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# Questions tagged [mathematics]

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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 $Tr(L^\dagger L\rho)-Tr(L^\dagger\rho L\rho)$?
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### 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 ...
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### 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 $\epsilon$-...
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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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: \text{Tr}(O\rho) \leq \...
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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 ...
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### 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>?
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### 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 ...
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### What are the expected measurement results in the diagram below? [closed]

I ask you to give a mathematical solution to this problem
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### 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 ...
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1 vote
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### 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 ...
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