可微量子架构搜索#
概述#
本教程演示了如何利用 TensorCircuit 提供的高级计算功能，例如 jit 和 vmap 来超级有效地模拟可微量子架构搜索（DQAS）算法，其中具有不同结构的量子电路的集合可以同时编译模拟。 [WIP note]
设置#
[1]:
import numpy as np
import tensorcircuit as tc
import tensorflow as tf
[2]:
K = tc.set_backend("tensorflow")
ctype, rtype = tc.set_dtype("complex128")
问题描述#
任务是找到 GHZ 状态的状态准备电路 \(\vert \text{GHZ}_N\rangle = \frac{1}{\sqrt{2}}\left(\vert 0^N\rangle +\vert 1^N\rangle \right)\)。我们为 \(N=2\) 演示准备了一个包含 rx0、rx1、ry0、ry1、rz0、rz1、cnot01、cnot10 的门池。 在八个门中，有六个是参数化的。
[3]:
def rx0(theta):
return K.kron(
K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._x_matrix, K.eye(2)
)
def rx1(theta):
return K.kron(
K.eye(2), K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._x_matrix
)
def ry0(theta):
return K.kron(
K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._y_matrix, K.eye(2)
)
def ry1(theta):
return K.kron(
K.eye(2), K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._y_matrix
)
def rz0(theta):
return K.kron(
K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._z_matrix, K.eye(2)
)
def rz1(theta):
return K.kron(
K.eye(2), K.cos(theta) * K.eye(2) + 1.0j * K.sin(theta) * tc.gates._z_matrix
)
def cnot01():
return K.cast(K.convert_to_tensor(tc.gates._cnot_matrix), ctype)
def cnot10():
return K.cast(
K.convert_to_tensor(
np.array([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
),
ctype,
)
ops_repr = ["rx0", "rx1", "ry0", "ry1", "rz0", "rz1", "cnot01", "cnot10"]
[4]:
n, p, ch = 2, 3, 8
# 量子比特数、层数、操作池大小
target = tc.array_to_tensor(np.array([1, 0, 0, 1.0]) / np.sqrt(2.0))
# 目标波函数，我们这里使用 GHZ2 状态作为目标函数
def ansatz(params, structures):
c = tc.Circuit(n)
params = K.cast(params, ctype)
structures = K.cast(structures, ctype)
for i in range(p):
c.any(
0,
1,
unitary=structures[i, 0] * rx0(params[i, 0])
+ structures[i, 1] * rx1(params[i, 1])
+ structures[i, 2] * ry0(params[i, 2])
+ structures[i, 3] * ry1(params[i, 3])
+ structures[i, 4] * rz0(params[i, 4])
+ structures[i, 5] * rz1(params[i, 5])
+ structures[i, 6] * cnot01()
+ structures[i, 7] * cnot10(),
)
s = c.state()
loss = K.sum(K.abs(target - s))
return loss
vag1 = K.jit(K.vvag(ansatz, argnums=0, vectorized_argnums=1))
概率系综方法#
这种方法更加实用和实验相关，并且与参考文献 1 中描述的算法相同，尽管我们在这里使用高级 vmap 来加速具有不同结构的电路的仿真。
[5]:
def sampling_from_structure(structures, batch=1):
prob = K.softmax(K.real(structures), axis=-1)
return np.array([np.random.choice(ch, p=K.numpy(prob[i])) for i in range(p)])
@K.jit
def best_from_structure(structures):
return K.argmax(structures, axis=-1)
@K.jit
def nmf_gradient(structures, oh):
"""
根据朴素平均场概率模型计算蒙特卡洛梯度
"""
choice = K.argmax(oh, axis=-1)
prob = K.softmax(K.real(structures), axis=-1)
indices = K.transpose(K.stack([K.cast(tf.range(p), "int64"), choice]))
prob = tf.gather_nd(prob, indices)
prob = K.reshape(prob, [-1, 1])
prob = K.tile(prob, [1, ch])
return tf.tensor_scatter_nd_add(
tf.cast(-prob, dtype=ctype),
indices,
tf.ones([p], dtype=ctype),
)
nmf_gradient_vmap = K.vmap(nmf_gradient, vectorized_argnums=1)
[6]:
verbose = False
epochs = 400
batch = 256
lr = tf.keras.optimizers.schedules.ExponentialDecay(0.06, 100, 0.5)
structure_opt = tc.backend.optimizer(tf.keras.optimizers.Adam(0.12))
network_opt = tc.backend.optimizer(tf.keras.optimizers.Adam(lr))
nnp = K.implicit_randn(stddev=0.02, shape=[p, 6], dtype=rtype)
stp = K.implicit_randn(stddev=0.02, shape=[p, 8], dtype=rtype)
avcost1 = 0
for epoch in range(epochs): # 更新结构参数的迭代
avcost2 = avcost1
costl = []
batched_stuctures = K.onehot(
