Questions tagged [probability]
For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.
96
questions
2
votes
1
answer
41
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 ...
- 21
0
votes
1
answer
62
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 $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
- 11
4
votes
0
answers
30
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 ...
- 53
3
votes
2
answers
102
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 ...
- 57
4
votes
1
answer
67
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 (...
- 161
2
votes
2
answers
101
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.
...
- 510
2
votes
0
answers
72
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
1
answer
88
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:
...
- 145
0
votes
0
answers
33
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 ...
- 429
2
votes
0
answers
47
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
0
answers
43
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
...
- 429
0
votes
1
answer
70
views
Stabilizer States - Calculating measurement probabilities with the rank of the stabilizer table's X-block
Consider a $n$-Qubit stabilizer state $\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau.
Any Stabilizer State can be expressed as an equally weighted superposition
$$
\ket{...
- 37
1
vote
2
answers
90
views
Confusion on the probability of measuring first qubit of a separable mixed state
Let $\rho = \sum_{x \in \{0,1\}^n} P_x |x \rangle\langle x|$ be a separable mixed state over bit strings $x$ of size $n$. Suppose also that $U = U_1 \otimes \cdots \otimes U_n$ is a product of local ...
- 429
1
vote
1
answer
286
views
How do you find the possible measurement values of an observable?
$\newcommand{\ket}[1]{\left|#1\right>}$
Note: I considered posting this as an update to a prior question, but it seemed like it should be it's own post.
So this is a very basic question, but one I'...
-2
votes
2
answers
94
views
Why are probabilities represented with alpha^2 and beta^2?
To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the ...
user24663
1
vote
1
answer
51
views
How much of quantum computing is based on probability?
I've recently discovered an interest in quantum computing and technology. Essentially, this means that I am trying to learn as much as I possibly can, one question at a time.
I have heard that quantum ...
- 111
2
votes
1
answer
87
views
Bounding operator norm by total variation distance
Let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ denote the probability distribution of obtaining the bitstring $y \in \{0,1\}^n$ on a fixed input $x \in \{0,1\}^n$ w.r.t. the unitary $U$. For $n$-...
- 429
4
votes
0
answers
126
views
Probability inequality for Quantum Approximate Optimization Algorithm (QAOA)
In arXiv:2207.14734 the authors claim that it is "straightforward to show that" their equation 8 holds:
$$\mathrm{Pr}_{x\sim q}[x:f(x)\geq \mu] \geq \frac{1}{M}$$
where we have an objective ...
- 105
0
votes
1
answer
49
views
An inequality involving quantum channels
Consider two quantum circuits $\mathsf{C}$ and $\mathsf{D}$ applied to $|0^n\rangle$. Then, measure in the standard basis and, for $x \in \{0, 1\}^n$, consider two probabilities:
\begin{equation}
p_{x,...
- 1,251
7
votes
1
answer
159
views
Generating random, but non-uniform state
I would like an algorithm that generates a random state, sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I have at my disposal a uniform (...
- 540
3
votes
1
answer
86
views
Independence in state prepared by independently drawn Haar random gates
Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and
$$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by
$$\sigma = V \rho V^{*}, $$
...
- 1,251
2
votes
3
answers
175
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
2
votes
2
answers
113
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(\...
- 1,251
5
votes
2
answers
163
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$. ...
- 51
1
vote
0
answers
54
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
48
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 ...
- 429
2
votes
1
answer
70
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 ...
- 1,251
1
vote
0
answers
75
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
2
votes
0
answers
96
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
0
votes
1
answer
64
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 ...
- 169
1
vote
1
answer
62
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 ...
- 13
-1
votes
1
answer
91
views
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 ...
- 299
1
vote
1
answer
108
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 ...
- 67
0
votes
1
answer
46
views
Cannot derive probability graph for Hadamard gate given in Qiskit textbook
I am reading the Qiskit textbook(beta) and they have explained Hadamard gate using an amplitude tree. To show how two H-gates on a qubit give the output as 0 everytime they said to consider that it ...
- 3
3
votes
1
answer
99
views
How do I get correct measurement probabilities in ZX calculus?
I'm learning ZX-calculus, but I'm getting confused when trying to obtain some simple results to compute probabilities for different outcomes.
Here's a simple example where I'm getting lost. Here, <...
