Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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Is there a general parametric transformation matrix form in bloch-space corresponding to the unitary operations on qutrits?

I've been looking into the structure of the Bloch sphere for qudits, and I am wondering if there is a transformation matrix (or rotation matrix) formula corresponding to high-dimensional quantum ...
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Specific relation between the classical Fourier transform for finite abelian groups and the QFT for finite abelian groups

$\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|}$ I am a math undergrad (with admittedly minimal background in quantum ...
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Do unitary matrices acting on entangled states always give a quantum state?

I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations: $$U_1 = \frac{1}{\sqrt{2}} \...
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2 answers
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Can I learn quantum computing without math?

I think this is a bit confusing question, I'm really interested in learning quantum computing, I've been learning the basics for a couple of months now, and I've also started developing some simple ...
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11 votes
1 answer
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Is there a closure property for the entire Clifford hierarchy?

TL;DR Is the entire Clifford hierarchy (as opposed to any one level), a group? Background. The Clifford hierarchy (on $n$ qubits), is a collection of nested subsets $\mathcal C^{(1)} \subset \mathcal ...
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Why does the condition $\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$ called parallel transport condition?

I'm reading this paper about nonadiabatic holonomic quantum computation and met the condition eq.(2) states that $\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$ is the ...
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Is the Pauli group isomorphic to the Heisenberg group over a finite field?

Let $ p $ be prime and let $ P_n(p) $ denote the Pauli group on $ n $ qudits each of size $ p $. Then $ P_n(p) $ and $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ are both extraspecial $ p $ groups of order $ ...
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Convexity of cost function in quantum machine learning

I have a small confusion on convexity of the most typical cost function considered in many qml papers: $\text{Tr}(\rho O)$, where $O$ is a Hermitian operator and $\rho$ is a quantum state. This is ...
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math behind rotation gate

$R_Z(\theta) = e^{-i\frac{\theta}{2}Z}$ and $R_{ZX}(\theta) = e^{-i\frac{\theta}{2}XZ}$ My question is, why is the rotation gate defined in an exponential format? Where does the $\frac{1}{2}$ come ...
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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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What is a good relation between a matrix and its matrix exponential, for non-self-inverse matrices?

This is inspired by comments on another recent question on a matrix for the fourth-root of $X$. That question links to a paper of Muradian and Frias, who provide a number of nice equations and ...
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Is the Clifford group perfect (equals its own commutator subgroup)?

Let $ Cl_n $ be the Clifford group on $n$ qubits. What is the commutator subgroup of $ Cl_n $? It is definitely not all of $ Cl_n $ since $ Cl_n $ is not perfect. My guess is that the abelianization ...
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Investigating the scaling of the error of a Trotter-Suzuki-approximation

I am doing an assignment and I am being asked to investigate the scaling of the error with the number of repetions $n$ of a approximation of the Hadamard with $R_x$ and $R_y$. This is the ...
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What are the eigenstates of an operator?

Sorry if this is a silly question, I am new to quantum computing I was just reading this article that talked about the eigenstates of an operator. And I wonder, how can we find those eigenstates for a ...
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Given a non-Clifford quantum circuit $U$, is it possible to construct a commuting Clifford circuit $C$?

Given a non-Clifford circuit $U$, say $U = \prod_{i=1}^k e^{i \theta_i P_i} $ for $P_i \in \{ I,X,Y,Z\}^{\otimes n} $ and $\theta_i \in \mathbb{R}$. Is it possible to construct a non-trivial Clifford ...
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Comparing Hilbert spaces of coupled and uncoupled qubits

Imagine two situations. In one, there are two qubits that are next to each other, that is, they have non-zero coupling terms in their Hamiltonian, and thus suffer from cross-talk and energy can leak ...
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Verification of local unitary equivalence between two pure states

This might be a non-trivial and hard problem. I've been thinking about this for days but couldn't find a good answer, so I hope any of you could give me a good answer/intuition for me to move forward. ...
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2 answers
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Tapering off qubits

Suppose you have a Hamiltonian of the form $$ H = ZXXX + YXXX + XXXX $$ where $Z,X,Y$ are the usual Pauli matrices with $ZXXX = Z \otimes X \otimes X \otimes X$ and similar for the other two terms. ...
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How to mathematically determine the final state of a system after some Gaussian microwave pulse?

