Questions tagged [mathematics]
Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.
497
questions
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Are permutations of the Pauli strings unitary operations?
Consider the set of Pauli strings $P_N=\{\tau \}$,
composed out of tensor products of Pauli matrices $\sigma_i^\alpha$
acting on $N$ or qubits, e.g.
$\tau=\sigma^x_1 \otimes \mathbb{1}_2 \otimes \...
- 5,159
4
votes
2
answers
340
views
Exotic transversal gate group for stabilizer code
What are examples of interesting $ [[n,1,d]] $ or $ [[n,2,d]] $ stabilizer codes, $ d \geq 2 $, whose group of transversal gates is not isomorphic to a subgroup of the Clifford group (on 1 and 2 ...
- 2,556
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2
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64
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what is square root of a density matrix power two?
I know that in algebra for a variable we have
$ \sqrt {x^2} = |x|$
What if $x$ is a density matrix?
Please share resource for your answer.
- 13.1k
12
votes
1
answer
4k
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
0
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1
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76
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What are some good resources for learning quantum math?
I'm new to Quantum dynamics as a whole and everytime i read an article on arxiv.org or watch a video on youtube and they introduce an equation like Shrodinger or other equations to show the logic and ...
glS♦
- 21.7k
1
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1
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38
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What does Pauli's $Y$ matrix represent?
It is easy to see that Pauli's $X$ matrix represents the bit flip operation, i.e. $X \lvert 0 \rangle = \lvert 1 \rangle$ and $X \lvert 1 \rangle = \lvert 0 \rangle$.
Similarly, Pauli's $Z$ matrix ...
- 1,078
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0
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31
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Improving operator norm bound on total variation distance
Let $U$ be an $n$-qubit unitary and $P_U(x) = |\langle x |U|0^n\rangle|$ the probability of measuring $x$ after acting $U$ on $|0^n\rangle$. For two $n$-qubit unitaries $U$ and $V$, one can prove that
...
- 397
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0
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27
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Tighter upper bound of $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$
I am wondering about an upper bound of the trace function $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$ (we assume that $\rho$ is the $N\times N$ density matrix representing the quantum ...
glS♦
- 21.7k
0
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0
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3
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Is the inequality $E[( tan(x))^{-1} ] \leq (tan(E[x]))^{-1}$ true? [migrated]
Given a random variable $x$ and expectation E, does the inequality
$E\left[\frac{1}{tan(x)} \right] \leq \frac{1}{tan(E[x])}$ hold?
- 21
1
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1
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Is every diagonal gate whose non-zero entries are $2^k$th roots of unity in the two qubit Clifford hierarchy?
Does the two qubit Clifford hierarchy contain all diagonal gates whose entries are $ 2^k $ roots of unity?
In particular, is it true that every $ 4 \times 4 $ diagonal matrix whose diagonal entries ...
- 2,556
6
votes
1
answer
350
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Simplify the tensor product of two exponentials
If I have a 2-qubits circuit with a Ry rotation gate acting on each one :
My unitary transformation performed on the 2-qubits state is written as :
$$e^{-i\theta_{1} \sigma_{y}} \otimes e^{-i\theta_{...
- 1,000
4
votes
2
answers
114
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Does the gradient commute with the partial trace?
Suppose I have a parameterized quantum state: $\rho(\theta) = U(\theta) \rho U^\dagger(\theta)$. I am curious to know whether the following holds:
$\frac{\partial \text{Tr}_A (\rho(\theta))}{\partial \...
- 7,559
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3
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3k
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How to prove that antipodal points on the Bloch sphere are orthogonal?
I started by assuming two antipodal states
$$
|(\theta,\psi)\rangle = \cos\dfrac{\theta}{2}|0\rangle + \sin\dfrac{\theta}{2}e^{i\psi}|1\rangle\\
|(\theta+\pi,\psi+\pi)\rangle= \cos\dfrac{\theta+\pi}{2}...
- 11
-1
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1
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What will be the new state after the outcome of the measurement of the state of the first qubit as ǀ0>? [closed]
Two qubits are prepared in a superposition state of the form:
What will be the new state after the outcome of the measurement of the state of the first qubit as
ǀ0>?
- 557
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votes
3
answers
56
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How to prove that CNOT and Rz gates are permutable?
How to prove that CNOT and Rz gates are permutable?
I tried to equate their switch to zero and calculate it, but for this you need to multiply the matrices. But the 4x4 and 2x2 matrices cannot be ...
