Dirac Notation

September 16, 2018August 27, 2018 quantumultiversesLeave a comment

One of the things that intrigued me in my studies of quantum computing was that I did not know which notation was this: |>. How do you read that? How do you pronounce it? The flea became a tortoise behind my ear, and it corroded my soul, and after a while I discovered what it was, it was Dirac’s notation. Our post today will talk about it, how it reads and how it represents vector operations.

paul_dirac Paul Dirac

In quantum physics, the physical state of a system is all the information obtainable from it, and can be represented by a complex wave function or by a state vector contained in a complex vector space. However, it is known that the representation of three-dimensional vector spaces of some elements, such as spins, is not possible. At this point comes the Nobel Prize for Physics in 1933 and pioneer of quantum physics and mechanics, Paul Dirac.

He grounded and introduced complex vector spaces by calling them kets, which represent a physical state of quantum mechanics containing all the information of that state. Kets are represented by the symbol “|>” and have a dual element of that state called bra represented by the symbol “<|”. The scalar product of this state can be represented by the symbol “<|>”, being called the brakets. The Dirac notation is also known as bracket notation.

The braket has some particularities like:Read More »

Quantum Cloud – Alibaba

September 2, 2018August 27, 2018 quantumultiversesLeave a comment

In February of this year 2018, Chinese giant Alibaba, through its arm Alibaba Cloud (Aliyun) and in partnership with the Chinese Academy of Sciences (CAS), announced the launch of a quantum computing service using a quantum processor of 11 Qubits, becoming IBM’s second-largest service in the world. Other than that, they are offering a 32 Qubits simulator.

alibaba_cloud_logo Logo Alibaba Cloud

As we can see in the announcement of the official website of the group, available at this link, the service operates at -273 degrees Celsius, and similar to IBM, users can access the quantum platform to perform tests with codes and quantum algorithms. Yaoyun Shi, director of Alibaba quantum laboratory, referring to quantum supremacy, said:

“About physicists worrying about when to achieve supremacy, it’s like worrying if your baby will be smarter than your puppy. Just focus on taking care of the baby, it will happen, even if you’re not sure when” .

This same laboratory was responsible for the creation, in May 2017, of the first photon-based quantum computer.

The group’s expectation is that by 2025 they will have the world’s fastest quantum computers. They also expect that by 2030 they will have a quantum computer prototype that uses something between 50 and 100 Qubits.

We will continue to follow this story and see the great progress made by the giant Alibaba!

References:
https://www.alibabacloud.com/press-room/alibaba-cloud-and-cas-launch-one-of-the-worlds-most
https://www.wired.com/story/google-alibaba-spar-over-timeline-for-quantum-supremacy/
https://www.thedailystar.net/bytes/tech-happening/china-scientists-create-first-photon-quantum-fastest-computer-1400458

Quantum Logic Gates 6 – Phase Shift Gate, Square Root of Swap Gate & Ising Gate

August 26, 2018July 6, 2018 quantumultiversesLeave a comment

In this last post about quantum gates (ahhhh…), we will present you with: Phase Shift gate, Square Root of Swap gate and Ising gate.

Phase Shift Gate

This gate is from a family that acts on a single qubit, keeping the initial state unchanged and mapping the state to $e^{{i\phi }}|1\rangle$ . The probability of measuring the states |0> or |1> does not change after applying this port, but it modifies the phase of the quantum state. It is the equivalent of drawing a horizontal circle on the Bloch sphere, in Φ radians.

phasegate

It is represented by the matrix:

$R_{\phi }={\begin{bmatrix}1&0\\0&e^{{i\phi }}\end{bmatrix}}$

Where Φ is the phase change of the quantum state.

Square Root of Swap Gate

This port does half the exchange of an exchange of 2 Qubits. The port is drawn with the following diagram in quantum circuits:

half_square

Your matrix representation is as follows:

${\sqrt {\mbox{SWAP}}}={\begin{bmatrix}1&0&0&0\\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\\0&0&0&1\\\end{bmatrix}}$

Ising Gate

This port is implemented natively on some quantum computers based on trapped ions.

