What are Quantum Algorithms and Why do We Need Them?

    from math import sqrt, pi
    import matplotlib.pyplot as plt
    
    def phase_shift(phi):
    """
    phase shift quantum operation
    :param phi: the angle of phase shift in degree unit
    :return: quantum operation matrix
    """
    mat = [[1., 0.], [0., exp(1j*pi*(phi/360))] ]
    return mat
    
    def amplitude_damp(gamma):
    """
    amplitude damping quantum operation
    :param gamma: the decay rate coefficient for exponential falloff from 1 to 0
    :return: quantum operation matrix
    """
    mat = [[1., 0.], [(1.-sqrt(1-gamma**2)), -(gamma/sqrt(1-gamma**2))]]
    return mat
    
    def add_circuit_element(mats, op_func, op_args):
    """
    combine multiple quantum operations into one circuit element matrix
    :param mats: list of matrices representing quantum gates or states
    :param op_func: function that returns a single quantum operation matrix given its arguments
    :param op_args: tuple of arguments passed to op_func
    :return: combined matrix representing circuit element
    """
    n = len(mats[0])
    new_mat = np.eye(n)
    if callable(op_func):
        elem_mat = op_func(*op_args)
        assert elem_mat.shape == (n,n), "matrix dimensions must match"
        for i in range(n):
            for j in range(n):
                new_mat[i][j] = sum([elem_mat[i][k]*mats[k][j] for k in range(len(mats))])
    else:
        # assume op_func is an iterable containing a mix of strings and numeric values
        idx = 0
        for i in range(n):
            row_vals = []
            for j in range(idx, len(op_func), 2):
                val = complex(op_func[j+1], 0) if j+1 < len(op_func) else 1.0
                if op_func[j] == 'R':
                    row_vals += [val*np.exp(complex(0,phase)*op_args)]
                elif op_func[j] == 'T' or op_func[j] == 'U':
                    theta = float(op_args['theta'])
                    phi = float(op_args['phi'])
                    lamda = float(op_args['lamda'])
                    if op_func[j] == 'T':
                        circ_mat = np.array([[1, 0], [0, np.exp(1j*phase)]])
                    else:
                        circ_mat = np.array([[np.cos(theta/2), -np.exp(1j*lamda)*np.sin(theta/2)], 
                                             [-np.exp(1j*phi)*np.sin(theta/2), np.exp(1j*(phi+lamda))*np.cos(theta/2)]])
                    row_vals += [val*circ_mat]
                idx = j+2
            new_row = [sum([v[k] for v in row_vals]) for k in range(len(new_mat[0]))]
            new_mat[i] = new_row
    return new_mat
    
    def grover_search(psi, Uf, iterations):
    """
    perform search using Grover's iteration algorithm on unstructured database with query oracle Uf
    :param psi: initial superposition vector over all elements in database
    :param Uf: oracle quantum operator that returns True when queried value matches expected value
    :param iterations: number of times to repeat the search step
    :return: measured index of matching value or None if no match found after max iterations reached
    """
    def bitstring(state):
        """
        convert quantum state vector to binary string representation
        :param state: numpy array representing quantum state
        :return: corresponding binary string
        """
        s = ""
        for prob in state.flatten():
            if abs(prob) > threshold:
                s += "1"
            else:
                s += "0"
        return s
    
    def amplituide_projection(psi, idx):
        """
        project quantum state onto basis vector corresponding to given index
        :param psi: numpy array representing quantum state
        :param idx: integer index to which we want to project our quantum state
        :return: projected quantum state represented by its amplitude vector
        """
        dim = int(np.log2(len(psi)))
        vec = np.zeros(2**dim).astype('complex')
        vec[idx] = psi[idx]/sqrt(abs(psi[idx])**2)
        return vec
    
    alpha = beta = 1./sqrt(len(psi))    # set up weights for post-measurement probabilities
    db = {}   # dictionary to store frequencies of each measurement outcome
    best_result = None   # track result from previous round so we can avoid unnecessary measurements
    for it in range(iterations):
        psi = np.dot(Uf, psi)     # apply oracle transformation
        psi = np.dot(amplitude_damp(beta), psi)   # apply amplitude damping before measurement
    
        proj_vecs = [amplituide_projection(psi, i) for i in range(2**(int(np.log2(len(psi)))))]
        probs = [np.dot(alpha*vec.conj(), vec)**2 for vec in proj_vecs]
    
        meas_outcome = int("".join(["1" if p >= threshold else "0" for p in probs]), 2)  # make measurement decision
        print("Iteration", it, ", measurement outcome:", bin(meas_outcome)[2:], end="\r")
    
        if meas_outcome not in db:
            db[meas_outcome] = 1
        else:
            db[meas_outcome] += 1
    
        if best_result is None or db[best_result] <= len(db)//2:
            best_result = meas_outcome
    
        if db[best_result] > len(db)//2 or it==iterations-1:
            break
    
        alpha *= db[meas_outcome]**-0.5      # update pre-measurement weight based on most recent measurement frequency
    
    print("\nMeasurement statistics:")
    sorted_keys = sorted(db.keys())
    for key in sorted_keys:
        freq = db[key] / iterations
        prob = "{:.3f}".format(freq)
        print("Outcome", str(bin(key))[2:].zfill(int(np.log2(len(psi)))), ": frequency", freq, ", probability", prob)
    
    return best_result
    
    if __name__=="__main__":
    # prepare some example data
    N = 20   # number of input elements in database
    target_value = random.randint(0, N-1)
    data = [random.randint(0, N-1) for _ in range(N)]
    while target_value in data[:data.index(target_value)+1]:
        data[-1] = random.randint(0, N-1)
    
    # build Hamiltionian matrix describing ideal search problem
    ham = np.zeros((2**N, 2**N)).astype('complex')
    for i in range(2**N):
        for j in range(2**N):
            mask = format(i ^ j, '0{}b'.format(N))
            parity = sum([int(mask[k])*((-1)**((k*((~i&~j)>>k)&1))) for k in range(N)]) % 2
            if parity == 0:
                term = (-1)**i * ((-1)**j * pow(2,-N/2, N))
            else:
                term = 0
            ham[i][j] = term
    
    # build oracle Uf that returns True when queried value matches expected value
    def Uf(x):
        bits = format(x,'0{}b'.format(N))
        pivot_bit = "".join([bits[(N//2)-1-(N%2):][:N%2]])
        mask = "".join(['1']*N)
        flip = ""
        for k in range(N):
            if ((N//2)-1-(N%2)+k)%N!= 0:
                flip += '1'
            else:
                flip += '0'
        y = int(flip+pivot_bit+"0"*math.ceil(N/2), 2)
        return ham[x][y] == -pow(2,-N/2, N)
    
    # run search algorithm with two queries per step, stopping early if no more than half of outcomes have been observed
    psi = np.ones(2**N)/sqrt(2**N)   # start with uniform superposition
    best_result = grover_search(psi, Uf, 2**(N//2))
    print("Best guess at correct answer:", best_result, ", actual answer:", target_value)
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
    代码解读