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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Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number

This question has been asked here: "Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number", but no answer has been provided there yet, thus I move it here. The problem is stated ...
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1answer
81 views

Quantum Principal Component analysis by Seth Lloyd

I am currently reading the paper quantum principal component analysis from Seth Lloyd's article Quantum Principal Component Analysis There is the following equation stated. I know from the ...
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1answer
98 views

Non-unitary matrix decomposition as a sum of unitary matrices

Several quantum algorithms that deals with linear algebra and matrices that are not necessarily unitary circumvent the problem of non-unitary matrices by requiring a decomposition of the non-unitary ...
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54 views

Diagrammatic Quantum Reasoning: Proving the loop equation using yanking equations

I'm trying to study the book: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning, and would like some help with Exercise 4.12: The relevant equations are as ...
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Why does quantum computing generally use Heisenberg's Matrix Mechanics instead of Schrödinger's Wave Mechanics?

Having just studied the two formulations of quantum mechanics, it makes me wonder why the decision was used to use Heisenberg's Matrix Mechanics and not Schrödinger's Wave Mechanics? I know the two ...
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147 views

Proof that most Hamiltonian evolutions are not efficiently approximable by quantum circuits

How to rigorously prove that finite Hamiltonians (for $n$-qubit systems), in general, are not efficiently$\dagger$ simulable (in the Hamiltonian simulation sense) using $\mathrm{poly}(n)$ number of ...
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62 views

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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111 views

On the probability of preparing of a uniform superposition by performing a controlled-multiplication and post-selecting $0$

I take as a starting point Watrous's celebrated paper defining the Quantum Merlin-Arthur (QMA) class. He provides a protocol for Arthur to test whether an element $h$ is not in a group $\mathcal{H}$ ...
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92 views

Interpretation of specific Hamiltonian operator

In the paper https://arxiv.org/abs/1909.05820 the authors introduce several Hamiltonians. For example they define $$ H_G = A^\dagger \left( \mathbb{I} - \vert b \rangle \langle b \vert \right) A $$ in ...
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161 views

Do states with the same purity always have the same rank?

The purity of a state $\rho$ is $\newcommand{\tr}{\operatorname{Tr}}\tr(\rho^2)$, which is known to equal $1$ iff $\rho$ is pure. Another quantity that can be used to quantify how close a state $\rho$...
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53 views

How to generate a SWAP of $N$ qubits?

I know that SWAP2 (swaps 2 qubits) gate looks like: $$ SWAP2=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{...
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1answer
54 views

How do I apply a controlled gate to specific qbits in the register?

Say, I have a specific scheme, where I need to specify inputs for controlled R logical gate, which here is $$ R(\theta)=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 &...
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153 views

How to get specific state applying $e^{-i\phi \sigma_2/2}$ to $|0\rangle$ or $|1\rangle$?

I try to solve problems from Problems in Quantum Computing. I stuck with problem #3: I do the following: Because: $$ \sigma_2 = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$$ Then: $$ -i \...
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175 views

Why are $d^2$ dimensions required to describe a density matrix?

A density matrix is defined as: $$\sum p_i |\psi_i \rangle \langle \psi_i|$$ If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
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How's quantum noise and fault-tolerance related to symplectic geometry and geometric quantization?

Gil Kalai often speaks of the apparent connection between symplectic geometry, geometric quantization, and quantum noise. He is known to describe one of his focus areas as: (...) properties and ...
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1answer
99 views

Can quantum entanglement be expressed in terms of knot theory?

While writing this answer I was wondering whether the analogy of the nature of entanglement in the GHZ state with Borromean rings is more than a mere analogy (cf. Aaronson's lecture). The question ...
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Please clarify the following orthogonal property (quantum anonymous voting)

I am a beginner at QC, currently going through a paper on Quantum Anonymous Voting. Please clarify the orthogonal property described in the following scenario. Consider $n$ voters $V_{0}, V_{1}, V_{2}...
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Prove that the state $\sum_{S\in P_n}(-)^{\tau(S)}|S\rangle$ is invariant up to a phase when changing the basis

I am trying to prove that the $|S_{n}\rangle$ is $n$-lateral rotationally invariant, where $|S_{n}\rangle$ is defined as $$|S_{n}\rangle=\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}|S\rangle\equiv\sum_{S \in ...
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146 views

Implications of commuting within the code space

The question: I have a Hilbert space $\mathcal{H}=\mathcal{H}_A\otimes \mathcal{H}_B$, and a codespace $\mathcal{H}_{code}\subset \mathcal{H}$, so that $\mathcal{H}=\mathcal{H}_{code}\oplus\mathcal{...
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1answer
22 views

Simon's Algorithm - How to simulate second Hadamard operation on first register?

