Quantum Gate
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Quantum Gate

In quantum computing and specifically the quantum circuit model of computation, a quantum gate (or quantum logic gate) is a basic quantum circuit operating on a small number of qubits. They 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. However, 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 ancillary bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.

Quantum logic gates are represented by unitary matrices. The most common quantum gates operate on spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. As matrices, quantum gates can be described by ${\displaystyle 2^{n}\times 2^{n}}$ sized unitary matrices, where ${\displaystyle n}$ is the number of qubits that the gate act on. The variables that the gates act upon, the quantum states, are vectors in ${\displaystyle 2^{n}}$ complex dimensions, where ${\displaystyle n}$ again is the number of qubits of the variable: The base vectors are the possible outcomes if measured, and a quantum state is a linear combinations of these outcomes.

## Commonly used gates

Quantum gates are usually represented as matrices. A gate which acts on k qubits is represented by a 2k x 2k unitary matrix. The number of qubits in the input and output of the gate have to be equal. 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. In the following, the vector representation of a single qubit is

${\displaystyle v_{0}|0\rangle +v_{1}|1\rangle \rightarrow {\begin{bmatrix}v_{0}\\v_{1}\end{bmatrix}}}$,

and the vector representation of two qubits is

${\displaystyle v_{00}|00\rangle +v_{01}|01\rangle +v_{10}|10\rangle +v_{11}|11\rangle \rightarrow {\begin{bmatrix}v_{00}\\v_{01}\\v_{10}\\v_{11}\end{bmatrix}}}$,

where ${\displaystyle |ab\rangle }$ is the basis vector representing a state where the first qubit is in the state ${\displaystyle |a\rangle }$ and the second qubit in the state ${\displaystyle |b\rangle }$.

### Hadamard (H) gate

The Hadamard gate acts on a single qubit. It maps the basis state ${\displaystyle |0\rangle }$ to ${\displaystyle {\frac {|0\rangle +|1\rangle }{\sqrt {2}}}}$ and ${\displaystyle |1\rangle }$ to ${\displaystyle {\frac {|0\rangle -|1\rangle }{\sqrt {2}}}}$, which means that a measurement will have equal probabilities to become 1 or 0 (i.e. creates a superposition). It represents a rotation of ${\displaystyle \pi }$ about the axis ${\displaystyle ({\hat {x}}+{\hat {z}})/{\sqrt {2}}}$. Equivalently, it is the combination of two rotations, ${\displaystyle \pi }$ about the X-axis followed by ${\displaystyle \pi /2}$ about the Y-axis. It is represented by the Hadamard matrix:

Circuit representation of Hadamard gate
${\displaystyle H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}}$.

The hadamard gate is the one-qubit version of the quantum fourier transform.

Since ${\displaystyle HH^{*}=I}$ where I is the identity matrix, H is indeed a unitary matrix.

### Pauli-X gate (= NOT gate)

Quantum circuit diagram of a NOT-gate

The Pauli-X gate acts on a single qubit. It is the quantum equivalent of the NOT gate for classical computers (with respect to the standard basis ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$, which privileges the Z-direction) . It equates to a rotation of the Bloch sphere around the X-axis by ? radians. It maps ${\displaystyle |0\rangle }$ to ${\displaystyle |1\rangle }$ and ${\displaystyle |1\rangle }$ to ${\displaystyle |0\rangle }$. Due to this nature, it is sometimes called bit-flip. It is represented by the Pauli matrix:

${\displaystyle X={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$.

${\displaystyle NOT\,NOT=I}$ where I is the identity matrix

### Pauli-Y gate

The Pauli-Y gate acts on a single qubit. It equates to a rotation around the Y-axis of the Bloch sphere by ${\displaystyle \pi }$ radians. It maps ${\displaystyle |0\rangle }$ to ${\displaystyle i|1\rangle }$ and ${\displaystyle |1\rangle }$ to ${\displaystyle -i|0\rangle }$. It is represented by the Pauli Y matrix:

${\displaystyle Y={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}}$.

### Pauli-Z (${\displaystyle R_{\pi }}$) gate

The Pauli-Z gate acts on a single qubit. It equates to a rotation around the Z-axis of the Bloch sphere by ${\displaystyle \pi }$ radians. Thus, it is a special case of a phase shift gate (next) with ${\displaystyle \phi =\pi }$. It leaves the basis state ${\displaystyle |0\rangle }$ unchanged and maps ${\displaystyle |1\rangle }$ to ${\displaystyle -|1\rangle }$. Due to this nature, it is sometimes called phase-flip. It is represented by the Pauli Z matrix:

${\displaystyle Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$.

