def get_adj(left, right):
A = zeros2((len(left), len(right)))
for i, l in enumerate(left):
for j, r in enumerate(right):
lr = l.intersect(r)
A[i, j] = len(lr)
return A
def rel(self, a, b):
incidence = self.incidence
A = zeros2(len(a), len(b))
for i, aa in enumerate(a):
for j, bb in enumerate(b):
A[i, j] = (aa, bb) in incidence
return A
def syparse(decl):
for c in '0123456789':
decl = decl.replace(c, ' ')
decl = decl.strip().split()
n = len(decl[0])
m = len(decl)
H = zeros2(m, 2 * n)
for i, row in enumerate(decl):
for j, c in enumerate(row):
if c == 'X' or c == 'Y':
H[i, j] = 1
if c == 'Z' or c == 'Y':
H[i, n + j] = 1
return H
def symplectic(n):
F = zeros2(2 * n, 2 * n)
for i in range(n):
F[i, n + i] = 1
F[n + i, i] = 1
return F
def is_422_cat_selfdual(Hz, Hx, perm):
"concatenate with the [[4,2,2]] code and see if we get a self-dual code"
from numpy import alltrue, zeros, dot
import qupy.ldpc.solve
import bruhat.solve
qupy.ldpc.solve.int_scalar = bruhat.solve.int_scalar
from bruhat.solve import shortstrx, zeros2, dot2, array2, solve
from qupy.ldpc.css import CSSCode
Cout = CSSCode(Hz=Hz, Hx=Hx)
#print(Cout)
#Cin = CSSCode(Hz=array2([[1,1,1,1]]), Hx=array2([[1,1,1,1]]))
#print(Cin)
pairs = []
singles = []
for i in range(Cout.n):
j = perm[i]
if j < i:
continue
if i == j:
singles.append(i)
else:
pairs.append((i, j))
#print(singles, pairs)
M = len(singles) + 4 * len(pairs)
# encoding matrices
enc_z = zeros2(M, Cout.n)
enc_x = zeros2(M, Cout.n)
row = 0
for col in singles:
enc_z[row, col] = 1
enc_x[row, col] = 1
row += 1
H = []
for (i, j) in pairs:
enc_z[row, i] = 1 # 1010
enc_z[row + 2, i] = 1
enc_z[row, j] = 1 # 1100
enc_z[row + 1, j] = 1
enc_x[row, i] = 1 # 1100
enc_x[row + 1, i] = 1
enc_x[row, j] = 1 # 1010
enc_x[row + 2, j] = 1
h = array2([0] * M)
h[row:row + 4] = 1
H.append(h)
row += 4
assert row == M
#print(shortstrx(enc_z, enc_x))
Hz = dot2(enc_z, Cout.Hz.transpose()).transpose()
Hx = dot2(enc_x, Cout.Hx.transpose()).transpose()
assert alltrue(dot2(Hz, Hx.transpose()) == 0)
Hz = numpy.concatenate((Hz, H))
Hx = numpy.concatenate((Hx, H))
assert alltrue(dot2(Hz, Hx.transpose()) == 0)
C = CSSCode(Hz=Hz, Hx=Hx)
assert C.k == Cout.k
#print(C)
lhs = (solve(Hz.transpose(), Hx.transpose()) is not None)
rhs = (solve(Hx.transpose(), Hz.transpose()) is not None)
return lhs and rhs
def search():