- 52.3k
-3
votes
1
answer
30
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What are the expected measurement results in the diagram below? [closed]
I ask you to give a mathematical solution to this problem
- 52.3k
2
votes
0
answers
113
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Weakly transversal gates for the $ [[5,1,3]] $ code
For the $ [[5,1,3]] $ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code $ X^{\otimes 5} $ implements logical $ X $ and $ Z^{\otimes 5} $ implements logical $ Z $. Also logical $ HP $ ...
- 2,556
2
votes
1
answer
79
views
Same weight enumerator iff equivalent by permutations and local unitaries
A non-entangling gate on $ n $ qubits is an element of the group
$$
N\Big(\bigotimes_{i=1}^n U(2)\Big)=\bigotimes_{i=1}^n U(2) \rtimes S_n
$$
which is generated by $ U(2) $ acting locally on each ...
- 446
1
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2
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72
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Weakly transversal gates for the $ [[15,1,3]] $ quantum Reed-Muller code
The $ [[15,1,3]] $ quantum Reed-Muller code is a CSS code famous for implementing logical $ T $ (strongly) transversally. In particular, logical $ T $ is implemented using the physical unitary
$$
\...
- 2,556
5
votes
2
answers
964
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Maximally entangled state definition, and orthonormal basis of maximally entangled state
My questions are probably more about details but the answer will help me to precisely understand the mathematical structure.
My questions are related to page 14 of this pdf.
Mathematical context:
...
- 3
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1
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43
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Diagonal gates in qubit Clifford hierarchy are generated by $ C^i Z^{1/2^j} $
Let $ \mathcal{C}^{(t)} $ denote the $ t $ level of the $ n $ qubit Clifford hierarchy.
Let $ \mathcal{F}^{(t)} $ denote the collection of all diagonal gates in $ \mathcal{C}^{(t)} $. $ \mathcal{C}^{(...
- 405
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1
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1k
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How is Grover's operator represented as a rotation matrix?
I have seen that it is possible to represent the Grover iterator as a rotation matrix $G$. My question is, how can you do that exactly?
So we say that $|\psi\rangle$ is a superposition of the states ...
glS♦
- 21.7k
0
votes
0
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17
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Calculus and perturbing expectation values
Consider the following quantity:
$$
f_O(|\psi\rangle) = \langle \psi | O | \psi \rangle
$$
How would we study a perturbation on $|\psi\rangle$, given that it has to be a valid quantum state? What ...
- 1,620
5
votes
1
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159
views
How many $ \sqrt{X} $ are there?
I was reading Square root of Pauli operators: is there a common convention to define them uniquely? and it got me thinking about square roots.
Recall the Pauli $ X $ gate
$$
X=\begin{bmatrix}
0 & ...
- 1,078
2
votes
2
answers
72
views
Weight enumerators for Hermitian operator
Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,556
4
votes
1
answer
120
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Link between quantum computing and Lie theory?
I know only little thing about Lie theory but I would like to learn more about its link to quantum computing.
Has someone got some references explaining it well ?
- 411
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2
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195
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Does every code have transversal Pauli group?
A transversal logical gate for an $ n $ qubit code is a gate from the group of local unitaries
$$
\bigotimes_{i=1}^n U(2)
$$
which also preserves the codespace. For an $ ((n,K,d)) $ code we say a ...
- 558
4
votes
1
answer
269
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.
$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$
...
glS♦
- 21.7k
0
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1
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34
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Find min of a quantum state L2 norm
I have a problem minimizing this norm with respect to $\alpha$:
$\min_{\alpha}||e^{i\alpha}|\psi\rangle-|\phi\rangle ||^2$ (1)
The result is that this achieves min when $\alpha=-\measuredangle \langle\...
- 2,194
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votes
0
answers
34
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Dimension of local operators stabilizing the code space?
What is the maximum dimension of a connected group of local operators stabilizing an $ [[n,k,d]] $ code with $ d \geq 2 $?
Some background:
Consider an $ [[n,k,d]] $ quantum error correcting code with ...
- 2,556
6
votes
1
answer
1k
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Understanding a quantum algorithm to estimate inner products
While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation.
Consider ...
2
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1
answer
34
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Weight enumerators for Hermitian operator (wrong $ B_j $ definition)
Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,556
10
votes
3
answers
2k
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Why can all quantum circuits be converted into circuits that use only real matrices?
I know that you need to add an additional ancilla qubit to "keep track" of whether or not you are in real space or imaginary space, but how exactly does this work?
