def solve(self):
q = gen_symbols(BinaryPoly, *self.board.get_size(), len(self.pieces),
8)
# 制約(a) 重複する置き方のピースは除外する
for y in range(self.board.get_size()[0]):
for x in range(self.board.get_size()[1]):
for i in range(len(self.pieces)):
for j in range(self.pieces[i].placement_count, 8):
q[y][x][i][j] = BinaryPoly(0)
# 制約(b) ピースはボードから外に出ない
for y in range(self.board.get_size()[0]):
for x in range(self.board.get_size()[1]):
for i in range(len(self.pieces)):
for j in range(self.pieces[i].placement_count):
if len(self.pieces[i].get_blocks(j, (x, y)) -
self.board.get_blocks()) > 0:
q[y][x][i][j] = BinaryPoly(0)
# 制約(c) ピース同士は重ならずボードを全て埋める
s = dict()
for b in self.board.get_blocks():
s[b] = BinaryPoly()
for y in range(self.board.get_size()[0]):
for x in range(self.board.get_size()[1]):
for i in range(len(self.pieces)):
for j in range(self.pieces[i].placement_count):
for p in self.pieces[i].get_blocks(
j, (x, y)) & self.board.get_blocks():
s[p] += q[y][x][i][j]
board_constraints = [equal_to(q, 1) for q in s.values()]
# 制約(d) 全てのピースは一度ずつ使われる
piece_constraints = [
equal_to(
sum(q[y][x][i][j] for y in range(self.board.get_size()[0])
for x in range(self.board.get_size()[1])
for j in range(8)), 1) for i in range(len(self.pieces))
]
constraints = (sum(board_constraints) + sum(piece_constraints))
solver = Solver(self.client)
model = BinaryQuadraticModel(constraints)
result = solver.solve(model)
if len(result) == 0:
raise RuntimeError("Any one of constaraints is not satisfied.")
solution = result[0]
values = solution.values
q_values = decode_solution(q, values)
Visualizer().visualize(self.pieces, self.board, q_values)
def by_amplify_limited(list_dependent_variables, num_registers, limitation):
num_variables = len(list_dependent_variables)
q = gen_symbols(BinaryPoly, num_variables, num_registers)
# 各変数を1つのレジスタに割り当てるOne-het制約
const_onehot = [
equal_to(sum_poly([q[i][r] for r in range(num_registers)]), 1)
for i in range(num_variables)
]
# レジスタスピルを減らすために，依存関係のある変数同士が同一のレジスタに割り当てられない制約
const_spill = [
penalty(q[i][r] * q[j][r]) for i in range(num_variables)
for j in list_dependent_variables[i] if i < j
for r in range(num_registers)
]
# ある変数が割り当てられるレジスタがわかっている時，必ずそのレジスタに割り当てられるようにする制約
const_limit = [
penalty(q[i][r]) for i, x in limitation.items()
for r in range(num_registers) if r not in x
]
constraints = sum(const_onehot)
if len(const_spill) != 0:
constraints += sum(const_spill)
if len(const_limit) != 0:
constraints += sum(const_limit)
return {"qubits": q, "model": BinaryQuadraticModel(constraints)}
def make_hamiltonian(d, feed_dict):
# set the number of cities
N = len(d)
# set hyperparameters
lambda_1 = feed_dict['h1']
lambda_2 = feed_dict['h2']
# make variables
x = gen_symbols(BinaryPoly, N, N)
# set One-hot constraint for time
h1 = [equal_to(sum_poly([x[n][i] for n in range(N)]), 1) for i in range(N)]
# set One-hot constraint for city
h2 = [equal_to(sum_poly([x[n][i] for i in range(N)]), 1) for n in range(N)]
# compute the total of constraints
const = lambda_1 * sum(h1) + lambda_2 * sum(h2)
# set objective function
obj = sum_poly(N, lambda n: sum_poly(N, lambda i: sum_poly(N, lambda j: d[i][j]*x[n][i]*x[(n+1)%N][j]), ), )
# compute model
model = obj + const
return x, model
def by_amplify(list_dependent_variables, num_registers):
num_variables = len(list_dependent_variables)
q = gen_symbols(BinaryPoly, num_variables, num_registers)
# 各変数を1つのレジスタに割り当てるOne-het制約
const_onehot = [
equal_to(sum_poly([q[i][r] for r in range(num_registers)]), 1)
for i in range(num_variables)
]
# レジスタスピルを減らすために，依存関係のある変数同士が同一のレジスタに割り当てられない制約
const_spill = [
