19
votes
Accepted
What is a stabilizer state?
Let $\mathcal{G}_n$ denote the Pauli group on $n$ qubits. An $n$-qubit state $|\psi\rangle$ is called a stabilizer state if there exists a subgroup $S \subset \mathcal{G}_n$ such that $|S|=2^n$ and $A|...
- 20.8k
14
votes
Accepted
What are min and max overlaps of a maximally entangled state with a separable state?
The minimum overlap is zero and the maximum overlap is $\frac{1}{d}$.
The overlap is a linear function of $\rho$ and the set $S$ of separable states is convex, so the overlap is both minimized and ...
- 20.8k
12
votes
Accepted
What is the longest time a qubit has survived with 0.9999 fidelity?
Well, for the longest coherence time ever, I'm finding this Science from 2013 entitled Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28, which indicates ...
- 3,399
11
votes
Accepted
What does fidelity mean?
It might be worth mentioning the physical motivation for these definitions and the concept of fidelity itself.
Unlike the classical computers we all know and love, quantum computers are ...
- 356
11
votes
Accepted
How to calculate the average fidelity of an amplitude damping channel
An elementary method is to simply carry out the integration
$$
\begin{align}
\overline{F} &= \int\langle\psi|\mathcal{N_\gamma}(|\psi\rangle\langle\psi|)|\psi\rangle d\psi\\
&=\int\langle\psi|...
- 20.8k
10
votes
Accepted
What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?
Recall the law of cosines for two unit vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb R^2$:
$$
\|\mathbf{u}-\mathbf{v}\|^2 = 2-2\cos\theta,
$$
where $\theta$ is the angle between the vectors. ...
- 574
8
votes
Accepted
What would be an ideal fidelity measure to determine the closeness between two non unitary matrices?
When you ask about an 'ideal' fidelity measure, it assumes that there is one measure which inherently is the most meaningful or truest measure. But this isn't really the case.
For unitary operators, ...
- 11.9k
8
votes
Accepted
Closeness of purifications of states
No dimension-independent bound is possible.
Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume
$$
\|\rho_A - \...
- 5,158
7
votes
Accepted
Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
I'll give a couple of methods to do this:
(Using matrix inequalities) The idea is to use CS inequality in the form
$$\newcommand{\tr}{\operatorname{Tr}}\lvert \sum_{ij}A_{ij}^* B_{ij}\rvert\le\sqrt{\...
glS♦
- 23.4k
7
votes
How can I calculate the inner product of two quantum registers of different sizes?
I guess you're looking at equations (130) and (131)? So, here, you have $|\psi\rangle=(|0\rangle|a\rangle+|1\rangle|b\rangle)/\sqrt{2}$ and $|\phi\rangle=|a| |0\rangle+|b| |1\rangle$. When it says to ...
- 55.7k
7
votes
Accepted
Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$
Background
If $v_1, v_2, \dots, v_n$ is an orthonormal basis in the inner product space $V$, then any vector $u\in V$ can be expressed as a linear combination
$$
u = \alpha_1 v_1 + \alpha_2 v_2 + \...
- 20.8k
6
votes
How can I calculate the inner product of two quantum registers of different sizes?
Actually, there should be a minus.
There is a mistake in the paper.
Wittek uses a minus in his (expensive) book.
Indeed say :
$$ |\psi\rangle = \frac{1}{\sqrt{2}} (|0,a\rangle + |1,b\rangle) $$
$$ |...
- 4,644
6
votes
Purpose of using Fidelity in Randomised Benchmarking
Nielsen and Chuang in their book "Quantum Computation and Quantum Information" have section (Chapter 9) on distance measures for quantum information.
Surprisingly they say in Section 9.3 " How well ...
- 591
6
votes
Accepted
Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
Both definitions are used and authors usually make it clear which one they mean.
Wikipedia also points this out under the Alternative Defintion section.
- 2,597
6
votes
Accepted
How to limit the error probability in large scale quantum computers
It is true that fidelity decays exponentially in the course of quantum computation. This is indeed a major limitation of NISQ computers that imposes a stringent "depth budget". In order to ...
- 20.8k
6
votes
Accepted
How to find the distance between a given $\rho$ and the nearest pure state(s)?
