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What is the meaning of $Tr(L^\dagger L\rho)-Tr(L^\dagger\rho L\rho)$?

Consider the non-Hermitian matrix $L$ and the Hermitian positive semidefinite matrix $\rho\geq0$ with $Tr(\rho)=1$. What is the physical meaning of $Tr(L^\dagger L\rho)-Tr(L^\dagger\rho L\rho)$?
Kohei's user avatar
  • 1
3 votes
0 answers
38 views

Algorithm for computationally generating the single qudit clifford group

For a d dimensional single qudit, knowing the generators of the group being the Hadamard gate and the Phase gate, how would I generate the entire group computationally in python? Or for any finite ...
Son100's user avatar
  • 33
3 votes
1 answer
50 views

How to prove the inclusion relation $\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \otimes \rho[Y])$ about density operators?

For $\rho \in \mathrm{D}(\mathcal{X} \otimes \mathcal{Y})$ denoting an arbitrary state of the pair $(\mathrm{X}, \mathrm{Y})$, how to prove the fact $\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \...
Aimin Xu's user avatar
  • 129
1 vote
1 answer
98 views

If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

Let our Hilbert space be $H = (A \otimes B) \oplus (A \otimes B)^{\perp}$. If $\rho \in A \otimes B$, then we have $\text{tr}_B \rho \in A$. Is the converse true: if $\text{tr}_B \rho \in A$, then $\...
karavan's user avatar
  • 21
4 votes
1 answer
77 views

Bounds on local expectation values for two states close in trace distance

I feel like this should have been recorded somewhere but I could not find any result in the literature (except in very specific cases). Consider two states $\rho,\sigma$ such that they are $\epsilon$-...
Evangeline A. K. McDowell's user avatar
2 votes
1 answer
87 views

What is the relationship between gate fidelity and norm?

I've seen a lot of analyses on quantum circuit error bound based on the norm difference $\Vert U - V \Vert$. On the other hand, I've also seen a lot of papers that use the gate fidelity $\frac{1}{2^n}\...
user185671631's user avatar
4 votes
2 answers
421 views

What unitary commutes with all local Pauli operators?

I was thinking about this problem of identifying a set of unitary operations (other than the identity operation) that commute with local pauli $\sigma_X$ and $\sigma_Z$ matrices, i.e. find $U$ such ...
Mohan's user avatar
  • 161
1 vote
1 answer
46 views

conditions for two hermitians operators same up to unitary

Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$? The first ...
Jon Megan's user avatar
  • 497
5 votes
1 answer
35 views

Upper bound on $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$

Let $U_i$ and $V_i$ be unitaries that act on the same subsystems. Can we upper bound the difference between the tensor products of these unitaries, i.e. $\Vert U_1 \otimes U_2 \otimes \cdots \otimes ...
Mohan's user avatar
  • 161
2 votes
1 answer
75 views

Truncated Qumode States and Support

I am currently running numerical simulations of a single qumode state acted upon by a parameterised unitary. The qumode state is realised as a Fock state with a fixed cutoff dimension $(d)$ and is ...
Song of Physics's user avatar
6 votes
1 answer
153 views

Why is the orbit of a unitary t design a complex projective t design?

The paper Qubit stabilizer states are complex projective 3-designs states in the final paragraph that "any orbit of a unitary t-design is a complex projective t-design." Using this fact one ...
Ian Gershon Teixeira's user avatar
7 votes
1 answer
158 views

Which Clifford groups are 2-designs?

Let $ X $ be the $ q \times q $ shift matrix sending $ |y \rangle \mapsto |y+1 \rangle $ where the ket index $ y=0,\dots, q-1 $ is taken mod $ q $. Let $ Z $ be the diagonal $ q \times q $ clock ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
56 views

unitary that transforms one Hilbert space to another Hilbert space

Let $H = A \otimes B$. If there exists a unitary operator $U$ that transforms the Hilbert space $H$ into another Hilbert space $H' = A' \otimes B'$ (meaning that $U$ maps each basis of $H$ to each ...
Mohan's user avatar
  • 161
1 vote
1 answer
78 views

general way to decompose a CZ gate on $n$-qubit system

Suppose I have a CZ(i,j) gate (or CNOT) acting on qubit $i$ and $j$, on a $n$-qubit system. Is there a general way to decompose this gate into a set of gates that only involve single-qubit gates and ...
user185671631's user avatar
6 votes
2 answers
175 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
94 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
  • 161
4 votes
2 answers
356 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
64 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
44 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
85 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
261 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
71 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
122 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
  • 161
5 votes
2 answers
147 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
  • 93
5 votes
1 answer
75 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
25 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
1 vote
0 answers
47 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
183 views

Can every unitary be approximated by gates from the Clifford Hierarchy?

For $k > 1$, we recursively define $\mathcal C^{(r)}(n)$ as $$ \mathcal C^{(r)}(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
42 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
38 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
89 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
57 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
79 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
  • 507
1 vote
0 answers
149 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
  • 2,895
1 vote
1 answer
40 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
174 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
  • 391
2 votes
2 answers
168 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
  • 733
1 vote
1 answer
131 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
46 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
31 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
68 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
432 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
  • 426
4 votes
2 answers
230 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
  • 497
-1 votes
1 answer
43 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
273 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
34 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
176 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
1 vote
0 answers
21 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
  • 1,736
2 votes
1 answer
82 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
2 votes
2 answers
237 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

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