In [1]:
from numba import jit
from numpy import cos, sin, pi
from random import random
from scipy.optimize import minimize
from scipy.special import binom
import collections
import functools
import itertools
import matplotlib.pyplot as plt
import multiprocessing
import numpy as np
import time

In [2]:
print("Starting time:", time.ctime())

Starting time: Tue Dec 29 04:07:41 2020


# Performance of 2-step threshold algorithm¶

In [3]:
def make_qfn(T):
def f(x):
"""This function can handle vectorized operations"""
return (x < T)*2 - 1
return f
#     return lambda x: 1 if x < T else -1

qfns = [make_qfn(T) for T in range(10000)]

In [4]:
# This now inputs an array, so it cannot be cached with functools.
# @functools.lru_cache(maxsize=int(1e7))
def binompmf(tot, times, p):
return p**times * (1 - p)**(tot-times) * binom(tot, times)

In [5]:
@functools.lru_cache(maxsize=4096)
def Z1plus(qf, n):
return 2**(-n)*np.sum([qf(m+1)*binom(n, m) for m in range(0, n+1)])

@functools.lru_cache(maxsize=4096)
def Z1minus(qf, n):
return (-1)*2**(-n)*np.sum([binom(n, m)*qf(m) for m in range(0, n+1)])

In [6]:
# Now unused; using np.meshgrid to generate all bigQ functions at once.
# def bigQ(qf, n, l, r, a, b):
#     return qf((n-l-r)*(1-a)/2 + (l+r)*(1+a)/2 + (1+b)/2)

def bigQ_all(qf, n, l_max, r_max, a, b):
vals = np.meshgrid(range(l_max), range(r_max))
def bigQ_f(lr):
l = lr[0]
r = lr[1]
return qf((n-l-r)*(1-a)/2 + (l+r)*(1+a)/2 + (1+b)/2)
return bigQ_f(vals)

In [7]:
@functools.lru_cache(maxsize=int(1e7))
def Z2(Z1, Z0, match, q2, n, k, p_plus, p_minus):
bp_minus = binompmf(n-k, np.arange(n-k+1), p_minus)
bp_plus = binompmf(k, np.arange(k+1), p_plus)

bq_all = bigQ_all(q2, n, k+1, n-k+1, Z1*Z0, match)
inter = np.sum(bq_all*bp_plus, axis=1)
out = np.sum(bp_minus*inter)

return Z1*out

In [8]:
def final(n, T1, T2):
"""
Calculates -1/2 * <Z_i Z_j>.
It should be odd function over the midpoint. Both T=0 and T> n+1 should be 0 (i.e. no improvement).
Recall that n = D-1, so this expression is for a (n+1)-regular graph.
"""
out = 0
q1 = qfns[T1]
q2 = qfns[T2]
p_plus = (1 + Z1plus(q1, n))/2
p_minus = (1 + Z1minus(q1, n))/2

binom_n = np.array([binom(n, k) for k in range(n+1)])

med = []
for k in range(n+1):
inter = []
for u in range(n+1):
equal = Z2(q1(k+1), 1, q1(k+1)*q1(u+1), q2, n, k, p_plus, p_minus) * Z2(q1(u+1), 1, q1(k+1)*q1(u+1), q2, n, k, p_plus, p_minus)
unequal = Z2(q1(k), 1, -q1(k)*q1(u),q2, n, k, p_plus, p_minus) * Z2(-q1(u), -1, -q1(k)*q1(u), q2, n, k, p_plus, p_minus)
inter.append( 0.5 * (equal + unequal))

med.append(np.sum(binom_n * inter))

out = np.sum(binom_n * med)
return -0.5 * 2**(-2*n) * out


### Finding the best threshold¶

Let's confirm the best range of threshold. Notice the pattern in optimal threshold value in the below plot. This gives me confidence that the optimal threshold value is between $D/2$ and $D/2 + \sqrt{D}$.

