def eq(self, other):
"Two codes are equal if their generating matrices have the same span."
G1, G2 = self.G, other.G
if len(G1) != len(G2):
return False
A = dot2(self.H, other.G.transpose())
B = dot2(other.H, self.G.transpose())
assert (A.sum()==0) == (B.sum()==0)
return A.sum() == 0
def main_fail():
print()
print("==" * 70)
H = """
012345678901234
0 YXZZ...........
1 X..X.XX.X......
2 ZZ..ZZ.......Z.
3 .X..Y....XZ....
4 .ZX....Y.Z.....
5 .....ZZZ.Z....Z
6 ..Z...ZZ....ZZ.
7 ...Z....Y.XX...
8 ..XX.......ZY..
9 .........ZXX..Y
10 .....X.....XXXX
11 ....X.X.X.X..X.
"""
H = syparse(H)
print()
print(shortstr(H))
m = len(H)
n = H.shape[1] // 2
assert n == 15
F = symplectic(n)
C = dot2(H, dot2(F, H.transpose()))
print(C.shape)
for i in range(m):
for j in range(i + 1, m):
if C[i, j]:
print("fail:", i, j)
print(rank(H))
H = linear_independent(H)
print("H=", H.shape)
print(shortstr(H))
HF = dot2(H, F)
K = array2(find_kernel(HF))
print("K=", K.shape)
print(shortstr(K))
HK = numpy.concatenate((H, K))
L = linear_independent(HK)
print()
print(shortstr(L))
def gen():
r = argv.get("r", None) # degree
m = argv.get("m", None)
if r is not None and m is not None:
code = reed_muller(r, m)
#print(code)
#print("d =", code.get_distance())
#code.dump()
#code = code.puncture(3)
#print(code)
code = code.puncture(0)
print(code)
for g in code.G:
print(shortstr(g), g.sum())
print()
#code.dump()
#print("d =", code.get_distance())
return
for m in range(2, 8):
for r in range(0, m+1):
code = reed_muller(r, m)
print(code, end=" ")
if code.is_selfdual():
print("is_selfdual", end=" ")
if code.is_morthogonal(2):
print("is_biorthogonal", end=" ")
if code.is_morthogonal(3):
print("is_triorthogonal", end=" ")
if dot2(code.H, code.H.transpose()).sum()==0:
print("***", end=" ")
p = code.puncture(0)
if p.is_morthogonal(3):
print("puncture.is_triorthogonal", end=" ")
if p.is_selfdual():
print("puncture.is_selfdual", end=" ")
if dot2(p.H, p.H.transpose()).sum()==0:
print("***", end=" ")
print()
if p.is_triorthogonal() and p.k < 20:
G = p.G
#print(shortstr(G))
A = list(span(G))
A = array2(A)
print(is_morthogonal(A, 3))
def main_torus():
n = 8
# ZZZZZZZZ|XXXXXXXX
# 12345678|12345678
H = parse("""
111..1..|........
1..1....|1.1.....
........|11.11...
.1..1...|.1...1..
..1...1.|...1..1.
...11.11|........
.....1.1|....1..1
........|..1..111
""".replace("|", ""))
print()
print("H=")
print(shortstr(H))
F = symplectic(n)
C = dot2(H, dot2(F, H.transpose()))
for i in range(n):
for j in range(i + 1, n):
if C[i, j]:
print("fail:", i + 1, j + 1)
print(rank(H))
H = linear_independent(H)
print("H=")
print(shortstr(H))
HF = dot2(H, F)
K = array2(find_kernel(HF))
print("K=")
print(shortstr(K))
HK = numpy.concatenate((H, K))
L = linear_independent(HK)
print()
print(shortstr(L))
def check_toric():
global toric # arff !
