7 votes
Accepted

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

Measurement postulate The statement you are asking about is a postulate of quantum mechanics, so it cannot be mathematically derived from other facts in the theory. Instead, it is justified by its ...
  • 14.7k
6 votes
Accepted

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

Given $\rho$ and a fixed ensemble $\{ |\psi_i \rangle \}$ it might not be possible to write $\rho$ as $\sum_i p_i |\psi_i \rangle \langle \psi_i |$. For example, let $| + \rangle = \frac{1}{\sqrt{2}} ...
  • 886
5 votes
Accepted

Confusion about the output distribution of Haar random quantum states

The two facts are connected in that they both arise as a result of rotational invariance of the Haar measure. We will derive them in the case of large $n$ since this is when the Porter-Thomas ...
  • 14.7k
5 votes
Accepted

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

This only holds if the two distributions are independent. In this case $$ \begin{aligned} H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\ &= \...
  • 4,252
5 votes
Accepted

Question regarding integration of Haar random state

The issue that easily leads to confusion is the dual role played by output bitstring probability. It enters the computation of the average in two ways. On one hand, it determines how often one sees ...
  • 14.7k
5 votes
Accepted

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

Since the classical algorithm samples "uniformly and independently $𝑘$ times from the search space", equation $(5)$ should be, $P(X_i=1, X_j=1)= P(X_i=1)P(X_j=1)=\frac{M^2}{N^2}$ instead. ...
  • 311
5 votes

Negative Probability — Reality vs Description

"I understand that quantum physics supports the concept that the probability of a qubit collapsing into (say) 1, can be negative or positive [...] If that is wrong, please say so; I understand ...
  • 12.1k
4 votes
Accepted

Question about Haar random quantum states

In the following, I'll show the evaluation of the probability densities of the transition probabilities: $|\langle \psi | z\rangle^2$ and their pairwise independence. I didn't work out the full mutual ...
4 votes

How to interpret complex probability of superposition state?

Just to note that the complex number also specify a quantum phase of a qubit. In this particular case $$ \frac{1+i}{2} = \frac{1}{\sqrt{2}}\mathrm{e}^{i\frac{\pi}{4}}, $$ so the relative phase is $\...
4 votes
Accepted

How to interpret complex probability of superposition state?

You forgot to take the absolute value. The Born rule for computing measurement outcome probability from the state vector amplitudes says that the probability is the square of the magnitude of the ...
  • 14.7k
4 votes
Accepted

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

These are classical conditional probabilities used extensively in Bayesian probability. Suppose Alice can prepare any of a number of pure states $|\psi_i\rangle$. She chooses $i$ randomly from ...
  • 14.7k
4 votes

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?

First, recall that $\mathrm{tr} A = \sum_i \langle i|A|i \rangle$. Each equation is then a sum where all terms are products of $P(z)$ and three other quantities. Further, the sum in the first equation ...
  • 14.7k
3 votes
Accepted

Quantum supremacy: shallow depth Haar random circuits and unitary designs

First of all, that does not imply anything for shorter (constant/logarithmic) depths. Moreover, the 2-design property does not imply that the outcome distribution is the same as for Haar-random ...
3 votes

Spoofing XQUATH with the Feynman method

The paper does not specify the exact algorithm or class of distributions $\mathcal{D}$ for which such algorithm fails to refute XQUATH, and some classes of distributions $\mathcal{D}$ do not satisfy ...
  • 326
3 votes
Accepted

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

The optimal probability of guessing correctly is $$ \frac12 + \frac12 \Big\|\frac23 \rho_0 - \frac13 \rho_1 \Big\|_1 $$ where $\| X \|_1 = \mathrm{Tr}[\sqrt{X^* X}]$ is the Schatten 1-norm. This ...
  • 4,252
3 votes
Accepted

Estimating output amplitudes of quantum circuits as GapP functions

Given a description of $ U_x $ you can efficiently find a decription of $ \textbf{B} $ and $ \phi $ by iterating over the gates of $U_x$ and adding one "free" variable every time you have a ...
  • 1,336
3 votes
Accepted