In [9]:
def plot(minvals, maxvals, Ds, ax):
for idx, D in enumerate(Ds):
xs = np.arange(minvals[idx], maxvals[idx])
ax.scatter((xs - D/2)*D**-0.5, [D**0.5 * final(D-1, t, t) for t in xs], label="D = %i" % D)
ax.grid()
ax.set_xlabel("($τ$ - 0.5 D) /  $\sqrt{D}$")
ax.set_ylabel("Scaled performance")
ax.legend()

In [10]:
fig, ax = plt.subplots(1, 1, figsize=(10, 5))
plot([0,]*5, [5, 10, 20, 30, 40], [5, 10, 20, 30, 40], ax)

In [11]:
fig, ax = plt.subplots(1, 1, figsize=(10, 5))
plot([int(D/2) for D in [50, 100, 150, 200]], [int(D/2) + int(D**0.5) + 1 for D in [50, 100, 150, 200]], [50, 100, 150, 200], ax)


From the last plot, I expect the optimal threshold to be approximately $D/2 + 0.4 \sqrt{D}$ at large values of $D$.

### Computing for D < 500¶

In [12]:
def get_max_final(d):
"""
This finds the optimal threshold assuming tau_1 = tau_2.
To speed this up, I limit my search to roughly (D/2 + k_min \sqrt{D}, D/2 + k_max \sqrt{D}).
I expect the optimal k to be approximately 0.4 - 0.5.
"""
k_min, k_max = 0, 1.5
if d > 150:
k_min, k_max = 0.35, 0.52
elif d > 60:
k_min, k_max = 0.3, 0.6

inps = range(int((d-1)/2 + k_min*(d-1)**0.5 ), int((d-1)/2 + max(10, k_max*(d-1)**0.5 + 1)))
vals = [final(d-1, t, t) for t in inps]
return max(vals), inps[vals.index(max(vals))]

In [13]:
# The multiprocessing library will make this run much faster.
pool = multiprocessing.Pool(4)
ds = range(2, 500 + 1)

In [14]:
%%time
deltas = pool.map(get_max_final, ds)
pool.close()

CPU times: user 683 ms, sys: 227 ms, total: 910 ms
Wall time: 15min 49s

In [15]:
for _ in [Z2, Z1minus, Z1plus]:
print(_.__name__, _.cache_info())

Z2 CacheInfo(hits=4443293, misses=47207, maxsize=10000000, currsize=47207)
Z1minus CacheInfo(hits=0, misses=152, maxsize=4096, currsize=152)
Z1plus CacheInfo(hits=0, misses=152, maxsize=4096, currsize=152)

In [16]:
plt.figure(figsize=(10, 6))
deltas2 = [d**0.5 * delt[0] for d, delt in zip(ds, deltas)]
plt.plot(ds[:len(deltas2)], deltas2, label="2-step, girth > 5")
plt.grid()

In [17]:
# the last few values, to get a sense of the value at large D
deltas2[-5:]

Out[17]:
[0.4171768586295556,
0.4168077842636685,
0.4171577452407655,
0.4168492886224562,
0.4171364666581488]
In [18]:
ts = [i[1] for i in deltas]
# t = d/2 + ksqrtd
# (t - d/2)**2 = k**2 d
plt.figure(figsize=(10, 5))
plt.plot(ds, ((np.array(ts) - np.array(ds)/2)* (np.array(ds))**-.5))
plt.ylim(0)
plt.xlabel("D (of a D-regular graph)")
plt.ylabel("Optimal $k$, given $τ= D/2 + k \sqrt{D}$")
plt.grid()


### Plot for the paper¶

In [19]:
fig, ((ax0, ax1), (ax2, ax3)) = plt.subplots(2, 2, figsize=(16, 10))
plot([0,]*5, [5, 10, 20, 30, 40], [5, 10, 20, 30, 40], ax0)
plot([int(D/2) for D in [50, 100, 150, 200]], [int(D/2) + int(D**0.5) + 1 for D in [50, 100, 150, 200]], [50, 100, 150, 200], ax1)

gs = ax2.get_gridspec()
ax2.remove()
ax3.remove()

axbig.plot(ds, ((np.array(ts) - np.array(ds)/2)* (np.array(ds))**-.5))
axbig.set_ylim(0)
axbig.set_xlabel("D (of a D-regular graph)")
axbig.set_ylabel("Optimal $k$, given $τ= D/2 + k \sqrt{D}$")
axbig.grid()

plt.savefig("best_threshold.png")


