Questions tagged [mathematics]
DO NOT use this tag. Use more specific tags such as [linear-algebra] instead.
507
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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 ...
- 331
2
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1
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287
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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 ...
- 65
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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 ...
- 33
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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 \...
- 169
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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 ...
- 93
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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\...
- 169
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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 ...
- 167
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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 ...
- 499
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1
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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 \...
- 4,500
3
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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
0
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1
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41
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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 ...
- 335
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3
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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 ...
- 1,853
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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 ...
- 146
4
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83
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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 ...
- 559
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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 ...
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