Questions tagged [mathematics]
DO NOT use this tag. Use more specific tags such as [linear-algebra] instead.
513
questions
5
votes
2
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84
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Decomposition of a $4 \times 4$ unitary matrix
I am currently studying the paper "Decomposition of unitary matrices and quantum gates (2012)" and referring to the textbook Quantum Computation and Quantum Information. Among the topics, I ...
- 53
2
votes
1
answer
84
views
$\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$
Is there a difference between the following two Hilbert spaces: $H_1 = \mathbb{C}^2 \otimes \mathbb{C}^2$ and $H_2 = \mathbb{C}^4$?
Here's my confusion. For the following bases, $H_1 = H_2$ holds: $\...
- 43
4
votes
2
answers
277
views
closeness between two unitaries on the bloch sphere
The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this ...
- 127
0
votes
1
answer
63
views
What are necessary and sufficient conditions for the output of a parametrized unitary $U(\theta)$ to be smooth?
Let us consider a unitary $U$ parameterised by $\theta \in \mathbb{R}$, i.e, $U(\theta)$. What are the necessary and sufficient conditions for the output states of this unitary to be smooth? One ...
- 168
2
votes
1
answer
35
views
Math Behind $X$ Gate With Arbitrary Phase is equivalent to $ZXZ$ Gate
An X gate where there is a phase shift $\phi$ to the applied sinusoidal wave $U = e^{-i\frac{\theta}{2}(cos(\phi)\sigma_x+sin(\phi)\sigma_y)}$ is equivalent to a series of gates $Z_{-\phi}X_{\theta}Z_{...
1
vote
0
answers
76
views
Representing networks with qubits as edges
I am looking to take a classical non-negative real valued network and generalize it to the quantum case for processing. A network is given by an adjacency matrix, essentially edge weights $e_{ij}$ for ...
- 279
2
votes
1
answer
172
views
How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?
I read this blog https://pennylane.ai/qml/demos/tutorial_haar_measure#deguise2018 regarding a basic introduction to haar measure. In the "show me more math" section, they said $SU(3)$ can be ...
- 35
3
votes
1
answer
46
views
Simulating Sparse Hamiltonians: help understanding query complexity bounds
tl;dr: How can I show that $e^k/k^k$ is less than $\epsilon^2/2$ when $k=\Omega\left(\frac{\log(1/\epsilon)}{\log \log(1/\epsilon)}\right)$, where $k,\epsilon\in \mathbb{R}$ and > 0?
Context:
Berry ...
- 33
2
votes
1
answer
69
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Given an observable $O$, what's the achievable maximum value of $\operatorname{Tr}(O\rho)$?
The maximum value of expectation value of an observable $O$ with respect to a density matrix $\rho$ can be computed by using Holder's inequality as follows:
\begin{equation}
\text{Tr}(O\rho) \leq \...
- 43
3
votes
1
answer
69
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Are quantum channels bounded linear maps?
I've been reading about quantum channels from a couple of sources and have some doubts regarding some mathematical perspectives and properties of quantum channels. I've listed them below:
It is known ...
- 73
5
votes
1
answer
62
views
Existence of Hamiltonians such that the time evolution unitary becomes identity
Can we always find a set of coefficients ${k_i}$ (where not every $k_i = 0$) for a given Hamiltonian $H = \sum k_i H_i$, such that the unitary operator becomes the identity operation: $e^{-iH} = e^{i\...
- 127
3
votes
0
answers
22
views
Question when deriving quantum differential privacy?
I met some problems when trying to derive proposition 4 in the paper Gentle measurement of quantum states and differential privacy.
I know that intuitively, if we act on a single register of ρ, and ...
- 141
0
votes
0
answers
33
views
Saturating an inequality relating the operator norm and the total variation distance
Let $U$ be an $n$-qubit unitary, and let $p_U(x) = |\langle x | U | 0\rangle |^2$ be the probability of obtaining $x \in \{0,1\}^n$ on the all zero input. Given two $n$-qubit unitaries $U$ and $V$, it ...
- 429
4
votes
1
answer
151
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Can every unitary be approximated by gates from the Clifford Hierarchy?
For $k > 1$, we recursively define $\mathcal C^{(k)}(n)$ as
$$
\mathcal C^{(k)}(n) = \Bigl\{ U \in \mathbf U(2^n)
\mathrel{\Big\vert} \forall P \in \mathcal C^{(1)}(n) : U P U^\dagger
\in \...
- 2,920
3
votes
1
answer
35
views
On unitary matrix form suggested in the Elementary gates paper
In the Elementary gates for quantum computation paper by Barenco et al authors start their proofs by defining a generic form of 2x2 unitary matrix of $\mathbb{C}$ as follows:
Can you help me with the ...