What is the proof for this?...
glS♦
- 21.7k
5
votes
1
answer
129
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Building universal gate set for $SU(d^n)$ from universal gate set for $SU(d)$
Let $G$ be a universal gate set for $SU(d)$. Then the words $\langle G \rangle$ of $G$ form a dense subset of $SU(d)$ with respect to some reasonable norm, and so every element of $SU(d)$ can be ...
- 2,556
2
votes
1
answer
64
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Codes with codewords that aren't uniform modulus superposition
All stabilizer codes and also all non stabilizer codes that I am aware of, for example the ones here,
Example non-stabilizer code?
have a basis of codewords which are all uniform modulus ...
- 2,556
1
vote
0
answers
47
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Eastin Knill Theorem and global phase
In quantum we don't care about global phases, but I want to ask a question about global phases anyway.
The original Eastin-Knill Theorem paper https://arxiv.org/abs/0811.4262 says
$$
CP = \Pi_{i=1}^k ...
- 2,556
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1
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32
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Where am I going wrong in my understanding of qubit associativity?
I am studying the basics of quantum computing math and am confused about qubit associativity. As I understand it, in quantum math, multiple qubits are represented as the tensor product of the qubits ...
- 303
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votes
1
answer
22
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What is the difference between Gate.power() and Gate.repeat()?
Why are the gates a and b in this code not the same?
a = UGate(0,0,0.9*np.pi).power(2)
b = UGate(0,0,0.9*np.pi).repeat(2)
I thought that unitary gates function ...
- 7,559
2
votes
1
answer
47
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Bounding operator norm by total variation distance
Let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ denote the probability distribution of obtaining the bitstring $y \in \{0,1\}^n$ on a fixed input $x \in \{0,1\}^n$ w.r.t. the unitary $U$. For $n$-...
- 4,906
5
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2
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376
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Map a 4-body Ising Hamiltonian to a 2-body Ising Hamiltonian
I wonder if there exists a way to map the square of a 2-body Ising Hamtiltonian (which will make it 4-body) back to a 2-body Hamiltonian that has the same ground state? Let me explain what I mean by ...
- 9,366
0
votes
2
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78
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Modular Addition general explanation
This is an incredibly basic question, but basically I'm really struggling to understand what the "addition modulo 2" is and why is it used in quantum computing.
I've tried Wikipedia, endless ...
- 633
2
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1
answer
40
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time evolution of Hamiltonian to generate the Bell pair
Consider two different Hamiltonians: $H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2$ and $H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2$, where $\alpha(t)$ and $\beta(t)$ are time-dependent functions. Starting ...
- 52.3k
3
votes
1
answer
105
views
Can Clifford gates be diagonalized using a gate from the third level of the Clifford hierarchy?
Is it always possible to diagonalize a Clifford gate $ g $ using a gate $ V $ from the third level $\mathcal{C}^{(3)}$ of the Clifford hierarchy? In other words can every Clifford gate be written as
$...
- 2,556
2
votes
1
answer
87
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Are all powers $g^m$ in the Clifford hierarchy if $g$ is?
It is already known that the Clifford hierarchy is not closed under arbitrary products, see this post which shows that the product $ THT $ is not in any level of the hierarchy.
What about products of ...
- 19.3k
2
votes
1
answer
65
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What are the elements of quotienting the Pauli group $\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$, and how to do calculations with it?
Let $\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$ together with all the phases $\{\pm 1, \pm i\}$ the regular Pauli group, and $N = \langle \pm i I\rangle $. I would ...
- 2,247
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votes
1
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466
views
Is every Clifford gate conjugate to a diagonal Clifford gate?
Let $ C $ be a Clifford gate. Let $ D $ be the diagonalization of $ C $. In other words $ D $ is a diagonal gate and
$$
C=VDV^{-1}
$$
for some $ V $. Is $ D $ also a Clifford gate?
Update: Filling in ...
- 2,556
4
votes
2
answers
119
views
Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as
$$
P=VDV^{-1}
$$
for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily ...
- 2,556
0
votes
1
answer
94
views
How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
- 1,504
1
vote
1
answer
213
views
Matrix representation of any conditioned gate
Is there an algorithm explaining how to represent any gate in the matrix form?
Suppose, the circuit is the following:
where operator
$ U = e^{iA\pi/4} =
\begin{bmatrix}
0.35-0.85i & -0.35-0.15i ...
- 1,300
3
votes
1
answer
54
views
Clarification defining/finding the relative phase of a qubit
Let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$.
According to p. 22 of ...
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