Quantum logic gate

In quantum computing and specifically the quantum circuit model of computation, a quantum logic 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.
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 the orthonormal basis vectors are labeled 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 qubits is represented by a unitary matrix, and the set of all such gates with the group operation of matrix multiplication is the unitary group U. The quantum states that the gates act upon are unit vectors in complex dimensions, with the complex Euclidean norm. The basis vectors 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
Here, and are the complex probability amplitudes of the qubit. These values determine the probability of measuring a 0 or a 1, when measuring the state of the qubit. See [|measurement] below for details.
The value zero is represented by the ket and the value one is represented by the ket
The tensor product is used to combine quantum states. The combined state for a qubit register is the tensor product of the constituent qubits. The tensor product is denoted by the symbol
The vector representation of two qubits is:
The action of the gate on a specific quantum state is found by multiplying the vector, which represents the state by the matrix representing the gate. The result is a new quantum state

Relation to the time evolution operator

The Schrödinger equation describes how quantum systems that are not observed evolve over time, and is When the system is in a stable environment, so it has a constant Hamiltonian, the solution to this equation is If the time is always the same it may be omitted for simplicity, and the way quantum states evolve can be described as just as in the above section.
That is, a quantum gate is how a quantum system that is not observed evolves over some specific time, or equivalently, a gate is the unitary time evolution operator acting on a quantum state for a specific duration.

Notable examples

There exists an uncountably infinite number of gates. Some of them have been named by various authors, and below follow some of those most often used in the literature.

Identity gate

The identity gate is the identity matrix, usually written as I, and is defined for a single qubit as
where I is basis independent and does not modify the quantum state. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits.

Pauli gates (''X'',''Y'',''Z'')

The Pauli gates are the three Pauli matrices and act on a single qubit. The Pauli X, Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by radians.
The Pauli-X gate is the quantum equivalent of the NOT gate for classical computers with respect to the standard basis which distinguishes the z axis on the Bloch sphere. It is sometimes called a bit-flip as it maps to and to. Similarly, the Pauli-Y maps to and to. Pauli Z leaves the basis state unchanged and maps to Due to this nature, Pauli Z is sometimes called phase-flip.
These matrices are usually represented as
The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the identity matrix.
The Pauli matrices also anti-commute, for example
The matrix exponential of a Pauli matrix is a rotation operator, often written as

Controlled gates

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation. For example, the controlled NOT gate acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is and otherwise leaves it unchanged. With respect to the basis it is represented by the Hermitian unitary matrix:
The [|CNOT] gate can be described as the gate that maps the basis states, where is XOR.
The CNOT can be expressed in the Pauli basis as:
Being a Hermitian unitary operator, CNOT has the property that and, and is involutory.
More generally if U is a gate that operates on a single qubit with matrix representation
then the controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.
The matrix representing the controlled U is
When U is one of the Pauli operators, X,''Y, Z'', the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used. Sometimes this is shortened to just CX, CY and CZ.
In general, any single qubit unitary gate can be expressed as, where H is a Hermitian matrix, and then the controlled U is
Control can be extended to gates with arbitrary number of qubits and functions in programming languages. Functions can be conditioned on superposition states.

Classical control

Gates can also be controlled by classical logic. A quantum computer is controlled by a classical computer, and behaves like a coprocessor that receives instructions from the classical computer about what gates to execute on which qubits. Classical control is simply the inclusion, or omission, of gates in the instruction sequence for the quantum computer.

Phase shift gates

The phase shift is a family of single-qubit gates that map the basis states and. The probability of measuring a or is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle, or a rotation about the z-axis on the Bloch sphere by radians. The phase shift gate is represented by the matrix:
where is the phase shift with the period. Some common examples are the T gate where , the phase gate where and the Pauli-Z gate where
The phase shift gates are related to each other as follows:
Note that the phase gate is not Hermitian. These gates are different from their Hermitian conjugates:. The two adjoint gates and are sometimes included in instruction sets.

Hadamard gate

The Hadamard or Walsh-Hadamard gate, named after Jacques Hadamard and Joseph L. Walsh, acts on a single qubit. It maps the basis states and . The two states and are sometimes written and respectively. The Hadamard gate performs a rotation of about the axis at the Bloch sphere, and is therefore involutory. It is represented by the Hadamard matrix:
Image:Hadamard gate.svg|upright=0.4|thumb|Circuit representation of Hadamard gate
If the Hermitian Hadamard gate is used to perform a change of basis, it flips and. For example, and

Swap gate

The swap gate swaps two qubits. With respect to the basis,,,, it is represented by the matrix
The swap gate can be decomposed into summation form:

Toffoli (CCNOT) gate

The Toffoli gate, named after Tommaso Toffoli and also called the CCNOT gate or Deutsch gate, is a 3-bit gate that is universal for classical computation but not for quantum computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If we limit ourselves to only accepting input qubits that are and then if the first two bits are in the state it applies a Pauli-X on the third bit, else it does nothing. It is an example of a CC-U gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.
The Toffoli gate is related to the classical AND and XOR operations as it performs the mapping on states in the computational basis.
The Toffoli gate can be expressed using Pauli matrices as

Universal quantum gates

A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible with anything less than an uncountable set of gates since the number of possible quantum gates is uncountable, whereas the number of finite sequences from a finite set is countable. To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently. Checking if a set of quantum gates is universal can be done using group theory methods and/or relation to unitary t-designs
Some universal quantum gate sets include:

The rotation operators,,, the phase shift gate and CNOT are commonly used to form a universal quantum gate set.
The Clifford set + T gate. The Clifford set alone is not a universal quantum gate set, as it can be efficiently simulated classically according to the Gottesman–Knill theorem.
The Toffoli gate + Hadamard gate. The Toffoli gate alone forms a set of universal gates for reversible Boolean algebraic logic circuits, which encompasses all classical computation.