penalty(q[i][r] * q[j][r]) for i in range(num_variables)
for j in list_dependent_variables[i] if i < j
for r in range(num_registers)
]
constraints = sum(const_onehot) + sum(const_spill)
return {"qubits": q, "model": BinaryQuadraticModel(constraints)}
def make_hamiltonian(type_matrix, weak_matrix, resist_matrix, enemies, num_party, feed_dict):
# set the number of types
N = len(type_matrix)
# set the number of enemies
M = len(enemies)
# set hyperparameters
lambda_1 = feed_dict['h1']
lambda_2 = feed_dict['h2']
lambda_3 = feed_dict['h3']
# make variables
x = gen_symbols(BinaryPoly, num_party, N)
# set one-hot constraint for types
h1 = [equal_to(sum_poly([x[i][j] for j in range(N)]), 1) for i in range(num_party)]
# set weak constraint
h2 = [less_equal(sum_poly(N, lambda j: sum_poly(num_party, lambda l: sum_poly(N, lambda k: enemies[i][j]*weak_matrix[j][k]*x[l][k]))), 2) for i in range(M)]
# set resist constraint
h3 = [greater_equal(sum_poly(N, lambda j: sum_poly(num_party, lambda l: sum_poly(N, lambda k: enemies[i][j]*resist_matrix[j][k]*x[l][k]))), 1) for i in range(M)]
# compute the total of constraints
const = lambda_1 * sum(h1) + lambda_2 * sum(h2) + lambda_3 * sum(h3)
# set objective function
obj = sum_poly(num_party, lambda i: sum_poly(N, lambda j: sum_poly(N, lambda k: sum_poly(M, lambda l: x[i][j]*type_matrix[j][k]*enemies[l][k]))))
# compute model
model = - obj + const
return x, model
solver = Solver(client)
# 四色の定義
colors = ["red", "green", "blue", "yellow"]
num_colors = len(colors)
num_region = len(jm.pref_names) - 1 # 都道府県数を取得
# 都道府県数 x 色数 の変数を作成
q = gen_symbols(BinaryPoly, num_region, num_colors)
# 各領域に対する制約
# 一つの領域に一色のみ（one-hot）
# sum_{c=0}^{C-1} q_{i,c} = 1 for all i
reg_constraints = [
equal_to(sum_poly([q[i][c] for c in range(num_colors)]), 1)
for i in range(num_region)
]
# 隣接する領域間の制約
adj_constraints = [
# 都道府県コードと配列インデックスは1ずれてるので注意
penalty(q[i][c] * q[j - 1][c]) for i in range(num_region)
for j in jm.adjacent(i + 1) # j: 隣接している都道府県コード
if i + 1 < j for c in range(num_colors)
]
constraints = sum(reg_constraints) + sum(adj_constraints)
model = BinaryQuadraticModel(constraints)
result = solver.solve(model)
def quantum_solver_approx(N, M,
query): # solve with Amplify (approximate version)
q = gen_symbols(BinaryPoly, M, N, N) # represent the solution
########## constraints ##########
# each layer doesn't have 2+ same values
one_hot_constraints_layer = [
# m -> layer
# n -> qubit
# v -> value of qubit
equal_to(sum(q[m][n][v] for n in range(N)), 1) for m in range(M)
for v in range(N)
]
# each qubit doesn't have 2+ values
one_hot_constraints_num = [
# m -> layer
# n -> qubit
# v -> value of qubit
equal_to(sum(q[m][n][v] for v in range(N)), 1) for m in range(M)
for n in range(N)
]
# every CX gate must be applied for 2 adjacent qubits
CXgate_constraints = []
for m in range(M):
for g0 in range(0, len(query[m]), 2):
v0, v1 = query[m][g0], query[m][g0 + 1]
# v0 and v1 must be adjacent each other
for i in range(N):
for j in range(i + 2, N):
CXgate_constraints.append(
penalty(q[m][i][v0] * q[m][j][v1]))
CXgate_constraints.append(
penalty(q[m][i][v1] * q[m][j][v0]))
constraints = (sum(one_hot_constraints_layer) +
sum(one_hot_constraints_num) + sum(CXgate_constraints))
cost = sum_poly(
M - 1, lambda m: sum_poly(
N, lambda i: sum_poly(
N, lambda j: sum_poly(N, lambda v: q[m][i][v] * q[m + 1][j][v])
* ((N - 1) * (i + j) - 2 * i * j) / N)))
########## solve ##########
solver = Solver(client)
model = BinaryQuadraticModel(constraints * constraintWeight + cost)
result = solver.solve(model)
if len(result) == 0:
raise RuntimeError("Any one of constraints is not satisfied.")