### Code optimization¶

On Dec 24 2020, I worked on optimizing this code. A few notes:

• cython doesn't work with functools
• I couldn't get numba_scipy to work; I couldn't find the right versions of np, scipy, numba that were all compatible.
• In the binompmf using np.arange is incompatible with lru_cache, but the np.arange is a bit faster
• I tried an scipy.optimize.minimize guess, but it wasn't that accurate.
In [20]:
# # these code profilers really helped.
# %lprun -f Z2 get_max_final(21)
# %prun get_max_final(508)


### Unequal thresholds¶

In [21]:
def get_max_final_2(d):
"""
This finds the optimal threshold for any tau_1 and tau_2.
This looks over all tau > (d-1)/2 to find the best threshold.
"""
assert d < 50, "this takes a very long time"

inps = range(int((d-1)/2), int(d+2))
vals = [final(d-1, t, t2) for t in inps for t2 in inps]

best = vals.index(max(vals))
return max(vals), inps[best // len(inps)], inps[best % len(inps)]


Let's find the optimal threshold for D < 50.

In [22]:
%%time
unequal_thresholds = [get_max_final_2(i) for i in range(2, 50)]

CPU times: user 6min 22s, sys: 827 ms, total: 6min 22s
Wall time: 6min 23s


Printing out the best results for D < 20:

In [23]:
for i in range(2, 20):
print(i, get_max_final(i), get_max_final_2(i))

2 (0.3125, 2) (0.3125, 2, 2)
3 (0.191162109375, 3) (0.24609375, 2, 3)
4 (0.1854366660118103, 3) (0.21282511949539185, 3, 4)
5 (0.18511548652895726, 4) (0.18511548652895726, 4, 4)
6 (0.13510421220398067, 5) (0.18319273736744357, 4, 5)
7 (0.16065630997630004, 5) (0.16065630997630004, 5, 5)
8 (0.1360233197931599, 6) (0.15986870158956498, 5, 6)
9 (0.1387957076629388, 6) (0.13992264564882453, 5, 7)
10 (0.12910557091839034, 7) (0.14089705236762443, 6, 7)
11 (0.12146098243509729, 7) (0.13005917848121065, 7, 8)
12 (0.12066098522000336, 8) (0.1254439622991265, 7, 8)
13 (0.10771186239759818, 8) (0.1217790052677211, 8, 9)
14 (0.11253840954798257, 9) (0.1162867106705756, 8, 10)
15 (0.10215029772247765, 10) (0.1143392767342369, 9, 10)
16 (0.10514090614606042, 10) (0.11097302099077844, 9, 11)
17 (0.09841351953270736, 11) (0.10755692892858511, 10, 11)
18 (0.09848615729090915, 11) (0.10563976189122315, 10, 12)
19 (0.09454371276543241, 12) (0.1013607439626949, 11, 12)


# Performance of QAOA$_2$¶

In [24]:
# the just-in-time compilation massively improves the runtime of this function.
@jit(nopython=True, nogil=True)
def qaoa2(beta_2, gamma_2, beta_1, gamma_1, D):
"""Cut fraction with p=2 QAOA on D-regular girth > 5 graphs"""
c = cos(2*beta_2)
s = sin(2*beta_2)
m = cos(gamma_2)
n = sin(gamma_2)
r = cos(2*beta_1)
t = sin(2*beta_1)
y = cos(gamma_1)
z = sin(gamma_1)

A = -2*c*c*r*t*z*y**(D-1)
bpt1 = 0.5*s*c*(
(1 + r)*(-m*r*z - n*y)*((m*y - n*r*z)**(D-1))
+ (1 - r)*(m*r*z - n*y)*((m*y + n*r*z)**(D-1))
)
bpt2 = 0.5*s*c*t*(
(m*t*(y**(D-1))*z + (1j)*n)*((m + (1j)*n*t*(y**(D-1))*z)**(D-1))
+ (m*t*(y**(D-1))*z - (1j)*n)*((m - (1j)*n*t*(y**(D-1))*z)**(D-1))
)
Ea = s*s*t*z * 0.5 * \
((1+r)*(m*y - n*z*r)**(D-1) - (1-r)*(m*y + n*z*r)**(D-1))
Eb = (m + (1j)*n * y**(D-1) * z*t)**(D-1) + \
(m - (1j)*n * y**(D-1) * z*t)**(D-1)
return 0.5 - 0.5 * (A + 2*(bpt1 + bpt2) + Ea*Eb)

