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In
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
and specifically the
quantum circuit In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly ...
model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes h ...
, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical
logic gate A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic ga ...
s 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 basis. Usually we use the ''computational basis'', which unless we compare it with something, just means that for a ''d''-level quantum system (such as a qubit, a quantum register, or qutrits and qudits) we have labeled the orthogonal basis vectors 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 John A. Smolin (born 1967) is an American physicist and Fellow of the American Physical Society at IBM's Thomas J. Watson Research Center. Smolin is best known for his work in quantum information theory, where, with collaborators, he introduced ...
, and Harald Weinfurter, building on notation introduced by
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superf ...
in 1986.


Representation

Quantum logic gates are represented by unitary matrices. A gate which acts on n qubits is represented by a 2^n \times 2^n unitary matrix, and the set of all such gates with the group operation of
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
is the symmetry group U(2''n''). The
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s that the gates act upon are unit vectors in 2^n complex dimensions, with the complex Euclidean norm (the
2-norm In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
). The basis vectors (sometimes called '' eigenstates'') are the possible outcomes if
measured Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared t ...
, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s of one or two qubits, just like the common classical logic gates operate on one or two
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
s. 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 decreases over time. If error correction is used, the usable gates are further restricted to a finite set. Later in this article, this is sometimes ignored as the focus is on the quantum gates' mathematical properties. Quantum states are typically represented by "kets", from a notation known as bra-ket. The vector representation of a single qubit is :, a\rangle = v_0 , 0 \rangle + v_1 , 1 \rangle \rightarrow \begin v_0 \\ v_1 \end , Here, v_0 and v_1 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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
(or Kronecker 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: :, a b \rangle = , a \rangle \otimes , b \rangle = v_ , 00 \rangle + v_ , 0 1 \rangle + v_ , 1 0 \rangle + v_ , 1 1 \rangle \rightarrow \begin v_ \\ v_ \\ v_ \\ v_ \end , The action of the gate on a specific quantum state is found by
multiplying Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition ...
the vector , \psi_1\rangle which represents the state by the matrix U representing the gate. The result is a new quantum state :U, \psi_1\rangle = , \psi_2\rangle.


Notable examples

There exists an
uncountably infinite In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
, usually written as ''I'', and is defined for a single qubit as : I = \begin 1 & 0 \\ 0 & 1 \end , 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 (X,Y,Z) are the three Pauli matrices (\sigma_x,\sigma_y,\sigma_z) 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 \pi 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 , 0\rangle to , 1\rangle and , 1\rangle to , 0\rangle. Similarly, the Pauli-''Y'' maps , 0\rangle to i, 1\rangle and , 1\rangle to . Pauli ''Z'' leaves the basis state , 0\rangle unchanged and maps , 1\rangle to Due to this nature, Pauli ''Z'' is sometimes called phase-flip. These matrices are usually represented as : X = \sigma_x =\operatorname = \begin 0 & 1 \\ 1 & 0 \end , : Y = \sigma_y = \begin 0 & -i \\ i & 0 \end, : Z = \sigma_z = \begin 1 & 0 \\ 0 & -1 \end. The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
. :I^2 = X^2 = Y^2 = Z^2 = -iXYZ = I The Pauli matrices also
anti-commute In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
, for example ZX=iY=-XZ. The matrix exponential of a Pauli matrix \sigma_j is a rotation operator, often written as e^.