values = result[0].values
q_values = decode_solution(q, values, 1)
# print(q_values_main)
########## decode the result into string ##########
ans = [[-1 for n in range(N)] for m in range(M)]
for m in range(M):
for n in range(N):
for v in range(N):
if (q_values[m][n][v] > 0.5):
ans[m][n] = v
cost = 0
for m in range(M - 1):
cost += calcCost(ans[m], ans[m + 1])
return cost, ans
def quantum_solver_strict(N, M, query): # solve by Amplify (strict version)
q_all = gen_symbols(BinaryPoly,
M * N * N + (M - 1) * N * N * N + (M - 1) * N * N)
q = q_all[:M * N * N] # represent the solution
q_sub = q_all[M * N * N:M * N * N + (M - 1) * N * N *
N] # q_sub[m][i][j][v] = q[m][i][v] * q[m+1][j][v]
q_C_matrix = q_all[
M * N * N + (M - 1) * N * N *
N:] # q_C_matrix[m][i][j] = sum(q_sub[m][i][j][v] for v)
########## constraints ##########
# each layer doesn't have 2+ same values
one_hot_constraints_layer = [
# m -> layer
# n -> physical qubit
# v -> logical qubit
equal_to(sum(q[(m * N + n) * N + v] for n in range(N)), 1)
for m in range(M) for v in range(N)
]
# each qubit doesn't have 2+ values
one_hot_constraints_num = [
# m -> layer
# n -> physical qubit
# v -> logical qubit
equal_to(sum(q[(m * N + n) * N + v] for v in range(N)), 1)
for m in range(M) for n in range(N)
]
# every CX gate must be applied for 2 adjacent qubits
CXgate_constraints = []
for m in range(M):
for g0 in range(0, len(query[m]), 2):
v0, v1 = query[m][g0], query[m][g0 + 1]
# v0 and v1 must be adjacent each other
for i in range(N):
for j in range(i + 2, N):
CXgate_constraints.append(
penalty(q[(m * N + i) * N + v0] *
q[(m * N + j) * N + v1]))
CXgate_constraints.append(
penalty(q[(m * N + i) * N + v1] *
q[(m * N + j) * N + v0]))
# q_sub[m][i][j][v] = q[m][i][v] * q[m+1][j][v]
sub_gate_constraints = []
for _idx in range((M - 1) * N**3):
idx = _idx
m = idx // (N**3)
idx %= N**3
i = idx // (N**2)
idx %= N**2
j = idx // N
idx %= N
v = idx
sub_gate_constraints.append(
penalty(3 * q_sub[((m * N + i) * N + j) * N + v] +
q[(m * N + i) * N + v] * q[((m + 1) * N + j) * N + v] -
2 * q_sub[((m * N + i) * N + j) * N + v] *
(q[(m * N + i) * N + v] + q[((m + 1) * N + j) * N + v])))
# q_C_matrix[m][i][j] = sum(q_sub[m][i][j][v] for v)
C_matrix_sum_constraints = []
for _idx in range((M - 1) * N**2):
idx = _idx
m = idx // (N**2)
idx %= N**2
i = idx // N
idx %= N
j = idx
C_matrix_sum_constraints.append(
equal_to(
q_C_matrix[(m * N + i) * N + j] -
sum(q_sub[((m * N + i) * N + j) * N + v] for v in range(N)),
0))
constraints = (sum(one_hot_constraints_layer) +
sum(one_hot_constraints_num) + sum(CXgate_constraints) +
sum(sub_gate_constraints) + sum(C_matrix_sum_constraints))
cost = []
for m in range(M - 1):
for i1 in range(N):
for j1 in range(i1): # i1 > j1
for i2 in range(N):
for j2 in range(i2 + 1, N): # i2 < j2
cost.append(q_C_matrix[(m * N + i1) * N + j1] *
q_C_matrix[(m * N + i2) * N + j2])
for j1 in range(i1 + 1, N): # i1 < j1
for i2 in range(N):
for j2 in range(i2): # i2 > j2
cost.append(q_C_matrix[(m * N + i1) * N + j1] *