import qupy.ldpc.solve
import bruhat.solve
qupy.ldpc.solve.int_scalar = bruhat.solve.int_scalar
from bruhat.solve import shortstr, zeros2, dot2, array2, solve
from numpy import alltrue, zeros, dot
l = argv.get("l", 2)
from qupy.ldpc.toric import Toric2D
toric = Toric2D(l)
Hx, Hz = toric.Hx, toric.Hz
assert alltrue(dot2(Hx, Hz.transpose()) == 0)
from qupy.condmat.isomorph import Tanner, search
src = Tanner.build2(Hx, Hz)
#tgt = Tanner.build2(Hx, Hz)
tgt = Tanner.build2(Hz, Hx) # weak duality
mx, n = Hx.shape
mz, n = Hz.shape
fns = []
perms = []
for fn in search(src, tgt):
assert len(fn) == mx + mz + n
bitmap = []
for i in range(n):
bitmap.append(fn[i + mx + mz] - mx - mz)
perm = tuple(bitmap)
#print(bitmap)
fixed = [i for i in range(n) if bitmap[i] == i]
print("perm:", perm)
print("fixed:", fixed)
g = Perm(perm, list(range(n)))
assert g.order() == 2
perms.append(perm)
for hx in Hx:
print(toric.strop(hx))
print("--->")
hx = array2([hx[i] for i in perm])
print(toric.strop(hx))
print("--->")
hx = array2([hx[i] for i in perm])
print(toric.strop(hx))
print()
check_dualities(Hz, Hx.transpose(), perms)
def test_rm():
params = [(r, m) for m in range(2, 8) for r in range(1, m)]
r = argv.get("r", None) # degree
m = argv.get("m", None)
if r is not None and m is not None:
params = [(r, m)]
for (r, m) in params:
#code = reed_muller(r, m)
# for code in [ reed_muller(r, m), reed_muller(r, m).puncture(0) ]:
for code in [reed_muller(r, m)]:
if argv.puncture:
print(code, end=" ", flush=True)
code = code.puncture(0)
code = code.get_even()
if argv.puncture==2:
code = code.puncture(0)
code = code.get_even()
G = code.G
k, n = G.shape
#code = Code(G)
#d = code.get_distance()
d = "."
print("puncture [%d, %d, %s]" % (n, k, d), end=" ", flush=True)
else:
G = code.G
print(code, end=" ", flush=True)
i = 1
while i<8:
if (is_morthogonal(G, i)):
print("(%d)"%i, end="", flush=True)
i += 1
else:
break
if i > code.k:
print("*", end="")
break
print()
if argv.show:
print(G.shape)
print(shortstr(G))
print(dot2(G, G.transpose()).sum())
if len(G) >= 14:
continue
A = array2(list(span(G)))
for i in [1, 2, 3]:
assert strong_morthogonal(G, i) == strong_morthogonal(A, i)
def test():
for idx, H in enumerate(items):
H = array2(H)
#print(H.shape)
print(names[idx])
print(shortstr(H))
assert (dot2(H, H.transpose()).sum()) == 0 # orthogonal code
G = H
for genus in range(1, 4):
print(strong_morthogonal(G, genus), end=" ")
print()
keys = [0, 4, 8, 12, 16, 20, 24]
counts = {0: 0, 4: 0, 8: 0, 12: 0, 16: 0, 20: 0, 24: 0}
for v in span(G):
counts[v.sum()] += 1
print([counts[k] for k in keys])
print()
def intersect(G1, G2):
G = numpy.concatenate((G1, G2))
#print("intersect")
#print(G1, G2)
#print(G)
G = G.transpose()
#print("find_kernel", G.shape)
K = find_kernel(G)
if not K:
K = numpy.array(K)
K.shape = 0, G.shape[1]
else:
K = numpy.array(K)
#print("K:")
#print(K, K.shape)