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 (or CNOT or CX) 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: : \mbox = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end . The CNOT (or controlled Pauli-''X'') gate can be described as the gate that maps the basis states , a,b\rangle \mapsto , a,a \oplus b\rangle, where \oplus is XOR. The CNOT can be expressed in the Pauli basis as: : \mbox = e^=e^. Being a Hermitian unitary operator, CNOT has the property e^=(\cos \theta)I+(i\sin \theta) U and U =e^=e^, and is involutory. More generally if ''U'' is a gate that operates on a single qubit with matrix representation : U = \begin u_ & u_ \\ u_ & u_ \end , 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. : , 0 0 \rangle \mapsto , 0 0 \rangle : , 0 1 \rangle \mapsto , 0 1 \rangle : , 1 0 \rangle \mapsto , 1 \rangle \otimes U , 0 \rangle = , 1 \rangle \otimes (u_ , 0 \rangle + u_ , 1 \rangle) : , 1 1 \rangle \mapsto , 1 \rangle \otimes U , 1 \rangle = , 1 \rangle \otimes (u_ , 0 \rangle + u_ , 1 \rangle) The matrix representing the controlled ''U'' is : \mboxU = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_ & u_ \\ 0 & 0 & u_ & u_ \end. 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 C''X'', C''Y'' and C''Z''. In general, any single qubit unitary gate can be expressed as U = e^ , where ''H'' is a Hermitian matrix, and then the controlled ''U'' is CU = e^. 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 behave 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 , 0\rangle \mapsto , 0\rangle and , 1\rangle \mapsto e^, 1\rangle. The probability of measuring a , 0\rangle or , 1\rangle is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude), or a rotation along the z-axis on the Bloch sphere by \varphi radians. The phase shift gate is represented by the matrix: :P(\varphi) = \begin 1 & 0 \\ 0 & e^ \end where \varphi is the ''phase shift'' with the period . Some common examples are the ''T'' gate where \varphi = \frac (historically known as the \pi /8 gate), the phase gate (also known as the S gate, written as ''S'', though ''S'' is sometimes used for SWAP gates) where \varphi= \frac and the Pauli-''Z'' gate where \varphi = \pi. The phase shift gates are related to each other as follows: : Z = \begin 1 & 0 \\ 0 & e^ \end = \begin 1 & 0 \\ 0 & -1 \end = P\left(\pi\right) : S = \begin 1 & 0 \\ 0 & e^ \end = \begin 1 & 0 \\ 0 & i \end = P\left(\frac\right)=\sqrt : T = \begin 1 & 0 \\ 0 & e^ \end =P\left(\frac\right) = \sqrt = \sqrt /math> Note that the phase gate P(\varphi) is not Hermitian (except for all \varphi = n\pi, n \in \mathbb). These gates are different from their Hermitian conjugates: P^\dagger(\varphi)=P(-\varphi). The two adjoint (or
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
) gates S^\dagger and T^\dagger are sometimes included in instruction sets.


Hadamard gate

The Hadamard or Walsh-Hadamard gate, named after
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a tea ...
() and
Joseph L. Walsh __NOTOC__ Joseph Leonard Walsh (September 21, 1895 – December 6, 1973) was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szeg ...
, acts on a single qubit. It maps the basis states , 0\rangle \mapsto \frac and , 1\rangle \mapsto \frac (it creates an equal superposition state if given a computational basis state). The two states (, 0\rangle + , 1\rangle)/\sqrt and (, 0\rangle - , 1\rangle)/\sqrt are sometimes written , +\rangle and , -\rangle respectively. The Hadamard gate performs a rotation of \pi about the axis (\hat+\hat)/\sqrt at the Bloch sphere. It is represented by the Hadamard matrix: : H = \frac \begin 1 & 1 \\ 1 & -1 \end . If the Hermitian (so H^=H^=H) Hadamard gate is used to perform a change of basis, it flips \hat and \hat. For example, HZH=X and H\sqrt\;H=\sqrt=S.


Swap gate

The swap gate swaps two qubits. With respect to the basis , 00\rangle, , 01\rangle, , 10\rangle, , 11\rangle, it is represented by the matrix: : \mbox = \begin 1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end . The swap gate can be decomposed into summation form: :\mbox=\frac


Toffoli (CCNOT) gate

The Toffoli gate, named after Tommaso Toffoli and also called the CCNOT gate or Deutsch gate D(\pi/2), is a 3-bit gate which 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 , 0\rangle and then if the first two bits are in the state , 1\rangle it applies a Pauli-''X'' (or NOT) on the third bit, else it does nothing. It is an example of a CC-U (controlled-controlled Unitary) 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 (\land) and XOR (\oplus) operations as it performs the mapping , a, b, c\rangle \mapsto , a, b, c\oplus (a \land b)\rangle on states in the computational basis. The Toffoli gate can be expressed using Pauli matrices as : \mbox = e^= e^.


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 In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2), then that set can be used to approximate any desired quantum gate with a relatively ...
guarantees that this can be done efficiently. The rotation operators , , , the phase shift gate and CNOT form a widely used universal set of quantum gates. A common universal gate set is the
Clifford Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford *Baron Clifford *Baron Clifford of Chudleigh *Baron de Clifford *Clifford baronets *Clifford fami ...
+ ''T'' gate set, which is composed of the CNOT, ''H'', ''S'' and ''T'' gates. The Clifford set alone is not universal and can be efficiently simulated classically by the Gottesman–Knill theorem. The Toffoli gate forms a set of universal gates for reversible
boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
ic logic circuits. A two-gate set of universal quantum gates containing a Toffoli gate can be constructed by adding the Hadamard gate to the set.