- 221
1
vote
1
answer
50
views
Formal measurement of the probability of an outcome of a Qubit in the hadamard base
I am fairly new to Quantum Computing and have a question which might be trivial for all of you, but I am really struggling with it.
From Quantum Computation and Quantum Information I have learned, ...
- 11
0
votes
1
answer
139
views
Exact Probabilities of Outcomes for Clifford Circuits with Mid-Circuit Measurements Using Stim
I am trying to find the exact probabilities of specific measurement outcomes for Clifford circuits with mid-circuit measurements. Essentially, I am looking for a function that takes an arbitrary ...
- 55
4
votes
0
answers
36
views
Is there a general framework that allows us to compare probabilistic and deterministic algorithms fairly?
Many popular QC algorithms are probabilistic in nature, like Grover's, Shor's, QAOA ..etc
For some of these we have formulas that give probabilities of success (like for Grover's and Shor's), and for ...
- 271
1
vote
0
answers
128
views
Finding the Exact Probability Distribution for the Outcomes of a Quantum Circuit with Mid-Circuit Measurements
I would like to find the exact probabilities of the possible outcomes of a circuit that includes mid-circuit measurements.
So, as a specific example, consider the following circuit:
I would like to ...
- 55
2
votes
0
answers
101
views
Distinguishing $n$ pure states in an $n$ dimensional Hilbert space
Suppose we have $n$ pure states in an $n$ dimensional Hilbert space, and we would like to distinguish them using POVM or PVM. We get any one of the pure states with equal probability, and we may set ...
- 21
0
votes
1
answer
63
views
What is the actual probability of not losing information (in a depolarizing channel)
The probability that a depolarizing channel doesn't affect the information is usually assumed to be $1-3p$, while, for convenience, it is affected with same probability $p$ by any Pauli operator $X,Y,...
- 1,566
-2
votes
1
answer
126
views
Probability outcome $0$? Post measurement state?
Does anyone know how to solve this exercise? Here is the question:
Let $|\psi\rangle$ be an arbitrary pure $n$-qubit state, i.e.
$$|\psi\rangle=\sum_{x_1,\ldots,x_n=0,1}\alpha_{x_1\cdots x_n}|x_1\...
- 9
4
votes
3
answers
388
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 $\varphi_{00}$ $\varphi_{01}$ $\varphi_{10}$ $\varphi_{11}$...
- 149
5
votes
1
answer
81
views
What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$
A standard trick in probability manipulation is to take some joint distribution $P_{XY}$ and express it as $P_{Y|X}P_X$. This trick is useful because when one looks at things like the ratio of $\frac{...
- 2,597
3
votes
1
answer
50
views
Relating upper bound on the total variation distance with a bound on a Pauli observable
Consider an $n$ qubit state $|\psi\rangle$. Let us say we have the following upper bound on the total variation distance between the distribution generated by measuring $|\psi\rangle$ in the standard ...
- 1,251
1
vote
1
answer
232
views
Implementing Quantum Walks at IBM
a question about quantum walks, would this circuit be correct to start a quantum walk in a hypercube? I saw something about increment and decrement, but I didn't quite understand how they would work ...
- 79
0
votes
1
answer
144
views
probability and post-measurement state with observable 𝐼⊗𝑋
Today I learned about these but I can't know how to solve this probem.
$|𝜓⟩={\frac1 {√3}}(|01⟩+|10⟩+|11⟩) $
I want to measure observable 𝐼⊗𝑋 . What is the probability of measuring |+⟩ on the ...
4
votes
1
answer
115
views
Question regarding the output probability of a quantum circuit
Consider a quantum circuit $\text{Q}$, run on $|0^{n}\rangle$. For a specific $x \in \{0, 1\}^{n}$, let's say we are interested in the probability
$$p_x = |\langle x|~\text{Q}~|0^{n}\rangle|^{2}.$$
...
- 315
4
votes
1
answer
229
views
Why is sampling from probability distributions generated by specific quantum circuits classically intractable?
I was reading a paper by Benedetti et al. titled Parameterized quantum circuits as machine learning models. Its authors state the following:
We also know that sampling from the probability ...
- 269
1
vote
1
answer
45
views
Statistical 'process' behind quantum gates
Thinking about gate creating entangled state, it looks to me that the inputs to the gate look like marginal/independent distributions, and the output looks like their joint distribution. Does this ...
- 373