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
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0
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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 \...
- 348
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votes
2
answers
64
views
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.
- 181
1
vote
1
answer
38
views
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 ...
- 159
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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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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
0
votes
0
answers
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 ...
- 21
1
vote
1
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60
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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
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350
views
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_{...
- 266
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 \...
- 455
-1
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1
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31
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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>?
- 1
0
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 ...
- 5
-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
- 5
2
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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
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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
$$
How would we study a perturbation on $|\psi\rangle$, given that it has to be a valid quantum state? What ...
- 1,620
2
votes
1
answer
43
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Diagonal gates in qubit Clifford hierarchy are generated by $ C^i Z^{1/2^j} $
Let $ \mathcal{C}^{(t)} $ denote the $ t $ level of the $ n $ qubit Clifford hierarchy.
Let $ \mathcal{F}^{(t)} $ denote the collection of all diagonal gates in $ \mathcal{C}^{(t)} $. $ \mathcal{C}^{(...
- 2,556
1
vote
2
answers
72
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Weakly transversal gates for the $ [[15,1,3]] $ quantum Reed-Muller code
The $ [[15,1,3]] $ quantum Reed-Muller code is a CSS code famous for implementing logical $ T $ (strongly) transversally. In particular, logical $ T $ is implemented using the physical unitary
$$
\...
- 2,556
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 ?
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4
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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 ...
- 2,556
0
votes
1
answer
34
views
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\...
- 13
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 ...
- 2,556
2
votes
2
answers
72
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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
2
votes
1
answer
34
views
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
views
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?...
- 309
1
vote
0
answers
47
views
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
0
votes
1
answer
32
views
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 ...
- 11
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 ...
- 397
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votes
1
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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 ...
- 15
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$-...
- 397
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2
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78
views
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 ...
2
votes
1
answer
40
views
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 ...
- 455
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1
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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
5
votes
1
answer
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 & ...
- 2,556
2
votes
1
answer
87
views
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 ...
- 2,556
4
votes
1
answer
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
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
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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2
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119
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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
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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 ...
- 790
1
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1
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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 ...
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 ...
- 133
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0
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47
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Close in operator norm imply close in weak multiplicative sense?
Fix $\epsilon > 0$, and suppose $U$ and $S$ are $n$ qubit unitaries such that $\| U - S \| \leq \epsilon$ (operator norm). Furthermore, let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ be the ...
- 397
2
votes
1
answer
92
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Notation for Lindblad operators
I was reading the paper Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation
. The claim is that universal quantum computation can be achieved using the ...
- 2,194
2
votes
1
answer
331
views
How to calculate the log of a density matrix?
In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
- 149
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votes
0
answers
54
views
How many gates are in the $ k $ level of single qubit Clifford hierarchy?
Define the single qubit Clifford hierarchy recursively by
$$
\mathcal{C}^1:=<iX,iZ>
$$
the determinaint 1 subgroup of the Pauli group. Define the rest of the the hierarchy inductively by
$$
\...
- 2,556
3
votes
1
answer
62
views
Does the real Clifford group contain all real diagonal gates? all permutation gates?
The real Pauli group is the subgroup of $ O_{2^n}(\mathbb{R}) $ generated by products and tensor products of $ X $ and $ Z $ (this deviates from the usual Pauli group in that only real Paulis are ...
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6
votes
1
answer
136
views
Is the Clifford hierarchy finite?
This question is inspired by
Is the Clifford group finite?
Which shows that that the Clifford group (the second level of the Clifford hierarchy) is finite. (finite meaning finite mod global phases) ...
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3
votes
3
answers
343
views
Realizing a swap gate using a commutator sequence and an auxiliary qudit
Say I have two qudits $1$ and $2$, each of which has Hilbert space of dimension $m$. Is it possible to introduce an auxiliary qudit $a$ (of any dimension $d_a\in \mathbb{Z}_{\geq 2}$) and find quantum ...
- 176
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votes
1
answer
108
views
Is Shor demonstration wrong?
in Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer by Peter W. Shor (also in Algorithms for quantum computation: discrete logarithms and factoring).
In ...
0
votes
0
answers
16
views
How do we know what angle formula X1 is encoded into qml.MottonenStatePreparation?
Known that X1 is a quantum state, it is prepared by qml.MottonenStatePreparation. How do we know what angle formula X1 is encoded into qml.MottonenStatePreparation? can be interpreted in python code. ...
- 390
3
votes
0
answers
47
views
Expectation value over random $k$-local Pauli operators for two random quantum states
Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} ...
- 307