Deutsch gate

A single-gate set of universal quantum gates can also be formulated using the parametrized three-qubit Deutsch gate D(\theta), named after physicist David Deutsch. It is a general case of ''CC-U'' or ''controlled-controlled-Unitary'' gate, and is defined as : , a,b,c\rangle \mapsto \begin i \cos(\theta) , a,b,c\rangle + \sin(\theta) , a,b,1-c\rangle & \mboxa=b=1 \\ , a,b,c\rangle & \mbox \end Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. There are some proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms. A universal logic gate for reversible classical computing, the Toffoli gate, is reducible to the Deutsch gate, D(\begin \frac \end), thus showing that all reversible classical logic operations can be performed on a universal quantum computer. There also exists a single two-qubit gate sufficient for universality, given it can be applied to any pairs of qubits (k,k+1)\bmod n on a circuit of width n.


Circuit composition


Serially wired gates

Assume that we have two gates ''A'' and ''B'', that both act on n qubits. When ''B'' is put after ''A'' in a series circuit, then the effect of the two gates can be described as a single gate ''C''. : C = B \cdot A Where \cdot is
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
. The resulting gate ''C'' will have the same dimensions as ''A'' and ''B''. The order in which the gates would appear in a circuit diagram is reversed when multiplying them together. For example, putting the Pauli ''X'' gate after the Pauli ''Y'' gate, both of which act on a single qubit, can be described as a single combined gate ''C'': : C = X \cdot Y = \begin 0 & 1 \\ 1 & 0 \end \cdot \begin 0 & -i \\ i & 0 \end = \begin i & 0 \\ 0 & -i \end = iZ The product symbol (\cdot) is often omitted.


Exponents of quantum gates

All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices. Positive integer exponents are equivalent to sequences of serially wired gates (e.g. and the real exponents is a generalization of the series circuit. For example, X^\pi and \sqrt=X^ are both valid quantum gates. U^0=I for any unitary matrix U. The
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
(I) behaves like a NOP and can be represented as bare wire in quantum circuits, or not shown at all. All gates are unitary matrices, so that U^\dagger U = UU^\dagger = I and where \dagger is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts: For example, some negative exponents of the phase shift gates are T^=T^ and Note that for a Hermitian matrix H^\dagger=H, and because of unitarity, HH^\dagger=I, so H^2 = I for all Hermitian gates. They are involutory. Examples of Hermitian gates are the Pauli gates, Hadamard, CNOT, SWAP and Toffoli. Hermitian unitary matrices H has the property e^=(\cos \theta)I+(i\sin \theta) H.


Parallel gates

The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
(or Kronecker product) of two quantum gates is the gate that is equal to the two gates in parallel. If we, as in the picture, combine the Pauli-''Y'' gate with the Pauli-''X'' gate in parallel, then this can be written as: : C = Y \otimes X = \begin 0 & -i \\ i & 0 \end \otimes \begin 0 & 1 \\ 1 & 0 \end = \begin 0 \begin 0 & 1 \\ 1 & 0 \end & -i \begin 0 & 1 \\ 1 & 0 \end \\ i \begin 0 & 1 \\ 1 & 0 \end & 0 \begin 0 & 1 \\ 1 & 0 \end\end = \begin 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \end Both the Pauli-''X'' and the Pauli-''Y'' gate act on a single qubit. The resulting gate C act on two qubits. Sometimes the tensor product symbol is omitted, and indexes are used for the operators instead. Example in eq. 2.


