Questions tagged [probability]

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

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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 ...
4
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1answer
118 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=|\...
2
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1answer
81 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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1answer
55 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 ...
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1answer
65 views

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 ...
4
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1answer
122 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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0answers
79 views

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, ...
4
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1answer
153 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 ...
2
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1answer
95 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 ...
2
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35 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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1answer
59 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 ...
2
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1answer
48 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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0answers
46 views

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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1answer
127 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 ...
2
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2answers
110 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^{...
4
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1answer
81 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 ...
2
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1answer
77 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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1answer
95 views

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 ...
2
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1answer
156 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 ...
3
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1answer
81 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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1answer
190 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 \...
4
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1answer
50 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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2answers
384 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 ...
4
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1answer
84 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|\...
6
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1answer
195 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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0answers
46 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 ...
2
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2answers
220 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 ...
2
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1answer
87 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{...
3
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2answers
241 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 |...
2
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2answers
72 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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0answers
31 views

RAC for XOR functions

I need the optimal encoding protocol for 3 $\rightarrow$ 1 Classical RAC such that the receiver is able to retrieve any one of the initial bits, as well as the XOR combinations of those bits. ( If a, ...
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0answers
54 views

In QAOA why do we need $m \log(m)$ repititions 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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1answer
62 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 ...
3
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1answer
86 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 ...
4
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1answer
139 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\...
2
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1answer
59 views

Hamiltonian simulation: how can I incorporate the constant before each term?

I got another follow-up question about Hamiltonian simulation from the previous post: if I perform the controlled time-evolution of the Hamiltonian: $$ H_{3} = \alpha\ X_1\otimes Y_2 + \beta \ Z_1\...
2
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0answers
82 views

What is the relation between density matrices and phase-space probability distributions?

According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical ...
2
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1answer
67 views

Relating quantum max-relative entropy to classical maximum entropy

The quantum max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$ where $\rho\leq \sigma$ should be read as $\sigma - \...
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1answer
110 views

Dirichlet distribution: posteriors and priors of distribution

Let $|\psi\rangle \in \mathbb{C}^{2n}$ be a random quantum state such that $ |\langle z| \psi \rangle|^{2} $ is distributed according to a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution, for $z \in \...
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1answer
223 views

What is the probability $\Pr(\|U-I\|_{\rm op}<\varepsilon)$ of a Haar-random unitary being close to the identity?

If one generates an $n\times n$ Haar random unitary $U$, then clearly $\Pr(U=I)=0$. However, for every $\epsilon>0$, the probability $$\Pr(\|U-I\|_{\rm op}<\varepsilon)$$ should be positive. How ...
2
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3answers
361 views

Find probability of a single qubit's measurement results from a 5 qubit state

I have a tensor product of a 5 qubit state $|h\rangle$. From this I want to calculate the probability of the 2nd qubit being in state $|1\rangle$. Can someone show me how I can do this? I know I can ...