Your matrix representation is as follows:

$XX_{\phi }={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&-ie^{i\phi }\\0&1&-i&0\\0&-i&1&0\\-ie^{-i\phi }&0&0&1\\\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&e^{i(\phi -\pi /2)}\\0&1&-i&0\\0&-i&1&0\\e^{i(-\phi -\pi /2)}&0&0&1\\\end{bmatrix}}$

Well… everything that is good last a little. We hope you enjoy this serie. Stay with us, cause soon we will have more good news… see ya!

References:
Série: Portas lógicas quânticas 11 – Phase shift gates
Série: Portas lógicas quânticas 12 – Square root of Swap gate (√SWAP)
Série: Portas lógicas quânticas 13 – Ising gate

Quantum Logic Gates 5 – Hadamard & (√NOT) Gate

August 19, 2018August 12, 2018 quantumultiversesLeave a comment

Hey duds! In our fifth post, we will talk about Hadamard and (√NOT) gates.

Hadamard Gate

This port has no analog equivalents, being a purely “quantum” logic gate. Michael Nielsen, author of the quantum computing reference book “Quantum Computation and Quantum Information,” published in his official youtube channel the video below, explaining a little more about this type of quantum logic gate:

This port acts in a single qubit. It maps the initial state |0> to (|0> + |1>) / √2 and |1> to (|0> – |1>) / √2, as shown in the figure below. The measure will have equal probabilities of becoming 1 or 0, creating the superposition.

had1

In the next figure, we can see how the gate act.

haa

The port is drawn with the following diagram in quantum circuits:

hadamard_gate

Your matrix representation is as follows:

$H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

(√NOT) Gate

This port acts on a single Qubit, being the square root of the logical gate NOT gate. The port is drawn with the following diagram in quantum circuits:

Qcircuit_SqrtNot

Your matrix representation is as follows:

${\sqrt {NOT}}={\frac {1}{2}}{\begin{bmatrix}1+i&1-i\\1-i&1+i\end{bmatrix}}$

Because it is the square root, it has the following form:

${\sqrt {NOT}}\,{\sqrt {NOT}}=NOT$

For more information on quantum logic gate NOT gate, enter this link, which contains the corresponding post from our blog. That’s it, people! Don’t lose our last post about this quantum gates series: Phase Shift Gate, Square Root of Swap Gate & Ising Gate.

References:
Série: Portas lógicas quânticas 9 – Hadamard gate
Série: Portas lógicas quânticas 10 – (√NOT) gate

Quantum Logic Gates 4 – CNOT & SWAP Gate

August 12, 2018July 6, 2018 quantumultiversesLeave a comment

Following the series “Quantum logic gates”, let’s talk today about CNOT and SWAP gates!

CNOT Gate

Controlled Not Gate, also known as CNOT gate or C-NOT Gate. Michael Nielsen, author of the quantum computing reference book “Quantum Computation and Quantum Information,” published in his official youtube channel the video below, explaining a little more about this type of quantum logic gate:

The CNOT port operates in 2 qubits, and uses the NOT operation on the second Qubit, only when the first is |1>, otherwise leaves it unchanged. The first Qubit is control, and the second is inverted with the classic NOT operation, hence the CNOT name, Controled Not Gate.

The port is drawn with the following diagram in quantum circuits:

CNOT_gate

It also has the symbology below in its representations:

Controlled_gate

Your matrix representation is as follows:

$\mbox{CNOT} = \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}$

The logic gate Toffoli gate inherits from the CNOT gate, its basic structure, adding just one more control Qubit.

SWAP Gate

This port has a very simple action to understand, basically changing the positions of the Qubits. It transforms |00> into |00>, |01> into |10>, |10> into |01| and |11| into |11|.

kisekae___lower_body_swap_by_drwigglewiggle-dbh0s33 Swap gate swaps the Qubits states as the people nowadays exchange … clothes 🙂

The port is drawn with the following diagram in quantum circuits:

Swap_gate

It also has the symbology below in its representations:

swap

Your matrix representation is as follows:

$\mbox{SWAP} = \begin{bmatrix} 1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}$

That’s all for now, folks! The next gate will lead us to Hadamard and (√NOT) gates. See ya!