Hadamard transform

The gate H_2 = H \otimes H is the Hadamard gate applied in parallel on 2 qubits. It can be written as: :H_2 = H \otimes H = \frac \begin 1 & 1 \\ 1 & -1 \end \otimes \frac \begin 1 & 1 \\ 1 & -1 \end = \frac \begin 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end This "two-qubit parallel Hadamard gate" will when applied to, for example, the two-qubit zero-vector create a quantum state that have equal probability of being observed in any of its four possible outcomes; and We can write this operation as: :H_2 , 00\rangle = \frac \begin 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end \begin 1 \\ 0 \\ 0 \\ 0 \end = \frac \begin 1 \\ 1 \\ 1 \\ 1 \end = \frac , 00\rangle + \frac , 01\rangle +\frac , 10\rangle +\frac , 11\rangle = \frac Here the amplitude for each measurable state is . The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases. See measurement for details. H_2 performs the Hadamard transform on two qubits. Similarly the gate \underbrace_ = \bigotimes_1^n H = H^ = H_n performs a Hadamard transform on a register of n qubits. When applied to a register of n qubits all initialized to the Hadamard transform puts the quantum register into a superposition with equal probability of being measured in any of its 2^n possible states: :\bigotimes_0^H, 0\rangle = \Big( \bigotimes_0^ H \Big)\Big( \bigotimes_0^ , 0\rangle \Big) = \frac \begin 1 \\ 1 \\ \vdots \\ 1 \end = \frac \Big( , 0\rangle + , 1\rangle + \dots + , 2^n-1\rangle \Big)= \frac\sum_^, i\rangle This state is a ''uniform superposition'' and it is generated as the first step in some search algorithms, for example in amplitude amplification and phase estimation.
Measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
this state results in a random number between , 0\rangle and How random the number is depends on the
fidelity Fidelity is the quality of faithfulness or loyalty. Its original meaning regarded duty in a broader sense than the related concept of ''fealty''. Both derive from the Latin word ''fidēlis'', meaning "faithful or loyal". In the City of London ...
of the logic gates. If not measured, it is a quantum state with equal probability amplitude \frac for each of its possible states. The Hadamard transform acts on a register , \psi\rangle with n qubits such that , \psi\rangle = \bigotimes_^ , \psi_i\rangle as follows: :\bigotimes_0^H, \psi\rangle = \bigotimes_^\frac = \frac\bigotimes_^\Big(, 0\rangle + (-1)^, 1\rangle\Big) = H, \psi_0\rangle \otimes H, \psi_1\rangle \otimes \cdots \otimes H, \psi_\rangle


Application on entangled states

If two or more qubits are viewed as a single quantum state, this combined state is equal to the tensor product of the constituent qubits. Any state that can be written as a tensor product from the constituent subsystems are called separable states. On the other hand, an entangled state is any state that cannot be tensor-factorized, or in other words: ''An entangled state can not be written as a tensor product of its constituent qubits states.'' Special care must be taken when applying gates to constituent qubits that make up entangled states. If we have a set of ''N'' qubits that are entangled and wish to apply a quantum gate on ''M'' < ''N'' qubits in the set, we will have to extend the gate to take ''N'' qubits. This application can be done by combining the gate with an
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
such that their tensor product becomes a gate that act on ''N'' qubits. The identity matrix is a representation of the gate that maps every state to itself (i.e., does nothing at all). In a circuit diagram the identity gate or matrix will often appear as just a bare wire. For example, the Hadamard gate acts on a single qubit, but if we feed it the first of the two qubits that constitute the entangled
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a fo ...
we cannot write that operation easily. We need to extend the Hadamard gate H with the identity gate I so that we can act on quantum states that span ''two'' qubits: :K = H \otimes I = \frac \begin 1 & 1 \\ 1 & -1 \end \otimes \begin 1 & 0 \\ 0 & 1\end = \frac \begin 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end The gate K can now be applied to any two-qubit state, entangled or otherwise. The gate K will leave the second qubit untouched and apply the Hadamard transform to the first qubit. If applied to the Bell state in our example, we may write that as: :K \frac = \frac \begin 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end \frac \begin1 \\ 0 \\ 0 \\ 1\end = \frac \begin 1 \\ 1 \\ 1 \\ -1 \end = \frac


Computational complexity and the tensor product

The time complexity for multiplying two n \times n-matrices is at least if using a classical machine. Because the size of a gate that operates on q qubits is 2^q \times 2^q it means that the time for simulating a step in a quantum circuit (by means of multiplying the gates) that operates on generic entangled states is For this reason it is believed to be
intractable Intractable may refer to: * Intractable conflict, a form of complex, severe, and enduring conflict * Intractable pain, pain which cannot be controlled/cured by any known treatment * Intractable problem In theoretical computer science and mathema ...
to simulate large entangled quantum systems using classical computers. Subsets of the gates, such as the Clifford gates, or the trivial case of circuits that only implement classical boolean functions (e.g. combinations of X, CNOT, Toffoli), can however be efficiently simulated on classical computers. The state vector of a quantum register with n qubits is 2^n complex entries. Storing the probability amplitudes as a list of
floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
values is not tractable for large n.