In [25]:
qaoa2_d_deg = lambda i, d: -qaoa2(*i, d).real

def get_best(d):
"""This runs the optimization 100 times, or 150 times if d < 20, and takes the best value."""
best = None
val = 0
i = 0
while i < 100 or (d < 20 and i < 150):
i += 1
init_val = [random()*(i % 2 + 1)*pi for i in range(4)]
result = minimize(qaoa2_d_deg, init_val, args=(d), options={'fatol': 1e-20}, method='Nelder-Mead')
if not best or result.fun < best.fun:
best = result
val = d**0.5 * (-best.fun - 0.5)
return best

In [26]:
%%time
pool = multiprocessing.Pool(4)
results = pool.map(get_best, ds)
pool.close()

CPU times: user 99.9 ms, sys: 92.2 ms, total: 192 ms
Wall time: 2min 51s

In [27]:
out = {d: -r.fun - 0.5 for d, r in zip(ds, results)}
qaoavals = np.array([out[k]*k**0.5 for k in sorted(out)])

print(sum (qaoavals[:-1] < qaoavals[1:]), "values that are not descending")
assert sum ( qaoavals < 0.407) == 0, "all values should be at least 0.407"
# the last few values, to get a sense of the value at large D
print(qaoavals[-5:])

0 values that are not descending
[0.40771208 0.40771175 0.40771141 0.40771108 0.40771075]

In [28]:
import matplotlib.pyplot as plt
plt.figure(figsize=(12, 5))
plt.plot([k for k in sorted(out)], qaoavals)
plt.grid()
plt.xlabel("D (of a D-regular graph)")
plt.ylabel("Scaled performance of QAOA$_2$")
plt.ylim(0.4)

Out[28]:
(0.4, 0.47458920953799794)

### Confirming known results¶

I confirm the results for $D=2$ (5/6 cut fraction) and $D=3$ (0.7559 cut fraction):

In [29]:
for D in range(2, 4):
print("D=%i" % D, "cut fraction:", -results[ds.index(D)].fun, "input angles in degrees:", 180/pi*results[ds.index(D)].x)

D=2 cut fraction: 0.8333333333333335 input angles in degrees: [ 53.05994804  36.94005106 108.4700259  106.11989664]
D=3 cut fraction: 0.755906458453232 input angles in degrees: [ 16.75218244 308.55760401 121.7936679  152.04908305]


For $D=3$, the input angles match the values listed in Table I of Wurtz 2020, up to degeneracy in the input ($\pi/2$ in $\beta$). Other input angles will also achieve this maximum; for example, consider the angles $(\beta_2, \gamma_2, \beta_1, \gamma_1) = (16.8^{\circ},51.4^{\circ}, 31.8^{\circ}, 28.0^{\circ}$):

In [30]:
qaoa2(*np.array([16.8, 51.4, 31.8, 28.0])*pi/180, 3).real

Out[30]:
0.7559043193525249

For $D=2$, I verify with SymPy that the formulas are in fact equal:

In [31]:
def f2(beta_1, gamma_1, beta_2, gamma_2):
c = cos(2*beta_2)
s = sin(2*beta_2)
m = cos(gamma_2)
n = sin(gamma_2)
r = cos(2*beta_1)
t = sin(2*beta_1)
y = cos(gamma_1)
z = sin(gamma_1)
pt1 = c*c*r*t*y*z
pt2 = -s*c*m*n*(r*r*z*z - y*y)
pt3 = -s*c*y*z*(t*t - r*r)*(m*m - n*n)
pt4 = -s*s*t*z*m*r*(m*y - n*z)
return 0.5 + pt1 + pt2 + pt3 + pt4