# Bravyi, Haah, 1209.2426v1 sec IX.
# https://arxiv.org/pdf/1209.2426.pdf
verbose = argv.get("verbose")
m = argv.get("m", 6) # _number of rows
k = argv.get("k", None) # _number of odd-weight rows
# these are the variables N_x
xs = list(cross([(0, 1)]*m))
maxweight = argv.maxweight
minweight = argv.get("minweight", 1)
xs = [x for x in xs if minweight <= sum(x)]
if maxweight:
xs = [x for x in xs if sum(x) <= maxweight]
N = len(xs)
lhs = []
rhs = []
# bi-orthogonality
for a in range(m):
for b in range(a+1, m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == x[b] == 1:
v[i] = 1
if v.sum():
lhs.append(v)
rhs.append(0)
# tri-orthogonality
for a in range(m):
for b in range(a+1, m):
for c in range(b+1, m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == x[b] == x[c] == 1:
v[i] = 1
if v.sum():
lhs.append(v)
rhs.append(0)
# # dissallow columns with weight <= 1
# for i, x in enumerate(xs):
# if sum(x)<=1:
# v = zeros2(N)
# v[i] = 1
# lhs.append(v)
# rhs.append(0)
if k is not None:
# constrain to k _number of odd-weight rows
assert 0<=k<m
for a in range(m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == 1:
v[i] = 1
lhs.append(v)
if a<k:
rhs.append(1)
else:
rhs.append(0)
A = array2(lhs)
rhs = array2(rhs)
#print(shortstr(A))
B = pseudo_inverse(A)
soln = dot2(B, rhs)
if not eq2(dot2(A, soln), rhs):
print("no solution")
return
if verbose:
print("soln:")
print(shortstr(soln))
soln.shape = (N, 1)
rhs.shape = A.shape[0], 1
K = array2(list(find_kernel(A)))
#print(K)
#print( dot2(A, K.transpose()))
#sols = []
#for v in span(K):
best = None
density = 1.0
size = 99*N
trials = argv.get("trials", 1024)
count = 0
for trial in range(trials):
u = rand2(len(K), 1)
v = dot2(K.transpose(), u)
#print(v)
v = (v+soln)%2
assert eq2(dot2(A, v), rhs)
if v.sum() > size:
continue
size = v.sum()
Gt = []
for i, x in enumerate(xs):
if v[i]:
Gt.append(x)
if not Gt:
continue
Gt = array2(Gt)
G = Gt.transpose()
assert is_morthogonal(G, 3)
if G.shape[1]<m:
continue
if 0 in G.sum(1):
continue
if argv.strong_morthogonal and not strong_morthogonal(G, 3):
continue
#print(shortstr(G))
# for g in G:
# print(shortstr(g), g.sum())
# print()
_density = float(G.sum()) / (G.shape[0]*G.shape[1])
#if best is None or _density < density:
if best is None or G.shape[1] <= size:
best = G
size = G.shape[1]
density = _density
if 0:
#sols.append(G)
Gx = even_rows(G)
assert is_morthogonal(Gx, 3)
if len(Gx)==0:
continue
GGx = array2(list(span(Gx)))
assert is_morthogonal(GGx, 3)
count += 1
print("found %d solutions" % count)
if best is None:
return
G = best
#print(shortstr(G))
for g in G:
print(shortstr(g), g.sum())
print()
print("density:", density)
print("shape:", G.shape)
G = linear_independent(G)
A = list(span(G))
print(strong_morthogonal(A, 1))
print(strong_morthogonal(A, 2))
print(strong_morthogonal(A, 3))
#print(shortstr(dot2(G, G.transpose())))
if 0:
B = pseudo_inverse(A)
v = dot2(B, rhs)
print("B:")
print(shortstr(B))
print("v:")
print(shortstr(v))
assert eq2(dot2(B, v), rhs)
def search_selfdual():
verbose = argv.get("verbose")
m = argv.get("m", 6) # _number of rows
k = argv.get("k", None) # _number of odd-weight rows
maxweight = argv.get("maxweight", m)
minweight = argv.get("minweight", 1)
# these are the variables N_x
print("building xs...")