Unitary inversion of gates

Because all quantum logical gates are reversible, any composition of multiple gates is also reversible. All products and tensor products (i.e. series and parallel combinations) of unitary matrices are also unitary matrices. This means that it is possible to construct an inverse of all algorithms and functions, as long as they contain only gates. Initialization, measurement, I/O and spontaneous decoherence are side effects in quantum computers. Gates however are purely functional and
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
. If U is a
unitary matrix In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, esp ...
, then U^\dagger U = UU^\dagger = I and The dagger (\dagger) denotes the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
. It is also called the Hermitian adjoint. If a function F is a product of m gates, the unitary inverse of the function F^\dagger can be constructed: Because (UV)^\dagger = V^\dagger U^\dagger we have, after repeated application on itself :F^\dagger = \left(\prod_^ A_i\right)^\dagger = \prod_^ A^\dagger_ = A_m^\dagger \cdot \dots \cdot A_2^\dagger \cdot A_1^\dagger Similarly if the function G consists of two gates A and B in parallel, then G=A\otimes B and Gates that are their own unitary inverses are called Hermitian or self-adjoint operators. Some elementary gates such as the Hadamard (''H'') and the Pauli gates (''I'', ''X'', ''Y'', ''Z'') are Hermitian operators, while others like the phase shift (''S'', ''T'', ''P'', CPhase) gates generally are not. For example, an algorithm for addition can be used for subtraction, if it is being "run in reverse", as its unitary inverse. The inverse quantum fourier transform is the unitary inverse. Unitary inverses can also be used for
uncomputation Uncomputation is a technique, used in reversible circuits, for cleaning up temporary effects on ancilla bits so that they can be re-used. Uncomputation is a fundamental step in quantum computing Quantum computing is a type of computation who ...
. Programming languages for quantum computers, such as
Microsoft Microsoft Corporation is an American multinational corporation, multinational technology company, technology corporation producing Software, computer software, consumer electronics, personal computers, and related services headquartered at th ...
's Q#,Operations and Functions (Q# documentation)
/ref> Bernhard Ömer's QCL, and IBM's Qiskit, contain function inversion as programming concepts.


Measurement

Measurement (sometimes called ''observation'') is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the basis vectors, with a likelihood equal to the square of the vector's length (in the
2-norm In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
) along that basis vector. This is known as the Born rule and appears as a
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. At the instant of measurement, the state is said to " collapse" to the definite single value that was measured. Why and how, or even if the quantum state collapses at measurement, is called the measurement problem. The probability of measuring a value with probability amplitude \phi is where , \cdot, is the modulus. Measuring a single qubit, whose quantum state is represented by the vector will result in , 0\rangle with probability and in For example, measuring a qubit with the quantum state \frac = \frac\begin 1 \\ -i \end will yield with equal probability either , 0\rangle or A quantum state , \Psi\rangle that spans qubits can be written as a vector in 2^n complex dimensions: This is because the tensor product of qubits is a vector in 2^n dimensions. This way, a register of qubits can be measured to 2^n distinct states, similar to how a register of classical
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
s can hold 2^n distinct states. Unlike with the bits of classical computers, quantum states can have non-zero probability amplitudes in multiple measurable values simultaneously. This is called ''superposition''. The sum of all probabilities for all outcomes must always be equal to . Another way to say this is that the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
generalized to \mathbb C^ has that all quantum states , \Psi\rangle with qubits must satisfy 1 = \sum_^, a_x, ^2, where a_x is the probability amplitude for measurable state A geometric interpretation of this is that the possible value-space of a quantum state , \Psi\rangle with qubits is the surface of the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A u ...
in \mathbb C^ and that the unitary transforms (i.e. quantum logic gates) applied to it are rotations on the sphere. The rotations that the gates perform is in the symmetry group U(2n). Measurement is then a probabilistic projection of the points at the surface of this complex sphere onto the basis vectors that span the space (and labels the outcomes). In many cases the space is represented as a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
\mathcal rather than some specific complex space. The number of dimensions (defined by the basis vectors, and thus also the possible outcomes from measurement) is then often implied by the operands, for example as the required state space for solving a
problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
. In Grover's algorithm, Lov named this generic basis vector set ''"the database"''. The selection of basis vectors against to measure a quantum state will influence the outcome of the measurement. See change of basis and Von Neumann entropy for details. In this article, we always use the ''computational basis'', which means that we have labeled the 2^n basis vectors of an -qubit register or use the binary representation In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, the basis vectors constitute an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
. An example of usage of an alternative measurement basis is in the BB84 cipher.


