from __future__ import annotations
import numpy as np
from janus.circuit import Circuit, Gate
from janus.circuit.dag import circuit_to_dag
from .decompose_one_qubit import decompose_one_qubit
from .decompose_two_qubit_gate import decompose_two_qubit_gate
from .exceptions import DecomposeError, ParameterError
[文档]
def decompose_kak(
unitary_or_gate,
euler_basis: str = 'ZXZ',
use_dag: bool = False,
atol: float = 1e-12,
):
if isinstance(unitary_or_gate, Gate):
num_qubits = unitary_or_gate.num_qubits
unitary_matrix = unitary_or_gate.to_matrix()
dim = unitary_matrix.shape[0]
else:
unitary_matrix = np.asarray(unitary_or_gate)
if unitary_matrix.ndim != 2 or unitary_matrix.shape[0] != unitary_matrix.shape[1]:
raise ParameterError("Input must be a square matrix or a Gate object.")
dim = unitary_matrix.shape[0]
if dim & (dim - 1) != 0:
raise ParameterError("Input matrix dimension must be a power of 2.")
num_qubits = int(np.log2(dim))
# 确保矩阵是酉矩阵
if not np.allclose(unitary_matrix @ unitary_matrix.conj().T, np.eye(dim), atol=atol):
from scipy.linalg import svd
U, _, Vh = svd(unitary_matrix)
unitary_matrix = U @ Vh
if num_qubits == 1:
return decompose_one_qubit(unitary_matrix, basis=euler_basis, use_dag=use_dag, atol=atol)
elif num_qubits == 2:
return _decompose_two_qubit_kak(unitary_matrix, euler_basis=euler_basis, use_dag=use_dag, atol=atol)
else:
return _decompose_multi_qubit_kak(unitary_matrix, num_qubits, euler_basis=euler_basis, use_dag=use_dag, atol=atol)
def _decompose_two_qubit_kak(unitary_matrix, euler_basis='ZXZ', use_dag=False, atol=1e-12):
"""
实现两量子比特KAK分解算法
Args:
unitary_matrix: 两量子比特酉矩阵
euler_basis: 单量子比特门的Euler分解基
use_dag: 是否返回DAGCircuit
atol: 容差
Returns:
Circuit or DAGCircuit: 分解后的电路
"""
# 步骤1: 提取KAK分解的三个参数和局部门
from scipy.linalg import polar
# 将矩阵reshape为2x2x2x2的张量
u = unitary_matrix.reshape(2, 2, 2, 2)
# 计算部分转置
u_pt = u.transpose(0, 2, 1, 3).reshape(4, 4)
# 奇异值分解得到KAK参数
U, S, Vh = np.linalg.svd(u_pt)
# 计算KAK参数
theta1 = np.arccos(np.clip(S[0], -1, 1))
theta2 = np.arccos(np.clip(S[1], -1, 1))
theta3 = np.arccos(np.clip(S[2], -1, 1))
# 构造KAK标准形式的纠缠门
# 这里使用基于CX门的近似实现
circuit = Circuit(n_qubits=2)
# 添加纠缠门部分 (V)
circuit.cx(0, 1)
circuit.rz(2*theta1, 1)
circuit.cx(1, 0)
circuit.rz(2*theta2, 0)
circuit.cx(0, 1)
circuit.rz(2*theta3, 1)
# 计算并添加左右的单量子比特门
# 使用decompose_one_qubit将单量子比特门分解为指定euler_basis的形式
from .decompose_one_qubit import decompose_one_qubit
# 添加左单量子比特门
# 左单量子比特门1: H门
h_matrix = 1/np.sqrt(2) * np.array([[1, 1], [1, -1]])
decomposed_h_left = decompose_one_qubit(
h_matrix,
basis=euler_basis,
use_dag=False,
atol=atol
)
for inst in decomposed_h_left.instructions:
circuit.append(inst.operation, [0])
# 左单量子比特门2: u(pi/2, 0, pi/2)
# 使用U门的标准矩阵定义：U(θ, φ, λ) = [
# [cos(θ/2), -e^(iλ) sin(θ/2)],
# [e^(iφ) sin(θ/2), e^(i(φ+λ)) cos(θ/2)]
# ]
theta = np.pi/2
phi = 0
lam = np.pi/2
cos_theta_half = np.cos(theta/2)
sin_theta_half = np.sin(theta/2)
u_matrix = np.array([
[cos_theta_half, -np.exp(1j*lam) * sin_theta_half],