np.stack([sampling_from_structure(stp) for _ in range(batch)]), num=8
)
infd, gnnp = vag1(nnp, batched_stuctures)
gs = nmf_gradient_vmap(stp, batched_stuctures) # \nabla lnp
gstp = [K.cast((infd[i] - avcost2), ctype) * gs[i] for i in range(infd.shape[0])]
gstp = K.real(K.sum(gstp, axis=0) / infd.shape[0])
avcost1 = K.sum(infd) / infd.shape[0]
nnp = network_opt.update(gnnp, nnp)
stp = structure_opt.update(gstp, stp)
if epoch % 40 == 0 or epoch == epochs - 1:
print("----------epoch %s-----------" % epoch)
print(
"batched average loss: ",
np.mean(avcost1),
)
if verbose:
print(
"strcuture parameter: \n",
stp.numpy(),
"\n network parameter: \n",
nnp.numpy(),
)
cand_preset = best_from_structure(stp)
print("best candidates so far:", [ops_repr[i] for i in cand_preset])
print(
"corresponding weights for each gate:",
[K.numpy(nnp[j, i]) if i < 6 else 0.0 for j, i in enumerate(cand_preset)],
)
WARNING:tensorflow:Using a while_loop for converting GatherNd
WARNING:tensorflow:Using a while_loop for converting TensorScatterAdd
----------epoch 0-----------
batched average loss: 1.486862041224946
best candidates so far: ['rz1', 'cnot01', 'rx0']
corresponding weights for each gate: [0.04850114379068718, 0.0, 0.05130625869908137]
----------epoch 40-----------
batched average loss: 1.0129558433713033
best candidates so far: ['rx0', 'rx1', 'cnot01']
corresponding weights for each gate: [0.027262624897770482, 0.027810247234826772, 0.0]
----------epoch 80-----------
batched average loss: 0.05192747113248927
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7929784361217006, 0.028373579732963325, 0.0]
----------epoch 120-----------
batched average loss: 0.031656973667466226
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7917033798904415, 0.02709852348238091, 0.0]
----------epoch 160-----------
batched average loss: 0.028017594123095527
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7898273038100041, 0.02406457071696495, 0.0]
----------epoch 200-----------
batched average loss: 0.029086134952175734
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7878800008021832, 0.020169898669812416, 0.0]
----------epoch 240-----------
batched average loss: 0.02272153644755242
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7860795359107594, 0.016568960455492835, 0.0]
----------epoch 280-----------
batched average loss: 0.019205161854778285
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7854269350973763, 0.013422528595912621, 0.0]
----------epoch 320-----------
batched average loss: 0.015424930560900666
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7854255410759101, 0.010762703562019423, 0.0]
----------epoch 360-----------
batched average loss: 0.012287067999120332
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7854212965432693, 0.008565353750462018, 0.0]
----------epoch 399-----------
batched average loss: 0.009789006724779316
best candidates so far: ['ry0', 'rx1', 'cnot01']
corresponding weights for each gate: [-0.7922338874583758, -4.046275452326831e-05, 0.0]
直接优化结构参数#
无论如何，由于我们是用数值模拟，所以可以直接优化结构参数，省略超级电路是否是幺正的，这种方法在某些场景下可以更快更可靠。
[7]:
def ansatz2(params, structures):
c = tc.Circuit(n)
params = K.cast(params, ctype)
structures = K.softmax(structures, axis=-1)
structures = K.cast(structures, ctype)
for i in range(p):
c.any(
0,
1,
unitary=structures[i, 0] * rx0(params[i, 0])
+ structures[i, 1] * rx1(params[i, 1])
+ structures[i, 2] * ry0(params[i, 2])
+ structures[i, 3] * ry1(params[i, 3])
+ structures[i, 4] * rz0(params[i, 4])