The effect of measurement on entangled states

If two
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s (i.e. qubits, or registers) are entangled (meaning that their combined state cannot be expressed as a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. This effect can be used for computation, and is used in many algorithms. The Hadamard-CNOT combination acts on the zero-state as follows: :\operatorname(H \otimes I), 00\rangle = \left( \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end \left( \frac \begin 1 & 1 \\ 1 & -1 \end \otimes \begin 1 & 0 \\ 0 & 1 \end \right) \right) \begin 1 \\ 0 \\ 0 \\ 0 \end = \frac \begin 1 \\ 0 \\ 0 \\ 1 \end = \frac This resulting state is the
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a fo ...
It cannot be described as a tensor product of two qubits. There is no solution for :\begin x \\ y \end \otimes \begin w \\ z \end = \begin xw \\ xz \\ yw \\ yz \end = \frac\begin 1 \\ 0 \\ 0 \\ 1 \end because for example needs to be both non-zero and zero in the case of and . The quantum state ''spans'' the two qubits. This is called ''entanglement''. Measuring one of the two qubits that make up this Bell state will result in that the other qubit logically must have the same value, both must be the same: Either it will be found in the state or in the state If we measure one of the qubits to be for example then the other qubit must also be because their combined state ''became'' Measurement of one of the qubits collapses the entire quantum state, that span the two qubits. The GHZ state is a similar entangled quantum state that spans three or more qubits. This type of value-assignment occurs ''instantaneously over any distance'' and this has as of 2018 been experimentally verified by QUESS for distances of up to 1200 kilometers. That the phenomena appears to happen instantaneously as opposed to the time it would take to traverse the distance separating the qubits at the speed of light is called the EPR paradox, and it is an open question in physics how to resolve this. Originally it was solved by giving up the assumption of local realism, but other interpretations have also emerged. For more information see the
Bell test experiments A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the ex ...
. The no-communication theorem proves that this phenomenon cannot be used for faster-than-light communication of classical information.


Measurement on registers with pairwise entangled qubits

Take a register A with qubits all initialized to and feed it through a parallel Hadamard gate Register A will then enter the state \frac \sum_^ , k\rangle that have equal probability of when measured to be in any of its 2^n possible states; , 0\rangle to Take a second register B, also with qubits initialized to , 0\rangle and pairwise CNOT its qubits with the qubits in register A, such that for each the qubits A_ and B_ forms the state If we now measure the qubits in register A, then register B will be found to contain the same value as A. If we however instead apply a quantum logic gate on A and then measure, then where F^\dagger is the unitary inverse of . Because of how unitary inverses of gates act, For example, say F(x)=x+3 \pmod, then The equality will hold no matter in which order measurement is performed (on the registers A or B), assuming that has run to completion. Measurement can even be randomly and concurrently interleaved qubit by qubit, since the measurements assignment of one qubit will limit the possible value-space from the other entangled qubits. Even though the equalities holds, the probabilities for measuring the possible outcomes may change as a result of applying , as may be the intent in a quantum search algorithm. This effect of value-sharing via entanglement is used in Shor's algorithm, phase estimation and in quantum counting. Using the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
to amplify the probability amplitudes of the solution states for some
problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
is a generic method known as " Fourier fishing".


