Questions tagged [probability]
For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.
123 questions
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Total number of (unique) moments of the Haar distribution
This is probably a standard fact but I cannot find it in my usual references. Let $G$ be one of the classic matrix Lie groups $\mathrm{U}(N), \mathrm{SU}(N), \mathrm{O}(N), \mathrm{SO}(N)$, equipped ...
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1
vote
1
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How computationally advantageous are e-bits compared to probabilistically dependent classic bits
Let a "classical entangled bit" be defined as 2 classical bits put in a dependent probabilistic state like $\left| E\right> = \frac{1}{2} \left|00\right> + \frac{1}{2} \left|11\right&...
2
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0
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Expectation value of an ensemble composed of multiplying the same Haar random unitaries
Let $U$ be an $n$ qubit Haar random unitary and $\mathbb{I}_n$ is the $n$ qubit identity operator. I want to find the density matrix corresponding to the following:
$$
\rho = \underset{U}{\mathbb{E}}\...
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1
vote
1
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Property of Haar random state
Let $|\psi\rangle$ be a Haar random state and let $|\psi^{\perp}\rangle$ be any state that is perpendicular to $|\psi\rangle$. Let us define
$$p_x = |\langle x| \psi \rangle|^2,$$
and $$q_x = |\...
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2
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1
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How does counting in classical 32-bit systems compare to quantum systems with 32 qubits in superposition? [closed]
I'm trying to understand how counting in a classical 32-bit system compares to a 32-qubit quantum system.
In a classical 32-bit system, we can count from 0 to the maximum value (i.e., ) using a CPU ...
1
vote
1
answer
75
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Independence of two systems
This is probably a very noob question, but I've just started learning about quantum computing.
In IBM's lesson on multiple systems they say that two classical systems $X$ and $Y$ with classical state ...
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0
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Reconstruction of the probability of the uncut state fails
I tried to follow the example from the paper "CutQC: Using Small Quantum Computers for Large Quantum Circuit Evaluations" (https://arxiv.org/abs/2012.02333).
Some hints were also given in ...
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0
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Alice and Bob play a Multi-Qubit game
Well I am quite new to this so excuse me if the question is absurd
Alice and Bob each can "measure" variables A and B respectively: Alice can use $a_1$ and $a_2$ methods of measurement while ...
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0
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112
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Does Levy's lemma hold for unitary/spherical designs?
Let $\mathcal{H}$ be a $d$-dimensional Hilbert space equipped with the Haar measure. Levy's lemma says that, for an $L$ -Lipschitz function $f$ on $\mathcal{H}$, the probability that $f(x)$ for a ...
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How do these two notions of "multiplicative approximation" relate?
While reading a few quantum supremacy results, I have come across two ostensibly different notions of "multiplicative approximation". I am wondering how they relate. In the following, $p$ ...
2
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0
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Probability that a quantum state is in the typical subspace of another quantum state
From the properties of the Typical subspace we already have the following theorem [1]:
Theorem (Unit Probability, see [1] page 467): Suppose that we perform a typical subspace measurement of a state $...
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How to get partial probability from measurements in qiskit?
Is there a way to get partial probability distributions on Qiskit?
Consider a quantum circuit measuring all the qubits. I want to retrieve the probability of, say, outcome ...
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vote
1
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40th Question IBM Sample test
Given this code fragment, what is the probability that the measurement would result in 1?
qc = QuantumCircuit(1)
qc.rz(3 * math.pi/4, 0)
A] 0
B] 0.14645
C] 0.85355
...
2
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0
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69
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Grover with randomized oracle
I'm sorry if this is a stupid question. I want to know about the behavior of Grover's algorithm with oracle having a low one-sided probability of error. So if $f(x)=0$ my oracle returns $0$ and if $f(...
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1
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1
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Matrix representation of the symmetric subspace for two copies
Consider two copies of an $n$ qubit Haar random state, given by:
\begin{equation}
\rho = \mathbb{E}_{U \sim \mathsf{Haar}}\left[U |0^n\rangle \langle 0^n| U^{*}\otimes U |0^n\rangle \langle 0^n| U^{*}\...
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4
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1
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Bound on success Probability for Regev's factoring algorithm
Theorem 4.1 in Regev's paper talks about a theorem due to Pomerance as follows:
Theorem 4.1: Suppose G is a finite abelian group with minimal number of generators $r$. Then, when choosing elements ...
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2
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1
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Existence of a two-outcome measurement $M$ such that the induced distributions differs between different density matrices
Let $\rho \neq \sigma$ be density matrices.
