7
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 ...
- 20.7k
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 ...
- 20.7k
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}} ...
- 936
5
votes
Accepted
Computing expectation value of $|\langle z|C|0^n\rangle|^2$ over Haar random circuit
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 ...
- 20.7k
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. ...
- 321
5
votes
Accepted
Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over 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 ...
- 2,485
5
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 ...
- 20.7k
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
$$
\begin{aligned}
H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\
&= \...
- 5,345
5
votes
Negative Probability — Reality vs Description
"I understand that quantum physics supports the concept that the probability of a qubit collapsing into (say) 1, can be negative or positive [...] If that is wrong, please say so; I understand ...
- 13.5k
5
votes
Accepted
Generating random, but non-uniform state
Rejection sampling is a good fit and works without any changes, simply by plugging the desired distribution $p(\psi)$ into the standard algorithm.
Let$^1$ $M:=\max_{\psi\in\mathbb{CP}^1} p(\psi)$. To ...
- 20.7k
4
votes
How to interpret complex probability of superposition state?
Just to note that the complex number also specify a quantum phase of a qubit. In this particular case
$$
\frac{1+i}{2} = \frac{1}{\sqrt{2}}\mathrm{e}^{i\frac{\pi}{4}},
$$
so the relative phase is $\...
- 13.6k
4
votes
Accepted
Quantum Amplitude Estimation vs Quantum Phase Estimation
Ok let's break this down. Firstly the success probability for QPE and QAE are defined slightly differently.
With QPE there are two error bounds $|\hat{x} - x| < \epsilon$ to consider
$\epsilon < ...
- 939
4
votes
Accepted
What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?
These are classical conditional probabilities used extensively in Bayesian probability.
Suppose Alice can prepare any of a number of pure states $|\psi_i\rangle$. She chooses $i$ randomly from ...
- 20.7k
4
votes
How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?
First, recall that $\mathrm{tr} A = \sum_i \langle i|A|i \rangle$. Each equation is then a sum where all terms are products of $P(z)$ and three other quantities. Further, the sum in the first equation ...
- 20.7k
4
votes
Accepted
Properties of frames in quasiprobability representation
The authors are certainly thinking about finite frames. In this case, your statement is correct, since the number of elements in every spanning set is at least the vector space dimension.
As glS ...
- 4,477
4
votes
Applying Hadamard gate to $\sqrt{3/4}|0\rangle + \sqrt{1/4}|1\rangle$
It seems like you are trying to apply a Hadamadard gate to the state $\frac{\sqrt{3}}{2}|0\rangle + \frac{1}{2}|1\rangle$. This would go as follows:
$$
\begin{align}
H\left[\frac{\sqrt{3}}{2}|0\rangle ...
- 2,865
4
votes
How do I get correct measurement probabilities in ZX calculus?
OK, I made two mistakes. Both were corrected by a closer reading of the paper "Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions".
First, as was pointed out by ...
- 221
4
votes
Accepted
Confusion on the probability of measuring first qubit of a separable mixed state
They're the same because
$$\sum_z |\langle z|\tilde U|\tilde x \rangle|^2=1$$
for any unitary $\tilde U$ and input $|\tilde x\rangle$. This follows directly from the normalization of probabilities: $|\...
glS♦
- 23.3k
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
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
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 ...
- 4,477
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 ...
- 5,345
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 ...
- 1,406
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 ...
- 5,359
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\...
- 3,649
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' ...
- 5,345
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 ...
- 612
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 ...
- 20.7k
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 ...
- 6,701
3
votes
Accepted
Why is $|P_0- P_1|=1$?
If you are promised that you will receive a qubit in a classical state and you also know the basis it is in, then it is certain that when you perform a measurement in that basis, the resulting ...
- 2,359
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