Logic function synthesis

Functions and routines that only use gates can themselves be described as matrices, just like the smaller gates. The matrix that represents a quantum function acting on q qubits has size For example, a function that acts on a "qubyte" (a register of 8 qubits) would be represented by a matrix with 2^8 \times 2^8 = 256 \times 256 elements. Unitary transformations that are not in the set of gates natively available at the quantum computer (the primitive gates) can be synthesised, or approximated, by combining the available primitive gates in a
circuit Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circu ...
. One way to do this is to factor the matrix that encodes the unitary transformation into a product of tensor products (i.e. series and parallel circuits) of the available primitive gates. The group U(2''q'') is the symmetry group for the gates that act on q qubits. Factorization is then the
problem Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
of finding a path in U(2''q'') from the generating set of primitive gates. The
Solovay–Kitaev theorem In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2), then that set can be used to approximate any desired quantum gate with a relatively ...
shows that given a sufficient set of primitive gates, there exist an efficient approximate for any gate. For the general case with a large number of qubits this direct approach to circuit synthesis is
intractable Intractable may refer to: * Intractable conflict, a form of complex, severe, and enduring conflict * Intractable pain, pain which cannot be controlled/cured by any known treatment * Intractable problem In theoretical computer science and mathema ...
. Because the gates unitary nature, all functions must be reversible and always be
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
mappings of input to output. There must always exist a function F^ such that Functions that are not invertible can be made invertible by adding ancilla qubits to the input or the output, or both. After the function has run to completion, the ancilla qubits can then either be uncomputed or left untouched. Measuring or otherwise collapsing the quantum state of an ancilla qubit (e.g. by re-initializing the value of it, or by its spontaneous decoherence) that have not been uncomputed may result in errors, as their state may be entangled with the qubits that are still being used in computations. Logically irreversible operations, for example addition modulo 2^n of two n-qubit registers a and b, can be made logically reversible by adding information to the output, so that the input can be computed from the output (i.e. there exist a function In our example, this can be done by passing on one of the input registers to the output: The output can then be used to compute the input (i.e. given the output a+b and we can easily find the input; a is given and and the function is made bijective. All
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
ic expressions can be encoded as unitary transforms (quantum logic gates), for example by using combinations of the Pauli-X, CNOT and Toffoli gates. These gates are functionally complete in the Boolean logic domain. There are many unitary transforms available in the libraries of Q#, QCL, Qiskit, and other
quantum programming Quantum programming is the process of assembling sequences of instructions, called quantum circuits, that are capable of running on a quantum computer. Quantum programming languages help express quantum algorithms using high-level constructs. Th ...
languages. It also appears in the literature. For example, \mathrm(, x\rangle) = , x + 1 \pmod\rangle, where x_\text is the number of qubits that constitutes the register is implemented as the following in QCL:QCL 0.6.4 source code, the file "lib/examples.qcl"
/ref> cond qufunct inc(qureg x) In QCL, decrement is done by "undoing" increment. The prefix ! is used to instead run the unitary inverse of the function. !inc(x) is the inverse of inc(x) and instead performs the operation The cond keyword means that the function can be
conditional Conditional (if then) may refer to: * Causal conditional, if X then Y, where X is a cause of Y * Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a ...
. In the
model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes h ...
used in this article (the
quantum circuit In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly ...
model), a classic computer generates the gate composition for the quantum computer, and the quantum computer behaves as a coprocessor that receives instructions from the classical computer about which primitive gates to apply to which qubits. Measurement of quantum registers results in binary values that the classical computer can use in its computations. Quantum algorithms often contain both a classical and a quantum part. Unmeasured I/O (sending qubits to remote computers without collapsing their quantum states) can be used to create networks of quantum computers. Entanglement swapping can then be used to realize distributed algorithms with quantum computers that are not directly connected. Examples of distributed algorithms that only require the use of a handful of quantum logic gates is superdense coding, the quantum Byzantine agreement and the BB84 cipherkey exchange protocol.


See also

* Adiabatic quantum computation *
Cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tess ...
* Cloud-based quantum computing * Counterfactual definiteness * Counterfactual quantum computation * Landauer's principle *
Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
* One-way quantum computer * Quantum algorithm * Quantum cellular automaton * Quantum channel * Quantum finite automata * Quantum logic *
Quantum memory In quantum computing, quantum memory is the quantum-mechanical version of ordinary computer memory. Whereas ordinary memory stores information as binary states (represented by "1"s and "0"s), quantum memory stores a quantum state for later ...
* Quantum network * Quantum Zeno effect *
Reversible computation Reversible computing is any model of computation where the computational process, to some extent, is time-reversible. In a model of computation that uses deterministic transitions from one state of the abstract machine to another, a necessary c ...
* Unitary transformation (quantum mechanics)


Notes


References


Sources

* * * {{DEFAULTSORT:Quantum Gate Quantum information science Logic gates Australian inventions