I want to show that there exists a two-outcome measurement $M$ such that the induced distributions $M(\rho)$ and $M(\sigma)$ differ.
From what I learned, ...
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2
votes
1
answer
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How to compute the measurement probability in the Hadamard test?
In the Hadamard test (e.g., page 40 of these lecture notes) we have:
But if you look at standard textbook reference, like Nielsen and Chuang, there's an example for how to compute the measurement ...
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1
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0
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How does Chernoff's bound help to solve Exercise 6.4.2 in Kaye et al.'s textbook? [duplicate]
I was wondering if anyone could help me with this question, I'm kind of new to quantum computing in general. I understand the Deutsch Josza Algorithm, but I'm not really sure where to even begin with ...
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3
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1
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201
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Uncertainty of estimates computed by stim/sinter
The sinter.plot_error_rate function handles the plotting of the error rates sampled by sinter. Along with the estimated error rates, it highlights a region within ...
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Where is my mistake in using a measurement operator instead of Born’s rule to calculate the probability of detecting photons at an arbitrary angle?
As I asked in this question: How can I calculate the measuring probabilities of a two qubit state along a certain axis?
From here I know how to calculate the probability of measuring a general state ...
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How can I find the probabilities corresponding to measurement results of an observable of the GHZ states?
I'm working on a problem involving the calculation of probabilities for outcomes of a measurement on a quantum state perturbed by an error. The state in question is a GHZ state $|\text{GHZ}\rangle = \...
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1
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54
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Measuring one register of the state $\frac{1}{2^{m}} \sum_{x}\sum_{k} (-1) ^ {x\cdot k} |k\rangle |f(x)\rangle$
I'm reading the article which contains lemma 1. Its proof contains the statement, that probability of getting $|0\rangle$ (denote it as $\text{Pr}\left[|0\rangle\right]$) after measuring the first ...
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1
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Why this QC probability drop to $|01⟩$ and $|11⟩$ instead of $|10⟩$ and $|11⟩$? [duplicate]
I have QC like this
The final state before measure should be $|1⟩ ⊗ |-⟩$, and it will be $\frac{1}{\sqrt{2}}(|10⟩ - |11⟩)$, so the probability of this QC should be a half is $|10⟩$ and other half is $...
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0
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Semi-Definite Program to maximise $P(X)$ with a fixed CHSH value
This question should be theoretically simple, yet I'm struggling as something may be incorrect about my code. I am trying to plot a graph of the maximum probability ($P(x)$) of a given system against ...
0
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1
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150
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How to find $p_x$ and $p_y$ components on the Bloch sphere?
Consider an arbitrary state:
$$|\psi\rangle = a|0\rangle+b|1\rangle,$$
where $a=\cos\left(\frac{\theta}{2}\right), b=\sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is ...
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1
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Calculate of theoretical probabilities for the outcomes
I have a $|+\rangle$ state qubit and I measure it in a random basis. The random basis is made with random $\theta$, $\varphi$ and $\lambda$ of $U3$ gate. How can I calculate the theoretical ...
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2
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1
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59
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quantum generalisation of random variables
What is the quantum information equivalent of a classical probability random variable ? Is it a density matrix or an observable ? If so can someone show me how to write a random variable that follows ...
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1
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90
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Is there a general method for calculating expectation values for time-dependent wavefunctions?
Is there a general method for calculating expectation value? I have a workshop question, and I'm sure what a good process to follow is. It is given that $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
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Kolmogorov dilations of positive definite kernels and Donsker's Delta
I'm having problems understanding a part of the proof of Kolmogorov’s dilation theorem (Theorem 3.2 given in this).
We define a map $V_K:S\to H$ given by $V_K(x):=\delta_x+N$. Then, we compute the ...
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2
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How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?
I'm working through the book "Introduction to the Theory of Quantum Information Processing" by Bergou and Hillary, and I've encountered a scenario that I'm not sure how to approach. In ...
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1
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Minimum probability of measuring marked state in Grover's algorithm is 1/8
I recently came across the proposition that for a database containing $N$ elements with $m<\sqrt{N}$ marked elements, applying Grover's algorithm with any $T$ $(0<T<\sqrt{N}-1)$ iterations (...
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2
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How to compute marginal probabilities of Alice's qubit (in density operator language)?
Let $| \psi \rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{2}|01\rangle + \frac{\sqrt{3}}{4} |10\rangle + \frac{1}{4}|11\rangle$ be a state vector describing a closed quantum mechanical system.
