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 ...
- 20k
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♦
- 23k
3
votes
Confusion on the probability of measuring first qubit of a separable mixed state
Another reason why they are the same, equivalent to @glS's answer: resolution of identity
$$\sum_{z \in \{0,1\}^{n-1}}|z\rangle\langle z|=I_{n-1}.$$
We can expand the absolute square in the second ...
- 3,504
3
votes
Accepted
An inequality involving quantum channels
A simple counterexample: let $\mathsf{C}$ be the circuit that applies $H$ while $\mathsf{D}$ is the circuit that applies $HX$. That is, $\mathsf{C}$ creates $|+\rangle$ while $\mathsf{D}$ creates $|-\...
- 5,112
3
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 ...
- 211
3
votes
Simultaneous measurements and Bell basis measurements to estimate $\lvert\text{Tr}(\sigma \rho)\rvert^2$ in Huang et al. paper
Here, "simultaneous measurement" of an observable does not refer to simultaneously in time, but rather the ability to estimate several observables from the same measurement setting acting on ...
- 6,024
3
votes
Accepted
Conditional expectation for Haar random states
Intuitively, that's the case: the vector being random, there is no reason to prefer $|0\rangle$ over $|1\rangle$ on the first qubit.
I don't think it requires to compute some integrals other than ...
- 5,112
3
votes
Accepted
Survival probability quantum circuit
We can calculate this survival probability $P(t) = |\langle \psi | e^{-i \mathcal{H}t} | \psi \rangle|^2$ in two ways. The first corresponds to the so-called Hadamard test, as shown in the figure ...
- 932
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 ...
- 2,339
2
votes
Accepted
How to write post-measurement states when the measurement apparatus measures one of two observables?
Let's assume that that both observables $A$ and $B$ have projectors $P^{A/B}_i$. I'm going to assume that the set of outcomes for the two is the same (and therefore, implicitly, that I don't know ...
- 54.5k
2
votes
Accepted
Computing a ratio involving Haar random unitaries
Here is a simpler way.
Since the Haar measure is left and right invariant, we are free to pull out a unitary to the left or right of $U$ and the expectation is left invariant.
In particular, let $U \...
- 500
2
votes
Computing a ratio involving Haar random unitaries
Since the numerator and the denominator are (apparently) not independent, I'm not convinced that their expectation can be computed separately.
First of all, note that:
$$\DeclareMathOperator{Tr}{Tr}\...
- 5,112
2
votes
How much of quantum computing is based on probability?
Probability plays the important role in quantum computing.
Unlike classical computing, a result is obtained by sampling from multiple shots of execution rather than just a single run.
The result of ...
- 65
2
votes
Accepted
Bounding operator norm by total variation distance
No you cannot, here's a counterexample.
Let $U=I$ be the identity matrix and let $S = \sum_{i} (-1)^{\delta_{0,i}} |i\rangle \langle i|$ where $\delta_{i,j}$ is the Kronecker delta. That is, $S$ is ...
- 5,201
2
votes
Accepted
Independence in state prepared by independently drawn Haar random gates
This is not the case. In fact, this is not true even in the case of a true Haar-random unitary.
First of all, note that since $\sigma$ is a quantum state, it is hermitian. As such, $X_{a,b}=\overline{...
- 5,112
2
votes
How to get exact measurement probabilities when having intermediate measurements with Qiskit?
To do what you want in Qiskit, you could replace each measurement operation in your circuit by a simple call to the Statevector.probabilities(qargs) method, where ...
- 1,422
2
votes
Stabilizer States - Calculating measurement probabilities with the rank of the stabilizer table's X-block
The issue is that numpy.linalg.matrix_rank is assuming you want the rank over real numbers, when actually you want the rank over integers modulo 2.
For example, ...
- 32.5k
2
votes
How do you find the possible measurement values of an observable?
Unfortunately, some parts of this question are unclear to me as currently written. I will do my best to try and answer, in the sense of addressing some aspects of your below statement
Attempt at ...
- 653
1
vote
How to compute marginal probabilities of Alice's qubit (in density operator language)?
Using pure states and kets, as you described, you compute the marginal probabilities by computing the squared norm of the projection on the states you're interested in. In this case, this means ...
glS♦
- 23k
1
vote
How to compute marginal probabilities of Alice's qubit (in density operator language)?
Let's first see how projective measurements work in 'density operator language'.
If you perform a measurement on your density operator $\rho$, corresponding to a set of projective measurement ...
- 571
1
vote
Why are probabilities represented with alpha^2 and beta^2?
The issue is that coefficients $\alpha$ and $\beta$ are complex numbers representing both probability of measuring certain basis state and so-called quantum phase. Note that the phase can be used for ...
- 13.5k
1
vote
How to get exact measurement probabilities when having intermediate measurements with Qiskit?
You can use save_probabilities() function to save the measurement outcome probabilities anywhere in your quantum circuit when using ...
- 8,469
1
vote
How to get exact measurement probabilities when having intermediate measurements with Qiskit?
Alternatively, you can use Quirk for small circuits. Especially it's chance and amplitude displays are useful for visualizing measurement probabilities.
- 949
1
vote
Simultaneous measurements and Bell basis measurements to estimate $\lvert\text{Tr}(\sigma \rho)\rvert^2$ in Huang et al. paper
I am not sure what you mean with Bell measurement (haven't read the paper in detail).
Maybe this demo can help understand the concept of simultaneous measurement in the Pauli basis. This is a general ...
- 125
1
vote
Accepted
Sum of probability in non-orthogonal basis
In Quantum Computing, a measurement projects a vector onto an eigenspace of an observable $A$, which is an Hermitian matrix.
Since it is Hermitian, its eigenvectors form an orthonormal basis of the ...
- 5,112
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