Referências:
Série: Portas lógicas quânticas 7 – CNOT gate
Série: Portas lógicas quânticas 8 – SWAP gate

Quantum Logic Gates 3 – Fredkin & Tofolli Gate

August 5, 2018July 6, 2018 quantumultiversesLeave a comment

Following the series “Quantum logic gates”, let’s talk today about the Fredkin and Toffoli gates!

Fredkin Gate

The Fredkin gate, also called the CSWAP gate, is named for being developed by Edward Fredkin, a teacher at Carnegie Mellon University. Edward was also a professor at MIT, Boston University, besides working with the legendary Richard Feynman at Caltech.

fredkin_ed Edward Fredkin

The logic gate that take his name has the characteristic of being universal, which means that any logical or arithmetic operation can be constructed entirely using Fredkin logic gates. This gate is composed of a circuit with 3 inputs and 3 outputs, which transmits the first bit without changes and inverts the other two if and only if the first bit is 1. If the first bit is 0, the other bits are not inverted. The video below demonstrates how it works:

It is drawn with the following diagram, in the quantum circuits:

150px-fredkin_gate-svg

It’s matrix representation is as follows:

${\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\\end{bmatrix}}$

This logic gate is part of the so-called “reversible computing”, which means its forming circuit is reversible because we can undo its transformations if we use it backwards.

In 2016 Australian scientists reported having built the first Fredkin quantum logic gate using tangled photons. The article can be read on this link, and the researchers’ scientific article is available at this link.

Toffoli Gate

The Toffoli gate, also known as the CCNOT gate, is named for being created by Tommaso Toffoli, an Italian professor of electrical and computer engineering at Boston University. He worked alongside Edward Fredkin on artificial life theory.

toffoli Tommaso Toffoli

The Toffoli gate has 3 input Qubits and 3 output Qubits. If the first two qubits are in the state |1>, it applies a Pauli X (NOT) port, in the third qubit, that is, it reverses its value, otherwise it does nothing.

It is drawn with the following diagram, in the quantum circuits:

1200px-toffoli_gate-svg

Your matrix representation is as follows:

${\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\end{bmatrix}}$

In January 2009, researchers from the University of Innsbruck in Austria were able to create a Toffoli gate, this study is available at this link.

That’s all for now, folks! The next gate will lead us to CNOT and SWAP gates. See ya!

References:
Série: Portas lógicas quânticas 5 – Fredkin gate
Série: Portas lógicas quânticas 6 – Toffoli gate

Quantum Logic Gates 2 – Pauli Gate

July 29, 2018July 6, 2018 quantumultiversesLeave a comment

Following the series “Quantum Logic Gates”, let’s talk today about the most intuitive of all quantum gates, the quantum logic gate created by Wolfgang Pauli that has his name. There are 3 kinds of them: Pauli X, Pauli Y and Pauli Z.

wolfgang_pauli Wolfgang Pauli, creator of Pauli’s matrix

Pauli X Gate or NOT Gate

In essence, the quantum logic gate NOT, also known as Pauli X gate acts identical to the classic logic gate NOT. As we can see in the diagram below, two inputs, with their respective values of A|0> and B|1>, have their outputs inverted, going to A|1> and B|0>.

Quantum-NOT-gate

This gate, which acts on only a single Qubit, has the diagram below used to represent it in a quantum circuit. The actuation of this gate is equivalent to a rotation of 180º, or π radians, on the x axis of the Bloch sphere, which is a representation for Qubits that we have already covered in this post and in this one as well.

