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```def make_Majorana(N):

# Make 'N' Majorana fermions. Set of N Hermitian matrices psi_i, i=1,..N
# obeying anti-commutation relations {psi_i,psi_j} = δ_{ij}

X = np.array([[0, 1], [1, 0]])
Y = np.array([[0, -1j], [1j, 0]])
Z = np.array([[1, 0], [0, -1]])
I = np.array([[1, 0], [0, 1]])

psi = dict()

if simpl == 0:
for i in range(1, N+1):

if (i % 2) == 1:
# For ex: On odd numbered Majorana with N = 8, we have --> XIII, ZXII, ZZXI, ZZZX
matlist = [Z] * int((i-1)/2)
matlist.append(X)
matlist = matlist + [I] * int((N/2 - (i+1)/2))
psi[i] = 1/np.sqrt(2)*reduce(np.kron, matlist)
else:
# On even numbered with N = 8 --> YIII, ZYII, ZZYI, ZZZY
matlist = [Z] * int((i - 2) / 2)
matlist.append(Y)
matlist = matlist + [I] * int((N/2 - i/2))
psi[i] = 1/np.sqrt(2)*reduce(np.kron, matlist)

if simpl == 1:
for i in range(1, N+1):

if (i % 2) == 1:
# For ex: On odd numbered Majorana with N = 8, we have --> XIII, IXII, IIXI, IIIX
matlist = [I] * int((i-1)/2)
matlist.append(X)
matlist = matlist + [I] * int((N/2 - (i+1)/2))
psi[i] = 1/np.sqrt(2)*reduce(np.kron, matlist)
else:
# On even numbered with N = 8 --> YIII, IYII, IIYI, IIIY
matlist = [I] * int((i - 2) / 2)
matlist.append(Y)
matlist = matlist + [I] * int((N/2 - i/2))
psi[i] = 1/np.sqrt(2)*reduce(np.kron, matlist)

for i in range(1, N+1):
for j in range(1, N+1):

if simpl == 0:
# Not checking this for simplified H yet.

if i != j:
if np.allclose(psi[i] @ psi[j], -psi[j] @ psi[i]) == False:
print ("Does not satisfy algebra")

if i == j:
if np.allclose(psi[i] @ psi[j] + psi[j] @ psi[i], np.eye(int(2**(N/2)))) == False:
print ("Does not satisfy algebra for i=j")

return psi

def make_Hamiltonian(psi, N, instances, J_squared):

# Creates multiple realisations of the SYK Hamiltonian
# Variance of couplings is given by 'J_squared * 3!/N^3'.

H = 0
J = dict()
sigma_sq = 6.*J_squared/(N**3)
sigma = math.sqrt(sigma_sq)

for i in range(1, N+1):
for j in range(i+1, N+1):
for k in range(j+1, N+1):
for l in range(k+1, N+1):
J[i, j, k, l] = np.random.normal(loc=0, scale=sigma,size=instances)
#print (J[i, j, k, l])
M = psi[i] @ psi[j] @ psi[k] @ psi[l]

if np.allclose(M, M.conj().T) == True:
H = H + np.array([element * M for element in J[i, j, k, l]])

else:
if np.allclose(M, -M.conj().T) == True:
M *= complex(0,1.)
H = H + np.array([element * M for element in J[i, j, k, l]])

return H```