Quantum Computing 2 -- Quantum gates (single qubit)

Quantum gates are the elementary operations for the qubits of a quantum computer. They are comparable to the logic gates for classical bits. In contrast to the Boolean operators, all quantum gates are reversible, so there is always an inverse operation that can undo all computation steps. There is also a much wider variety of possible operations for qubits.

A quantum gate for a single qubit is described by a unitary matrix: U=(abcd)a,b,c,dC U = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)\, \quad a,b,c,d \in \mathbb{C} The operation on a state ψ=α0+β1| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle corresponds to the multiplication of the matrix UU with the state vector: Uψ=U(α0+β1)=(abcd)(αβ)=(aα+bβcα+dβ)=(aα+bβ)0+(cα+dβ)1\begin{aligned} U |\psi \rangle &= U (\alpha |0\rangle + \beta |1\rangle)\\ &= \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \\ &= \left( \begin{array}{c} a \alpha + b \beta \\ c \alpha + d \beta \end{array} \right) \\ &= (a \alpha + b \beta) |0\rangle + (c \alpha + d \beta) |1\rangle \end{aligned} Executing two operators UU and \ (V ) one after the other is equivalent to an operator resulting from the matrix multiplication of UU and VV : VU=(abcd)(abcd)=(aa+bcab+bdca+dccb+dd)\begin{aligned} V \cdot U &= \left( \begin{array}{cc} a' & b' \\ c' & d' \end{array} \right) \cdot \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \\ &= \left( \begin{array}{cc} a' a + b' c & a' b + b' d \\ c' a + d' c & c' b + d' d \end{array} \right) \end{aligned} Since the operators are unitary matrices UU , the adjoint matrix UU ^ \dagger corresponds to the inverse operator. (The quantum gate UU ^ \dagger thus reverses the effect of the gate UU .) U=(acbd)UU=I=(1001)\begin{aligned} U^\dagger &= \left( \begin{array}{cc} a^\ast & c^\ast \\ b^\ast & d^\ast \end{array} \right) \\ U^\dagger \cdot U &= I = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \end{aligned} where II is the identity operator or unit matrix and zz^\ast denotes the complex conjungate of zz.

Hadamard-Gate

The Hadamard gate is one of the most fundamental operations in quantum computing. The Hadamard gate for single qubit is described in the {0,1}\{| 0 \rangle, | 1 \rangle \} basis by the following matrix:

H=12(1111)H0=12(0+1)+H1=12(01)\begin{aligned} H &= \frac{1}{\sqrt{2}} \left( \begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array} \right) \\ H|0\rangle &= \frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle) \equiv |+\rangle\\ H|1\rangle &= \frac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \equiv |-\rangle \end{aligned} The basis states 0| 0 \rangle and 1| 1 \rangle are converted by the Hadamard gate into a superposition. You can also understand this operation as a transition between the bases {0,1}\{| 0 \rangle, | 1 \rangle \} and {+,}\{| + \rangle, | - \rangle \} . The Hadamard matrix is self-adjoint (H=HH ^ \dagger = H ), so that the two-time application of the Hadamard gate gives the identity (HH=HH=IH \cdot H = H ^ \dagger \cdot H = I ).

Phase-Shift

The phase-shift RϕR_ \phi is a whole family of single-qubit gates. It is described by the following matrix in the {0,1}\{| 0 \rangle, | 1 \rangle \} basis:

Rϕ=(100exp(iϕ))Rϕ0=0Rϕ1=exp(iϕ)1\begin{aligned} R_\phi &= \left( \begin{array}{cc} 1 & 0 \\ 0 & \exp(i \phi) \end{array} \right) \\ R_\phi|0\rangle &= |0\rangle \\ R_\phi|1\rangle &= \exp(i \phi) |1\rangle \end{aligned} where exp(iϕ)=cos(ϕ)+isin(ϕ)\exp (i \phi) = \cos (\phi) + i \sin (\phi) . This operation leaves the amplitudes of the state vector unchanged and only introduces a phase shift between the vector components by the angle ϕ[π,π]\phi \in [- \pi, \pi] . The adjoint matrix yields Rϕ=RϕR_ \phi ^ \dagger = R _ {- \phi} such that RϕRϕ=IR _ {- \phi} \cdot R_ \phi = I .

Of particular importance are the phase shift gates T=Rπ/4T = R _ {\pi / 4} , S=Rπ/2S = R _ {\pi / 2} and Z=RπZ = R_ \pi .

Pauli-Gates

The Pauli gates are a family of three operators with the matrices X=(0110)Y=(0ii0)Z=(1001) \begin{aligned} X &= \left(\begin{array}{cc}0 & 1 \\ 1 & 0 \end{array}\right) \\ Y &= \left(\begin{array}{cc}0 & -i \\ i & 0 \end{array}\right) \\ Z &= \left(\begin{array}{cc}1 & 0 \\ 0 & -1 \end{array}\right) \end{aligned} The XX gate is the quantum mechanical equivalent to the classical NOT operator and interchanges the basis states so that X0=1X | 0 \rangle = | 1 \rangle and X1=0X | 1 \rangle = | 0 \rangle (bit-flip). The ZZ gate performs a special phase-shift operation around the angle π\pi (phase-flip). The YY gate is a combination of bit and phase flip. The Pauli matrices are self-adjoint and obey the algebra XX=YY=ZZ=iXYZ=IX \cdot X = Y \cdot Y = Z \cdot Z = -i X \cdot Y \cdot Z = I .

Note by Markus Michelmann
1 year, 1 month ago

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Thanks for posting these, as well as the problems. It's cool to realize that this stuff is actually somewhat accessible to non-specialists.

Steven Chase - 1 year, 1 month ago

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Can anyone please tell me the latex code to write example in notes as in wiki.

@Calvin Lin, @Markus Michelmann

Ram Mohith - 10 months, 1 week ago

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