# The Quantum Gate

## Quantum Munich : Collisional Quantum Gate Arrays

Some other key single-qubit gates are detailed below:. R1 theta : Double, qubit : Qubit. More generally:.

The SWAP gate unsurprisingly swaps the two input qubits around. Another class of multi-qubit gates is the controlled gates. Controlled gates require at least one control and one target qubit — the gate in question will only operate on the target qubit if the control qubit is in a specific state. You can construct the matrix for any controlled gate by taking the identity matrix and attaching it to the top-left corner of the desired gate, then filling in the blank entries with zeroes, like so:.

This means that any computation possible on our system must decompose back into a finite sequence of known gates. Universality is actually much more exciting than it looks at first.

It means that any transformation permitted by quantum physics can be implemented on a quantum computer. This not only means we can run any quantum program; we can also use universality to run any physics. Universality therefore allows us to use a quantum computer to mimic molecules, super conductors and all manner of weird and wonderful quantum systems.

In fact this ability of quantum computers to use universality simulate physics that can flummox even the best supercomputers promises to change the world. Not so boring right? Using these key gates and some others , we can now start implementing full-blown quantum circuits!

My next post will be all about how we can apply this new knowledge in the context of the Quantum Fourier Transform , a hugely important operation that has applications in all sorts of useful algorithms. See you there! Introduction Before we can move on to a discussion of quantum computing in real world quantum devices, we must first understand how to manipulate our qubits to perform computations. The Basics: Radians When talking about quantum computing, rotations are measured in radians.

## The Quantum Menace

NOT The act of measuring a qubit causes any superpositions to collapse and its quantum properties to vanish, returning classical information. Hence, the measurement element below is seen to take in a qubit and output a classical bit. Single-Qubit Gates Single-qubit gates are unsurprisingly the simplest, so we will start there! This can be seen on the Bloch Sphere: Because of this relationship, we say that the Pauli matrices are the square roots of the identity matrix. If you are brand new to the field, it will be a better place to start. Like everything in quantum computing, not to mention quantum mechanics, quantum gates are shrouded in an unfamiliar fog of jargon and matrix mathematics that reflects the quantum mystery.

My goal in this post is to peel off a few layers of that mystery. At least, not in All we can do today is reveal the striking similarities and alarming differences between classical gates and quantum gates, and explore the implications for the near and far future of computing. If nothing else, classical logic gates and quantum logic gates are both logic gates. A logic gate, whether classical or quantum, is any physical structure or system that takes a set of binary inputs whether 0s and 1s, apples and oranges, spin-up electrons and spin-down electrons, you name it and spits out a single binary output: a 1, an orange, a spin-up electron, or even one of two states of superposition.

What governs the output is a Boolean function. The gates are then combined into circuits, and the circuits into CPUs or other computational components. Classical gates operate on classical bits, while quantum gates operate on quantum bits qubits. This means that quantum gates can leverage two key aspects of quantum mechanics that are entirely out of reach for classical gates: superposition and entanglement.

### System Requirements

Simply put, quantum gates are reversible. This means that, at least in principle, quantum gates never lose information. Qubits that are entangled on their way into the quantum gate remain entangled on the way out, keeping their information safely sealed throughout the transition. Interestingly enough, that information is not ultimately lost to the universe, but rather seeps out into your room or your lap as the heat in your classical computer.

In the language of quantum mechanics and computing, vectors are depicted in an admittedly pretty weird package called a ket, which comes from the second half of the word braket.

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And they look the part. These vectors can span any number of numbers, so to speak. So instead of looking like towering column vectors, they just looked like numbers stacked two-high. This transformation of one vector into another takes place through the barely understood magic of matrix multiplication , which is completely different than the kind of multiplication we all learned in pre-quantum school.

If you need some more motivation, just remember that it was through the language of matrix mathematics that Heisenberg unlocked the secrets of the all-encompassing uncertainty principle. Suffice it to say, for now, that we generally use a matrix to stand-in for a quantum gate. But the size of the matrix balloons with two, three or more qubits.

Here, n is the number of qubits the quantum gate is operating on.

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As you can see, this number goes through the roof as the number of qubits n increases. The Pauli gates are named after Wolfgang Pauli, who not only has a cool name, but has managed to immortalize himself in two of the best-known principles of modern physics: the celebrated Pauli exclusion principle and the dreaded Pauli effect. The Pauli gates are based on the better-known Pauli matrices aka Pauli spin matrices which are incredibly useful for calculating changes to the spin of a single electron. At least the X and Z-gates are.

The Y is a little weird.

## Researchers successfully demonstrate a quantum gate in silicon

His gates act on one, and only one, qubit at a time. This translates to simple, 2 x 2 matrices with only 4 elements a piece. The Pauli X-gate is a dream come true for those that fear matrix math.