G = dot2(K[:, :len(G1)], G1)
#print("G:")
#print(G, G.shape)
#print()
G = normal_form(G)
return G
def make_bring():
# ------------------------------------------------
# Bring's curve rotation group
ngens = 2
a, ai, b, bi = range(2 * ngens)
rels = [(ai, a), (bi, b), (a, ) * 5, (b, ) * 2]
rels += [(a, a, a, b) * 3]
graph = Schreier(2 * ngens, rels)
#graph.DEBUG = True
graph.build()
assert len(graph) == 60 # == 12 * 5
# --------------- Group ------------------
G = graph.get_group()
#burnside(G)
assert len(G.gens) == 4
ai, a, bi, b = G.gens
assert a == ~ai
assert b == ~bi
assert a.order() == 5
assert b.order() == 2
H = Group.generate([a])
faces = G.left_cosets(H)
assert len(faces) == 12
#hom = G.left_action(faces)
ab = a * b
K = Group.generate([ab])
vertices = G.left_cosets(K)
assert len(vertices) == 12
J = Group.generate([b])
edges = G.left_cosets(J)
assert len(edges) == 30
from bruhat.solve import shortstr, zeros2, dot2
from numpy import alltrue, zeros, dot
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
Hz = get_adj(faces, edges)
Hxt = get_adj(edges, vertices)
#print(shortstr(Hz))
#print()
#print(shortstr(Hxt))
#print()
assert alltrue(dot2(Hz, Hxt) == 0)
# ------------ _oriented version ---------------
# Bring's curve reflection group
ngens = 3
a, ai, b, bi, c, ci = range(2 * ngens)
rels = [
(ai, a),
(bi, b),
(ci, c),
(a, ) * 2,
(b, ) * 2,
(c, ) * 2,
(a, b) * 5,
(b, c) * 5,
(a, c) * 2,
]
a1 = (b, a)
b1 = (a, c)
rels += [(3 * a1 + b1) * 3]
graph = Schreier(2 * ngens, rels)
graph.build()
assert len(graph) == 120 # == 12 * 10
# --------------- Group ------------------
G = graph.get_group()
assert len(G) == 120
a, ai, b, bi, c, ci = G.gens
a1 = b * a
b1 = a * c
assert a1.order() == 5
assert b1.order() == 2
bc = b * c
ba = b * a
assert bc.order() == 5
assert ba.order() == 5
L = Group.generate([a1, b1])
assert len(L) == 60, len(L)
orients = G.left_cosets(L)
assert len(orients) == 2
#L, gL = orients
orients = list(orients)
H = Group.generate([a, b])
assert len([g for g in H if g in L]) == 5
faces = G.left_cosets(H)
assert len(faces) == 12 # unoriented faces
#hom = G.left_action(faces)
o_faces = [face.intersect(orients[0]) for face in faces] # oriented faces
r_faces = [face.intersect(orients[1])
for face in faces] # reverse oriented faces
K = Group.generate([b, c])
vertices = G.left_cosets(K)
assert len(vertices) == 12 # unoriented vertices
o_vertices = [vertex.intersect(orients[0])
for vertex in vertices] # oriented vertices
J = Group.generate([a, c])
u_edges = G.left_cosets(J)
assert len(u_edges) == 30 # unoriented edges
J = Group.generate([c])
edges = G.left_cosets(J)
assert len(edges) == 60, len(edges) # oriented edges ?
# Here we choose an orientation on each edge:
pairs = {}
for e in u_edges:
pairs[e] = []
for e1 in edges:
if e.intersect(e1):
pairs[e].append(e1)
assert len(pairs[e]) == 2
def shortstr(A):
s = str(A)
s = s.replace("0", '.')