Qcircuit_NOT

In short, the quantum logic gate NOT or Pauli-X gate, inverts the logical value of an input |0> to |1>, and an Input |1> to |0>. It receives this “Pauli X gate” name, on account of the matrix used to represent it, which is the matrix of Pauli:

$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

The author of the bible of Quantum Computation, “Quantum Computation and Quantum Information”, Michael A. Nielsen, released a video on youtube telling us a little more about this logical gate, as we can see below:

Pauli Y Gate

The quantum logic gate Pauli Y, acts on a single Qubit. The actuation of this gate is equivalent to a rotation of 180º, or π radians, on the y-axis of the Bloch sphere, transforming a value of |0> to i|1> and a value of |1> to –i|0>. It receives the name of Pauli Y gate, by the matrix of Pauli that represents it:

$Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$

It is drawn with the following diagram, in the quantum circuits:

qcircuit_y

In this video below, we can see around 25 seconds the action of the Pauli Y gate in a Bloch sphere:

Pauli Z gate

Just like its sisters, Pauli Y and Pauli X, this gate acts on a single Qubit. The actuation of this gate is equivalent to a rotation of 180º, or π radians, in the Z axis of the Bloch sphere. It maintains the initial state |0> unchanged and maps |1> to -|1>. Because of its nature, it is also called a phase-flip gate / phase-shift gate, a special case of this quantum gate category. It receives the name of Pauli Z gate, because of the Pauli matrix that represents it:

$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

It is drawn with the following diagram, in the quantum circuits:

qcircuit_z

In this video below, we can see around 28 seconds the action of the Pauli Z port in a Bloch sphere:

That’s all for now folks! Next week we will talk about Fredkin and Toffoli gates.

Reference:
Série: Portas lógicas quânticas 2 – NOT gate
Série: Portas lógicas quânticas 3 – Pauli Y gate
Série: Portas lógicas quânticas 4 – Pauli Z gate

Quantum Logic Gates 1 – Genius At Work!

July 22, 2018July 1, 2018 quantumultiversesLeave a comment

In the classical world, so-called logic gates make up circuits, which are the basis of all computing. Logic gates are essentially transistors that allow or not the electric flow of energy, which associated in different ways implement a boolean logic that allows to create diverse operations.

portas Integrated Circuit

Before we enter the dimensional portal of quantum bizarre, we will review some concepts of classical computation. Let’s look at some of the main classic logic gates:

portinha Read More »

Book Review – Quantum Computers

July 15, 2018July 7, 2018 quantumultiversesLeave a comment

Jon Schiller’s Quantum Computers book, released in 2009, is a good study for those who want to learn more about quantum computing. The book, available only in English through Amazon in this link, brings several basic knowledge, and some advanced, within the thematic of quantum computing.

jon_schiller_book Jon Schiller Book

The book was born out of a lecture on quantum computing, which the author attended at a seminar at the California Institute of Technology, Caltech, which prompted curiosity, resulting in a survey of academic articles, and searches on google, resulting in the book.

The book is divided into 9 main chapters and 4 appendices, being:

Chapter 1: Introduction
Chapter 2: How a Quantum Computer Works
Chapter 3: Timeline and references to quantum computers
Chapter 4: How Quantum Computers Differ from Classic Computers
Chapter 5: What problems can a quantum computer solve
Chapter 6: What dangers exist for privacy with quantum computers
Chapter 7: How Quantum Computers Store Data
Chapter 8: When you can buy a quantum computer
Chapter 9: IBM’s efforts in quantum computing
Appendix A: Definitions used in quantum computing: Shor Algorithms
Appendix B: Glossary of terms of quantum computing: Entaglement, Heisenberg’s uncertainty principle and Qubit
Appendix C: The Einstein-Podolsky-Rosen Paradox (EPR)
Appendix D: What’s new in the internet about quantum computers

The topics of this book will be further detailed in future posts. This book is very rich in diverse information, however in some points it goes deep in the mathematical aspect that can take some lay people off the subject. In some aspects the book is dated, due to developments in the area, but for the most part the book brings many relevant topics and a relatively accessible language.

Look forward to the next posts in which we will use this book (and others) as a reference for the study of this grand and intriguing universe of quantum computing!

Quantum hugs to all!

Reference:
Resenha do Livro “Quantum Computers”