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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
17 views

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 \...
Nichola's user avatar
  • 348
0 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.
reza's user avatar
  • 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 ...
3nondatur's user avatar
  • 159
1 vote
0 answers
31 views

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 ...
trillianhaze's user avatar
0 votes
0 answers
3 views

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?
Kochan's user avatar
  • 21
0 votes
0 answers
27 views

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 ...
Kochan's user avatar
  • 21
1 vote
1 answer
60 views

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 ...
Ian Gershon Teixeira's user avatar
6 votes
1 answer
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_{...
Duen's user avatar
  • 266
4 votes
2 answers
114 views

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 \...
Jon Megan's user avatar
  • 455
-1 votes
1 answer
31 views

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>?
Flaplap's user avatar
0 votes
3 answers
56 views

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 ...
Creative's user avatar
-3 votes
1 answer
30 views

What are the expected measurement results in the diagram below? [closed]

I ask you to give a mathematical solution to this problem
Creative's user avatar
2 votes
0 answers
113 views

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 $ ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
17 views

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 ...
C. Kang's user avatar
  • 1,620
2 votes
1 answer
43 views

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}^{(...
Ian Gershon Teixeira's user avatar
1 vote
2 answers
72 views

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 $$ \...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
120 views

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 ?
Duen's user avatar
  • 266
4 votes
2 answers
195 views

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 ...
Ian Gershon Teixeira's user avatar
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\...
Việt Nguyễn's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
Ian Gershon Teixeira's user avatar
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 $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
Ian Gershon Teixeira's user avatar
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?...
Rydberg's user avatar
  • 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 ...
Ian Gershon Teixeira's user avatar
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 ...
IsuzuGawa's user avatar
5 votes
1 answer
129 views

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 ...
trillianhaze's user avatar
0 votes
1 answer
22 views

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 ...
QuantumAudit's user avatar
2 votes
1 answer
47 views

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 $n$-...
trillianhaze's user avatar
0 votes
2 answers
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 ...
user_confused's user avatar
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 ...
Jon Megan's user avatar
  • 455
2 votes
1 answer
64 views

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 ...
Ian Gershon Teixeira's user avatar
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 & ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 $...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
65 views

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 ...
R.W's user avatar
  • 2,247
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 ...
Ian Gershon Teixeira's user avatar
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 ...
M. Al Jumaily's user avatar
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 ...
Марина Лисниченко's user avatar
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 ...
RyRy the Fly Guy's user avatar
1 vote
0 answers
47 views

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 ...
trillianhaze's user avatar
2 votes
1 answer
92 views

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 ...
MonteNero's user avatar
  • 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 ...
Visipi's user avatar
  • 149
2 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 $$ \...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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) ...
Ian Gershon Teixeira's user avatar
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 ...
Lagrenge's user avatar
  • 176
0 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 ...
Philip.q.c's user avatar
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. ...
R-X Zhao's user avatar
  • 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} ...
userflux9674's user avatar

1
2 3 4 5
10