Questions tagged [mathematics]

DO NOT use this tag. Use more specific tags such as [linear-algebra] instead.

Filter by
Sorted by
Tagged with
5 votes
2 answers
84 views

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 ...
junghyunHa's user avatar
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: $\...
Mohan's user avatar
  • 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 ...
Hailey Han's user avatar
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 ...
Song of Physics's user avatar
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_{...
Esam El-khouly's user avatar
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 ...
Jackson Walters's user avatar
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 ...
Việt Nguyễn's user avatar
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 ...
muru's user avatar
  • 33
2 votes
1 answer
69 views

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 \...
Mohan's user avatar
  • 43
3 votes
1 answer
69 views

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 ...
Peeveey's user avatar
  • 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\...
Hailey Han's user avatar
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 ...
Zehong Fan's user avatar
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 ...
trillianhaze's user avatar
4 votes
1 answer
151 views

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 \...
Ian Gershon Teixeira's user avatar
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 ...
Grwlf's user avatar
  • 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 ...
userflux9674's user avatar
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 ...
Abdullah Khalid's user avatar
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 ...
Alan Whitteaker's user avatar
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 ...
SescoMath's user avatar
  • 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 ...
epelaez's user avatar
  • 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 ...
J.Doe's user avatar
  • 113
0 votes
1 answer
118 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
  • 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.
reza's user avatar
  • 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 ...
3nondatur's user avatar
  • 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 ...
trillianhaze's user avatar
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 ...
Kochan's user avatar
  • 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 ...
Ian Gershon Teixeira's user avatar
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_{...
Duen's user avatar
  • 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 \...
Jon Megan's user avatar
  • 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>?
Flaplap's user avatar
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 ...
Creative's user avatar
-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
Creative's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
C. Kang's user avatar
  • 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}^{(...
Ian Gershon Teixeira's user avatar
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 $$ \...
Ian Gershon Teixeira's user avatar
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 ?
Duen's user avatar
  • 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 ...
Ian Gershon Teixeira's user avatar
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\...
Việt Nguyễn's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
Ian Gershon Teixeira's user avatar
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 $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
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
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 ...
Ian Gershon Teixeira's user avatar
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 ...
IsuzuGawa's user avatar
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 ...
trillianhaze's user avatar
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 ...
QuantumAudit's user avatar
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 $n$-...
trillianhaze's user avatar
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 ...
user_confused's user avatar
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 ...
Jon Megan's user avatar
  • 465

1
2 3 4 5
11