+ structures[i, 5] * rz1(params[i, 5])
+ structures[i, 6] * cnot01()
+ structures[i, 7] * cnot10(),
)
s = c.state()
s /= K.norm(s)
loss = K.sum(K.abs(target - s))
return loss
vag2 = K.jit(K.value_and_grad(ansatz2, argnums=(0, 1)))
[8]:
verbose = True
epochs = 700
lr = tf.keras.optimizers.schedules.ExponentialDecay(0.05, 200, 0.5)
structure_opt = tc.backend.optimizer(tf.keras.optimizers.Adam(0.04))
network_opt = tc.backend.optimizer(tf.keras.optimizers.Adam(lr))
nnp = K.implicit_randn(stddev=0.02, shape=[p, 6], dtype=rtype)
stp = K.implicit_randn(stddev=0.02, shape=[p, 8], dtype=rtype)
for epoch in range(epochs):
infd, (gnnp, gstp) = vag2(nnp, stp)
nnp = network_opt.update(gnnp, nnp)
stp = structure_opt.update(gstp, stp)
if epoch % 70 == 0 or epoch == epochs - 1:
print("----------epoch %s-----------" % epoch)
print(
"batched average loss: ",
np.mean(infd),
)
if verbose:
print(
"strcuture parameter: \n",
stp.numpy(),
"\n network parameter: \n",
nnp.numpy(),
)
cand_preset = best_from_structure(stp)
print("best candidates so far:", [ops_repr[i] for i in cand_preset])
print(
"corresponding weights for each gate:",
[K.numpy(nnp[j, i]) if i < 6 else 0.0 for j, i in enumerate(cand_preset)],
)
----------epoch 0-----------
batched average loss: 1.3024341605187928
strcuture parameter:
[[ 0.00265054 0.04495954 0.05265605 0.04751008 0.03309468 0.02743368
0.03382795 -0.06647121]
[ 0.03544281 0.03207712 0.03629811 0.0266235 0.03264895 0.03198189
0.03505167 -0.03449981]
[ 0.0304648 0.07042194 0.03075206 0.02515865 0.02984363 0.00955019
0.07527341 -0.05831911]]
network parameter:
[[-0.0380125 0.0688923 0.04393423 0.04205065 0.06243917 0.03672062]
[-0.05277717 0.04834309 0.05176114 0.07030034 0.02983666 0.04821408]
[-0.04095011 0.0393773 0.03383929 0.06559557 0.03458135 0.02436751]]
best candidates so far: ['ry0', 'ry0', 'cnot01']
corresponding weights for each gate: [0.043934227235354874, 0.05176113831452516, 0.0]
----------epoch 70-----------
batched average loss: 1.0078726220234666
strcuture parameter:
[[ 0.34119556 0.37154559 0.2781548 0.37689646 1.79624262 1.78939153
1.79791154 -0.3958881 ]
[-1.04375011 -0.1280161 -0.98656309 0.35601588 1.79624062 1.7944131
1.80069633 -0.36391841]
[ 0.05278476 0.40486479 0.33074328 0.35455104 1.79289574 1.77260117
1.83987217 -0.38773815]]
network parameter:
[[ 0.03213363 -0.00113532 -0.02629044 0.44615806 -0.00711866 -0.03271277]
[ 0.01737624 -0.02169861 -0.01841347 0.47440825 0.01052075 -0.02137655]
[ 0.02919352 -0.03064528 -0.03628025 0.46970421 -0.03434029 -0.04519493]]
best candidates so far: ['cnot01', 'cnot01', 'cnot01']
corresponding weights for each gate: [0.0, 0.0, 0.0]
----------epoch 140-----------
batched average loss: 0.8974790925982725
strcuture parameter:
[[-0.60734424 -0.71178177 1.75016478 0.37059946 2.29304101 1.40087053
2.75041722 -0.4812161 ]
[-3.20848853 -2.62803529 -1.0799243 0.34964202 1.75222537 0.96340588
4.59441388 -0.44922864]
[-1.14052853 -0.10976557 0.9998582 0.34848638 3.81458301 3.38821681
2.51193413 -0.38172792]]
network parameter:
[[-0.03052873 -0.00288839 -1.08840853 0.51536679 -0.00833568 -0.03375284]
[ 0.0216315 -0.02917968 0.8216509 0.54353084 0.00946658 -0.02120069]
[ 0.02581133 -0.03281417 0.94900358 0.46842014 -0.0355813 -0.04637678]]
best candidates so far: ['cnot01', 'cnot01', 'rz0']
corresponding weights for each gate: [0.0, 0.0, -0.03558129647764995]
----------epoch 210-----------
batched average loss: 0.06833171337421318
strcuture parameter:
[[-1.46101866 -1.56562139 2.60460926 0.36697705 1.43861808 0.54649088
1.8959831 -0.6664916 ]
[-2.8599476 -3.71447893 -0.1444609 -0.31801788 0.70398108 0.12590137
5.44896172 -0.64032593]
[-0.29063004 0.46966802 1.85427644 0.44130728 4.56540152 4.13933904
1.26524942 -0.47396024]]
network parameter:
[[-0.80254223 -0.07223631 -1.72113169 0.65719836 -0.00717114 -0.03248183]
[ 0.86880067 -0.98161163 1.73279897 0.68543186 0.57173565 0.81901641]
[ 0.41384514 0.03460214 1.4409846 0.26020133 -0.03450978 -0.04528429]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7211316854515077, 0.0, -0.03450978363305436]
----------epoch 280-----------
batched average loss: 0.07287093721912785
strcuture parameter:
[[-1.46462962 -1.5694426 2.60832482 0.36325644 1.43489835 0.54276147
1.89226573 -0.66280103]
[-1.57089047 -3.66510778 -0.14837051 -0.32242832 0.70013882 0.1239224
5.45278654 -0.63670874]
[-0.28665073 0.50527464 1.85793113 0.43721505 4.56217177 4.13630294
1.26107282 -0.47016776]]
network parameter:
[[-0.80172748 -0.0712512 -1.71930255 0.6591615 -0.00531003 -0.030584 ]
[ 0.8833277 -1.26937984 1.73185803 0.68715766 0.57394473 0.8184381 ]
[ 0.41194181 0.03270603 1.4387577 0.26174242 -0.03266689 -0.04343292]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7193025540909423, 0.0, -0.032666892917796356]
----------epoch 350-----------
batched average loss: 0.0763633455796077
strcuture parameter:
[[-1.46759968 -1.57262214 2.61142979 0.36014718 1.43179046 0.53964649
1.88915951 -0.65972411]
[-1.34936013 -3.5935837 -0.15090151 -0.32639326 0.69677033 0.10483803
5.45606073 -0.63438681]
[-0.28282805 0.54032205 1.86109945 0.43217854 4.55947529 4.13345958
1.25730198 -0.46703484]]
network parameter:
[[-0.80018367 -0.06928467 -1.71726239 0.66128651 -0.00323225 -0.0284642 ]
[ 0.88958794 -1.36344637 1.73074888 0.68928184 0.57647947 0.81753049]
[ 0.40992401 0.03064141 1.43664452 0.26328958 -0.03061012 -0.04136483]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7172623928298167, 0.0, -0.030610123400026196]
----------epoch 420-----------
batched average loss: 0.07933778661926846
strcuture parameter:
[[-1.47009725 -1.57533451 2.61409128 0.35748195 1.42912701 0.53697792
1.88649718 -0.65709335]
[-1.19469005 -3.49291914 -0.15236124 -0.32995762 0.69380539 0.07513886
5.45890732 -0.63296266]
[-0.27952556 0.57900165 1.86385667 0.42698093 4.55720421 4.13093395
1.25401354 -0.46439162]]
network parameter:
[[-7.98493487e-01 -6.71330183e-02 -1.71528444e+00 6.63308869e-01
-1.20663132e-03 -2.63889444e-02]
[ 8.92122476e-01 -1.41065660e+00 1.72949508e+00 6.91390756e-01
5.79085213e-01 8.16347571e-01]
[ 4.07963360e-01 2.86428770e-02 1.43467895e+00 2.64831277e-01
-2.86101270e-02 -3.93481586e-02]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7152844386878365, 0.0, -0.028610127013962768]
----------epoch 490-----------
batched average loss: 0.08254545024351018
strcuture parameter:
[[-1.47222071e+00 -1.57769163e+00 2.61641114e+00 3.55158679e-01
1.42680584e+00 5.34652980e-01 1.88417675e+00 -6.54806876e-01]
[-9.73250463e-01 -3.16903830e+00 -3.18559749e-03 -3.33158739e-01
8.55776706e-01 5.24787309e-02 5.46137377e+00 -4.79944986e-01]
[-4.31555592e-01 6.29503046e-01 1.86558679e+00 4.22659216e-01
4.55614569e+00 4.13032405e+00 1.25003608e+00 -4.62133108e-01]]
network parameter:
[[-7.96868537e-01 -6.41205534e-02 -1.71348758e+00 6.65124861e-01
7.01296452e-04 -2.44123401e-02]
[ 8.95067083e-01 -1.46634629e+00 1.72823707e+00 6.93124762e-01
6.19045003e-01 8.16911575e-01]