...
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Density matrix and State vector give different result in mesolve in QuTiP
qutip mesolve gives me different population evolve depending on that initial state is state vector or density matrix. And, in some situation, it gives me negative population. It doesn't make sense...
...
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How to calculate probability of measuring $|1\rangle$ after application of $R_x$ gate
I was trying to understand how to calculate
the probability of measuring $|1\rangle$ when executing the following circuit in Qiskit:
...
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1
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0
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Saturating an inequality relating the operator norm and the total variation distance
Let $U$ be an $n$-qubit unitary, and let $p_U(x) = |\langle x | U | 0\rangle |^2$ be the probability of obtaining $x \in \{0,1\}^n$ on the all zero input. Given two $n$-qubit unitaries $U$ and $V$, it ...
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Given a unitary $U_p:|0\rangle\to\sum_\omega\sqrt{P(\omega)}|\omega\rangle$, what does $|0\rangle$ represent exactly?
Consider a random variable $X$ on a probability space $(\Omega, 2^\Omega, P)$. Let $H_\Omega$ be a Hilbert space with basis states ${| \omega \rangle}_{\omega \in \Omega}$, and fix a unitary $U_P$ ...
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Improving operator norm bound on total variation distance
Let $U$ be an $n$-qubit unitary and $P_U(x) = |\langle x |U|0^n\rangle|$ the probability of measuring $x$ after acting $U$ on $|0^n\rangle$. For two $n$-qubit unitaries $U$ and $V$, one can prove that
...
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Stabilizer States - Calculating measurement probabilities with the rank of the stabilizer table's X-block
Consider a $n$-Qubit stabilizer state $\newcommand{\ket}[1]{\vert#1\rangle}\newcommand{\bra}[1]{\langle#1\vert}\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau.
Any ...
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Confusion on the probability of measuring first qubit of a separable mixed state
Let $\rho = \sum_{x \in \{0,1\}^n} P_x |x \rangle\langle x|$ be a separable mixed state over bit strings $x$ of size $n$. Suppose also that $U = U_1 \otimes \cdots \otimes U_n$ is a product of local ...
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1
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How do you find the possible measurement values of an observable?
$\newcommand{\ket}[1]{\left|#1\right>}$
Note: I considered posting this as an update to a prior question, but it seemed like it should be it's own post.
So this is a very basic question, but one I'...
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2
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108
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Why are probabilities represented with alpha^2 and beta^2?
To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the ...
user24663
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1
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How much of quantum computing is based on probability?
I've recently discovered an interest in quantum computing and technology. Essentially, this means that I am trying to learn as much as I possibly can, one question at a time.
I have heard that quantum ...
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1
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Bounding operator norm by total variation distance
Let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ denote the probability distribution of obtaining the bitstring $y \in \{0,1\}^n$ on a fixed input $x \in \{0,1\}^n$ w.r.t. the unitary $U$. For $n$-...
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135
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Probability inequality for Quantum Approximate Optimization Algorithm (QAOA)
In arXiv:2207.14734 the authors claim that it is "straightforward to show that" their equation 8 holds:
$$\mathrm{Pr}_{x\sim q}[x:f(x)\geq \mu] \geq \frac{1}{M}$$
where we have an objective ...
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1
answer
58
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An inequality involving quantum channels
Consider two quantum circuits $\mathsf{C}$ and $\mathsf{D}$ applied to $|0^n\rangle$. Then, measure in the standard basis and, for $x \in \{0, 1\}^n$, consider two probabilities:
\begin{equation}
p_{x,...
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7
votes
1
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Generating random, but non-uniform state
I would like an algorithm that generates a random state, sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I have at my disposal a uniform (...
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3
votes
1
answer
123
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Independence in state prepared by independently drawn Haar random gates
Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and
$$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by
$$\sigma = V \rho V^{*}, $$
...
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votes
3
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How to get exact measurement probabilities when having intermediate measurements with Qiskit?
Suppose we have a circuit with two qubits, A and B. Both are initialized to $|0\rangle$. Over qubit A we apply a single rotation gate (e.g. $R_y$) with an angle given by $x_0$, and then we entangle ...
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2
votes
2
answers
149
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Computing a ratio involving Haar random unitaries
Consider an $n$-qubit Haar random unitary $U$.
I am trying to compute the expression
\begin{equation}
\mathbb{E}\left[ \frac{\text{Tr}\left(|0^n\rangle \langle 0^n | ~U\rho U^*\right)}{\text{Tr}\left(\...
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