Questions tagged [probability]
For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.
86
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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 $\rho = \ket{\psi}\bra{\psi}$ and its $n \times 2n$ boolean stabilizer tableau.
Any Stabilizer State can be expressed as an equally weighted superposition
$$
\ket{...
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Confusion on the probability of measuring first qubit of a separable mixed state
Let $\rho = \sum_{x \in \{0,1\}^n} P_x |x \rangle\langle x|$ be a separable mixed state over bit strings $x$ of size $n$. Suppose also that $U = U_1 \otimes \cdots \otimes U_n$ is a product of local ...
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How do you find the possible measurement values of an observable?
$\newcommand{\ket}[1]{\left|#1\right>}$
Note: I considered posting this as an update to a prior question, but it seemed like it should be it's own post.
So this is a very basic question, but one I'...
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Why are probabilities represented with alpha^2 and beta^2?
To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the ...
user24663
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How much of quantum computing is based on probability?
I've recently discovered an interest in quantum computing and technology. Essentially, this means that I am trying to learn as much as I possibly can, one question at a time.
I have heard that quantum ...
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Bounding operator norm by total variation distance
Let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ denote the probability distribution of obtaining the bitstring $y \in \{0,1\}^n$ on a fixed input $x \in \{0,1\}^n$ w.r.t. the unitary $U$. For $n$-...
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Probability inequality for Quantum Approximate Optimization Algorithm (QAOA)
In arXiv:2207.14734 the authors claim that it is "straightforward to show that" their equation 8 holds:
$$\mathrm{Pr}_{x\sim q}[x:f(x)\geq \mu] \geq \frac{1}{M}$$
where we have an objective ...
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An inequality involving quantum channels
Consider two quantum circuits $\mathsf{C}$ and $\mathsf{D}$ applied to $|0^n\rangle$. Then, measure in the standard basis and, for $x \in \{0, 1\}^n$, consider two probabilities:
\begin{equation}
p_{x,...
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Generating random, but non-uniform state
I would like an algorithm that generates a random state, sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I have at my disposal a uniform (...
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Independence in state prepared by independently drawn Haar random gates
Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and
$$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by
$$\sigma = V \rho V^{*}, $$
...
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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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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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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$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, <...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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\...
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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 $\varphi_{00}$ $\varphi_{01}$ $\varphi_{10}$ $\varphi_{11}$...
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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{...
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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 ...
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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 ...
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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 ...
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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
$$p_x = |\langle x|~\text{Q}~|0^{n}\rangle|^{2}.$$
...
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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 ...
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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 ...
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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 $|0>$ ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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=|\...
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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\...
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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 ...