return s
Hz = zeros((len(o_faces), len(u_edges)), dtype=int)
for i, l in enumerate(o_faces):
for j, e in enumerate(u_edges):
le = l.intersect(e)
if not le:
continue
e1, e2 = pairs[e]
if e1.intersect(le):
Hz[i, j] = 1
elif e2.intersect(le):
Hz[i, j] = -1
else:
assert 0
Hxt = zeros((len(u_edges), len(o_vertices)), dtype=int)
for i, e in enumerate(u_edges):
for j, v in enumerate(o_vertices):
ev = e.intersect(v)
if not ev:
continue
e1, e2 = pairs[e]
if e1.intersect(ev):
Hxt[i, j] = 1
elif e2.intersect(ev):
Hxt[i, j] = -1
else:
assert 0
#print(shortstr(Hz))
#print()
#print(shortstr(Hxt))
#print()
assert alltrue(dot(Hz, Hxt) == 0)
def check_dualities(Hz, Hxt, dualities):
from bruhat.solve import shortstr, zeros2, dot2, array2, solve, span
from numpy import alltrue, zeros, dot
import qupy.ldpc.solve
import bruhat.solve
qupy.ldpc.solve.int_scalar = bruhat.solve.int_scalar
from qupy.ldpc.css import CSSCode
Hz = Hz % 2
Hx = Hxt.transpose() % 2
code = CSSCode(Hz=Hz, Hx=Hx)
print(code)
n = code.n
Lx, Lz = code.Lx, code.Lz
# check we really do have weak dualities here:
for perm in dualities:
Hz1 = Hz[:, perm]
Hxt1 = Hxt[perm, :]
assert solve(Hxt, Hz1.transpose()) is not None
assert solve(Hz1.transpose(), Hxt) is not None
Lz1 = Lz[:, perm]
Lx1 = Lx[:, perm]
find_xz = solve(concatenate((Lx, Hx)).transpose(),
Lz1.transpose()) is not None
find_zx = solve(concatenate((Lz1, Hz1)).transpose(),
Lx.transpose()) is not None
#print(find_xz, find_zx)
assert find_xz
assert find_zx
# the fixed points live simultaneously in the homology & the cohomology
fixed = [i for i in range(n) if perm[i] == i]
if len(fixed) == 0:
continue
v = array2([0] * n)
v[fixed] = 1
v.shape = (n, 1)
find_xz = solve(concatenate((Lx, Hx)).transpose(), v) is not None
find_zx = solve(concatenate((Lz, Hz)).transpose(), v) is not None
#print(find_xz, find_zx)
assert find_xz
assert find_zx
from qupy.ldpc.asymplectic import Stim as Clifford
s_gates = []
h_gates = []
s_names = []
for idx, swap in enumerate(dualities):
fixed = [i for i in range(n) if swap[i] == i]
print(idx, fixed)
for signs in cross([(-1, 1)] *
len(fixed)): # <------- does not scale !!! XXX
v = [0] * n
for i, sign in enumerate(signs):
v[fixed[i]] = sign
ux = numpy.dot(Hx, v)
uz = numpy.dot(Hz, v)
if numpy.alltrue(ux == 0) and numpy.alltrue(uz == 0):
#print("*", end=" ")
break
#else:
# assert 0
#print(v)
#print()
# transversal S/CZ
g = Clifford.identity(n)
name = []
for i in range(n):
j = swap[i]
if j < i:
continue
if j == i:
assert v[i] in [1, -1]
if v[i] == 1:
op = Clifford.s_gate(n, i)
name.append("S_%d" % (i, ))
else:
op = Clifford.s_gate(n, i).inverse()
name.append("Si_%d" % (i, ))
else:
op = Clifford.cz_gate(n, i, j)
name.append("CZ_%d_%d" % (i, j))
g = op * g
name = "*".join(reversed(name))
s_names.append(name)
#print(g)
#print()
#assert g.is_transversal(code)
s_gates.append(g)
h = Clifford.identity(n)
for i in range(n):
h = h * Clifford.h_gate(n, i)
for i in range(n):
j = swap[i]
if j <= i:
continue
h = h * Clifford.swap_gate(n, i, j)
#print(g)
#print()
#assert h.is_transversal(code)
h_gates.append(h)
if 0:
print()
print("CZ:")
CZ = Clifford.cz_gate(2, 0, 1)
op = (1., [0, 0], [1, 1])
for i in range(4):
print(op)