I am implementing a simulation of Simon's Algorithm, and am at the part of applying the second n-qbit Hadamard transformation, $H^{\otimes n}$, to the first of the two n-qbit registers, leaving the ...
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1answer
73 views

Why does representation theory often arise in the context of quantum algorithms for the hidden subgroup problem?

I noticed that approaches for finding quantum algorithms the hidden subgroup problem for both Abelian groups ($(\Bbb Z_n\times \Bbb Z_n, +)$, $(\Bbb R, +)$, etc.) and non-Abelian finite groups like ...
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35 views

What does superposition do for quantum probabilistic sampling?

The idea of a qubit being able to exist for several values between 0 and 1 (superposition) makes it sound like it can do alot for probabilistic math problems, but for one task that comes instantly to ...
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Translating classical math and code, to quantum math and code

I am starting to see alot 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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49 views

How to construct Schur-Weyl decomposition of independent and identically distributed mixed qudit states?

Given a $d$-dimensional Hilbert space $\mathcal{H} \approx \mathbb{C}^{d}$ (i.e. a qudit system) if I have $N$ identical copies of a mixed state I can use Schur-Weyl duality to decompose the state as $...
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Why is the probability vector of a uniformly random state $\sum_i\alpha_i|i\rangle$ uniformly random only if $\alpha_i\in\mathbb C$?

In these lecture notes by Scott Aaronson, the author states the following (towards the end of the document, just before the Linearity section): There's actually another phenomenon with the same "...
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310 views

Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]

Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products? For example, I have a matrix 16x16 and want it to be decomposed into something ...
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93 views

Can we conclude that errors on Sycamore are Poisson-distributed Pauli errors?

In Martinis' recent Caltech lecture on the Sycamore paper, he appears to make much of the fact that FIG. 4 of the paper show straight-line fidelity - that is, the fidelity decreases log-linearly with ...
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1answer
106 views

What can we know about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix?

What information can we get out about the eigenvalues of a reduced density matrix knowing the eigenvalues of the original matrix? For example, it can be proved that if all the eigenvalues of a ...
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135 views

If a state is only “close to” an eigenstate of an operator, how many applications of the operator does it take to scramble the state?

Suppose we have an operator $U$, and a register $|\lambda\rangle$ in an eigenstate of $U$ with eigenvalue $\lambda=1$. Repeatedly applying $U$ to $|\lambda\rangle$ does not affect $|\lambda\rangle$ - ...
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127 views

Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

I've been trying to figure this out for a while and I'm totally lost. My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$ So far ...
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168 views

Probability of observing search string $\omega$ after $r$ iterations

Per Wikipedia, in Grover's algorithm the probability of observing search string $\omega$ after $r$ iterations is: $$\left|\begin{bmatrix}\langle \omega | \omega \rangle &\langle \omega | s \...
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Conditional probability between parameter and operator in quantum mechanics?

Background So I came across a question on conditional probability in quantum mechanics: There's an interesting comment which tells why this does not work for "the non-commutative case". I was ...
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141 views

How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

How can I demonstrate on the exponential part equality of the Hadamard matrix: $$H=\frac{X+Z}{\sqrt2}\equiv\exp\left(i\frac{\pi}{2}\frac{X+Z}{\sqrt2}\right).$$ In general, how can I demonstrate on: $\...
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1answer
63 views

Matrix Index and multiplication rules for Hermitian Pauli group products

Given the Hermitian Pauli group products $$ \Omega_{a,b}=\{\pm 1,\pm i\}_{a,b}\cdot \{I,X,Y,Z\}_{a,b}^{\otimes n} $$ composed of $n$ 2x2 pauli matrices $(I,X,Y,Z)$ in tensor product, such that they ...
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1answer
91 views

How to check if a two-qubit gate is entangling?