Recall that for any Hermitian operator $A$ and any unit vector $|\psi\rangle$ the real number $\langle \psi|A|\psi\rangle$, known as the Rayleigh quotient, is bounded by the largest eigenvalue $\...
- 20.8k
6
votes
If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?
Observe that, for any collection of matrices $A_i$, we have
$$\sqrt{\sum_i |i\rangle\!\langle i|\otimes A_i}
= \sum_i |i\rangle\!\langle i|\otimes \sqrt{A_i}, \\
{\rm Tr}\left(\sum_i |i\rangle\!\...
glS♦
- 23.4k
6
votes
Accepted
Approximate Cloning
Given the constraints you impose (why those are appropriate constraints is perhaps another discussion), I think you're over-complicating things.
Without loss of generality, you can assume
$$
|\alpha_0\...
- 55.7k
6
votes
Accepted
What use cases are there for 127 qubit QPUs?
I think that the main reason behind is to tackle technical difficulties connected with building huge number of qubits. Having hundred of qubits brings about issues with interconnection, connections to ...
- 13.6k
6
votes
Does a fidelity of $\mathcal{F}(U_1|0\rangle, U_2|0\rangle)=1$ imply that $U_1=U_2$?
I think you mean that if $\mathcal{F}=1$, the two states are identical.
Now to your question: is it true that
$$
\mathcal{F}(U_1|0\rangle,U_2|0\rangle)=0\implies U_1=U_2.
$$
This is not true. You need ...
- 55.7k
5
votes
Accepted
Quantum circuit for computing fidelity
Prohibited device
Such a circuit $C$ would enable faster-than-light communication and therefore does not exist.
Suppose Alice and Bob share a Bell pair $|\psi\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |...
- 20.8k
5
votes
Accepted
Intuitive role of the polar decomposition in proof of Uhlmann's theorem for fidelity
(I will give the argument with formulas for now, hopefully I find time for some pictures later.)
Let $|m\rangle$ be the (unnormalized) maximally entangled state. Then, a purification of $\rho$ is ...
- 5,682
5
votes
What does fidelity mean?
Simply it is the distance (similarity measure) between two quantum states, for example the fidelity between $|0\rangle$ and $|1\rangle$ is less than the fidelity between $|0\rangle$ and $\frac{1}{\...
- 483
5
votes
Are there disadvantages in using the inner product between states instead of the fidelity?
The quantity $\text{Tr}(\sqrt{A}\sqrt{B})$ that you defined there is actually referred to as the "just-as-good fidelity" (see 1801.02800) because it does have a relationship with the trace distance ...
- 81
5
votes
Are there disadvantages in using the inner product between states instead of the fidelity?
A few thoughts:
It mostly depends on what you are trying to quantify.
The inner product of states, $\text{Tr}(\rho\sigma)$, is used to quantify the distance in state space. More precisely, the ...
glS♦
- 23.4k
5
votes
Can we combine the square roots inside the definition of the fidelity?
This result is indeed correct and gets rediscovered from time to time. We rediscovered it last year and got it published at Phys Rev A 107, 012427 (2023) (free version at arXiv:2211.02623). But it's ...
5
votes
Accepted
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The answer is no, as the following counter-example reveals. Let $\varepsilon\in(0,1)$ and define
$$
\rho_0 =
\begin{pmatrix}
\frac{1+\varepsilon}{2} & 0 & 0\\
0 & 0 & 0\\
0 & 0 &...
- 5,158
5
votes
What does quantum gate fidelity mean?
Very briefly, gate fidelity refers to a way to compare how "close" two gates, or more generally operations, are to each other.
As discussed e.g. in (Magesan et al. 2012), if one wants to ...
glS♦
- 23.4k
5
votes
Accepted
Fidelity of extensions of states
Let's start with the second question. There is nothing special about an extension $\sigma_{AR}^{\ast}$ that allows it to be optimal for the right-hand side of (1); any extension $\sigma_{AR}$ of $\...
- 5,158
5
votes
Accepted
Simulate a quantum channel with a certain fidelity
In the first case, when the details of the physical process do not matter, you can choose any quantum channel type that can achieve your target average fidelity. Depolarizing channel may be a good ...
- 20.8k
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