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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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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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-2 votes
1 answer
70 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\...
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4 votes
1 answer
81 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 $\varphi_{00}$ $\varphi_{01}$ $\varphi_{10}$ $\varphi_{11}$...
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5 votes
1 answer
72 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{...
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3 votes
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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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0 answers
40 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 ...
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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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4 votes
1 answer
72 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 $$p_x = |\langle x|~\text{Q}~|0^{n}\rangle|^{2}.$$ ...
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4 votes
1 answer
90 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 ...
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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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2 votes
1 answer
70 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 ...
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3 votes
1 answer
176 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 ...
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1 answer
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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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6 votes
1 answer
162 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 ...
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2 votes
1 answer
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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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4 votes
1 answer
198 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=|\...
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2 votes
1 answer
128 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\...
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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 ...
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1 answer
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Negative Probability — Reality vs Description [closed]

I understand that quantum physics supports the concept that the probability of a qubit collapsing into (say) 1, can be negative or positive… and that quantum computing uses this as a feature, adding ...
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4 votes
1 answer
155 views

How to prepare a quantum state of the form $\frac1{2^{n/2}}\sum_{x \in \{0, 1\}^{n}} |x\rangle |y_x\rangle$ with $y_x$ random variables?

Let's say I am given an efficiently samplable probability distribution $D$, over $n$ bit strings. I want to efficiently prepare the following state \begin{equation} |\psi\rangle = \frac{1}{\sqrt{2^{n}}...
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2 votes
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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, ...
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4 votes
1 answer
159 views

Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
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2 votes
1 answer
139 views

How is the probability of success for Simon's algorithm determined?

In step 3 of Simon's algorithm, we are told to "Repeat until there are enough such $y$’s that we can classically solve for $s$." It then goes on: The above are from this course notes. I am ...
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2 votes
0 answers
42 views

Marginal output probability of first bit for constant-depth circuits

Consider a constant depth $1\text{D}$ quantum circuit, which is applied to the input state $|0^{n}\rangle$, and whose output is measured in the standard basis. You can assume that the gates of the ...
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1 vote
1 answer
62 views

Properties of frames in quasiprobability representation

Let $\mathbb{C}^{d}$ be a complex Euclidean space. Let $\mathsf{H}(\mathbb{C}^{d})$ be the set of all Hermitian operators, mapping vectors from $\mathbb{C}^{d}$ to $\mathbb{C}^{d}$. I had some ...
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2 votes
1 answer
60 views

How to get probability when the coefficient in wave function is a matrix?

Following this circuit: With $\mathcal{G}, A$ being unitary matrices and $|\psi\rangle$ the initial state. First, the system is: $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\;\otimes|\psi\rangle$ Next: $\...
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Relation between approximate counting and sampling

Consider the following statement of Stockmeyer counting theorem. Given as input a function $f:\{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$ and $y \in \{0, 1\}^{m}$, there is a procedure that runs in ...
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1 vote
1 answer
132 views

Intuitions about probabilities relating to evolving a two-qubit state through a CNOT gate

If the initial state of $|x_0\rangle = \alpha |0\rangle + \beta |1\rangle$ and $|x_1\rangle =|0\rangle$, and the final state at the barrier is $|10\rangle$ (in the form $|x_1x_0\rangle$), what would ...
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2 votes
2 answers
124 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^{...
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4 votes
1 answer
206 views

How to find the POVM that optimally distinguishes between two given states?

A quantum state preparation machine emits a state $\rho_0$ with probability $2/3$ and emits the state $\rho_1$ with probability $1/3$. We aim to make the best guess which one is it using a set of two ...
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2 votes
1 answer
117 views

Measurement probability of a state from three hilbert spaces

I'm curious how to find the probability measurement of a state when one qubit is measured. For example this state: $|\gamma\rangle = \frac{1}{\sqrt{2}}(| 010 \rangle + | 101 \rangle )$ Assuming these ...
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0 votes
1 answer
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Quantum supremacy: shallow depth Haar random circuits and unitary designs

I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only ...
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2 votes
1 answer
164 views

How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?

I have been given and expression for a probability distribution $$P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$$ and I have been asked to show that the above expression can be ...
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3 votes
1 answer
175 views

Probability of measuring one qubit from the state of two qubits

I am new to quantum information and I am trying to work on some problems but I have confused myself over a qubit problem. I have the state of two qubits $|\psi\rangle_{AB}=a_{00}|00\rangle+a_{01}|01\...
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3 votes
1 answer
299 views

Confusion about the output distribution of Haar random quantum states

Consider a Haar random quantum state $|\psi \rangle$. I was confused between two facts about $|\psi \rangle$, which appear related: Consider the output distribution of a particular $n$-qubit $|\psi \...
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4 votes
1 answer
55 views

Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?

Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...
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7 votes
2 answers
695 views

Quantum Amplitude Estimation vs Quantum Phase Estimation

Quick question concerning the probability of success after a phase estimation algorithm vs an amplitude estimation algorithm. Given the calculation on the wikipedia page, the probability of measuring ...
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4 votes
1 answer
92 views

What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?

Suppose we perform a measurement described by measurement operators $M_m$. If the initial state is $|{\psi_i}\rangle$, then the probability of getting result $m$ is $$ \begin{align} p(m|i)=\| M_m|\...
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6 votes
1 answer
201 views

What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?

I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the ...
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1 vote
0 answers
47 views

Is this inequality related to time-energy uncertainty true or testable?

Background It is known: In all physical systems in which energy is bounded below, there is no self-adjoint observable that tracks the time parameter t. However I don't think this forbids any ...
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2 votes
2 answers
248 views

How to interpret complex probability of superposition state?

I have one qubit and I apply two gates to it: H and T, which yields the following superposition: $$ \frac{1}{\sqrt{2}} |0\rangle + \frac{1+i}{2}|1\rangle $$ Now I want to calculate probability of 0 ...
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2 votes
1 answer
120 views

Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by \begin{equation} H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right), \end{...
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3 votes
2 answers
466 views

Question regarding integration of Haar random state

I am trying to understand the integration on page 4 of this paper. Consider a Haar random circuit $C$ and a fixed basis $z$. Each output probability of a Haar random circuit (given by $|\langle z | C |...
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2 votes
2 answers
82 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 ...
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6 votes
0 answers
63 views

In QAOA why do we need $m \log(m)$ repetitions to get at least $F_{p}(\beta , \gamma) - 1$ with probability of $1 - 1/m$?

In the original QAOA paper from Farhi https://arxiv.org/pdf/1411.4028.pdf, it is stated in chapter 2 last paragraph (page 6) that: when measuring $F_{p}(\beta , \gamma)$ we get an outcome of at least ...
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  • 469
0 votes
1 answer
68 views

Quantum Circuit to inverse the probability distribution

I'm using Qiskit and after running the circuit, as we all know, we get a count dictionary such as ...
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3 votes
1 answer
113 views

Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm

$\newcommand{\expectation}[1]{\mathop{\mathbb{E}} \left[ #1 \right] } \newcommand{\Var}{\mathrm{Var}}$ From Nielsen & Chuang 10th edition page 261: Consider a classical algorithm for the counting ...
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4 votes
1 answer
173 views

Question about Haar random quantum states

Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$, $$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
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