I would like to know if there's an analog for Schmidt rank that can tell me if a two-qubit unitary is entangling? Suppose I have a parametrized two-qubit unitary $U^{(2)}(\theta)$. I would like to ...
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1answer
92 views

Proving $\langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle =\operatorname{Tr}(M_j\rho)$

I'm trying to prove that: $$ \langle j_2|\langle j_1| U(|0\rangle\!\langle0|\otimes\rho)U^\dagger|j_2\rangle|j_1\rangle =\operatorname{Tr}(M_j\rho) $$ where $\rho$ is the density operator, $M_j=\...
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1answer
41 views

Quantum Fisher information for pure states query

Assume that a density matrix is given in its eigenbasis as $$\rho = \sum_{k}\lambda_k |k \rangle \langle k|.$$ On page 19 of this paper, it states that the Quantum Fisher Information is given as $$F_{...
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1answer
57 views

How to split a 2-local unitary operator through singular value decomposition?

I’m studying the paper Expressive power of tensor-network factorizations for probabilistic modeling by Glasser et al. In equation S6 (page 2 of supplementary material, excerpt of paper figure below) ...
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History of anyon theory, braidings and tensor categories

What was the first paper/who was the first person to phrase anyon theory in terms of tensor categories? Going through Wilczek's book on fractional statistics, some of the reprinted papers anticipate ...
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1answer
97 views

Understanding the action of operators on vectors in tensor product spaces

I'm studying Quantum Computing: A Gentle Introduction. On page 33, Section 3.1.2, after defining tensor product with 3 properties (distribution over addition on both left and right, scalar on both ...
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1answer
74 views

Trace of Hermitian Operator and Operator Function

I am having trouble understanding the following step. From: $$\operatorname{trace}\left(\sum_z |z\rangle\langle z| \rho_A |z\rangle\langle z| * \log( \sum_z |z\rangle\langle z| \sum_x |\langle x|z \...
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Is there an algorithm for determining if a given vector is separable or entangled?

I'm trying to understand if there is some sort of formula or procedural way to determine if a vector is separable or entangled – aka whether or not a vector of size $m$ could be represented by the ...
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1answer
78 views

Variational Quantum Eigensolver (VQE) - Question about finding the imaginary part of measurement

I've been reading this article in order to understand how to implement a VQE on a quantum computer. Equation 38 involves the imaginary part of $ \langle\psi_0 |V_k^{j\dagger}(t)O_iU(t)|\psi_0\rangle ...
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1answer
100 views

Evaluate the following teleportation equation for $U = ZX$ and $a = 0, b = 0$

When I evaluate the following equation using $U = ZX$ and $a = 0, b = 0$ [for Bell state], I am getting LHS not equal to RHS. Before describing the protocol, let us first review the teleportation ...
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42 views

Time Evolution Operator of Rabi Oscillations

I am referring to Exercise 7.18 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. The exercise wants me to show that the time evolution operator related to Rabi ...
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1answer
58 views

How to show whether two states are indistinguishable or not by measuring in a different basis?

I'm struggling with understanding a bit of basic quantum mechanics math that I was hoping someone could clarify. If I have two states such as these: $$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$$ and ...
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1answer
52 views

Can I find the states of individual qubits in a quantum register using only linear algebra?

Say I have a quantum register consisting of two qubits like this $\left| -,0\right>$ which as a vector would be $\frac{1}{\sqrt{2}}(1, 0, -1, 0)$. If I only started with this vector, would it ...
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2answers
117 views

How to prove that antipodal points on the Bloch sphere are orthogonal?

I started by assuming two antipodal states $$ |(\theta,\psi)\rangle = \cos\dfrac{\theta}{2}|0\rangle + \sin\dfrac{\theta}{2}e^{i\psi}|1\rangle\\ |(\theta+\pi,\psi+\pi)\rangle= \cos\dfrac{\theta+\pi}{2}...
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1answer
74 views

Can someone show the linear algebra calculations for X, H, and CNOT gates?

I am on Ch.1 of the Mike & Ike book. On page 18, the text shows an X gate that essentially flips the $\alpha$ and $\beta$ amplitudes. The text shows the $X$ matrix but it doesn't show those for ...
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172 views

Why use inner and outer product?

Inner product: how similar the vectors are Outer product: ??? For inner product I can find this explanation. "The inner product of two vectors therefore yields just a number. As we'll see, we can ...