if 0:
xs = cross([(0, 1)]*m)
xs = [x for x in xs if minweight <= sum(x) <= maxweight]
prune = argv.get("prune", 0.5)
xs = [x for x in xs if random() < prune]
xs = []
N = argv.get("N", m*100)
colweight = argv.get("colweight", maxweight)
assert colweight <= m
for i in range(N):
x = [0]*m
total = 0
while total < colweight:
idx = randint(0, m-1)
if x[idx] == 0:
x[idx] = 1
total += 1
xs.append(x)
N = len(xs)
lhs = []
rhs = []
# bi-orthogonality
for a in range(m):
for b in range(a+1, m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == x[b] == 1:
v[i] = 1
if v.sum():
lhs.append(v)
rhs.append(0)
k = 0 # all rows must have even weight
# constrain to k _number of odd-weight rows
assert 0<=k<m
for a in range(m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == 1:
v[i] = 1
lhs.append(v)
if a<k:
rhs.append(1)
else:
rhs.append(0)
logops = argv.logops
A = array2(lhs)
rhs = array2(rhs)
#print(shortstr(A))
print("solve...")
B = pseudo_inverse(A)
soln = dot2(B, rhs)
if not eq2(dot2(A, soln), rhs):
print("no solution")
return
if verbose:
print("soln:")
print(shortstr(soln))
soln.shape = (N, 1)
rhs.shape = A.shape[0], 1
K = array2(list(find_kernel(A)))
print("kernel:", K.shape)
if len(K)==0:
return
#print(K)
#print( dot2(A, K.transpose()))
#sols = []
#for v in span(K):
best = None
density = 1.0
size = 99*N
trials = argv.get("trials", 1024)
count = 0
print("trials...")
for trial in range(trials):
u = rand2(len(K), 1)
v = dot2(K.transpose(), u)
#print(v)
v = (v+soln)%2
assert eq2(dot2(A, v), rhs)
if v.sum() >= size:
continue
if v.sum() < m:
continue
if v.sum():
print(v.sum(), end=" ", flush=True)
size = v.sum()
if logops is not None and size != 2*m+logops:
continue
Gt = []
for i, x in enumerate(xs):
if v[i]:
Gt.append(x)
Gt = array2(Gt)
G = Gt.transpose()
if dot2(G, Gt).sum() != 0:
# not self-dual
print(shortstr(dot2(G, Gt)))
assert 0
return
#if G.shape[1]<m:
# continue
if 0 in G.sum(1):
print(".", end="", flush=True)
continue
#print(shortstr(G))
# for g in G:
# print(shortstr(g), g.sum())
# print()
_density = float(G.sum()) / (G.shape[0]*G.shape[1])
#if best is None or _density < density:
if best is None or G.shape[1] <= size:
best = G
size = G.shape[1]
density = _density
if 0:
#sols.append(G)
Gx = even_rows(G)
assert is_morthogonal(Gx, 3)
if len(Gx)==0:
continue
GGx = array2(list(span(Gx)))
assert is_morthogonal(GGx, 3)
count += 1
print("found %d solutions" % count)
if best is None:
return
G = best
#print(shortstr(G))
f = open("selfdual.ldpc", "w")
for spec in ["Hx =", "Hz ="]:
print(spec, file=f)
for g in G:
print(shortstr(g), file=f)
f.close()
print()
print("density:", density)
print("shape:", G.shape)
if 0:
B = pseudo_inverse(A)
v = dot2(B, rhs)
print("B:")
print(shortstr(B))
print("v:")
print(shortstr(v))
assert eq2(dot2(B, v), rhs)
def search_extend():
# Extend the checks of a random code to make it triorthogonal.
# Based on the search function above.
verbose = argv.get("verbose")
m = argv.get("m", 6)
n = argv.get("n", m+2)
k = argv.get("k") # odd _numbered rows ( logical operators)
code = argv.get("code", "rand")
if code == "rand":
while 1:
G0 = rand2(m, n)
counts = G0.sum(0)
if min(counts)==2 and rank(G0) == m:
cols = set()
for i in range(n):
cols.add(tuple(G0[:, i]))
if len(cols) == n: # no repeated cols
break
elif code == "toric":
G0 = parse("""
11.11...
.111..1.
