# Questions tagged [mathematics]

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### Equality condition for Hölder's inequality if $p=1$

The equality condition for Hölder's inequality, $|{\rm tr}(A^\dagger B)| \leq \| A\| _p\| B\| _q$ is $|A|^p = \lambda |B|^q$ for some scalar $\lambda > 0$; here $\|\cdot\|_p$ the usual Schatten ...
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### Identity for linear codes and their duals: why do we have $\sum_y (-1)^{x\cdot y}=|C|\delta_{x\in C^\perp}$?

I've come across this exercise plenty of times and I still don't understand how to do it. (Here it is from N.C. Ex.10.25) Let $C$ be a linear code (Lets suppose its a binary code, i.e. a $k$-...
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### Maximally entangled state definition, and orthonormal basis of maximally entangled state

My questions are probably more about details but the answer will help me to precisely understand the mathematical structure. My questions are related to page 14 of this pdf. Mathematical context: ...
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### Dimension of local operators stabilizing the code space?

What is the maximum dimension of a connected group of local operators stabilizing an $[[n,k,d]]$ code with $d \geq 2$? Some background: Consider an $[[n,k,d]]$ quantum error correcting code with ...
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### Understanding a quantum algorithm to estimate inner products

While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation. Consider ...
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### Map a 4-body Ising Hamiltonian to a 2-body Ising Hamiltonian

I wonder if there exists a way to map the square of a 2-body Ising Hamtiltonian (which will make it 4-body) back to a 2-body Hamiltonian that has the same ground state? Let me explain what I mean by ...
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### How to understand the Haar measure from a quantum information perspective?

I found it a little difficult to understand it using Wikipedia and some mathematical documents. How to understand the Haar measure from a quantum information theory perspective? Are there any ...
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### Realizing a swap gate using a commutator sequence and an auxiliary qudit

Say I have two qudits $1$ and $2$, each of which has Hilbert space of dimension $m$. Is it possible to introduce an auxiliary qudit $a$ (of any dimension $d_a\in \mathbb{Z}_{\geq 2}$) and find quantum ...
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### The Clifford hierarchy and $e^{2 \pi i/2^k}$

Could someone give me an example of a gate in the Clifford hierarchy which cannot be written as $$e^{i \theta} V$$ for some unitary $V$ with entries in terms of $\zeta_{2^k}$? If no such example ...
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### Which monomial matrices are in the Clifford hierarchy?

This is essentially a follow-up on the very interesting answer given here Is there a closure property for the entire Clifford hierarchy? I'm interested in sufficient conditions to conclude that a ...
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### Is geometric algebra/calculus used in quantum computing?

This is really a question out of curiosity. I am aware that geometric algebra and geometric calculus provide simplifications in many aspects of physics. I'm wondering if this framework's usefulness ...
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### Is the Clifford hierarchy finite?

This question is inspired by Is the Clifford group finite? Which shows that that the Clifford group (the second level of the Clifford hierarchy) is finite. (finite meaning finite mod global phases) ...
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### Translating classical math and code to quantum math and code

I am starting to see a lot of classical quantitative problems such as linear regression being represented in quantum math, which suggests that almost anything based on frequentist statistics could be ...
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### How to perform a basis change on a 2x2 density operator?

I have an ensemble described by following density operator: $$P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-|$$ I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis. I know that ...
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### Minimum number of 2 qubit gates to build any unitary

Any unitary $U$ acting on $N$ qubits can be decomposed in a finite product $U=U_1U_2...U_n$ where every $U_i$ acts on only 2 qubits, for example through decomposition in CNOT, phase shifts and 1 qubit ...
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### What are well-known orthogonal 2-designs, other than the real Clifford group?

The paper Real Randomized Benchmarking https://quantum-journal.org/papers/q-2018-08-22-85/ https://arxiv.org/abs/1801.06121 makes use of the fact that the real Clifford group is an orthogonal 2-design ...
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### Prove that the partial trace is a quantum operation, finding its Kraus representation

I am referring to Nielsen and Chuang Quantum Computation and Quantum Information 10th Anniversary Edition Textbook, Chapter 8.3. A linear operator $E_i:H_{QR}\longrightarrow H_Q$ is defined by: ...
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### Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
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### Is the Clifford group superperfect?

This is a follow up to Is the Clifford group perfect (equals its own commutator subgroup)? Motivation: Since global phase is unphysical in quantum mechanics we often consider projective ...
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### Understanding Shor algorithm fo Elliptic Curves Demonstration

I was reading Shor's discrete logarithm quantum algorithm for elliptic curves. And i have two questions. In page 7 they say that $x = (x0 - dy) mod q$, where $x0$ is between 0 and q-1, but then they ...