def ring(beta_1, gamma_1_input, beta_2, gamma_2_input):
gamma_1 = -gamma_1_input/2
gamma_2 = -gamma_2_input/2
Fbyn = 1/64 * (
-7*cos(4*beta_1 + 4*beta_2 + 4*gamma_1 + 4*gamma_2)
-6*cos(4*beta_1 + 4*beta_2 + 4*gamma_1)
+3*cos(4*beta_1 + 4*beta_2 - 4*gamma_1 + 4*gamma_2)
+4*cos(4*beta_1 + 4*beta_2 + 4*gamma_2)
+3*cos(4*beta_1 - 4*beta_2 + 4*gamma_1 + 4*gamma_2)
-6*cos(4*beta_1 - 4*beta_2 + 4*gamma_1)
-3*cos(4*beta_1 - 4*beta_2 - 4*gamma_1 + 4*gamma_2)
+4*cos(4*beta_1 + 4*gamma_1 + 4*gamma_2)
-4*cos(4*beta_1 + 4*gamma_1)
-4*cos(4*beta_1 + 4*gamma_2)
-3*cos(-4*beta_1 + 4*beta_2 + 4*gamma_1 + 4*gamma_2)
+6*cos(-4*beta_1 + 4*beta_2 + 4*gamma_1)
+3*cos(-4*beta_1 + 4*beta_2 - 4*gamma_1 + 4*gamma_2)
+7*cos(-4*beta_1 - 4*beta_2 + 4*gamma_1 + 4*gamma_2)
+6*cos(-4*beta_1 - 4*beta_2 + 4*gamma_1)
-3*cos(-4*beta_1 - 4*beta_2 - 4*gamma_1 + 4*gamma_2)
-4*cos(-4*beta_1 - 4*beta_2 + 4*gamma_2)
-4*cos(-4*beta_1 + 4*gamma_1 + 4*gamma_2)
+4*cos(-4*beta_1 + 4*gamma_1)
+4*cos(-4*beta_1 + 4*gamma_2)
-6*cos(4*beta_2 + 4*gamma_1 + 4*gamma_2)
-6*cos(4*beta_2 - 4*gamma_1 + 4*gamma_2)
-4*cos(4*beta_2 + 4*gamma_2)
+6*cos(-4*beta_2 + 4*gamma_1 + 4*gamma_2)
+6*cos(-4*beta_2 - 4*gamma_1 + 4*gamma_2)
+4*cos(-4*beta_2 + 4*gamma_2)
)
return 0.5 * (1 - Fbyn)

In [32]:
# evaluates to 0
from sympy import symbols, simplify, collect, expand, expand_trig, cos, sin
b1, g1, b2, g2 = symbols('b1 g1 b2 g2')
result = simplify(
collect(
expand(expand_trig(f2(b1, g1, b2, g2))) - expand(expand_trig(ring(b1, g1, b2, g2)))
, cos(b1))
)
print(result)
assert result == 0

0


# Performance of 1-step algorithms¶

This reproduces work done by Wang et al 2018 and Hastings 2019.

In [33]:
def q(k, threshold):
"""Works for numpy arrays"""
return (1-(k >= threshold)*2)

def calc_performance(D, T):
"""
This calculates the performance over random after 1 step of Hastings' thresholding algorithm.
Precisely, returns delta = -1/2 <Z_i Z_j> for D-regular graphs with threshold T.
"""
s1 = 0
s2 = 0
# go through # of agreeing neighbors, from 0 through D-1
for i in range(D):
# factor out two copies of 2^-n, n=D-1
s1 += binom(D-1, i)*q(i+1, T)
s2 += binom(D-1, i)*q(i, T)
return -0.5 * 0.5 * (s1**2 - s2**2) * 2**(-2*(D-1))

hrss_1 = [D**0.5*max([calc_performance(D, int((D-1)/2+k)) for k in range(0, max(int(D**0.5 + 1), 10))]) for D in ds]

In [34]:
def qaoa1(D):
"""Best QAOA1 performance on triangle-free graphs."""
d = D-1
return 0.5 * (d/(d+1))**(d/2)
qaoa_1 = [qaoa1(D) for D in ds]