1...11.1
""") # l=2 toric code X logops + X stabs
l = argv.get("l", 3)
G0 = build_toric(l)
m, n = G0.shape
else:
return
code = Code(G0, check=False)
print(shortstr(G0))
print("is_triorthogonal:", code.is_triorthogonal())
# these are the variables N_x
xs = list(cross([(0, 1)]*m))
N = len(xs)
lookup = {}
for i, x in enumerate(xs):
lookup[x] = i
lhs = []
rhs = []
taken = set()
for i in range(n):
x = G0[:, i]
idx = lookup[tuple(x)]
assert idx not in taken
taken.add(idx)
if verbose:
for idx in range(N):
print(idx, xs[idx], "*" if idx in taken else "")
for idx in taken:
v = zeros2(N)
v[idx] = 1
lhs.append(v)
rhs.append(1)
# bi-orthogonality
for a in range(m):
for b in range(a+1, m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == x[b] == 1:
v[i] += 1
assert v.sum()
lhs.append(v)
rhs.append(0)
# tri-orthogonality
for a in range(m):
for b in range(a+1, m):
for c in range(b+1, m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == x[b] == x[c] == 1:
v[i] += 1
assert v.sum()
lhs.append(v)
rhs.append(0)
# dissallow columns with weight <= 1
for i, x in enumerate(xs):
if sum(x)<=1:
v = zeros2(N)
v[i] = 1
lhs.append(v)
rhs.append(0)
if k is not None:
# constrain to k _number of odd-weight rows
assert 0<=k<m
for a in range(m):
v = zeros2(N)
for i, x in enumerate(xs):
if x[a] == 1:
v[i] = 1
lhs.append(v)
if a<k:
rhs.append(1)
else:
rhs.append(0)
A = array2(lhs)
rhs = array2(rhs)
if verbose:
print("lhs:")
print(shortstr(A))
print("rhs:")
print(shortstr(rhs))
B = pseudo_inverse(A)
soln = dot2(B, rhs)
if not eq2(dot2(A, soln), rhs):
print("no solution")
return
if verbose:
print("soln:")
print(shortstr(soln))
soln.shape = (N, 1)
rhs.shape = A.shape[0], 1
K = array2(list(find_kernel(A)))
best = None
density = 1.0
size = 9999*n
trials = argv.get("trials", 1024)
count = 0
for trial in range(trials):
u = rand2(len(K), 1)
v = dot2(K.transpose(), u)
#print(v)
assert dot2(A, v).sum()==0
#if v.sum() != n:
# continue
assert v[0]==0
v = (v+soln)%2
assert eq2(dot2(A, v), rhs)
Gt = list(G0.transpose())
for i, x in enumerate(xs):
if v[i] and not i in taken:
Gt.append(x)
if not Gt:
continue
Gt = array2(Gt)
G = Gt.transpose()
if verbose:
print("G0")
print(shortstr(G0))
print("solution:")
print(shortstr(G))
assert is_morthogonal(G, 3)
if G.shape[1]<m:
continue
if 0 in G.sum(1):
continue
#print(shortstr(G))
# for g in G:
# print(shortstr(g), g.sum())
# print()
_density = float(G.sum()) / (G.shape[0]*G.shape[1])
#if best is None or _density < density:
if best is None or G.shape[1] < size:
best = G
density = _density
size = G.shape[1]
if 0:
#sols.append(G)
Gx = even_rows(G)
assert is_morthogonal(Gx, 3)
if len(Gx)==0:
continue
GGx = array2(list(span(Gx)))
assert is_morthogonal(GGx, 3)
count += 1
print("found %d solutions" % count)
G = best
#print(shortstr(G))
for g in G:
print(shortstr(g), g.sum())
print()
print("density:", density)
def build_toric(l=3, allgen=True):