I'm trying to figure out the final state of a system after some arbitrary microwave pulse (let's say some physical Gaussian pulse with a duration $d$, amp $a$ and width $\sigma$). (I’m looking for how ...
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5 votes
1 answer
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Construction of unitary matrices built from linear combination of Pauli strings

Let's define $P_k \in \{ I, X, Y, Z \}^{\otimes n}$ and called each of these $P_k$ as a Pauli string (or word) then given that $$U = \sum_{k=1}^L c_kP_k $$ with the following conditions: $\sum_{k=1}...
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Verify that $\langle \sigma^x\rangle^2+\langle\sigma^y\rangle^2+\langle\sigma^z\rangle^2=1$ for $|\psi\rangle=\cos\theta|0\rangle+\sin\theta|1\rangle$

I am trying to solve an exercise, but I can't seem to get it to work. I get given this rule, $$\langle \sigma_x \rangle^2 + \langle \sigma_y \rangle^2+\langle \sigma_z \rangle^2 =1 $$ and I am asked ...
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How to represent the Hadamard gate as a rotations on the Bloch sphere?

I am new to Quantum Computing, and I have decided to try and learn the quantum gates. I am trying to understand how to represent some basic gates as rotations on the Bloch Sphere. I was able to ...
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2 votes
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385 views

Why is the angle of Rz(π/4) Rx(π/4) an irrational multiple of π

It is stated in the Qiskit tutorial section 2.4 that if you apply a rotation around the z-axis of π/4 and subsequently a rotation around the x-axis of π/4, the end result is an angle around some axis ...
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6 votes
1 answer
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How to show that the integral over all Haar states vanishes: $\int|\psi\rangle\,{\rm d}\psi = 0 $?

Can we show that the integral over all Haar states $|\psi \rangle $ is $$ \int |\psi \rangle \, \mathrm{d}\psi = 0~. $$ This is an integral over Haar vectors Reference to a post about what is Haar ...
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Sampling Haar over two systems

Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$. With $|\psi\rangle$ sampled from Haar, how can we show that $$ \int (...
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Can the Bloch sphere representation be applied to many-qubit states with an iterative approach?

By ignoring the global phase, we can represent a single qubit state as \begin{equation} |\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle \end{equation} which very much looks like ...
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2 votes
0 answers
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Equality condition on Holder's inequality for matrix for infinity norm

The equality condition for Holder's inequality, $\text{Tr}A^*B \leq ||A||_p||B||_q $ is $|A|^p = \lambda |B|^q$ for scaler $\lambda > 0$. What happens when $p$ or $q$ is $\infty$? I found out that ...
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4 votes
2 answers
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If two unitary operators commute, do their roots also commute?

This is probably a pretty basic linear algebra question, but suppose we have two unitary operators $A$ and $B$, acting on the same $n$ qubits of $|\psi\rangle$, with $[A,B]=0$ - that is, $A$ and $B$ ...
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5 votes
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maximization of trace between two operators with respect to different norm constraints

I want to maximize $\text{Tr}(XY)$ over $X$ for fixed $Y$, where $X$ and $Y$ are both hermitian (but doesn't necessarily positive) operators, and $X$ is constrained by its p-norm bounded by $1$, i.e. $...
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In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?

My question: Is the center of $ \overline{\text{Cl}_n} $ trivial? Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group must ...
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4 votes
1 answer
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Showing that two unitary matrices are equal up to a global phase

Let $U$ and $V$ be two $d × d$ unitary matrices, representing two reversible quantum processes on a $d$-dimensional quantum system. We say that the two processes “act in the same way” on the state $|ψ\...
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2 votes
2 answers
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How to prove that the trace of n-qubit matrices satisfies ${\rm Tr}(XY)=2^n\sum_{M\in\{I,X,Y,Z\}^n} x_M y_M$?

It is known that for n-qubit matrices X, Y $\in \mathbb{C}^{2^{n}\times 2^{n}}$ (and Pauli matrices $I, X, Y, Z$) such that $$ X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n} $...
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How to express $n$-qubit Hermitian operator with Pauli matrices

How can we prove that all $n$-qubit Hermitian matrices can be written in terms of Pauli matrices $I$, $X$, $Y$, and $Z$ as $$ \sum_{W_k \in \{I, X, Y, Z\}} a_{W_1,\dots,W_n}W_{1}\otimes ... \otimes W_{...
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Does the 4x4 matrix $|00\rangle\!\langle00|+|11\rangle\!\langle11|$ have a decomposition?

