def density_op_random(n, real_or_complex=2, rank=None):
r"""随机生成一个密度矩阵的 numpy 形式。
Args:
n (int): 量子比特数量
real_or_complex (int, optional): 默认为 2，即生成复数组，若为 1，则生成实数组
rank (int, optional): 矩阵的秩，默认为 :math:`2^n` （当 ``rank`` 为 ``None`` 时）
Returns:
numpy.ndarray: 一个形状为 ``(2**n, 2**n)`` 的 numpy 数组
"""
assert n > 0, 'qubit number must be positive'
rank = rank if rank is not None else 2**n
assert 0 < rank <= 2**n, 'rank is an invalid number'
if real_or_complex == 1:
psi = np_random.randn(2**n, rank)
else:
psi = np_random.randn(2**n, rank) + 1j * np_random.randn(2**n, rank)
psi_dagger = psi.conj().T
rho = np_matmul(psi, psi_dagger)
rho = rho / np_trace(rho)
return rho.astype('complex128')
def purity(rho):
r"""计算量子态的纯度。
.. math::
P = \text{tr}(\rho^2)
Args:
rho (numpy.ndarray): 量子态的密度矩阵形式
Returns:
float: 输入的量子态的纯度
"""
gamma = np_trace(np_matmul(rho, rho))
return gamma.real
def main():
# 生成用泡利字符串表示的特定的哈密顿量
H = [[-1.0, 'z0,z1'], [-1.0, 'z1,z2'], [-1.0, 'z0,z2']]
# 生成哈密顿量的矩阵信息
N_SYS_B = 3 # 用于生成吉布斯态的子系统B的量子比特数
hamiltonian = pauli_str_to_matrix(H, N_SYS_B)
# 生成理想情况下的目标吉布斯态 rho
beta = 1.5 # 设置逆温度参数 beta
rho_G = scipy.linalg.expm(-1 * beta * hamiltonian) / np_trace(scipy.linalg.expm(-1 * beta * hamiltonian))
# 设置成 Paddle quantum 所支持的数据类型
hamiltonian = hamiltonian.astype("complex128")
rho_G = rho_G.astype("complex128")
rho_B = Paddle_GIBBS(hamiltonian, rho_G)
print(rho_B)
def main():
# Generate Pauli string representing a specific Hamiltonian
H = [[-1.0, 'z0,z1'], [-1.0, 'z1,z2'], [-1.0, 'z0,z2']]
# Generate the marix form of the Hamiltonian
N_SYS_B = 3 # Number of qubits in subsystem B used to generate Gibbs state
hamiltonian = pauli_str_to_matrix(H, N_SYS_B)
# Generate the target Gibbs state rho
beta = 1.5 # Set inverse temperature beta
rho_G = scipy.linalg.expm(-1 * beta * hamiltonian) / np_trace(
scipy.linalg.expm(-1 * beta * hamiltonian))
# Convert to the data type supported by Paddle Quantum
hamiltonian = hamiltonian.astype("complex128")
rho_G = rho_G.astype("complex128")
rho_B = Paddle_GIBBS(hamiltonian, rho_G)
print(rho_B)
def gate_fidelity(U, V):
r"""计算两个量子门的保真度。
.. math::
F(U, V) = |\text{tr}(UV^\dagger)|/2^n
:math:`U` 是一个 :math:`2^n\times 2^n` 的 Unitary 矩阵。
Args:
U (numpy.ndarray): 量子门的酉矩阵形式
V (numpy.ndarray): 量子门的酉矩阵形式
Returns:
float: 输入的量子门之间的保真度
"""
assert U.shape == V.shape, 'The shape of two unitary matrices are different'
dim = U.shape[0]
fidelity = absolute(np_trace(np_matmul(U, V.conj().T))) / dim
return fidelity
def H_generator():
"""
Generate a Hamiltonian with trivial descriptions
Returns: A Hamiltonian
"""
# Generate Pauli string representing a specific Hamiltonian
H = [[-1.0, 'z0,z1'], [-1.0, 'z1,z2'], [-1.0, 'z0,z2']]
# Generate the matrix form of the Hamiltonian
N_SYS_B = 3 # Number of qubits in subsystem B used to generate Gibbs state
hamiltonian = pauli_str_to_matrix(H, N_SYS_B)
# Generate the target Gibbs state rho
beta = 1.5 # Set inverse temperature beta
rho_G = scipy.linalg.expm(-1 * beta * hamiltonian) / np_trace(
scipy.linalg.expm(-1 * beta * hamiltonian))
# Convert to the data type supported by Paddle Quantum
hamiltonian = hamiltonian.astype("complex128")
rho_G = rho_G.astype("complex128")
return hamiltonian, rho_G
def H_generator():
"""
Generate a Hamiltonian with trivial descriptions
Returns: A Hamiltonian
"""
# 生成用泡利字符串表示的特定的哈密顿量
H = [[-1.0, 'z0,z1'], [-1.0, 'z1,z2'], [-1.0, 'z0,z2']]
# 生成哈密顿量的矩阵信息
N_SYS_B = 3 # 用于生成吉布斯态的子系统B的量子比特数
hamiltonian = pauli_str_to_matrix(H, N_SYS_B)
# 生成理想情况下的目标吉布斯态 rho
beta = 1.5 # 设置逆温度参数 beta
rho_G = scipy.linalg.expm(-1 * beta * hamiltonian) / np_trace(
scipy.linalg.expm(-1 * beta * hamiltonian))
# 设置成 Paddle quantum 所支持的数据类型
hamiltonian = hamiltonian.astype("complex128")
rho_G = rho_G.astype("complex128")
return hamiltonian, rho_G