q_C_matrix[(m * N + i2) * N + j2])
# print(constraints)
# print(cost)
########## solve ##########
solver = Solver(client)
model = BinaryQuadraticModel(constraints * constraintWeight + sum(cost))
result = solver.solve(model)
if len(result) == 0:
raise RuntimeError("Any one of constraints is not satisfied.")
values = result[0].values
q_values = decode_solution(q_all, values, 1)
# print(q_values_main)
########## decode the result into string ##########
ans = [[-1 for n in range(N)] for m in range(M)]
for m in range(M):
for n in range(N):
for v in range(N):
if (q_values[(m * N + n) * N + v] > 0.5):
ans[m][n] = v
cost = 0
for m in range(M - 1):
cost += calcCost(ans[m], ans[m + 1])
return cost, ans
def find_feasible_solution(self):
"""find a feasible locations with makespan, found -> set self.used_edges
"""
# create variables
q = []
index = 0
for t in range(self.makespan):
q.append([])
for v in range(self.field["size"]):
l = len(self.field["adj"][v])+1 # +1 -> stay at the current location
q[-1].append(
amplify.gen_symbols( amplify.BinaryPoly, index, (1, l) )
)
index += l
# set starts
constraints_starts = [
equal_to(sum_poly( q[0][v][0] ), 1) # q[timestep][node][0]
for v in self.instance["starts"]
]
for v in range(self.field["size"]):
if v in self.instance["starts"]:
continue
# other locations
for i in range(len(q[0][v][0])):
q[0][v][0][i] = amplify.BinaryPoly(0)
# set goals
constraints_goals = [
equal_to(sum_poly([ q[-1][u][0][ self.field["adj"][u].index(v) ]
for u in self.field["adj"][v] ] +
[ q[-1][v][0][ len(self.field["adj"][v]) ] ]),
1)
for v in self.instance["goals"]
]
for v in range(self.field["size"]):
# other locations
for i in range(len(self.field["adj"][v])):
if self.field["adj"][v][i] not in self.instance["goals"]:
q[-1][v][0][i] = amplify.BinaryPoly(0)
if v not in self.instance["goals"]:
q[-1][v][0][-1] = amplify.BinaryPoly(0)
# upper bound, in
constraints_in = [
less_equal(sum_poly([ q[t][u][0][ self.field["adj"][u].index(v) ]
for u in self.field["adj"][v] ] +
[ q[t][v][0][ len(self.field["adj"][v]) ] ]),
1)
for v, t in product(range(self.field["size"]), range(0, self.makespan-1))
]
# upper bound, out
constraints_out = [
less_equal(sum_poly( q[t][v][0] ),
1)
for v, t in product(range(self.field["size"]), range(1, self.makespan))
]
# continuity
constraints_continuity = [
equal_to(sum_poly([ q[t][u][0][ self.field["adj"][u].index(v) ]
for u in self.field["adj"][v] ] +
[ q[t][v][0][ len(self.field["adj"][v]) ] ])
-
sum_poly( q[t+1][v][0] ),
0)
for v, t in product(range(self.field["size"]), range(0, self.makespan-1))
]
# branching
for v in range(self.field["size"]):
if not self.field["body"][v]:
continue
# unreachable vertexes from starts
for t in range(0, min(self.DIST_TABLE_FROM_STARTS[v], self.makespan)):
for i in range(len(q[t][v][0])):
q[t][v][0][i] = amplify.BinaryPoly(0)
# unreachable vertexes to goals
for t in range(max(self.makespan - self.DIST_TABLE_FROM_GOALS[v] + 1, 0), self.makespan):
for i in range(len(q[t][v][0])):
