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:
[[ 0.232-0.153j  0.427+0.587j -0.287+0.552j  0.005-0.09j ]
 [ 0.718+0.145j -0.216-0.247j  0.159+0.175j -0.276-0.473j]
 [ 0.071-0.18j   0.355-0.448j -0.7  -0.245j -0.29 -0.033j]
 [-0.411-0.425j -0.197+0.012j -0.073+0.03j   0.142-0.765j]]
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=0.013 seconds] loss=0.7230776691889222
[step=5][time=0.039 seconds] loss=0.13019596904965303
[step=10][time=0.030 seconds] loss=0.0006483885335120121
[step=15][time=0.031 seconds] loss=2.688432843012123e-08
[round=0] min(f)=4.66817251520979e-10, current(f)=4.66817251520979e-10

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 = 
[[-0.016+0.015j -0.671-0.741j]
 [ 0.671-0.741j -0.016-0.015j]]
U1 = 
[[ 0.023+0.422j  0.33 -0.844j]
 [-0.33 -0.844j  0.023-0.422j]]
U2 = 
[[-0.184-0.07j  -0.531-0.824j]
 [ 0.531-0.824j -0.184+0.07j ]]

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:
[[ 0.056-0.273j  0.717+0.113j  0.188+0.593j -0.061-0.067j]
 [ 0.61 -0.405j -0.327-0.021j  0.236+0.011j -0.53 -0.14j ]
 [-0.077-0.178j -0.066-0.568j -0.668+0.322j -0.229+0.182j]
 [-0.591-0.01j  -0.131+0.148j -0.03 +0.073j -0.441-0.642j]]
check unitary (V U^dag):
[[ 7.0711e-01-7.0711e-01j -1.0000e-05-0.0000e+00j  2.0000e-05-2.0000e-05j
   1.0000e-05+0.0000e+00j]
 [-0.0000e+00-1.0000e-05j  7.0711e-01-7.0710e-01j  0.0000e+00+0.0000e+00j
   0.0000e+00-0.0000e+00j]
 [-2.0000e-05+2.0000e-05j  0.0000e+00+0.0000e+00j  7.0710e-01-7.0711e-01j
  -0.0000e+00-0.0000e+00j]
 [ 0.0000e+00+1.0000e-05j -0.0000e+00+0.0000e+00j -0.0000e+00-0.0000e+00j
   7.0711e-01-7.0711e-01j]]