Gate Decomposition¶

Adapted from qgopt/docs/gate-decomposition

It's known that any two-qubits gate can be decomposed as follows:

u1 u2
cnot
u3 u4
cnot
u5 u6
cnot
u7 u8

where each line represent one layer of either single-qubit gates or CNOT gates. In this tutorial, we will show how to perform gate such decomposition for any two-qubits gate.

In [1]:

import numpy as np
import torch

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

import numpy as np import torch try: import numqi except ImportError: %pip install numqi import numqi

In [2]:

class GateDecompositionModel(torch.nn.Module):
    def __init__(self):
        super().__init__()
        self.manifold = numqi.manifold.SpecialOrthogonal(dim=2, batch_size=8, dtype=torch.complex128)
        self.unitary_dag = None #we will set this in .set_unitary()
        self.cnot = torch.tensor([[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]], dtype=torch.complex128)

    def set_unitary(self, np0):
        assert np0.shape==(4,4)
        assert np.abs(np0@np0.T.conj() - np.eye(4)).max() < 1e-10, 'unitary matrix required'
        self.unitary_dag = torch.tensor(np0.T.conj().copy(), dtype=torch.complex128)

    def get_unitary(self, return_numpy=True):
        ulist = self.manifold()
        matU = torch.kron(ulist[0], ulist[1])
        for i in range(3):
            matU = self.cnot @ matU
            matU = torch.kron(ulist[2*i+2], ulist[2*i+3]) @ matU
        ret = ulist, matU
        if return_numpy:
            ret = ret[0].detach().numpy(), ret[1].detach().numpy()
        return ret

    def forward(self):
        _,matU = self.get_unitary(return_numpy=False)
        tmp0 = torch.trace(self.unitary_dag @ matU)
        # unitary infidelity https://docs.q-ctrl.com/references/qctrl/Graphs/Graph/unitary_infidelity.html
        loss = 1 - (tmp0.real**2 + tmp0.imag**2) / 16
        return loss

class GateDecompositionModel(torch.nn.Module): def __init__(self): super().__init__() self.manifold = numqi.manifold.SpecialOrthogonal(dim=2, batch_size=8, dtype=torch.complex128) self.unitary_dag = None #we will set this in .set_unitary() self.cnot = torch.tensor([[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]], dtype=torch.complex128) def set_unitary(self, np0): assert np0.shape==(4,4) assert np.abs(np0@np0.T.conj() - np.eye(4)).max() < 1e-10, 'unitary matrix required' self.unitary_dag = torch.tensor(np0.T.conj().copy(), dtype=torch.complex128) def get_unitary(self, return_numpy=True): ulist = self.manifold() matU = torch.kron(ulist[0], ulist[1]) for i in range(3): matU = self.cnot @ matU matU = torch.kron(ulist[2*i+2], ulist[2*i+3]) @ matU ret = ulist, matU if return_numpy: ret = ret[0].detach().numpy(), ret[1].detach().numpy() return ret def forward(self): _,matU = self.get_unitary(return_numpy=False) tmp0 = torch.trace(self.unitary_dag @ matU) # unitary infidelity https://docs.q-ctrl.com/references/qctrl/Graphs/Graph/unitary_infidelity.html loss = 1 - (tmp0.real**2 + tmp0.imag**2) / 16 return loss

Different from QGopt, we choose unitary infidelity as loss function qctrl/unitary_infidelity

$$ \mathcal{L}=1 - \lvert\mathrm{Tr}[V(\theta) U^\dagger]\rvert^2 / d^2 $$

for that we ignore the global phase of the target unitary $V$. Above $d$ denotes the dimension of the Hilbert space $d=4$, $U$ for the target untiary matrix and $V(\theta)$ is the parametrized unitary matrix composed of single-qubit gates and CNOT gates.