# Comparison plots¶

In [35]:
plt.figure(figsize=(10, 6))
diff_thresholds = [x[0]*d**0.5 for x, d in zip(unequal_thresholds, ds[:len(unequal_thresholds)])]
plt.plot(ds[:len(diff_thresholds)], diff_thresholds, label="Best of 2-step threshold")
plt.plot(ds, deltas2, label="Best of 2-step threshold where $τ_1 = τ_2$")
plt.plot(ds, qaoavals, label="Best of QAOA$_2$")
plt.plot(ds, hrss_1, label="Best of 1-step threshold")
plt.plot(ds, qaoa_1, label="Best of QAOA$_1$")
plt.grid()
plt.xlim(0, 500)
plt.xlabel("D (of a D-regular graph)")
plt.ylabel("Scaled performance")
plt.legend()
plt.savefig("all_results.png")


QAOA$_2$ does win in a few cases:

In [36]:
print("D where qaoa wins")
weirds = np.array(ds)[qaoavals > deltas2]
print(weirds)
print("total:", len(weirds))

D where qaoa wins
[ 2  3  4  5  6  8  9 10 11 13 15 17 22 24 26 28 39 41]
total: 18

In [37]:
print("D where qaoa wins even with unequal thresholds")
weirds = np.array(ds)[:len(diff_thresholds)][qaoavals[:len(diff_thresholds)] > diff_thresholds]
print(weirds)
print("total:", len(weirds))

D where qaoa wins even with unequal thresholds
[2 3 4 5]
total: 4


# Small-degree local classical algorithms¶

In [38]:
print("improvement above random quantum vs classical")
for i in range(4):
print("D =", 2+i, qaoavals[i] * (2+i)**-0.5, "vs", diff_thresholds[i] * (2+i)**-0.5)

improvement above random quantum vs classical
D = 2 0.33333333333333354 vs 0.31250000000000006
D = 3 0.25590645845323196 vs 0.24609374999999997
D = 4 0.21609163550938715 vs 0.21282511949539185
D = 5 0.1907089419367839 vs 0.18511548652895726


### Reproducing the 1-step local classical algorithms¶

In [39]:
def local_1(c, n1, n2, s1, s2):
"""Implements a 1-step local linear algorithm given the description in Hastings 2019."""
v1 = n1 - c*s1 - c*n2
v2 = n2 - c*s2 - c*n1

# if any of (v1, v2) is 0 it (correctly) gives a 1/2 chance of success
return 0.5 - 0.5*np.sign(v1)*np.sign(v2)


Hastings describes the improvement over random of $D=3$ at $c=0.599$ with initial values $[-1, -1/3, 1/3, 1]$, noting that it was at least better than QAOA$_1$'s value of 0.1925 but not as good as a uniform distribution at 0.1980.

In [40]:
c = 0.599
d = 3
poss = np.array([-1, -1/3, 1/3, 1])
nbriter = list(itertools.product(*[poss,]*(d-1)))
nbrsums = np.sum(nbriter, axis=1)

def t(args):
return local_1(c, *args)

ct = np.sum ( t(np.meshgrid(poss, poss, nbrsums, nbrsums)) )
improvement = ct / len(poss)**(2*d) - 0.5
print("poss = ", poss, "c = ", c, "success = ", improvement)
assert improvement > 0.1925, "should be better than QAOA1"
assert improvement < 0.1980, "should be less than uniform dist"

poss =  [-1.         -0.33333333  0.33333333  1.        ] c =  0.599 success =  0.1953125


### Implementing the 2-step local classical algorithm¶

The code that searches is extremely optimized. Initially, it would have taken 1.5s for D=3; now it takes 5ms for D=3. It groups different possibilities and multiplies them by their occurrence probability, greatly reducing the $O(m^{D^2})$ runtime given $m$ initial values.