- 133
1
vote
1
answer
35
views
Derivative of cost function with respect to the unitary matrix
Suppose I have a cost function $C = \langle \psi \rvert U^\dagger O U \rvert \psi \rangle$ for a fixed observable $O$ and a fixed state $\rvert \psi \rangle$. I know that usually people take the ...
- 337
2
votes
3
answers
63
views
References for homology, suitable as background for quantum codes
Quantum codes are often related to the concepts in homology, such as chain complexes.
Is there an introduction to homology suitable for building a strong understanding of these results?
I am looking ...
- 1,715
1
vote
1
answer
44
views
Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
- 147
4
votes
1
answer
64
views
Definition of quantum junta is not basis independent: isn't this a problem?
A quantum $k$-junta is defined as a unitary matrix $U$ acting on $n$ qubits which has the form $U = V \otimes \mathbb I$ where $V$ is a unitary acting some $k < n$ of the qubits. The fact that a ...
- 485
1
vote
0
answers
89
views
How is the definition of $n$-qubit Pauli group derived?
The authors give the following definition for the Pauli group in the paper Averaged circuit eigenvalue sampling.
The n-qubit Pauli group $P_n$ consists of n-fold tensor products of single-qubit Pauli ...
- 2,865
1
vote
1
answer
38
views
A conceptual Query regarding measurement during a Quantum Algorithm
I am new to Quantum Computing and my original background is in Computer Science thus this possible trivial query.
Case 1: Given a set of $N$ Q-bit System in some superposition state $I_0$. Let us ...
- 113
0
votes
1
answer
118
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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 \...
- 360
2
votes
2
answers
128
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.
- 461
1
vote
1
answer
59
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 ...
- 173
1
vote
0
answers
43
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
...
- 429
1
vote
0
answers
28
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 ...
- 31
1
vote
1
answer
62
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 ...
- 2,920
6
votes
1
answer
384
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_{...
- 296
4
votes
2
answers
168
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 \...
- 465
-1
votes
1
answer
36
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>?
- 1
0
votes
3
answers
136
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 ...
- 5
-3
votes
1
answer
31
views
What are the expected measurement results in the diagram below? [closed]
I ask you to give a mathematical solution to this problem
- 5
3
votes
1
answer
156
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 $. A less common gate ...
- 2,920
0
votes
0
answers
20
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 ...
- 1,684
2
votes
1
answer
60
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}^{(...
- 2,920
1
vote
2
answers
141
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
$$
\...
- 2,920
4
votes
1
answer
353
views
Link between quantum computing and Lie theory?
I know only little thing about Lie theory but I would like to learn more about its link to quantum computing.
Has someone got some references explaining it well ?
- 296
4
votes
2
answers
227
views
Does every code have transversal Pauli group?
A transversal logical gate for an $ n $ qubit code is a gate from the group of local unitaries
$$
\bigotimes_{i=1}^n U(2)
$$
which also preserves the codespace. For an $ ((n,K,d)) $ code we say a ...
- 2,920
0
votes
1
answer
37
views
Find min of a quantum state L2 norm
I have a problem minimizing this norm with respect to $\alpha$:
$\min_{\alpha}||e^{i\alpha}|\psi\rangle-|\phi\rangle ||^2$ (1)
The result is that this achieves min when $\alpha=-\measuredangle \langle\...
- 35
2
votes
1
answer
89
views
Same weight enumerator iff equivalent by permutations and local unitaries
A non-entangling gate on $ n $ qubits is an element of the group
$$
N\Big(\bigotimes_{i=1}^n U(2)\Big)=\bigotimes_{i=1}^n U(2) \rtimes S_n
$$
which is generated by $ U(2) $ acting locally on each ...
- 2,920
2
votes
2
answers
80
views
Weight enumerators for Hermitian operator
Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,920
2
votes
1
answer
37
views
Weight enumerators for Hermitian operator (wrong $ B_j $ definition)
Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients
$$
A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2
$$
...
- 2,920
10
votes
3
answers
2k
views
Why can all quantum circuits be converted into circuits that use only real matrices?
I know that you need to add an additional ancilla qubit to "keep track" of whether or not you are in real space or imaginary space, but how exactly does this work?
What is the proof for this?...
- 309
4
votes
0
answers
85
views
Eastin Knill Theorem and global phase
In quantum we don't care about global phases, but I want to ask a question about global phases anyway.
The original Eastin-Knill theorem paper https://arxiv.org/abs/0811.4262 says
$$
CP = \Pi_{i=1}^k ...
- 2,920
0
votes
1
answer
36
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
195
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 ...
- 429
0
votes
1
answer
25
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
89
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 $n$-...
- 429
1
vote
2
answers
154
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
50
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 ...
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