In [3]:

target_unitary = numqi.random.rand_special_orthogonal_matrix(4, tag_complex=True)
print(f'target unitary:\n{np.around(target_unitary, 3)}')

tmp0 = target_unitary @ target_unitary.T.conj()
print(f'unitary check (U U^dag): \n{np.around(tmp0, 3)}')

target_unitary = numqi.random.rand_special_orthogonal_matrix(4, tag_complex=True) print(f'target unitary:\n{np.around(target_unitary, 3)}') tmp0 = target_unitary @ target_unitary.T.conj() print(f'unitary check (U U^dag): \n{np.around(tmp0, 3)}')

target unitary:
[[-0.188+0.864j  0.036+0.082j  0.014+0.455j -0.033-0.027j]
 [ 0.041-0.012j -0.18 +0.527j  0.071-0.01j   0.745-0.357j]
 [ 0.216+0.294j -0.516-0.311j -0.45 -0.384j -0.06 -0.388j]
 [-0.023-0.287j -0.24 -0.512j  0.218+0.624j  0.102-0.387j]]
unitary check (U U^dag): 
[[ 1.+0.j -0.+0.j  0.-0.j -0.+0.j]
 [-0.-0.j  1.+0.j -0.+0.j  0.+0.j]
 [ 0.+0.j -0.-0.j  1.+0.j -0.+0.j]
 [-0.-0.j  0.-0.j -0.-0.j  1.+0.j]]

In [4]:

model = GateDecompositionModel()
model.set_unitary(target_unitary)
theta_optim = numqi.optimize.minimize(model, theta0='uniform', num_repeat=1, print_freq=5, tol=1e-10)

model = GateDecompositionModel() model.set_unitary(target_unitary) theta_optim = numqi.optimize.minimize(model, theta0='uniform', num_repeat=1, print_freq=5, tol=1e-10)

[step=0][time=0.018 seconds] loss=0.7808158115126216
[step=5][time=0.059 seconds] loss=0.20156773680584394
[step=10][time=0.032 seconds] loss=0.03328672718606285
[step=15][time=0.036 seconds] loss=9.866100433364444e-05
[step=20][time=0.032 seconds] loss=1.6803563362977059e-07

[step=25][time=0.036 seconds] loss=1.5783325757467992e-09

[round=0] min(f)=2.0062840278001204e-12, current(f)=2.0062840278001204e-12

In [5]:

ulist, matU = model.get_unitary()

print('first several single-qubit unitary matrices:')
for i in range(3):
    print(f'U{i} = \n{np.around(ulist[i], 3)}')

ulist, matU = model.get_unitary() print('first several single-qubit unitary matrices:') for i in range(3): print(f'U{i} = \n{np.around(ulist[i], 3)}')

first several single-qubit unitary matrices:
U0 = 
[[ 0.296-0.667j -0.307+0.611j]
 [ 0.307+0.611j  0.296+0.667j]]
U1 = 
[[ 0.354+0.622j  0.113+0.689j]
 [-0.113+0.689j  0.354-0.622j]]
U2 = 
[[ 0.491+0.466j  0.204-0.707j]
 [-0.204-0.707j  0.491-0.466j]]

In [6]:

print(f'final 2-qubit unitary matrix:\n{np.around(matU, 3)}')

tmp0 = matU @ target_unitary.T.conj()
print(f'check unitary (V U^dag):\n{np.around(tmp0,5)}')

print(f'final 2-qubit unitary matrix:\n{np.around(matU, 3)}') tmp0 = matU @ target_unitary.T.conj() print(f'check unitary (V U^dag):\n{np.around(tmp0,5)}')

final 2-qubit unitary matrix:
[[ 0.478+0.744j  0.084+0.033j  0.332+0.312j -0.042+0.005j]
 [ 0.021-0.038j  0.245+0.5j    0.043-0.057j  0.275-0.78j ]
 [ 0.36 +0.055j -0.585+0.145j -0.59 +0.047j -0.317-0.232j]
 [-0.219-0.187j -0.531-0.192j  0.596+0.287j -0.202-0.346j]]
check unitary (V U^dag):
[[ 0.70711-0.70711j -0.     -0.j       0.     +0.j      -0.     +0.j     ]
 [-0.     -0.j       0.70711-0.70711j  0.     +0.j       0.     -0.j     ]
 [ 0.     +0.j       0.     +0.j       0.70711-0.70711j  0.     +0.j     ]
 [ 0.     -0.j      -0.     +0.j       0.     +0.j       0.70711-0.70711j]]