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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 $n$-...
trillianhaze's user avatar
1 vote
2 answers
242 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
53 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
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2 votes
1 answer
70 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
482 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
129 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
507 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
167 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
150 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
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4 votes
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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 ...
Ian Gershon Teixeira's user avatar
0 votes
1 answer
486 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
245 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
130 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
57 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
113 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
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2 votes
1 answer
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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
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2 votes
0 answers
73 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
161 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
207 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
4 votes
3 answers
420 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
  • 185
0 votes
1 answer
135 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
1 vote
0 answers
34 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. ...
Ren-Xin Zhao's user avatar
4 votes
0 answers
79 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
2 votes
1 answer
230 views

The Clifford hierarchy and $ e^{2 \pi i/2^k} $

Could someone give me an example of a gate in the Clifford hierarchy which cannot be written as $$ e^{i \theta} V $$ for some unitary $ V $ with entries in terms of $ \zeta_{2^k} $? If no such example ...
Ian Gershon Teixeira's user avatar
7 votes
1 answer
248 views

Is this single qubit gate in the Clifford hierarchy?

For a single qubit, the Clifford hierarchy is defined to be $$ \mathcal C^{(k)} = \Bigl\{ U \in \mathbf U(2) \mathrel{\Big\vert} \forall P \in \mathcal C^{(1)} : U P U^\dagger \in \mathcal C^{(...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
36 views

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 ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
187 views

Which monomial matrices are in the Clifford hierarchy?

This is essentially a follow-up on the very interesting answer given here Is there a closure property for the entire Clifford hierarchy? I'm interested in sufficient conditions to conclude that a ...
Ian Gershon Teixeira's user avatar
1 vote
2 answers
339 views

How to perform a basis change on a 2x2 density operator?

I have an ensemble described by following density operator: $$ P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-| $$ I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis. I know that ...
mohaddese's user avatar
5 votes
2 answers
417 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 ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
304 views

What are well-known orthogonal 2-designs, other than the real Clifford group?

The paper Real Randomized Benchmarking https://quantum-journal.org/papers/q-2018-08-22-85/ https://arxiv.org/abs/1801.06121 makes use of the fact that the real Clifford group is an orthogonal 2-design ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
58 views

Understanding Shor algorithm fo Elliptic Curves Demonstration

I was reading Shor's discrete logarithm quantum algorithm for elliptic curves. And i have two questions. In page 7 they say that $x = (x0 - dy) mod q$, where $x0$ is between 0 and q-1, but then they ...
Philip.q.c's user avatar
3 votes
1 answer
345 views

Is there an expression for the partial trace of a vectorized density matrix?

Is there an expression for the partial trace of vectorized density matrix? I did some literature review but didn't find not much relevant information.
Will Yang's user avatar
  • 187
2 votes
0 answers
74 views

Simple Maths Operators implementation in Quantum

I am a newbie in quantum programming and trying to learn implementing some simple classical programs in quantum, just as a starting point. The thing I am kind of struggling in is availability of ...
aneela's user avatar
  • 233
13 votes
2 answers
620 views

What are the possible non-entangling two-qubit gates?

The non-entangling gates in $ SU_4 $ contains the entire group of gates of the form $$ SU_2 \otimes SU_2. $$ It also contains $$ \zeta_8 SWAP= \zeta_8 \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 &...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
187 views

Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)

I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define: ...
silgon's user avatar
  • 167
1 vote
0 answers
173 views

How to implement an unitary operator expressed as a linear combination of unitaries without qubits ancilla

Let's say that I know the decomposition of a unitary operator $\hat{A}$ in terms of other unitary operators $U_{k=0, \dots, M}$, i.e: $$ \hat{A} = \sum_k \alpha_k U_k$$ I know how to implement in ...
Andrés Ruiz's user avatar
2 votes
1 answer
210 views

Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
PGibbon's user avatar
  • 472
2 votes
1 answer
82 views

Meaning of the notation $[d]$ in scientific paper

Here in this paper (and several others), the notation [d] is used. For instance here is Lemma 9 of the paper: Let $T_d\in R[x]$ be the d-th Chebyshev polynomial of the first kind. Let $Φ\in R_d$ be ...
Jadzia's user avatar
  • 131
3 votes
2 answers
696 views

Prove the triangle inequality for the trace norm: $\|M+N\|_1\le \|M\|_1+\|N\|_1$

I have been trying to show that $$||M+N|| \le ||M|| + ||N||$$ However, I seem to be missing some fundamental property of either how the trace or square root acts on these sums of matrices, or how the ...
GaussStrife's user avatar
  • 1,115
1 vote
1 answer
99 views

Do the linear operators $M\otimes I$ and $I\otimes N$ commute?

If not, does that mean that when doing partial measurements on two different shares of an entangled state, the results (expressed as a proability mass function) can depend on the order (i.e who ...
eternalstudent's user avatar
2 votes
1 answer
244 views

What is the tensor product expression for the following quantum circuit? [duplicate]

Qiskit generates the following matrix for this 3-qubit CNOT circuit. Can anyone explain how do we get this mathematically ? This is the Quantum Circuit This is the Output of Unitary Simulator
Adityashu's user avatar
2 votes
1 answer
88 views

Confusion about Rodeo algorithm "spectral weight suppression" argument

In this first paper on the Rodeo algorithm, there is an argument on the second page about the suppression of "spectral weights" that I don't really understand. In short, the algorithm is ...
tomdodd4598's user avatar
0 votes
1 answer
125 views

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 ...
PsOom's user avatar
  • 3
0 votes
0 answers
58 views

Usefulness of Heisenberg Uncertainty Principle

$ \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} $The Heisenberg Uncertainty principle as formulated in Nielsen and Chuang is $$ \Delta (C) \Delta (D) \geq \...
Techmaster21's user avatar
0 votes
1 answer
356 views

Question about proof that non-orthogonal states can't be reliably distinguished in QCQI

$$ \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} $$ The beginning portion of Box 2.3 on page 87 of QCQI is as follows: "Suppose such a measurement is ...
Techmaster21's user avatar
3 votes
2 answers
371 views

Generators for single qudit Clifford, d=4

The generators for single qubit Clifford are phase $ P $ and Hadamard $ H $. The generators for single qutrit Clifford can be found for example here What is the set of generators for the qutrit ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
58 views

Is there a general parametric transformation matrix form in bloch-space corresponding to the unitary operations on qutrits?

I've been looking into the structure of the Bloch sphere for qudits, and I am wondering if there is a transformation matrix (or rotation matrix) formula corresponding to high-dimensional quantum ...
Waing's user avatar
  • 9
4 votes
0 answers
49 views

Specific relation between the classical Fourier transform for finite abelian groups and the QFT for finite abelian groups

$\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|}$ I am a math undergrad (with admittedly minimal background in quantum ...
Kenanski Bowspleefi's user avatar
1 vote
1 answer
211 views

Do unitary matrices acting on entangled states always give a quantum state?

I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations: $$U_1 = \frac{1}{\sqrt{2}} \...
Counterband's user avatar
1 vote
2 answers
231 views

Can I learn quantum computing without math?

I think this is a bit confusing question, I'm really interested in learning quantum computing, I've been learning the basics for a couple of months now, and I've also started developing some simple ...
Zakaria Halloumi's user avatar

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