is a classical bipartite state written in quantum notation?

As in the title, is a classical bipartite state on $AA'$ given by $$\sum_{ij} p_{ij} \vert i\rangle\langle i\vert_A \otimes \vert j\rangle\langle j\vert_{A'}$$ with $\sum_{ij}p_{ij} = 1$. In ...
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How to generate a quantum circuit from the quantum state $|1000\rangle+|0100\rangle+|0010\rangle+|0001\rangle$?

I am trying to understand the steps of how make a state preparation circuit from a quantum state. For making my question more clearer, for example, for the state is $\frac{|00\rangle+|11\rangle}{\...
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2answers
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What are the constraints for the coefficents of the basis states in quantum computing?

$\newcommand{\ket}[1]{\left|#1\right>}$ It's known that the Kolmogorov axioms characterise a probability distribution: Probability of an event is a non-negative real number. The sum of all ...
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1answer
106 views

How to normalise when the probability of measurement is zero?

In one of the answers to this question on measuring one qubit it is explained that given a general two-qubit state $$ |\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \...
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Can one always find purifications which preserve equality of statistical mixtures?

When pure states $|\psi_1⟩$, $|\psi_2⟩$ and $|\phi_1⟩$, $|\phi_2⟩$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ have identical statistical mixtures $$\frac{1}{2}(|\psi_1⟩⟨\psi_1| + |\psi_2⟩⟨\psi_2|) = \...
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1answer
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How do get $\rho_{BA}$ if I have $\rho_{AB}$

If Alice and Bob share the state: $$\left| {{\psi _{AB}}} \right\rangle = \sin \theta \left| {10} \right\rangle + \cos \theta \left| {01} \right\rangle $$ then $\rho_{AB}$ can be obtained as: $${\...
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Meaning of a pound sign (#) on a Bloch sphere

For the following Bloch sphere representation of a qubit, what does the highlighted symbol mean? I'm not sure if it means anything or it's just for showing that it's a sphere, not a circle.
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158 views

Is there an algorithm that can decide if a state is a mixed state or a pure state?

Given we are using the computational basis,is there a quantum algorithm that can decide if an arbitrary input state $\vert A\rangle$ ( using $N$ qubits) is a pure state or a mixed state? $\vert A\...
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1answer
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What is the complexity of loading $n$ inputs using a qRAM?

I am interested in reading data from a real database and I find qRAM has the effect of loading data: $\sum\phi\left|i\right>\left|0\right>\rightarrow\sum\phi\left|i\right>\left|d_i\right>$,...
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2answers
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What are the conditions ensuring a two-qubit density matrix is positive semidefinite?

I've seen some papers writing $$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
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1answer
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What is the "quantum phase" of a quantum state?

On this page IMBQ docs, until the sentence '..and since the global phase of a quantum state is not detectable..' I follow everything. However 'quantum phase' is introduced without any explaination? ...
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1answer
107 views

Unitary Transformations for Schmidt Decomposition

$\newcommand{\ket}[1]{|#1\rangle}$ Suppose a pure state $\ket{\psi}$ has a Schmidt decomposition given by $\ket{\psi^{SD}}$, which can be obtained via the diagonalization of the reduced density matrix ...
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How to understand combination states vs pure/mixed states?

I've learned that representing a combination of two states, I simply need to take the tensor product of the states. For example: $$\left|\Psi\right>=\alpha_0\left|0\right>+\beta_0\left|1\right&...
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Error correction on $n$ qubits all in same state except for a few

This might be a straightforward problem for you guys; it would be helpful if you can explain it in simple language. I have $n$-qubits given as $$\frac{1}{\sqrt{2}} \left(|0\rangle+ e^{\iota\theta_{1}}|...
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1answer
60 views

How to perform a plot histogram for a circuit?

I have created a circuit and I don't know how to plot a histogram. I tried to plot a histogram but it gives me output for 0000 case only, how to get to know the probability for all of the cases. The ...
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3answers
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What is the best way to extend a state $\rho_S$ to a tensor product of spaces ${\cal H}_S\otimes{\cal H}_A?$

Let $\Phi_S$ be an operator acting on a space $\mathcal H_S$. If we introduce an ancilla $A$, the total space becomes $\mathcal H_S\otimes \mathcal H_A$ and I can naturally extend the operator $\Phi_S$...
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On existence of orthonormal basis for each subsystem in Separable state [closed]

A separable state in $\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is given by $$\rho_{s}=\sum_{\alpha,\beta}p(\alpha,\beta)|\alpha\rangle\!\langle\alpha|\otimes|\beta\rangle\!\langle\beta|.$$ Now, my ...
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What are the conditions under which an unknown quantum state is learnable with arbitrary precision?

Assume that we have an unknown quantum state and we need to learn that unknown state with arbitrary precision. Under what conditions can we learn the unknown state with arbitrary precision? One ...
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1answer
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Restrictions on Quantum Gates and Pure Partially Traced Output States

Suppose we have a quantum circuit that contains an arbitrary number of quantum gates and takes as an input more than a single qubit, say three. What are the restrictions on the quantum gates and the ...
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1answer
108 views

Find the conditions under which the state $|\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle$ is unentangled

Show that the state $ |\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle $ is unentangled if $a \in \{ 0,1,...,2^n - 1\} $ and $|\phi\rangle$ can be expressed in the form $ \...
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1answer
151 views

Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
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1answer
66 views

How can I simulate the Avg CNOT Error on IBMQ Backends?

I want to know exactly how to estimate the Avg CNOT Error rate for IBMQ Backends? For instance, I tried to estimate the Avg CNOT Error rate for Belem Backend; I randomly prepared the 00,01,10,11 ...
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1answer
34 views

What is this pulse experiment showing?

The following results in the figure were achieved using Qiskit pulse by doing the following pulse sequence; { $\frac{\pi}{2}$, delay $\tau$, $\pi$, delay $\tau$, $\frac{\pi}{2}$} The figure is the ...
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1answer
631 views

What is a stabilizer state?

I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'. "The number of repetitions depends on the ...
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Postselection and hardness of estimating amplitudes

Let $A$ be a class of quantum circuits such that \begin{equation} \text{Post}A = \text{Post}BQP, \end{equation} where $\text{Post}$ indicates post-selection. Is only this amount of information ...
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3answers
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Do sequences of operations (including measurements) applied to different halves of an entangled pair always commute?

Let us say $A$ has one half of an entangled qubit pair, and $B$ has the other half. $A$ may be able to perform any type of operation on their half of the pair, such as unitary operations, entangling ...
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How do you quantify the figurative 'cost' of a quantum circuit

Many gates are not available on a real computer and therefore the circuit must be transpiled into a specific set of gates. I have seen this equation below which is used to to determine the 'cost' of a ...
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1answer
41 views

Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be $$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$ where $D(\rho\|\sigma)$...
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82 views

Circuit state preparation using amplitude encoding

I am following an example of preparing an input state using amplitude encoding from this book. How to calculate $\beta_1^1$ using given formula above? In my understanding, $\beta_1^1 = 2\arcsin(\frac{\...
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2answers
102 views

How do I find the state of each qubit at the end of the circuit?

I have this Quantum Fourier Transform (QFT) and I want to know how to find the final state of each qubit if q0, q1, ...

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