[ 3.68577179e-01 -1.06859075e-02 1.43203849e+00 2.65575517e-01
-2.67782067e-02 -3.74965029e-02]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7134875797200289, 0.0, -0.026778206681234634]
----------epoch 560-----------
batched average loss: 0.08507962845391319
strcuture parameter:
[[-1.47404605e+00 -1.57976616e+00 2.61845726e+00 3.53109374e-01
1.42475889e+00 5.32603322e-01 1.88213024e+00 -6.52796788e-01]
[-9.74391719e-01 -2.60249357e+00 -5.45311032e-03 -3.36090881e-01
8.87457853e-01 5.09795396e-02 5.46352386e+00 -4.77818480e-01]
[-4.31981604e-01 7.01277774e-01 1.86614168e+00 4.17460986e-01
4.55685554e+00 4.13219105e+00 1.24506389e+00 -4.60189080e-01]]
network parameter:
[[-0.7953961 -0.05746862 -1.7119146 0.66669705 0.00291374 -0.0224405 ]
[ 0.90083188 -1.50990732 1.72710663 0.69459133 0.61788545 0.82003669]
[ 0.37017675 -0.00904591 1.42870116 0.26514657 -0.02513608 -0.03583994]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.711914597657612, 0.0, -0.025136080292754256]
----------epoch 630-----------
batched average loss: 0.0874301658835076
strcuture parameter:
[[-1.47562848 -1.58159874 2.62027778 0.35128611 1.42293816 0.53078116
1.88030961 -0.65101441]
[-0.97538969 -2.02211517 -0.00750621 -0.33876198 0.90002101 0.04975063
5.46545544 -0.47588949]
[-0.43177273 0.78495553 1.86618092 0.41182654 4.55815983 4.13468705
1.24131582 -0.45851101]]
network parameter:
[[-0.79411737 -0.03262083 -1.71056973 0.66801607 0.00931137 -0.02061058]
[ 0.90559687 -1.52299082 1.72605402 0.69584073 0.61673658 0.82270295]
[ 0.37157164 -0.00763778 1.42576784 0.26501218 -0.02367246 -0.03437195]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.7105697331825656, 0.0, -0.02367245788118412]
----------epoch 699-----------
batched average loss: 0.07954282768319362
strcuture parameter:
[[-1.38724436 -1.4934299 2.53210121 0.43945868 1.51111454 0.61895662
1.96848611 -0.7392194 ]
[-0.88692788 -1.19022046 0.08057521 -0.25156948 0.81347111 -0.04509805
5.3775582 -0.56406862]
[-0.34354263 0.94817808 1.77696399 0.48982178 4.64871759 4.22638916
1.32761078 -0.54681732]]
network parameter:
[[-0.78310707 0.01573153 -1.69951606 0.67901289 0.0447718 -0.00935101]
[ 0.91920164 -1.51646333 1.7152693 0.70682227 0.60494688 0.8352184 ]
[ 0.38342071 0.00423554 1.41389299 0.27478806 -0.01255172 -0.02324825]]
best candidates so far: ['ry0', 'cnot01', 'rz0']
corresponding weights for each gate: [-1.6995160647885235, 0.0, -0.01255171733214709]
最后的微调#
对于获得的电路布局，我们可以进一步调整电路权重，使目标函数更接近于零。
[9]:
chosen_structure = K.onehot(np.array([2, 4, 6]), num=8)
chosen_structure = K.reshape(chosen_structure, [1, p, ch])
chosen_structure
[9]:
<tf.Tensor: shape=(1, 3, 8), dtype=float32, numpy=
array([[[0., 0., 1., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 1., 0.]]], dtype=float32)>
[10]:
network_opt = tc.backend.optimizer(tf.keras.optimizers.Adam(1e-3))
nnp = K.implicit_randn(stddev=0.02, shape=[p, 6], dtype=rtype)
verbose = True
epochs = 600
for epoch in range(epochs):
infd, gnnp = vag1(nnp, chosen_structure)
nnp = network_opt.update(gnnp, nnp)
if epoch % 60 == 0 or epoch == epochs - 1:
print(epoch, "loss: ", K.numpy(infd[0]))
0 loss: 1.004872758871745
60 loss: 0.9679200431227549
120 loss: 0.9091748060127385
180 loss: 0.8302632245154631
240 loss: 0.73297561645977
300 loss: 0.6183436622858383
360 loss: 0.4874390992051161
420 loss: 0.34168453047914643
480 loss: 0.18299822849737177
540 loss: 0.013923344772669728
599 loss: 0.0016133833518836463