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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28 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) it is claimed that a 2-local ...
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24 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 ...
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1answer
81 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
64 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
44 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
94 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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2answers
32 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
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1answer
48 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
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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 ...
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2answers
56 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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61 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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92 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
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1answer
57 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}...
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2answers
54 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=...
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15 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→=...
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1answer
46 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(\...
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19 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....
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1answer
56 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 ...
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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 ...
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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 ...
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36 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)...
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1answer
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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 \...
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121 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(...
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2answers
92 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 ...
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1answer
22 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{\...
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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 ...
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Motivation for the definition of k-distillability

Definition of k-distillability For a bipartite state $\rho$, $H=H_A\otimes H_B$ and for an integer $k\geq 1$, $\rho$ is $k$-distillable if there exists a (non-normalized) state $|\psi\rangle\in ...
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1answer
54 views

Construction of Phase Shift Operation used in Quantum Search

In Chapter 6 of "Quantum Computation and Quantum Information" Textbook by Nielsen and Chuang, Box 6.1 gives a circuit example of Quantum Search Algorithm done on a two-bit sized search space. The ...
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1answer
64 views

Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
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49 views

What is the motivation for Weyl matrices in quantum information theory?

Quantum Entanglement and Geometry — Andreas Gabriel (2010) — Sec: 2.3.4 ~p. 11 Another basis for $d\times d$-dimensional matrices that has proven to be quite useful in quantum information theory is ...
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1answer
45 views

Clarification of a procedure to compute the product of the exponential of two matrices

In trying to understand a method outlined here (page 3, subroutine 1). Consider $$R_3 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} .$$ Let $A$ be a ...
4
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1answer
251 views

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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3answers
117 views

How are orthogonal sets of pure states arranged in state space?

It is well known that the state of a (pure) qubit can be described as a point on a two-dimensional sphere, the so-called Bloch sphere. The mapping $\lvert\psi\rangle\mapsto \boldsymbol r_\psi$ that ...
4
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1answer
119 views

What is the HOG test and how would it help proving quantum supremacy?

Proposed experiments in achieving quantum supremacy, such as with BosonSampling or using random circuits, have been described as using a (not necessarily Turing complete) quantum computer to perform ...
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1answer
53 views

Structural Physical Approximation of Partial Transpose

To make the partial transpose a complete positive and therefore physical map, one has to mix it with enough of the maximally mixed state to offset the negative eigenvalues. The most negative ...
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3answers
212 views

What is the spectral decomposition of the Pauli $X$ gate?

The definition of spectral decomposition is as follows: Assume the eigenvectors of $\hat{A}$ define a basis $\beta=\{|\psi_j\rangle\}$. Then $$A_{kj}=\langle\psi_k|\hat{A}|\psi_j\rangle=\alpha_j\...
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1answer
75 views

Trace distance of two classical-quantum states

I have these two classical-quantum states: $$\rho = \sum_{a} \lvert a\rangle \langle a\lvert \otimes q^a \\ \mu = \sum_{a} \lvert a\rangle \langle a\lvert \otimes r^a $$ Where $a$ are the classical ...
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How does a map being “only” positive reflect on its Choi representation?

We know that a map $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ being completely positive is equivalent to its Choi representation being positive: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal ...
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1answer
58 views

Direct derivation of the Kraus representation from the natural representation, using SVD

$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL( \X)...
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1answer
57 views

Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem

In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following. $$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$ ...
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1answer
55 views

Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$

In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
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2answers
249 views

Can an isometry leave entropy invariant?

Consider two finite dimensional Hilbert spaces $A$ and $B$. If I have an isometry $V:A\rightarrow A\otimes B$, under what condition can I find a unitary $U:A\otimes B\to A\otimes B$ such that $$U\rho_{...
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2answers
80 views

Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation: $$\rho=\frac{1}{d}\left(\mathbb I +\...
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60 views

Writing the transformation matrix for the following in terms of Kronecker products of elementary 2-qubit gates

I have a set of transformations that transforms $|11001\rangle\to |10101\rangle$ which is basically keeping the leftmost qubit as it is and then it is just the CNOT between the successive qubits, I ...
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2answers
282 views

Square root of CNOT and spectral decomposition of the Hadamard gate

I'm trying to compute the spectral decomposition of the Hadamard gate but I'm making a mistake somewhere. Note: I believe (though I may be wrong so correct me if I am) that spectral decomposition is ...
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79 views

Encoding bosonic degrees of freedom

A well-known way of encoding $N$ levels of a harmonic (bosonic) oscillator is as follows: \begin{equation} |n\rangle = |1\rangle^{\otimes n} \otimes |0\rangle^{\otimes N-n+1} \quad,\qquad ...
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2answers
554 views

How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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1answer
60 views

Eigenstate of unitary operator used for Order-Finding

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem. (Eqn 5.36) states ...
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1answer
82 views

Correct way of expressing a measurement in a different computational basis

Sometimes we find that the result we want from a quantum algorithm is expressed in terms of a basis that is different from the usual computational basis, which I will call $$ B_C = \left\{ \lvert 0 \...