Can the diagonal matrix $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0& 0 \\0&0&0&0 \\ 0&0&0&1 \end{pmatrix}$$ be written as a tensor product $A\otimes B$...
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What does $ A - \langle A \rangle $ mean?

I've seen the uncertainty of $A$ written as $$ (\Delta A)^2 = \langle (A - \langle A \rangle)^2 \rangle. $$ But what does this even mean since $ A $ is an operator and $ \langle A \rangle $ is a ...
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1 vote
3 answers
112 views

How to show that the QFT satisfies $\frac1{\sqrt N}\sum_j\prod_le^{2\pi i j_l k/2^l}|j_1...j_n⟩=\bigotimes_l \frac1{\sqrt2}(|0⟩+e^{2\pi i k/2^l}|1⟩)$?

I'm reading Ronald de Wolf's lecture notes, and in chapter 4.5 he writes that $$ \frac{1}{\sqrt N}\sum\limits_{j=0}^{N-1}\prod\limits_{l=1}^{n}e^{2\pi i j_l k / 2^l}|j_1...j_n\rangle = \bigotimes\...
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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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2 votes
1 answer
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how to obtain partial transpose of a Tripartite operator?

i know for a bipartite system with elements |ij><kl| elements of its partial transpose are |kj><il| now suppose a ...
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Expectation value of an observable containing a single projector vs Born rule for the projector

Suppose I have a state $|\psi\rangle$ and I want to estimate the probability of obtaining a computational basis state $|x\rangle$. Then by Born rule: $$ p(x) = |\langle x|\psi\rangle|^2 = Tr[|x\rangle ...
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3 answers
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Why do we need Hilbert spaces when talking about qubits and quantum computation?

I was just curious to know why do we need Hilbert Spaces when talking about the qubits and quantum computation in general. I mean why can't we just work with inner product spaces, rather than going ...
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3 votes
3 answers
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Are complex amplitudes really needed?

Qubit amplitudes are defined as complex numbers. But in all tutorials I have recently read, only real numbers are used and everything works. So, if I completely forget the official 'complex' ...
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1 vote
1 answer
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Two-qubit Bell measurement matrix where the two qubits are not contiguouis

In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
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6 votes
1 answer
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Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
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4 votes
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How to decompose Bloch sphere rotations $e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ in terms of $R_x,R_y,R_z$?

I learned a formula to represent the rotation around bloch sphere: $\theta_{\phi} = e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ So that $\pi_0$ is the gate $X$ and $\pi_{\frac{\pi}{...
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3 votes
1 answer
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How to compute derivatives of partial traces of the form $\frac{\partial \operatorname{Tr}_B(F(\mathbf{X}))}{\partial \mathbf{X}}$?

The Matrix Cookbook says that for any differentiable matrix function $F(\cdot)$, it holds that $$\frac{\partial \operatorname{Tr}(F(\mathbf{X}))}{\partial \mathbf{X}}=f(\mathbf{X})^{T},$$ where $f(\...
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2 votes
1 answer
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How to find the eigenstates of a general $2\times 2$ Hermitian matrix?

Given a measurement operator in the general Hemitian form $$ M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix}, $$ where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are $$ ...
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2 votes
1 answer
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Why does $H^2=X^2 =I$ not imply $H=X$?

if $HH = I$ and $XX =I$, then is $H=X$? $HH = I = XX$ or, $HH = XX$ then, taking under root, is $H = X$? This is absurd but how to disprove it?
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6 votes
1 answer
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
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3 votes
1 answer
88 views

Why are orthogonal spins $(1,0)$ and $(0,1)$ represented as collinear vectors in the Bloch sphere?

I'm reading the book "Quantum Computing 4 real IT people" by Chris Bernhardt and I have a question about the following phrase in chapter 3 which says that An ordered orthonormal basis ...
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4 votes
1 answer
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How to exactly implement Trotter-Suzuki formula on quantum computer

Recently, I am studying some topics related to product formula, and I am curious about how to implement such formula on real quantum devices. The $(2k)$-th order product formula can be witten as \...
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