q[t][v][0][i] = amplify.BinaryPoly(0)
# set occupied vertex
for v in range(self.field["size"]):
if self.field["body"][v]:
continue
for t in range(0, self.makespan):
q[t][v][0][-1] = amplify.BinaryPoly(0)
# create model
model = sum(constraints_starts)
model += sum(constraints_goals)
if len(constraints_in) > 0:
model += sum(constraints_in)
if len(constraints_out) > 0:
model += sum(constraints_out)
if len(constraints_continuity) > 0:
model += sum(constraints_continuity)
# setup client
client = FixstarsClient()
client.token = os.environ['TOKEN']
client.parameters.timeout = self.timeout
# solve
solver = amplify.Solver(client)
result = solver.solve(model)
if len(result) > 0:
self.used_edges = amplify.decode_solution(q, result[0].values)
# CS3: Omata, Ando, Hattori, Watanabe
q2020[labs.index("Omata")][grps.index("CS3")] = BinaryPoly(1)
q2020[labs.index("Ando")][grps.index("CS3")] = BinaryPoly(1)
q2020[labs.index("Hattori")][grps.index("CS3")] = BinaryPoly(1)
q2020[labs.index("Watanabe")][grps.index("CS3")] = BinaryPoly(1)
# CS4: Suzuki, Iwanuma, Kinoshita, Nabeshima
q2020[labs.index("Suzuki")][grps.index("CS4")] = BinaryPoly(1)
q2020[labs.index("Iwanuma")][grps.index("CS4")] = BinaryPoly(1)
q2020[labs.index("Kinoshita")][grps.index("CS4")] = BinaryPoly(1)
q2020[labs.index("Nabeshima")][grps.index("CS4")] = BinaryPoly(1)
##################################################################################
# 行（研究室）に対する制約： one-hot制約（1つの研究室が属するグループは1つだけ）
row_constraints = [
equal_to(sum_poly([q[i][j] for j in range(ngrp)]), 1) for i in range(nlab)
]
##################################################################################
# 制約
constraints = sum(row_constraints)
# モデル
model = cost + 5 * constraints
# ソルバの生成
solver = Solver(client)
# solver.filter_solution = False # 実行可能解以外をフィルタリングしない
##################################################################################
# ソルバ起動
g = (sum_poly(q) - 1)**2 # one-hot 制約に対応したペナルティ関数
print("One-Hot")
print(f"g = {g}")
# 問題を解いて結果を表示
result = solver.solve(g)
for sol in result:
energy = sol.energy
values = sol.values
print(f"energy = {energy}, {q} = {decode_solution(q, values)}")
###########################################################
# 等式制約：q0 * q1 + q2 = 1
g = equal_to(q[0] * q[1] + q[2], 1) # 等式制約
print("Equal")
print(f"g = {g}")
# 問題を解いて結果を表示
result = solver.solve(g)
for sol in result:
energy = sol.energy
values = sol.values
print(f"energy = {energy}, {q} = {decode_solution(q, values)}")
###########################################################
# 不等式制約 less_equal：q0 + q1 + q2 <= 1
g = less_equal(sum_poly(q), 1) # 不等式制約
from amplify import BinaryQuadraticModel, Solver, decode_solution
from amplify.client import FixstarsClient
from amplify import BinaryPoly, gen_symbols, sum_poly
from amplify.constraint import equal_to
q = gen_symbols(BinaryPoly, 3)
H = equal_to(sum_poly(q), 3)
model = BinaryQuadraticModel(H)
solver = Solver(client)
result = solver.solve(model)
values = result[0].values
print(f"q = {decode_solution(q, values)}")