Properties of frames in quasiprobability representation

The authors are certainly thinking about finite frames. In this case, your statement is correct, since the number of elements in every spanning set is at least the vector space dimension. As glS ...
3 votes
Accepted

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

Note that if $p_{i}q_{i} = 0\,\,\forall i$, then for all $i$ either $p_{i} = 0$, $q_{i} = 0$, or both are $0$. Divide $\{i\} = \{1,\ldots,N\}$ into those $i$ for which these three different things ...
  • 4,898
3 votes
Accepted

Quantum Amplitude Estimation vs Quantum Phase Estimation

Ok let's break this down. Firstly the success probability for QPE and QAE are defined slightly differently. With QPE there are two error bounds $|\hat{x} - x| < \epsilon$ to consider $\epsilon < ...
3 votes

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

$U-I$ is a normal matrix so $||U-I||_{op}$ is its eigenvalue with the largest magnitude. The eigenvalue equation for this matrix is $$(U-I)|\psi\rangle=\lambda|\psi\rangle,$$ so $$|\lambda|^2=(\cos\...
3 votes
Accepted

Relating quantum max-relative entropy to classical maximum entropy

As far as I'm aware there isn't much of a meaningful connection. The corresponding entropy for $D_{\max}$ is the min-entropy (written $H_{\min}$ or $H_{\infty}$). It measures a sort of `worst case' ...
  • 4,252
3 votes

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

It seems like you are trying to apply a Hadamadard gate to the state $\frac{\sqrt{3}}{2}|0\rangle + \frac{1}{2}|1\rangle$. This would go as follows: $$ \begin{align} H\left[\frac{\sqrt{3}}{2}|0\rangle ...
  • 2,655
3 votes
Accepted

Why is sampling from probability distributions generated by specific quantum circuits classically intractable?

The problem that led to an important role of IQP circuits is Boson Sampling. Boson sampling algorithm has to sample from a distribution based on the permanent of some complex matrix. Computing complex ...
  • 610
3 votes
Accepted

Question regarding the output probability of a quantum circuit

Let $\mathcal{H}_A$ denote the Hilbert space of qubits in partition $A$ and similarly for $\mathcal{H}_\bar{A}$. Define the operator $P:=Q|0^n\rangle\langle 0^n|Q^\dagger$ and write its Schmidt ...
  • 14.7k
3 votes
Accepted

What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$

The classical conditional distribution of $Y$ given $X$ is defined as the joint distribution divided by the marginal distribution. Note that we can reconstruct the joint distribution if we know all ...
  • 5,978
2 votes
Accepted

Dirichlet distribution: posteriors and priors of distribution

Let $ p_i = |\langle i | \phi \rangle|^2 \sim Dir(a_1, .., a_{2^n}) = Dir(1, .., 1) $ and $ m_i $ the occurences of outcome $ |i\rangle $ on samples $z_1, .. z_k$. Since the Dirichlet distribution is ...
  • 1,336
2 votes

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

So probability of the second qubit being in state $|1\rangle$ is the probability of the 5 qubit system being in a state that has $|1\rangle$ as the second qubit. So among all the 32 states, find the ...
2 votes
Accepted

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

You'll place the phase within the CRz gate. The approach you've taken essentially argues that: $$ e^{it H_3} \approx e^{it \alpha X_1 \otimes Y_2} e^{it \beta Z_1 \otimes Z_2} $$ So, when you're ...
  • 1,531
2 votes

Quantum Circuit to inverse the probability distribution

There is no a way in Qiskit to get the results in that format. So, you will have to go for the pure Python way: ...
  • 4,229
2 votes
Accepted

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

The statement made in the research paper is right. The initialization of the $\psi$ state is more than one qubit. For n qubits, it's state is in $2^n$ dimensions. Coming back to your question, compute ...
  • 148

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