keys = []
keymap = {}
for i in range(l):
for j in range(l):
for k in (0, 1):
m = len(keys)
keys.append((i, j, k))
for di in (-l, 0, l):
for dj in (-l, 0, l):
keymap[i+di, j+dj, k] = m
if l>2:
assert keys[keymap[2, 1, 0]] == (2, 1, 0)
if allgen:
m = l**2 # rows (constraints)
else:
m = l**2-1 # rows (constraints)
n = len(keys) # cols (bits)
assert n == 2*(l**2)
Lx = zeros2(2, n)
Lz = zeros2(2, n)
Hx = zeros2(m, n)
Tz = zeros2(m, n)
Hz = zeros2(m, n)
Tx = zeros2(m, n)
for i in range(l):
Lx[0, keymap[i, l-1, 1]] = 1
Lx[1, keymap[l-1, i, 0]] = 1
Lz[0, keymap[0, i, 1]] = 1
Lz[1, keymap[i, 0, 0]] = 1
row = 0
xmap = {}
for i in range(l):
for j in range(l):
if (i, j)==(0, 0) and not allgen:
continue
Hx[row, keymap[i, j, 0]] = 1
Hx[row, keymap[i, j, 1]] = 1
Hx[row, keymap[i-1, j, 0]] = 1
Hx[row, keymap[i, j-1, 1]] = 1
xmap[i, j] = row
i1 = i
while i1>0:
Tz[row, keymap[i1-1, j, 0]] = 1
i1 -= 1
j1 = j
while j1>0:
Tz[row, keymap[i1, j1-1, 1]] = 1
j1 -= 1
row += 1
row = 0
zmap = {}
for i in range(l):
for j in range(l):
if i==l-1 and j==l-1 and not allgen:
continue
Hz[row, keymap[i, j, 0]] = 1
Hz[row, keymap[i, j, 1]] = 1
Hz[row, keymap[i+1, j, 1]] = 1
Hz[row, keymap[i, j+1, 0]] = 1
zmap[i, j] = row
i1 = i
while i1<l-1:
Tx[row, keymap[i1+1, j, 1]] = 1
i1 += 1
j1 = j
while j1<l-1:
Tx[row, keymap[i1, j1+1, 0]] = 1
j1 += 1
row += 1
return Hx
def hecke(self):
bag0 = self.get_bag()
bag1 = self.get_bag()
points = [p for p in bag0 if p.desc == '0']
lines = [p for p in bag0 if p.desc == '1']
print(points)
print(lines)
flags = []
for p in points:
ls = [l for l in p.nbd if l != p]
print(p, ls)
for l in ls:
flags.append((p, l))
print("flags:", len(flags))
# for point in bag0:
# print point, repr(point.desc)
#items = [(point.idx, line.idx) for point in points for line in lines]
#items = [(a.idx, b.idx) for a in points for b in points]
items = [(a, b) for a in flags for b in flags]
# fixing two points gives the Klein four group, Z_2 x Z_2:
#for f in isomorph.search(bag0, bag1):
# if f[0]==0 and f[1]==1:
# print f
perms = []
for f in isomorph.search(bag0, bag1):
#print f
g = dict((bag0[idx], bag0[f[idx]]) for idx in list(f.keys()))
perm = {}
for a, b in items:
a1 = (g[a[0]], g[a[1]])
b1 = (g[b[0]], g[b[1]])
assert (a1, b1) in items
perm[a, b] = a1, b1
perm = Perm(perm, items)
perms.append(perm)
write('.')
print()
g = Group(perms, items)
print("group:", len(g))
orbits = g.orbits()
#print orbits
print("orbits:", len(orbits))
eq = numpy.allclose
dot = numpy.dot
n = len(flags)
basis = []
gen = []
I = identity2(n)
for orbit in orbits:
A = zeros2(n, n)
#print orbits
for a, b in orbit:
A[flags.index(a), flags.index(b)] = 1
print(shortstr(A))
print(A.sum())
#print
basis.append(A)
if A.sum() == 42:
gen.append(A)
assert len(gen) == 2
L = gen[0]
P = gen[1]
q = 2
assert eq(dot(L, L), L + q * I)
assert eq(dot(P, P), P + q * I)
assert eq(dot(L, dot(P, L)), dot(P, dot(L, P)))