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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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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 ...
trillianhaze's user avatar
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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 $\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau. Any Stabilizer State can be expressed as an equally weighted superposition $$ \ket{...
Coryn7's user avatar
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2 answers
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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 ...
trillianhaze's user avatar
1 vote
1 answer
106 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'...
quantumstudent's user avatar
-2 votes
2 answers
74 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 ...
user avatar
1 vote
1 answer
48 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 ...
Logan's user avatar
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2 votes
1 answer
47 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 $n$-...
trillianhaze's user avatar
4 votes
0 answers
107 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 ...
Juri V's user avatar
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0 votes
1 answer
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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,...
BlackHat18's user avatar
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7 votes
1 answer
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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 (...
nervxxx's user avatar
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3 votes
1 answer
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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 $$\sigma = V \rho V^{*}, $$ ...
BlackHat18's user avatar
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2 votes
3 answers
106 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 ...
dviqu's user avatar
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2 votes
2 answers
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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(\...
BlackHat18's user avatar
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5 votes
2 answers
104 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 $\rho$. ...
user8183310's user avatar
1 vote
0 answers
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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 ...
can kanaroğlu's user avatar
1 vote
0 answers
47 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 ...
trillianhaze's user avatar
2 votes
1 answer
54 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 ...
BlackHat18's user avatar
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0 votes
0 answers
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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 ...
JoJo's user avatar
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2 votes
0 answers
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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 ...
nervxxx's user avatar
  • 500
0 votes
1 answer
50 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 ...
FearlessVirgo's user avatar
1 vote
1 answer
48 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 ...
Gabe Ebag's user avatar
-1 votes
1 answer
89 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 ...
user206904's user avatar
1 vote
1 answer
97 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 ...
username9's user avatar
0 votes
1 answer
42 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 ...
Rai's user avatar
  • 3
3 votes
1 answer
80 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, <...
jjgoings's user avatar
  • 191
1 vote
1 answer
42 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, ...
dribble290's user avatar
0 votes
1 answer
92 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 ...
user206444's user avatar
4 votes
0 answers
35 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 ...
bubakazouba's user avatar
1 vote
0 answers
84 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 ...
user206444's user avatar
2 votes
0 answers
96 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 ...
Stan's user avatar
  • 21
0 votes
1 answer
48 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,...
Daniele Cuomo's user avatar
-2 votes
1 answer
110 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\...
Rodrigo's user avatar
4 votes
3 answers
303 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 $\varphi_{00}$ $\varphi_{01}$ $\varphi_{10}$ $\varphi_{11}$...
Vishakha Lall's user avatar
5 votes
1 answer
75 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{...
user1936752's user avatar
  • 2,367
3 votes
1 answer
40 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 ...
BlackHat18's user avatar
  • 1,119
1 vote
1 answer
172 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 ...
Ikky R's user avatar
  • 79
0 votes
1 answer
111 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 ...
user19382's user avatar
4 votes
1 answer
110 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 $$p_x = |\langle x|~\text{Q}~|0^{n}\rangle|^{2}.$$ ...
Tom Clancy's user avatar
4 votes
1 answer
174 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 ...
karolyzz's user avatar
  • 269
1 vote
1 answer
39 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 ...
sitems's user avatar
  • 363
1 vote
1 answer
33 views

How to change the probability of observation by some set amount when initial probability is unknown?

If I have some state $|\psi> = \alpha |0> + \beta|1>$, I know that the probability of observing $|0>$ is $p_1 = |\alpha|^2$. Is it possible to change the probability of observing $|0>$ ...
Dawson Beatty's user avatar
2 votes
1 answer
137 views

Why does measurement in computational basis result in classic probability?

I am working on a problem related to finding the limits on the joint probability distributions/correlations of three or more quantum systems who share entangled states, after measurement. I have been ...
Pegi's user avatar
  • 165
3 votes
1 answer
228 views

Schmidt vectors for random quantum states

Consider a random quantum circuit $U$ over $n$ qubits, drawn from the Haar measure. Consider the quantum state $$|\psi\rangle = U |0^{n}\rangle.$$ Now, partition $n$ into two and consider the Schmidt ...
BlackHat18's user avatar
  • 1,119
0 votes
1 answer
88 views

Find the Probability for a "+" outcome when making a Pauli-x Measurement

So, we apply Equation 3.28 (above) to our initial vector state, following the equation below, to get $|\psi(t)\rangle$. What I obtained was the basically the same equation, except now we have a $|up\...
Dwye's user avatar
  • 101
3 votes
0 answers
96 views

Schur transform and the outcome probabilities for a particular type of state

I was reading about the Schur transform and its applications in knowing about an unknown quantum state. Consider $\rho^{\otimes k}$, which means $k$ copies of an unknown $n$ qubit quantum density ...
BlackHat18's user avatar
  • 1,119
6 votes
1 answer
317 views

Do entangled measurements across multiple copies help in state distinguishability?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
BlackHat18's user avatar
  • 1,119
2 votes
2 answers
128 views

Random quantum circuits and general efficient POVM measurement

Let's consider a random quantum circuit $C$, applied to the $n$ qubit initial state $|0^{n}\rangle$, producing the state $|\psi\rangle$. Consider a general efficiently implementable $m$-outcome POVM ...
BlackHat18's user avatar
  • 1,119
4 votes
1 answer
351 views

Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?

States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\...
Balter 90s's user avatar
2 votes
1 answer
255 views

Average output state of random quantum circuits

Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \begin{equation} \rho_1 = \mathbb{E}[|\psi\...
BlackHat18's user avatar
  • 1,119
0 votes
1 answer
96 views

Applying Hadamard gate to $\sqrt{3/4}|0\rangle + \sqrt{1/4}|1\rangle$

[I am just transferring this from Stack Overflow. It might need editing.] ———— [The reader can skip to “It all sounds fine…”, before the spreadsheet representation.] I am trying to figure out quantum ...
Carsogrin's user avatar