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Questions tagged [probability]

For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.

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2 votes
0 answers
79 views

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 $...
1 vote
1 answer
147 views

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: ...
1 vote
1 answer
22 views

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 ...
2 votes
0 answers
61 views

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(...
0 votes
1 answer
114 views

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 ...
1 vote
1 answer
158 views

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 ...
1 vote
1 answer
94 views

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^{*}\...
3 votes
1 answer
66 views

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 ...
2 votes
1 answer
51 views

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, ...
2 votes
1 answer
57 views

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 ...
1 vote
0 answers
67 views

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 ...
2 votes
1 answer
84 views

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 ...
0 votes
0 answers
54 views

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 ...
0 votes
0 answers
24 views

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 = \...
2 votes
1 answer
50 views

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 ...
2 votes
1 answer
481 views

How to prove that the mutual information is subadditive?

Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
0 votes
1 answer
129 views

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 ...
2 votes
1 answer
65 views

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 $...
1 vote
0 answers
75 views

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 votes
1 answer
44 views

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 ...
2 votes
1 answer
47 views

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 ...
0 votes
1 answer
77 views

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 $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
4 votes
0 answers
38 views

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 ...
3 votes
2 answers
122 views

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 ...
4 votes
1 answer
110 views

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 (...
2 votes
2 answers
115 views

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. ...
2 votes
0 answers
138 views

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... ...
2 votes
2 answers
146 views

Given averages of powers of position and momentum in quantum mechanics what information can be secured about the wave-function?

Question If I tell you all the averages of powers of position and momentum in quantum mechanics can you tell me the value of the wave-function? What can you tell me about the wavefunction? Is there ...
1 vote
0 answers
51 views

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 ...
2 votes
0 answers
48 views

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 $U_P$ ...
2 votes
2 answers
191 views

Estimating output amplitudes of quantum circuits as GapP functions

Let's fix a universal gate set comprising of a Hadamard gate and a Toffoli gate. Consider an $n$ qubit quantum circuit $U_{x}$, made up of gates from that universal set, applied to initial state $|0^{...
1 vote
0 answers
47 views

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 ...
4 votes
3 answers
507 views

How do we achieve mathematically that the probability that Eve learns $x$ is $\cos^{2}\left ( \frac{\pi }{8} \right )$?

I'm trying to understand this problem. Alice I attempting to send a 2 classical bit message to Bob using 1 qubit such that there are 4 states $\varphi_{00}$ $\varphi_{01}$ $\varphi_{10}$ $\varphi_{11}$...
1 vote
2 answers
99 views

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 ...
1 vote
1 answer
529 views

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'...
1 vote
1 answer
128 views

How to write post-measurement states when the measurement apparatus measures one of two observables?

If I want to measure an observable $A$ but the measurement apparatus has $(1-p)$ probability of measuring the observable $B$ and probability $p$ that a measurement of $A$ would be done. So how can I ...
-2 votes
2 answers
101 views

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 ...
1 vote
1 answer
56 views

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 ...
5 votes
2 answers
259 views

Simultaneous measurements and Bell basis measurements to estimate $\lvert\text{Tr}(\sigma \rho)\rvert^2$ in Huang et al. paper

Theorem 2 of this paper says if one is able to prepare $\rho^{\otimes k}$ then it is possible to predict expectation values of all $n$-qubit Pauli observables using $O(n)$ number of copies of $\rho$. ...
2 votes
1 answer
148 views

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 $n$-...
4 votes
0 answers
132 views

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 ...
2 votes
2 answers
136 views

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(\...
0 votes
1 answer
55 views

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,...
3 votes
1 answer
110 views

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 $$\sigma = V \rho V^{*}, $$ ...
7 votes
1 answer
178 views

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 (...
2 votes
3 answers
358 views

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 ...
1 vote
0 answers
63 views

How to implement the Mach Zehnder Interferometer in Qutip?

I was trying to implement the Mach-Zehnder Interferometer with a phase shifter in Qutip but I couldn't nail it. I just want to give two number states as input and at the end see the probability ...
1 vote
0 answers
57 views

Close in operator norm imply close in weak multiplicative sense?

Fix $\epsilon > 0$, and suppose $U$ and $S$ are $n$ qubit unitaries such that $\| U - S \| \leq \epsilon$ (operator norm). Furthermore, let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ be the ...
2 votes
1 answer
83 views

Conditional expectation for Haar random states

Let $U$ be an $n$ qubit Haar random circuit applied to $|0^n \rangle$. Thereafter, the state is measured in the standard basis. Let $p_0$ be the probability of getting $0$ in the first qubit. We know ...
1 vote
0 answers
94 views

I am optimising a variational quantum circuit to learn a distribution $p(x)$, but it doesn't converge over a training set $\mathcal{X}$?

I am training a variational quantum circuit to learn distributions: given data $s(\vec{\lambda})$, what is the probability distribution for the parameterisation $\vec{\lambda}$, i.e. the posterior ...