In [41]:
# # Code profiling
# %lprun -f local_2 run(0.6, 1.1, 3, [-1, 0, 1])
# %timeit run(0.6, 1.1, 3, [-1,1])
# %prun run(0.6, 1.1, 3, [-1,-1/3,1/3,1])

In [42]:
def local_2(d, c, c2, n1, n2,  s1, s2, nbr_nbrsum1, nbr_nbrsum2):
"""
Implements a 2-step local linear algorithm given the description in Hastings 2019.
This has been optimized greatly.
A few notation changes: (c0, c1) are now (c, c2).
The s1, s2 correspond to sums of the neighbors of the nodes connected to the edge.
The nbr_nbrsum1, nbr_nbrsum2 values correspond to sums of the neighbors of the neighbors of the nodes connected to the edge.
"""
#     v1 = n1 - c*s1 - c*n2
#     v2 = n2 - c*s2 - c*n1
#     x1 = s1 - c*nbr_nbrsum1 - (d-1)*c*n1
#     x2 = s2 - c*nbr_nbrsum2 - (d-1)*c*n2
#     z1 = v1 - c2*x1 - c2*v2
#     z2 = v2 - c2*x2 - c2*v1

cc2 = c*c2
cpc2 = c+c2
cc2dp1 = 1+cc2*d
z1 = n1*cc2dp1 - (s1+n2)*cpc2 + cc2*(s2 + nbr_nbrsum1)
z2 = n2*cc2dp1 - (s2+n1)*cpc2 + cc2*(s1 + nbr_nbrsum2)

# if any of (v1, v2) is 0 it gives a 1/2 chance of success
return 0.5 - 0.5* np.sign(z1)*np.sign(z2)

In [43]:
def get_iters(*poss, d):
nbriter = np.array(list(itertools.product(*[list(poss),]*(d-1))))
nbr_nbriter = list(itertools.product(*[nbriter,]*(d-1)))
nbrsums = np.sum(nbriter, axis=1)
nbr_nbrsums = np.sum(nbr_nbriter, axis=(1, 2))
return nbrsums, nbr_nbrsums

In [44]:
def run(c, c2, d, poss):
"""
Note that the runtime is extremely poor: len(poss) ** (2*d(d-1) + 2)
Fortunately, using np.unique helps significantly.
"""

nbrsums, nbr_nbrsums = get_iters(*poss, d=d)

poss_u = np.unique(poss, return_counts=True)
nbr_u = np.unique(nbrsums, return_counts=True)
nbr_nbr_u = np.unique(nbr_nbrsums, return_counts=True)

mgs = list(zip(poss_u, poss_u, nbr_u, nbr_u, nbr_nbr_u, nbr_nbr_u))

ct = np.sum ( local_2(d, c, c2, *np.meshgrid(*mgs[0])) *  functools.reduce(np.multiply, np.meshgrid(*mgs[1])) )
perf = ct / len(poss)**(2*(1 + d-1 + (d-1)**2)) - 0.5

return perf


Here's an example grid-based search for $D=3$:

In [45]:
d=3
poss2 = np.array([1, -1])

maxx = 0
data = (0,0)
for c in np.linspace(0, 1.5, 100):
for c2 in np.linspace(0, 1, 100):
val = run(c, c2, d, poss2)
if val > maxx:
maxx = val
data = (c, c2)
print("best:", maxx, data)

best: 0.109375 (0.0, 0.33333333333333337)
best: 0.1875 (0.0, 0.3434343434343435)
best: 0.191162109375 (0.015151515151515152, 0.33333333333333337)
best: 0.191650390625 (0.015151515151515152, 0.9494949494949496)
best: 0.219970703125 (0.015151515151515152, 0.9797979797979799)
best: 0.262939453125 (0.015151515151515152, 1.0)
best: 0.26483154296875 (0.33333333333333337, 1.0)
best: 0.2666015625 (0.3484848484848485, 1.0)
best: 0.26953125 (0.6666666666666667, 1.0)
best: 0.270263671875 (1.1060606060606062, 0.9090909090909092)


This searches until it finds a value greater than the associated QAOA$_2$ value.

