# Questions tagged [probability]

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

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• 101
1 vote
94 views

### Matrix representation of the symmetric subspace for two copies

Consider two copies of an $n$ qubit Haar random state, given by: \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^{*}\...
• 1,363
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 ...
• 681
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, ...
• 229
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
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 ...
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 ...
• 578
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 ...
• 43
24 views

• 21
1 vote
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 ...
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 ...
• 255
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 ...
• 9
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 ...
• 127
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 $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
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 ...
• 51
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 ...
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 (...
• 165
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. ...
• 518
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... ...
• 21
1 vote
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
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 ...
• 499
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 $U_P$ ...
• 41
1 vote
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 ...
• 499
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 ...
• 37
1 vote
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 ...
• 499
1 vote
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'...
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
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 ...
• 111
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 $n$-...
• 499
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 ...
• 105
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: p_{x,...
• 1,363
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 (...
• 540
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 $$\sigma = V \rho V^{*},$$ ...
• 1,363
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 ...
• 21
136 views

### Computing a ratio involving Haar random unitaries

Consider an $n$-qubit Haar random unitary $U$. I am trying to compute the expression \mathbb{E}\left[ \frac{\text{Tr}\left(|0^n\rangle \langle 0^n | ~U\rho U^*\right)}{\text{Tr}\left(\...
• 1,363
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 $\rho$. ...
1 vote
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
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 ...
• 499
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 ...
• 1,363
1 vote
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 ...
• 99
112 views

### Distribution of partial trace of Haar unitary

I am sure this must have been covered in the mathematical literature, but hoping someone can direct me to the right place. Let us be given random unitaries $U$ on $n$ qubits (so dimension of the space ...
• 540
77 views

### Survival probability quantum circuit

Suppose say that I have a quantum state $\vert\psi\rangle$ at time $t = 0$, which is now evolved by a hamiltonian $H$ $$e^{-iHt}\vert\psi\rangle$$. I can ask the question, how much of initial state is ...
1 vote
84 views

### Sum of probability in non-orthogonal basis

On standard basis, the sum of the probability of a vector $\newcommand\ket[1]{\left|#1\right\rangle}\ket{v} = a \ket{0} + b \ket{1}$ is $a^2 + b^2 = 1$, right? What about the two states of the basis ...
### Why is $|P_0- P_1|=1$?
I have a question we have $|0 \rangle, |1 \rangle, |+ \rangle$ and $|- \rangle,$ defined as usual. Let $P_0$= probability that a state be in 0, $P_+$= probability that a state be in +, and same ...
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 ...