Quantum logic gate

Key takeaways

Quantum logic gate belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quantum logic gate to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quantum logic gate from memory before moving on to harder problems.

Reference excerpt

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. According to quantum mechanics, a quantum system can only either evolve unitarily according to the Schrödinger equation, or be measured (sometimes called "Observed"). Quantum gates describe these unitary transformations, that occur when the system is not being measured. The expression "quantum gate" appears in relation to quantum processors, and in this context they are the logical operations that the quantum computer at the assembly language-level of abstraction (e.g. OpenQASM) can perform on the quantum data (qubits or quantum states) that they process. They can also be whole algorithms (e.g. the Quantum Fourier transform) if such algorithms contain no measurement operations. When the quantum data is measured, it is usually transformed into binary bits, which is then sent to a normal ("classical") computer. The quantum processor behaves like a coprocessor to such classical binary processors. Unlike many classical logic gates, quantum logic gates are reversible. It is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits. Quantum gates are unitary operators, and are described as unitary matrices relative to some orthonormal basis. Usually the computational basis is used, which unless comparing it with something, just means that for a d-level quantum system (such as a qubit, a quantum register, or qutrits and qudits) the orthonormal basis vectors are labeled | 0 ⟩ , | 1 ⟩ , … , | d − 1 ⟩ {\displaystyle |0\rangle ,|1\rangle ,\dots ,|d-1\rangle } , or use binary notation.

History The current notation for quantum gates was developed by many of the founders of quantum information science including Adriano Barenco, Charles Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, building on notation introduced by Richard Feynman in 1986.

Representation

Quantum logic gates are represented by unitary matrices. A gate that acts on n {\displaystyle n} qubits (a register) is represented by a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} unitary matrix, and the set of all such gates with the group operation of matrix multiplication is the unitary group U(2n). The quantum states that the gates act upon are unit vectors in 2 n {\displaystyle 2^{n}} complex dimensions, with the complex Euclidean norm (the 2-norm). The basis vectors (sometimes called eigenstates) are the possible outcomes if the state of the qubits is measured, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on vector spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. Even though the quantum logic gates belong to continuous symmetry groups, real hardware is inexact and thus limited in precision. The application of gates typically introduces errors, and the quantum states' fidelities decrease over time. If error correction is used, the usable gates are further restricted to a finite set. Later in this article, this is ignored as the focus is on the ideal quantum gates' properties. Quantum states are typically represented by "kets", from a notation known as bra–ket. The vector representation of a single qubit is

| a ⟩ = v 0 | 0 ⟩ + v 1 | 1 ⟩ → [ v 0 v 1 ] . {\displaystyle |a\rangle =v_{0}|0\rangle +v_{1}|1\rangle \rightarrow {\begin{bmatrix}v_{0}\\v_{1}\end{bmatrix}}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Quantum logic gate

Start with the simplest possible case. Write down what Quantum logic gate claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quantum logic gate before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quantum logic gate ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quantum logic gate

In research: Quantum logic gate appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quantum logic gate in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quantum logic gate is common in secondary-school and first-year university syllabi. It links to neighbouring topics Australian inventions, Logic gates, Quantum gates, so understanding it makes those chapters shorter.
In everyday life: Look for Quantum logic gate outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Quantum logic gate in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quantum logic gate means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quantum logic gate out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quantum logic gate in simple terms?

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building blocks of quantum circuits, like classical logic gates are for convent…

Why does Quantum logic gate matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quantum logic gate?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quantum logic gate.