大量量子ビットでQAOAやってみます。まだツールとしては実装が終わってないのでスクラッチで実装してみます。まずはライブラリとblueqatを最新に。
pip install --no-deps -U git+https://github.com/jcmgray/quimb.git@develop autoray blueqat
Collecting git+https://github.com/jcmgray/quimb.git@develop
Cloning https://github.com/jcmgray/quimb.git (to revision develop) to /tmp/pip-req-build-re4i9utl
Running command git clone -q https://github.com/jcmgray/quimb.git /tmp/pip-req-build-re4i9utl
Requirement already satisfied: autoray in /opt/conda/lib/python3.8/site-packages (0.3.1)
Requirement already satisfied: blueqat in /opt/conda/lib/python3.8/site-packages (1.0.0)
Note: you may need to restart the kernel to use updated packages.
最初のツールを読み込みます。
import numpy as np
import scipy.optimize as optimize
from blueqat import Circuit
from blueqat.pauli import qubo_bit as q
量子ビットは100にします。
N = 100
次に解きたい社会問題をQUBO形式にします。今回は適当に隣接結合をループ状にします。
hamiltonian = 0
for i in range(N):
hamiltonian += np.random.rand()*q(i)
hamiltonian += np.random.rand()*q(i)*q((i+1)%N)
ここからは流れ作業です。QUBOを時間発展演算子に直します。blueqatにはそういう機能がついてました。
time_evolutions = [
term.get_time_evolution() for term in hamiltonian
]
今回はシンプルなXハミルトニアンをmixerにします。初期状態を重ね合わせにして準備し、その後QUBOの時間発展をかけます。最後に横磁場項を時間発展に直したRXを書けます。p=1のQAOAでは最適化すべき角度は2種類あります。今回はbeta/gammaを最適化します。
テンソルネットワークで直接期待値を計算します。
def f(params):
c = Circuit(N).h[:]
beta = params[0]
gamma = params[1]
for evo in time_evolutions:
evo(c, gamma)
c.rx(beta)[:]
return c.run(backend="quimb", hamiltonian=hamiltonian)
初期にランダムな角度を選び、scipyを使って、期待値を最小化するようにbeta/gammaを最適化します。
initial_guess = [np.random.rand()*2*np.pi for _ in range(2)]
result = optimize.minimize(f, initial_guess)
最適化が完了したら終了です。100量子ビットなのでめちゃくちゃ時間かかります。
if result.success:
fitted_params = result.x
print(fitted_params)
else:
raise ValueError(result.message)
[5.38861753 3.15076607]
最適化されたパラメータを使ってサンプリングしてみたいと思います。
c = Circuit(N).h[:]
beta = fitted_params[0]
gamma = fitted_params[1]
for evo in time_evolutions:
evo(c, gamma)
c.rx(beta)[:]
c.run(backend="quimb", shots=100)
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結構大きな問題が解けるのはかなりいいですね。以上です。