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.
480
questions
2
votes
1
answer
37
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 ...
- 2,114
0
votes
1
answer
26
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
1
vote
0
answers
48
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 ...
- 2,114
2
votes
2
answers
52
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
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,114
2
votes
1
answer
33
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
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,114
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
44
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 ...
- 2,114
0
votes
1
answer
27
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
97
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 ...
- 107
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 ...
- 15
2
votes
1
answer
36
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$-...
- 107
0
votes
2
answers
46
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
33
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 ...
- 413
2
votes
1
answer
60
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 ...
- 2,114
4
votes
1
answer
93
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 & ...
- 2,114
2
votes
1
answer
68
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 ...
- 2,114
4
votes
1
answer
449
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 ...
- 2,114
3
votes
1
answer
101
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
$...
- 2,114
2
votes
1
answer
54
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 ...
- 2,224
4
votes
2
answers
89
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 ...
- 2,114
0
votes
1
answer
64
views
How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
- 790
1
vote
1
answer
205
views
Matrix representation of any conditioned gate
Is there an algorithm explaining how to represent any gate in the matrix form?
Suppose, the circuit is the following:
where operator
$ U = e^{iA\pi/4} =
\begin{bmatrix}
0.35-0.85i & -0.35-0.15i ...
3
votes
1
answer
50
views
Clarification defining/finding the relative phase of a qubit
Let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$.
According to p. 22 of ...
- 133
1
vote
0
answers
45
views
Close in operator norm imply close in weak multiplicative sense?
Fix $\epsilon > 0$, and suppose $U$ and $S$ are $n$ qubit unitaries such that $\| U - S \| \leq \epsilon$ (operator norm). Furthermore, let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ be the ...
- 107
2
votes
1
answer
84
views
Notation for Lindblad operators
I was reading the paper Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation
. The claim is that universal quantum computation can be achieved using the ...
- 2,033
2
votes
1
answer
194
views
How to calculate the log of a density matrix?
In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
- 149
2
votes
0
answers
52
views
How many gates are in the $ k $ level of single qubit Clifford hierarchy?
Define the single qubit Clifford hierarchy recursively by
$$
\mathcal{C}^1:=<iX,iZ>
$$
the determinaint 1 subgroup of the Pauli group. Define the rest of the the hierarchy inductively by
$$
\...
- 2,114
3
votes
1
answer
50
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 ...
- 2,114
6
votes
1
answer
126
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) ...
- 2,114
3
votes
3
answers
324
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
0
votes
1
answer
99
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
13
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
35
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
2
votes
1
answer
154
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 ...
- 2,114
6
votes
1
answer
131
views
Is this single qubit gate in the Clifford hierarchy?
For a single qubit, the Clifford hierarchy is defined to be
$$
\mathcal C^{(k)} = \Bigl\{ U \in \mathbf U(2)
\mathrel{\Big\vert} \forall P \in \mathcal C^{(1)} : U P U^\dagger
\in \mathcal C^{(...
- 2,114
2
votes
0
answers
32
views
Dimension of local operators stabilizing the code space?
What is the maximum dimension of a connected group of local operators stabilizing an $ [[n,k,d]] $ code with $ d \geq 2 $?
Some background:
Consider an $ [[n,k,d]] $ quantum error correcting code with ...
- 2,114
3
votes
1
answer
117
views
Which monomial matrices are in the Clifford hierarchy?
This is essentially a follow-up on the very interesting answer given here
Is there a closure property for the entire Clifford hierarchy?
I'm interested in sufficient conditions to conclude that a ...
- 2,114
1
vote
2
answers
82
views
How to perform a basis change on a 2x2 density operator?
I have an ensemble described by following density operator:
$$
P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-|
$$
I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis.
I know that ...
- 37
4
votes
2
answers
275
views
Exotic transversal gate group
What are examples of interesting $ [[n,1,d]] $ or $ [[n,2,d]] $ stabilizer codes, $ d \geq 2 $, whose group of transversal gates is not isomorphic to a subgroup of the Clifford group (on 1 and 2 ...
- 2,114
3
votes
1
answer
261
views
Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$
As mentioned e.g. in this answer, if we compute the average
$$\Phi(X)\equiv \mathbb{E}_U[UXU^\dagger] \equiv \int_{\mathbf U(d)} d\mu(U) UXU^\dagger,$$
where $d\mu(U)$ is the Haar measure over the ...
glS♦
- 21.2k
5
votes
1
answer
142
views
What are well-known orthogonal 2-designs, other than the real Clifford group?
The paper Real Randomized Benchmarking
https://quantum-journal.org/papers/q-2018-08-22-85/
https://arxiv.org/abs/1801.06121
makes use of the fact that the real Clifford group is an orthogonal 2-design ...
- 2,114
0
votes
0
answers
62
views
How to represent the following regular quantum circuits with tensor and concatenation symbols?
I am fascinated by such a quantum structure as above, how should the regular distribution for Toffoli and SWAP gates be described by the formula? Is the following correct?
$\prod\limits_{n}{{{I}_{{{2}^...
- 390
0
votes
0
answers
45
views
Understanding Shor algorithm fo Elliptic Curves Demonstration
I was reading Shor's discrete logarithm quantum algorithm for elliptic curves. And i have two questions.
In page 7 they say that $x = (x0 - dy) mod q$, where $x0$ is between 0 and q-1, but then they ...
2
votes
1
answer
82
views
Is there an expression for the partial trace of a vectorized density matrix?
Is there an expression for the partial trace of vectorized density matrix? I did some literature review but didn't find not much relevant information.
- 155
1
vote
0
answers
50
views
Simple Maths Operators implementation in Quantum
I am a newbie in quantum programming and trying to learn implementing some simple classical programs in quantum, just as a starting point. The thing I am kind of struggling in is availability of ...
- 191
0
votes
0
answers
49
views
Equivalence check between rotational gates and Pauli gates
My question is highly related to this one.
I am trying to understand the relationship between rotational gates $R_P(\theta)$, where $P \in \{X,Y,Z\}$.
As stated here, $\exp(iPx)=\cos(x)I+i\sin(x)P$. ...
- 1,136
9
votes
2
answers
334
views
Non-entangling two-qubit gates
The non-entangling gates in $ SU_4 $ contains the entire group of gates of the form
$$
SU_2 \otimes SU_2.
$$
It also contains
$$
\zeta_8 SWAP= \zeta_8 \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &...
- 2,114
2
votes
1
answer
151
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:
...
- 167
2
votes
0
answers
143
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 ...
- 174
2
votes
1
answer
89
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 ...
- 440