Note the square roots of the identity matrix -

${\displaystyle I^{2}=X^{2}=Y^{2}=Z^{2}=I}$

### Square root of NOT gate (?NOT)

Quantum circuit diagram of a square-root-of-NOT gate

The NOT gate acts on a single qubit.

${\displaystyle {\sqrt {X}}={\sqrt {NOT}}={\frac {1}{2}}{\begin{bmatrix}1+i&1-i\\1-i&1+i\end{bmatrix}}}$.

${\displaystyle {\sqrt {NOT}}\,{\sqrt {NOT}}=NOT}$, so this gate is a square root of the NOT gate.

Similar squared root-gates can be constructed for all other gates by finding the unitary matrix that, multiplied by itself, yields the gate one wishes to construct the squared root gate of. All fractional exponents of all gates can be created in this way. (Only approximations of irrational exponents are possible to synthesize from composite gates whose elements are not themselves irrational, since exact synthesis would result in infinite gate depth.)

### Phase shift (${\displaystyle R_{\phi }}$) gates

This is a family of single-qubit gates that leave the basis state ${\displaystyle |0\rangle }$ unchanged and map ${\displaystyle |1\rangle }$ to ${\displaystyle e^{i\phi }|1\rangle }$. The probability of measuring a ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$ 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) on the Bloch sphere by ${\displaystyle \phi }$ radians.

${\displaystyle R_{\phi }={\begin{bmatrix}1&0\\0&e^{i\phi }\end{bmatrix}}}$

where ${\displaystyle \phi }$ is the phase shift. Some common examples are the ${\displaystyle {\frac {\pi }{8}}}$ gate where ${\displaystyle \phi ={\frac {\pi }{4}}}$, the phase gate where ${\displaystyle \phi ={\frac {\pi }{2}}}$ and the Pauli-Z gate where ${\displaystyle \phi =\pi }$.

### Swap (S) gate

Circuit representation of SWAP gate

The swap gate swaps two qubits. With respect to the basis ${\displaystyle |00\rangle }$, ${\displaystyle |01\rangle }$, ${\displaystyle |10\rangle }$, ${\displaystyle |11\rangle }$, it is represented by the matrix:

${\displaystyle {\mbox{SWAP}}=S={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}}$.

### Square root of Swap gate (?S)

Circuit representation of ${\displaystyle {\sqrt {\mbox{SWAP}}}}$ gate

The sqrt(swap) gate performs half-way of a two-qubit swap. It is universal such that any quantum many qubit gate can be constructed from only sqrt(swap) and single qubit gates.

${\displaystyle {\sqrt {S}}={\sqrt {\mbox{SWAP}}}={\begin{bmatrix}1&0&0&0\\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\\0&0&0&1\\\end{bmatrix}}}$.

### Controlled (cX cY cZ) gates

Circuit representation of controlled NOT gate

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 ${\displaystyle |1\rangle }$, and otherwise leaves it unchanged. It is represented by the matrix

${\displaystyle {\mbox{CNOT}}=cX={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}}$.

More generally if U is a gate that operates on single qubits with matrix representation

${\displaystyle U={\begin{bmatrix}u_{00}&u_{01}\\u_{10}&u_{11}\end{bmatrix}}}$,

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.

Circuit representation of controlled-U gate
${\displaystyle |00\rangle \mapsto |00\rangle }$
${\displaystyle |01\rangle \mapsto |01\rangle }$
${\displaystyle |10\rangle \mapsto |1\rangle \otimes U|0\rangle =|1\rangle \otimes \left(u_{00}|0\rangle +u_{10}|1\rangle \right)}$
${\displaystyle |11\rangle \mapsto |1\rangle \otimes U|1\rangle =|1\rangle \otimes \left(u_{01}|0\rangle +u_{11}|1\rangle \right)}$

The matrix representing the controlled U is

${\displaystyle {\mbox{C}}(U)={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&u_{00}&u_{01}\\0&0&u_{10}&u_{11}\end{bmatrix}}}$.
controlled X-, Y- and Z- gates
controlled-X gate
controlled-Y gate
controlled-Z gate

When U is one of the Pauli matrices, ?x, ?y, or ?z, the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used.[1]

The CNOT gate is generally used in quantum computing to generate entangled states.

