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Creating a Quantum toolbox

  • At this point, we have shown how to implement a simple circuit with two types of gates

  • Nonetheless, with this two gates, we can't model complex circuits

  • We need something more universal

Introduction to the Bloch Sphere

  • All states of a qubit \(|\psi\rangle\) can be drawn on the surface of an sphere. We can show it by going back to the definition of a single qubit: \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\)

  • We know that \(\alpha, \beta \in \mathbb{C}\) which means that \(\alpha=r_1e^{i\gamma_1}\) and \(\beta=r_2e^{i\gamma_2}\)

  • By Observing that \(|\alpha|^2 + |\beta|^2 = r_1^2 + r_2^2 = 1\) and \(0\leq r_1 \leq 1\), \(0\leq r_2 \leq 1\), we can therefore use the following change of variable: \(r_1=cos(\frac{\theta}{2})\) and \(r_2=sin(\frac{\theta}{2})\)

\[ \begin{aligned} |\psi\rangle & = cos\frac{\theta}{2}e^{i\gamma_1}|0\rangle + sin\frac{\theta}{2}e^{i\gamma_2}|1\rangle \end{aligned} \]

  • We can simplify this expression by noting that \(|\psi\rangle\) can be multiplied by any complex number or global phase \(z\) with \(|z|=1\) without changing its state

  • Let's take \(z=e^{-i\gamma_1}\) as global phase:

\[ \begin{aligned} e^{-i\gamma_1}|\psi\rangle & = cos\frac{\theta}{2}|0\rangle + sin\frac{\theta}{2}e^{i\gamma_2-\gamma_1}|1\rangle & = cos\frac{\theta}{2}|0\rangle + sin\frac{\theta}{2}e^{i\sigma}|1\rangle \end{aligned} \]
  • With \(\theta\) and \(\sigma\), representing respectively, the polar angle and an azimuthal angle.

$$ \theta=\frac{\pi}{4};\sigma=0 \hspace{5cm} \theta=\frac{\pi}{4};\sigma=\frac{\pi}{4} $$

Warning

If you did not already guess it, the Pauli X, Y and Z gates are rotations around their respective axis. The X gate transform \(|0\rangle\) to \(|1\rangle\) which corresponds to \(\pi\) rotation

  • More general rotation operators are generated by exponentiation of the Pauli matrices, i.e., \(e^{iAx} = cos(x)I+isin(x)A\)

    • The rotation around the \(X\) axis: \(R_X(\theta)=e^{-i\frac{\theta}{2}X}=\begin{pmatrix}cos\frac{\theta}{2} & -isin\frac{\theta}{2} \\\\ -isin\frac{\theta}{2} & cos\frac{\theta}{2}\end{pmatrix}\)
    • The rotation around the \(Y\) axis: \(R_Y(\theta)=e^{-i\frac{\theta}{2}Y}=\begin{pmatrix}cos\frac{\theta}{2} & -sin\frac{\theta}{2} \\\\ sin\frac{\theta}{2} & cos\frac{\theta}{2}\end{pmatrix}\)
    • The rotation around the \(Z\) axis: \(R_Z(\theta)=e^{-i\frac{\theta}{2}Z}=\begin{pmatrix}1 & 0 \\\\ 0 & e^{i\theta}\end{pmatrix}\)

Implementing the rotation gates

Rotations gates

  • Implement the 3 Rotation gates \(R_x\), \(R_y\) and \(R_z\)
  • To test the rotation gates, set the variable SOURCE_FILES as follows set(SOURCE_FILES src/test_rxryrz.cpp src/kernels.cpp src/blochSphere.cpp) in the CMakeLists.txt file

Building and the code

mkdir build-rotation-gates && cd build-rotation-gates
cmake .. -DBUILD=U -DUSER_FLAGS="-lGL -lGLU -lglut"
make fpga
LD_PRELOAD=${JEMALLOC_PRELOAD} ./quantum.fpga
  • You should see the following on your screen:

OpenGL Bloch Sphere for the resulting qubit

  • Add the following code in the void rx(...) function body 
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    double angle_2 = 0.5*angle;
double cos = std::cos(angle_2);
double sin = std::sin(angle_2);
std::complex<float> A (cos,0.0f);
std::complex<float> B (0.0f,-1.0*sin);
std::complex<float> C (0.0f,-1.0*sin);
std::complex<float> D (cos,0.0f);
apply_gate(queue,stateVector_d,std::pow(2,numQubits)/2,target,A,
                                                          B,
                                                          C,
                                                          D);
  • Add the following code in the void ry(...) function body 
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    double angle_2 = 0.5*angle;
double cos = std::cos(angle_2);
double sin = std::sin(angle_2);
std::complex<float> A (cos,0.0f);
std::complex<float> B (-1.0*sin,0.0f);
std::complex<float> C (sin,0.0f);
std::complex<float> D (cos,0.0f);
apply_gate(queue,stateVector_d,std::pow(2,numQubits)/2,target,A,
                                                          B,
                                                          C,
                                                          D);
  • Add the following code in the void rz(...) function body 
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    double angle_2 = 0.5*angle;
double cos = std::cos(angle_2);
double sin = std::sin(angle_2);
std::complex<float> A (cos,-sin);
std::complex<float> B (0.0f,0.0f);
std::complex<float> C (0.0f,0.0f);
std::complex<float> D (cos,sin);
apply_gate(queue,stateVector_d,std::pow(2,numQubits)/2,target,A,
                                                              B,
                                                              C,
                                                              D);