New answers tagged textbook-and-exercises
2
votes
Accepted
How to extract probabilities from Kraus representation?
We can indeed rewrite $\mathcal{E}(\rho)=\sum_iK_i\rho K_i^\dagger$ as $\mathcal{E}(\rho)=\sum_ip(i)\rho_i$ by setting $p(i):=\mathrm{tr}(K_i\rho K_i^\dagger)$ and $\rho_i:=\frac{K_i\rho K_i^\dagger}{...
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0
votes
Why for every state there is always a measurement that has a deterministic outcome?
This is just a simpler take of Franklin's answer but that response goes more in depth.
Any quantum circuit (without measurements) of $N$-qubits can be built from the state $|000\cdots0\rangle$ and set ...
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3
votes
Why for every state there is always a measurement that has a deterministic outcome?
How can we say this is true of all quantum states and gates – "Whatever state our quantum system is in, there is always a measurement that has a deterministic outcome.” Can someone explain this ...
0
votes
Why adding H-gate is referred as changing the basis of measurement?
Any unitary operator can be understood as a change of basis in which we describe operators in quantum theory.
Specifically, the Hadamard gate is a unitary that sends the basis $\{\vert 0\rangle , \...
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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 ...
- 606
2
votes
Understanding the Bernstein-Vazirani implementation from the qiskit website
The Bernstein-Vazirani algorithm is an oracular problem: you suppose that you are given an oracle that implements a function $f$ such that there is a secret $s$ such that $f(x)=x\cdot s$. Crucially, ...
- 4,278
2
votes
Does applying an operator to a state have nothing to do with taking a measurement?
Both quantum gates and measurements can be represented as matrices. Every quantum gate corresponds to a unitary matrix, and every measurement corresponds to a Hermitian matrix. Some matrices happen to ...
0
votes
Is this state entangled?
It may be worth noting that the states analogous to the Bell state $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$ for 3 qubits quantum system is W state: $\frac{1}{\sqrt{3}}|001\rangle +...
- 117
3
votes
Accepted
What physical quantity does a density operator represent as an observable?
Here's a possible way of thinking of the measurement defined by a general density operator $\rho$, not necessarily one representing a pure state $\lvert\psi\rangle$. If $\mathcal H\simeq \mathbb C^d$ ...
4
votes
Is this state entangled?
$$
\frac{{\sqrt 2 }}{4}\left| {\left. {000} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {001} \right\rangle } \right. + \frac{{\sqrt 2 }}{4}\left| {\left. {010} \right\rangle } \right....
- 606
0
votes
How to choose a suitable number of iterations for Grover's algorithm?
Indeed, the adequate number of iterations over Grover's iterator depends on the number of solutions. Various methods have been proposed over the years to overcome this difficulty when the number of ...
- 1,574
2
votes
Accepted
Is there a tutorial for error mitigation in qiskit that does not use the deprecated ignis?
Here is an up-to-date tutorial on measurement error mitigation using Qiskit experiments module which (partly) replaces Qiskit Ignis.
Furthermore, if you are using Qiskit Runtime primitives, you have ...
- 7,109
1
vote
How to modify the Deutsch-Jozsa algorithm to distinguish between two types of $n$-bit inputs
I prefer to think about it in terms of function (which is also the usual way of introducing the Deutsch-Jozsa algorithm). This is exactly the same formalism, with the exception that I will call $f(i)$ ...
- 4,278
2
votes
Calculate qubit state in terms of two states that are opposite points on Bloch sphere
a) First rewrite the equations in terms of $\left| 0 \right>$ and $\left| 1 \right>$ as follows (here is how to do so):
$$\begin{aligned}
\left| 0 \right> &= \frac{\sqrt{3}}{2}\left| a \...
- 167
3
votes
Accepted
What are the possible $\beta$ in a qubit state $\frac{e^{i\pi/8}}{\sqrt5}|0\rangle+\beta|1\rangle$?
The exercise asks for a possible value for $\beta$, which you've given. So your answer is correct.
The fact that you get an answer different from the textbook comes from your first equation:
$$\left(\...
- 4,278
4
votes
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
There's a couple of ways you can do this. Let $\{|u_i\rangle\}$ be any orthonormal basis. Then the definition of the trace is simply
$$
\text{Tr}(A|\psi\rangle\langle\psi|)=\sum_i\langle u_i|A|\psi\...
- 51.1k
4
votes
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
You can proceed in the following way: $\text{Tr}(A|\psi\rangle\langle\psi|) = \sum_i \langle i|A|\psi\rangle\langle\psi|i\rangle = \sum_i \langle\psi|i\rangle\langle i|a|\psi\rangle = \langle \psi|(\...
- 1,429
3
votes
How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
You can use the cyclic property of the trace, ${\rm Tr}(XY) = {\rm Tr}(YX)$.
Another way is to note that both sides are linear over $A$. Thus it's enough to prove it for $A = E_{ij} = |i\rangle\langle ...
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