op = CZ(*op)
return
for idx, sop in enumerate(s_gates):
print("idx =", idx)
#for hx in Hx:
perm = dualities[idx]
#for hx in span(Hx):
for hx in Hx:
#print("hx =", hx)
#print(s_names[idx])
phase, zop, xop = sop(1., None, hx)
assert numpy.alltrue(xop == hx) # fixes x component
print(phase, zop, xop, dot2(zop, xop))
for (i, j) in enumerate(perm):
if xop[i] and xop[j] and i < j:
print("pair", (i, j))
if toric is None:
continue
print("xop =")
print(toric.strop(xop))
print("zop =")
print(toric.strop(zop))
print()
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 make_surface_54(G_0):
"find Z/2 chain complex"
from bruhat.solve import shortstr, zeros2, dot2
from numpy import alltrue, zeros, dot
print("|G_0| =", len(G_0))
a, ai, b, bi, c, ci, d, di = G_0.gens
# 5,5 coxeter subgroup
G_1 = Group.generate([a, b, c])
assert G_0.is_subgroup(G_1)
# orientation subgroup
L_0 = Group.generate([a * b, a * d])
assert G_0.is_subgroup(L_0)
orients = G_0.left_cosets(L_0)
assert len(orients) == 2
H_1 = Group.generate([a, b])
faces = G_1.left_cosets(H_1)
#face_vertices = G_0.left_cosets(H_1) # G_0 swaps faces & vertices
K_1 = Group.generate([b, c])
vertices = G_1.left_cosets(K_1)
J_1 = Group.generate([a, c])
edges_1 = G_1.left_cosets(J_1)
print("faces: %d, edges: %d, vertices: %d" %
(len(faces), len(edges_1), len(vertices)))
J_0 = Group.generate([a, d])
assert len(J_0) == 8
edges_0 = G_0.left_cosets(J_0)
print("edges_0:", len(edges_0))
assert G_0.is_subgroup(J_0)
act_edges_0 = G_0.action_subgroup(J_0)
#print("here")
#while 1:
# sleep(1)
from bruhat.solve import shortstr, zeros2, dot2, array2, solve
from numpy import alltrue, zeros, dot
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) > 0
return A
Hz = get_adj(faces, edges_0)
Hxt = get_adj(edges_0, vertices)
Hx = Hxt.transpose()
#print(shortstr(Hz))
#print()
#print(shortstr(Hxt))
#print()
assert alltrue(dot2(Hz, Hxt) == 0)
# act_fold_faces = G_0.action_subgroup(H_1)
dualities = []
idx = 0
for i, g in enumerate(G_0):
assert g in G_0
h_edge_0 = act_edges_0[g]
n_edge = len(h_edge_0.fixed())
# h_fold_faces = act_fold_faces[g]
# n_fold_faces = len(h_fold_faces.fixed())
count = 0
#perms = [] # extract action on the fixed edges
for e0 in edges_0:
e1 = g * e0
if e0 != e1:
continue
count += 1
# perm = e0.left_mul_perm(g)
# #print(perm, end=" ")
# perms.append(perm)
assert count == n_edge
if g in G_1:
continue
elif g.order() != 2:
continue
#elif g in L_0:
# continue
#elif g.order() == 2 and g not in G_1 and g not in L_0:
perm = h_edge_0.get_idxs()
dualities.append(perm)
#if g not in L_0:
# check_432(Hz, Hx, perm)
# return
result = is_422_cat_selfdual(Hz, Hx, perm)
print(
"%d: [|g|=%s,%s.%s.%s.%s]" % (
len(dualities),
g.order(),
n_edge or ' ', # num fixed edges
# n_fold_faces or ' ', # num fixed faces+vertexes
[" ", "pres"
][int(g in G_1)], # preserves face/vertex distinction
["refl", "rot "][int(g in L_0)], # oriented or un-oriented
[" ", "selfdual"][result],
),
end=" ",
flush=True)
print()
print()
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 check(self):
G, H = self.G, self.H
assert rank(G) == len(G)
A = dot2(H, G.transpose())
assert A.sum()==0
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)