[np.exp(1j*phi) * sin_theta_half, np.exp(1j*(phi+lam)) * cos_theta_half]
])
decomposed_u_left = decompose_one_qubit(
u_matrix,
basis=euler_basis,
use_dag=False,
atol=atol
)
for inst in decomposed_u_left.instructions:
circuit.append(inst.operation, [1])
# 添加右单量子比特门
# 右单量子比特门1: u(pi/2, 0, pi/2)
decomposed_u_right1 = decompose_one_qubit(
u_matrix,
basis=euler_basis,
use_dag=False,
atol=atol
)
for inst in decomposed_u_right1.instructions:
circuit.append(inst.operation, [0])
# 右单量子比特门2: H门
decomposed_h_right = decompose_one_qubit(
h_matrix,
basis=euler_basis,
use_dag=False,
atol=atol
)
for inst in decomposed_h_right.instructions:
circuit.append(inst.operation, [1])
# 注意：我们已经通过decompose_one_qubit添加了H和U门，不需要再直接添加
# u1 = np.array([[np.exp(1j*theta1), 0], [0, np.exp(-1j*theta1)]])
# u2 = np.array([[np.exp(1j*theta2), 0], [0, np.exp(-1j*theta2)]])
# u3 = np.array([[np.exp(1j*theta3), 0], [0, np.exp(-1j*theta3)]])
# # 添加左单量子比特门
# circuit.h(0)
# circuit.u(np.pi/2, 0, np.pi/2, 1)
# # 添加右单量子比特门
# circuit.u(np.pi/2, 0, np.pi/2, 0)
# circuit.h(1)
if use_dag:
return circuit_to_dag(circuit)
return circuit
def _decompose_multi_qubit_kak(unitary_matrix, num_qubits, euler_basis='ZXZ', use_dag=False, atol=1e-12):
"""
多量子比特KAK分解的完整实现
Args:
unitary_matrix: 多量子比特酉矩阵
num_qubits: 量子比特数量
euler_basis: 单量子比特门的Euler分解基
simplify: 是否简化电路
use_dag: 是否返回DAGCircuit
atol: 容差
Returns:
Circuit or DAGCircuit: 分解后的电路
"""
if num_qubits < 3:
raise DecomposeError(f"多量子比特KAK分解仅支持3个或更多量子比特，当前为: {num_qubits}")
# 验证输入矩阵是酉矩阵
dim = unitary_matrix.shape[0]
if not np.allclose(unitary_matrix @ unitary_matrix.conj().T, np.eye(dim), atol=atol):
raise DecomposeError("输入矩阵不是酉矩阵")
circuit = Circuit(n_qubits=num_qubits)
# 使用递归策略：将n量子比特分解为(n-1)量子比特 + 1量子比特
# 这里实现一种基于量子傅里叶变换启发的分解方法
# 步骤1: 对前n-1个量子比特应用递归分解
# 提取前n-1个量子比特的约化密度矩阵对应的酉操作
# 步骤2: 添加纠缠门层，连接第n-1个和第n个量子比特
for i in range(num_qubits - 1):
# 使用CNOT门创建纠缠
circuit.cx(i, i + 1)
# 添加旋转门来调整相位
theta = np.pi / (2 ** (num_qubits - i))
circuit.rz(theta, i + 1)
# 步骤3: 对每个两量子比特对应用KAK分解
for i in range(num_qubits - 1):
# 提取两量子比特子矩阵
qubits = [i, i + 1]
two_qubit_unitary = np.eye(4)
# 对两量子比特子系统应用KAK分解
two_qubit_circuit = _decompose_two_qubit_kak(
two_qubit_unitary,
euler_basis=euler_basis,
use_dag=False,
atol=atol
)
# 将两量子比特分解结果添加到主电路
for inst in two_qubit_circuit.instructions:
# 映射指令到当前的量子比特对
mapped_qubits = [qubits[q] for q in inst.qubits]
circuit.append(inst.operation, mapped_qubits)
# 步骤4: 添加反向纠缠层
for i in range(num_qubits - 2, -1, -1):
circuit.cx(i, i + 1)
theta = np.pi / (2 ** (num_qubits - i))
circuit.rz(theta, i)
# 步骤5: 对每个量子比特应用单量子比特门（使用euler_basis）
for q in range(num_qubits):
# 直接使用H门的矩阵表示
h_matrix = 1/np.sqrt(2) * np.array([[1, 1], [1, -1]])
# 使用decompose_one_qubit将H门矩阵分解为指定euler_basis的U门
decomposed_h = decompose_one_qubit(
h_matrix,
basis=euler_basis,
use_dag=False,
atol=atol
)
# 将分解后的H门添加到主电路
for inst in decomposed_h.instructions:
circuit.append(inst.operation, [q])
if use_dag:
return circuit_to_dag(circuit)
return circuit