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