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.

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

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 ...
3
votes
0answers
39 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 ...
2
votes
0answers
43 views

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}...
4
votes
2answers
77 views

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 ...
3
votes
1answer
139 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{...
2
votes
1answer
17 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 ...
5
votes
1answer
67 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 ...
2
votes
1answer
32 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 ...
2
votes
2answers
55 views

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 ...
2
votes
0answers
47 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 $...
6
votes
1answer
121 views

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 "...
4
votes
2answers
239 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 ...
3
votes
1answer
79 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 ...
3
votes
1answer
101 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 ...
3
votes
2answers
126 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 $|\lambda\rangle$ - ...
6
votes
2answers
123 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 ...
3
votes
2answers
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 \...
1
vote
0answers
23 views

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 ...
6
votes
3answers
138 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: $\...
4
votes
1answer
61 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 ...
4
votes
1answer
75 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 ...
4
votes
1answer
91 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=\...
2
votes
1answer
33 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_{...
4
votes
1answer
56 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) ...
3
votes
0answers
26 views

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 ...
2
votes
1answer
92 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 ...
1
vote
1answer
68 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 \...
6
votes
2answers
81 views

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 ...
1
vote
1answer
59 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 ...
0
votes
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 ...
3
votes
2answers
38 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 ...
3
votes
1answer
56 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 ...
3
votes
1answer
51 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 ...
2
votes
2answers
84 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}...
2
votes
1answer
68 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 ...
2
votes
2answers
120 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 ...
4
votes
1answer
62 views

Extensions of product states

Given a product state $\rho_{AC} = \rho_A\otimes \rho_C$, what can we say about the structure of states $\rho_{ABC}$ that are extensions of $\rho_{A}\otimes \rho_C$? By extension I mean that $\text{tr}...
2
votes
2answers
62 views

How do I prove that $\sum_{y=0}^{N-1}e^{2\pi i xy/N}=N\delta_{x,0}$?

I am trying to prove the following relation related to the Quantum Fourier Transform: $$\sum_{y=0}^{N-1}e^{2\pi i\frac{x}{N}y} = \begin{cases}0 & \text{if } x\neq 0\mod N \\ N & \text{if } x=...
1
vote
0answers
19 views

Error message for Classical Probability coding in Python

strong textHi guys I am doing a coding exercise for probabilities in vectors. Exercise 2 (1 point). As you recall, we may also write the probability distribution as a stochastic vector p⃗ =[p0p1]p→=...
2
votes
1answer
49 views

Does the general form of a unitary operator define strict signs for the second column?

As per IBM's documentation for quantum circuits, the general unitary operator is defined as: $$\hat{U}=\begin{bmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\e^{i\phi}\sin(\...
0
votes
0answers
21 views

Help with understanding Unitary Operator of Quantum Optical Fredkin Gate

I'm referring to the textbook "Quantum Computation and Quantum Information" 10th Anniversary Edition by Nielsen and Chuang. Chapter 7.4 has a Box 7.4 which introduces the Quantum Optical Fredkin Gate....
2
votes
1answer
61 views

Explain matrix multiplication math operations of a Hadamard gate after a phase gate

Starting with $|0\rangle$, I would like to understand how the probability values of 85.4% $|0\rangle$ and 14.6% $|1\rangle$ are derived from the payload circuit below? After applying the 45 degree ...
5
votes
1answer
1k views

Why isn't there a contradiction between the existence of CNOT gate/entanglement and the no-cloning theorem?

Of course I am not implying that I am right and the no cloning theorem is wrong, but I am trying to figure out what is wrong with my reasoning and yet I couldn't find the mistake. Based on Wikipedia ...
3
votes
0answers
29 views

Estimating errors in Hamiltonian Simulation paper

I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read $$ ||PA |0\rangle |\psi \rangle - |0\rangle ...
5
votes
1answer
43 views

Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evoultion operator which implents the operation

I was looking back over an old assignment and I came across a question I wasn't quite sure how to do the problem statement is as follows: The Hadamard rotation is an element of the group $U(2)$. (i)...
0
votes
1answer
24 views

Minimum Multi-Degree Polynomials representing Boolean Functions

In the 10th Anniversary Edition of Nielsen and Chuang Quantum Computation and Quantum Information textbook, Chapter 6.7 talks about Black Box algorithm limits. It is given: $f:\{0,1\}^n \...
1
vote
1answer
129 views

given an EPR pair, how do I calculate expectation value?

given A and B share EPR pairs $ (|00⟩+|11⟩)/√2$ both are free to measure their own qubit with the following measurement settings A measures with $[ |0⟩, |1⟩ ]$ B measures with $[ sin(3π/8)|0⟩ + cos(...
2
votes
2answers
96 views

Angular Error associated with Quantum Search Algorithm

Chapter 6.3 of "Quantum Computation and Quantum Information 10th Anniversary Edition" textbook by Nielsen and Chuang talks about using the Quantum Counting Algorithm to find the number of solutions to ...
1
vote
1answer
23 views

What is the probability that measurement finds it in the $|0\rangle$ state?

Suppose that there is an ensemble with 60% of the states prepared in $$|a\rangle=\sqrt{\frac{2}{5}}|+\rangle-\sqrt{\frac{3}{5}}|-\rangle$$ and 40% in: $$|b\rangle=\sqrt{\frac{5}{8}}|+\rangle+\sqrt{\...
3
votes
0answers
37 views

Geometric interpretation of 1-distillability

This is a sequel to Motivation for the definition of k-distillability Geometrical interpretation from the definition of 1-distillability The eigenstate $|\psi\rangle$ of the partially ...