# Phase gate (cuStateVec)
[[p-gate|Phase gate]] implementation using cuStateVec. The example applies $P(\pi/2) = S$ to $\lvert +\rangle$, rotating it to $\lvert i\rangle = \frac{1}{\sqrt{2}}(\lvert 0\rangle + i\lvert 1\rangle)$.

```c
// Compile: nvcc main.cu -o main -lcustatevec
// Run:     ./main

#include <stdio.h>
#include <math.h>
#include <cuda_runtime.h>
#include <custatevec.h>

int main() {
    const int nQubits = 1;
    const int dim = 1 << nQubits;

    cuDoubleComplex h_sv[2] = {{1,0},{0,0}};  // |0>
    cuDoubleComplex *d_sv;
    cudaMalloc(&d_sv, dim * sizeof(cuDoubleComplex));
    cudaMemcpy(d_sv, h_sv, dim * sizeof(cuDoubleComplex), cudaMemcpyHostToDevice);

    custatevecHandle_t handle;
    custatevecCreate(&handle);

    int32_t targets[] = {0};

    // Apply H: |0> -> |+>
    double s = M_SQRT1_2;
    cuDoubleComplex h_gate[4] = {{s,0},{s,0},{s,0},{-s,0}};
    custatevecApplyMatrix(
        handle, d_sv, CUDA_C_64F, nQubits,
        h_gate, CUDA_C_64F, CUSTATEVEC_MATRIX_LAYOUT_ROW, 0,
        targets, 1, NULL, NULL, 0,
        CUSTATEVEC_COMPUTE_64F, NULL, 0);

    // Apply P(pi/2) = S: |+> -> |i>
    cuDoubleComplex p_gate[4] = {{1,0},{0,0},{0,0},{0,1}};
    custatevecApplyMatrix(
        handle, d_sv, CUDA_C_64F, nQubits,
        p_gate, CUDA_C_64F, CUSTATEVEC_MATRIX_LAYOUT_ROW, 0,
        targets, 1, NULL, NULL, 0,
        CUSTATEVEC_COMPUTE_64F, NULL, 0);

    cudaMemcpy(h_sv, d_sv, dim * sizeof(cuDoubleComplex), cudaMemcpyDeviceToHost);
    for (int i = 0; i < dim; i++)
        printf("|%d>: (%.4f, %.4f)\n", i, cuCreal(h_sv[i]), cuCimag(h_sv[i]));
    // |0>: (0.7071, 0.0000)
    // |1>: (0.0000, 0.7071)  <- |i> = (|0> + i|1>) / sqrt(2)

    custatevecDestroy(handle);
    cudaFree(d_sv);
    return 0;
}
```