### Toffoli (CCNOT) gate

Circuit representation of Toffoli gate

The Toffoli gate, also CCNOT gate or Deutsch ${\displaystyle D(\pi /2)}$ gate, is a 3-bit gate, which is universal for classical computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If the first two bits are in the state ${\displaystyle |1\rangle }$, it applies a Pauli-X (or NOT) on the third bit, else it does nothing. It is an example of a controlled gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table.

Truth table Matrix form
INPUT OUTPUT
0   0   0   0   0   0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0

${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\end{bmatrix}}}$

It can be also described as the gate which maps ${\displaystyle |a,b,c\rangle }$ to ${\displaystyle |a,b,c\oplus ab\rangle }$.

### Fredkin (CSWAP) gate

Circuit representation of Fredkin gate

The Fredkin gate (also CSWAP or cS gate) is a 3-bit gate that performs a controlled swap. It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.

Truth table Matrix form
INPUT OUTPUT
C I1 I2 C O1 O2
0   0   0   0   0   0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1

${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\\end{bmatrix}}}$

### Ising (XX) gate

The Ising gate (or XX gate) is a 2-bit gate that is implemented natively in some trapped-ion quantum computers.[2][3] It is defined as

${\displaystyle XX_{\phi }={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&-ie^{i\phi }\\0&1&-i&0\\0&-i&1&0\\-ie^{-i\phi }&0&0&1\\\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&0&e^{i(\phi -\pi /2)}\\0&1&-i&0\\0&-i&1&0\\e^{i(-\phi -\pi /2)}&0&0&1\\\end{bmatrix}}}$

### Deutsch (${\displaystyle D_{\theta }}$) gate

Deutsch (or ${\displaystyle D_{\theta }}$) gate is a 3 qubit gate. It is defined as

${\displaystyle |a,b,c\rangle \mapsto {\begin{cases}i\cos(\theta )|a,b,c\rangle +\sin(\theta )|a,b,1-c\rangle &{\mbox{for }}a=b=1\\|a,b,c\rangle &{\mbox{otherwise.}}\end{cases}}}$

## Universal quantum gates

Both CNOT and ${\displaystyle {\sqrt {\mbox{SWAP}}}}$ are universal two-qubit gates and can be transformed into each other.

Informally, 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 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.

One simple set of two-qubit universal quantum gates is the Hadamard gate ${\displaystyle H}$, the ${\displaystyle \pi /8}$ gate ${\displaystyle R_{Z}(\pi /4)=diag(1,e^{\frac {i\pi }{4}})}$, and the controlled-NOT gate ${\displaystyle cX}$.[4]

A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate ${\displaystyle D(\theta )}$, which performs the transformation[5]

${\displaystyle |a,b,c\rangle \mapsto {\begin{cases}i\cos(\theta )|a,b,c\rangle +\sin(\theta )|a,b,1-c\rangle &{\mbox{for }}a=b=1\\|a,b,c\rangle &{\mbox{otherwise.}}\end{cases}}}$

The universal classical logic gate, the Toffoli gate, is reducible to the Deutsch gate, ${\displaystyle D({\begin{matrix}{\frac {\pi }{2}}\end{matrix}})}$, thus showing that all classical logic operations can be performed on a universal quantum computer.

Another set of universal quantum gates consists of the Ising gate and the phase-shift gate. These are the set of gates natively available in some trapped-ion quantum computers.[3]

## Measurement

Circuit representation of measurement. The two lines on the right hand side represents a classical bit, the single line on the left hand side represents a qubit.

Measurement appears as similar to a quantum gate even though it is not a gate, because measurement actively alters the observed variable. Measurement takes a quantum state and projects it to one of the base vectors, with a likelihood equal to the square of the vectors depth along that base vector. This is a non-reversible operation as it sets the quantum state equal to the base vector that represents the measured state (the state "collapses" to a definite singular value). Why and how this is so is called the measurement problem.

If two different quantum registers are entangled (they are not linearly independent), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. An example of such a lineary inseparable state is the EPR pair, which can be constructed with the CNOT and the Hadamard gates, described above. This effect is used in many algorithms: If two variables A and B are maximally entangled (the bell state is the simplest example of this), a function F is applied to A such that A is updated to the value of F(A), followed by measurement of A, then B will when measured be a value such that F(B) = A. This way, measurement of one register can be used to assign properties to some other registers. As shown by the Bell test experiments, this type of assignment happens instantanously over any distance.

