# Tag Info

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 ...
• 20.8k
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 ...
• 20.8k
Accepted

• 23.4k

### 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
Accepted

### Probability of measuring one qubit from the state of two qubits

If we have the state $|\psi \rangle = a_{00}|00\rangle +a_{01}|01\rangle +a_{10}|10\rangle +a_{11}|11\rangle$ then the probability of the second qubit being in the state $|1\rangle$ is the ...
• 13.7k
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 ...
• 4,497
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 ...
• 5,355
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 ...
• 1,406
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 ...
• 5,369

### 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\...
• 3,674
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' ...
• 5,355
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 ...
• 612
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 ...
• 20.8k
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 ...
• 6,711