## Circuit composition and 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 (The Kronecker product in standard basis). An entangled state is any state that can not be tensor-factorized (the state can not be separated into its constituent qubits). The CNOT, Ising and Toffoli gates are examples of gates that act on states constructed of multiple qubits.

The tensor product of two n-qubit quantum gates generates the gate that is equal to the two gates in parallel. This gate will act on ${\displaystyle 2n}$ qubits. For example, the gate ${\displaystyle G=H\otimes H}$ is the hadamard gate (${\displaystyle H}$) applied in parallel on 2 qubits. It can be written as

${\displaystyle G=H\otimes H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}}$

This "two-qubit parallel hadamard gate" will when applied to, for example, the two-qubit zero-vector (${\displaystyle |00\rangle }$) create a quantum state that have equal probability of being observed in any of its four possible outcomes; 00, 01, 10 and 11. We can write this operation as:

${\displaystyle G|00\rangle ={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}{\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1\\1\\1\\1\end{bmatrix}}={\frac {1}{2}}|00\rangle +{\frac {1}{2}}|01\rangle +{\frac {1}{2}}|10\rangle +{\frac {1}{2}}|11\rangle ={\frac {|00\rangle +|01\rangle +|10\rangle +|11\rangle }{2}}}$

The probability to observe any state is the absolute value of the amplitude squared, which in the above example means that there is one in four that we observe any of the four values. (Amplitudes are complex values, so their squares are not always positive.)

If we have a set of N qubits that are entangled (their combined state can not be tensor-factorized into an expression of the individual qubits) 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 can be done by combining the gate with an identity matrix such that their tensor product becomes a gate that act on N qubits. The identity matrix (${\displaystyle I}$) 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 appear as just a wire.

For example, the hadamard transform (${\displaystyle H}$) acts on a single qubit, but if we for example feed it the second of the two qubits that constitute the entangled Bell state ${\displaystyle {\frac {|00\rangle +|11\rangle }{\sqrt {2}}}}$, we can not write that operation easily. We need to extend the hadamard transform with the do-nothing gate ${\displaystyle I}$ so that we can act on quantum states that span two qubits:

${\displaystyle M=H\otimes I={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&0\\0&1\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1&0&0\\1&-1&0&0\\0&0&1&1\\0&0&1&-1\end{bmatrix}}}$

The gate ${\displaystyle M}$ can now be applied to any two-qubit state, entangled or otherwise. The M-gate will leave the first qubit untouched and apply the hadamard transform to the 2nd qubit. If applied to the bell state in our example, we may write that as:

${\displaystyle M{\frac {|00\rangle +|11\rangle }{\sqrt {2}}}={\frac {1}{2}}{\begin{bmatrix}1&1&0&0\\1&-1&0&0\\0&0&1&1\\0&0&1&-1\end{bmatrix}}{\begin{bmatrix}1\\0\\0\\1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1\\1\\1\\-1\end{bmatrix}}={\frac {|00\rangle +|01\rangle +|10\rangle -|11\rangle }{2}}}$

Because the number of elements in the matrices is ${\displaystyle 2^{2x}}$, where x is the number of qubits the gates act on, it is intractable to simulate large quantum systems using classical computers.

## History

The current notation for quantum gates was developed by Barenco et al.,[6] building on notation introduced by Feynman.[7]

## Notes

1. ^ M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000
2. ^ http://online.kitp.ucsb.edu/online/mbl_c15/monroe/pdf/Monroe_MBL15Conf_KITP.pdf
3. ^ a b http://iontrap.umd.edu/wp-content/uploads/2012/12/nature18648.pdf
4. ^ M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2016, p. 189; ISBN 978-1-107-00217-3
5. ^ Deutsch, David (September 8, 1989), "Quantum computational networks", Proc. R. Soc. Lond. A, 425 (1989): 73-90, Bibcode:1989RSPSA.425...73D, doi:10.1098/rspa.1989.0099
6. ^ Phys. Rev. A 52 3457-3467 (1995), doi:10.1103/PhysRevA.52.3457; e-print arXiv:quant-ph/9503016
7. ^ R. P. Feynman, "Quantum mechanical computers", Optics News, February 1985, 11, p. 11; reprinted in Foundations of Physics 16(6) 507-531.

## References

• M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000

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