In [46]:
qaoa2_best = {
3: 0.2559,
4: .2161,
5:.1907
}

maxx = {3: 0, 4:0, 5:0}
data = {}
for d in range(3, 6):
vals = [1, -1]
f = lambda x: -run(*x, d, vals)
i = 0
while maxx[d] < qaoa2_best[d] and i < 100:
i += 1
st = (random(), random())
out = minimize(f, st, method='Powell')
if -out.fun > qaoa2_best[d]:
maxx[d] = -out.fun
data[d] = (out, st)
print(d, maxx[d], data[d][0].x)
break

3 0.270263671875 [2.09942807 0.64388207]
4 0.23085129261016846 [11.12630289  0.38793097]
5 0.20331271992745314 [4.64256915 0.34832495]


Let's check the success with rounded $(c_0, c_1)$. I'm using values from when I first ran the code, which may be different than the output above.

In [47]:
print("D=3", run(7.22, 0.58, 3, [-1, 1]) )
print("D=4", run(0.38, 11.20, 4, [-1, 1]) )
print("D=5", run(5.44, 0.33, 5, [-1, 1]) )

D=3 0.270263671875
D=4 0.23085129261016846
D=5 0.20453225069923064


### For D=2¶

The above strategy did not work for D=2!

Look at the distribution of performance across various $(c_0, c_1)$:

In [48]:
# why does it peak at 2?
poss = [1, -1]
g = np.linspace(0, 5, 100)
plt.figure(figsize=(10, 6))
plt.plot(g, [1/3]*len(g), 'gray', linewidth=5, label='cutoff')
for x in np.linspace(0, 5, 10):
plt.plot(g, [run(x, i, 2,  poss) for i in g], label=str(x))
plt.grid()
plt.legend()

Out[48]:
<matplotlib.legend.Legend at 0x7f3702e00a10>

It gets close to 1/3, but doesn't reach it. But consider the same plot for a different set of initial values:

In [49]:
# why does it peak at 2?
poss = [1, -0.7, -0.3, 0.3, 0.7, 1]
g = np.linspace(0, 5, 100)
plt.figure(figsize=(10, 6))
plt.plot(g, [1/3]*len(g), 'gray', linewidth=5, label='cutoff')
for x in np.linspace(0, 5, 10):
plt.plot(g, [run(x, i, 2,  poss) for i in g], label=str(x))
plt.grid()
plt.legend()

Out[49]:
<matplotlib.legend.Legend at 0x7f370287cb90>

It seems like the peak performance is around 1/3 ! This is really strange!

I tried many possible input values, eventually trying random samples from the triangular distribution, and focusing on the $(c_0, c_1)$ combinations that typically had peak performance.

In [50]:
maxx = 0
data = 0
for _ in range(3000):
#     s = np.random.normal(size=10)
s = np.random.triangular(-1, 0, 1, size=6)
out = run(100, 0.57, 2, s)
if out > maxx:
maxx = out
data = s
print(maxx)
print(data, maxx)

0.20520404663923186
0.31185699588477367
0.3301183127572016
0.3310185185185185
0.3324331275720165
0.3329475308641975
0.3330761316872428
0.3331189986282579
0.3332475994513031
0.33333333333333337
0.33371913580246915
0.33389060356652944
[ 0.67513125 -0.602384   -0.02149638 -0.08816768 -0.0102057   0.03811386] 0.33389060356652944


This did eventually find something! This is approximately what I got when I first ran the code:

In [51]:
run(10, 0.58, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45])

Out[51]:
0.33406207133058985

Optimizing with this set of initial values:

In [52]:
%%time
best = None
poss = [0.85, 0.29, 0.03, 0.01, -0.49, -0.45]

for _ in range(100):
f = lambda x: 0-run(*x, 2, poss)
o = minimize(f, [random()*2 for i in range(2)], method='Nelder-Mead')
if not best or -best.fun < -o.fun:
best = o
print(-best.fun)
print(best.x, -best.fun)

0.284593621399177
0.31918724279835387
0.32270233196159126
0.33423353909465026
0.33427640603566533
0.3343192729766804
[ 0.56348509 18.66995185] 0.3343192729766804
CPU times: user 13.9 s, sys: 3.96 ms, total: 14 s
Wall time: 14 s

In [53]:
run(17.96, 0.56, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45])

Out[53]:
0.3343192729766804

Weird! There's something special about this 1/3 value (i.e. 5/6 cut fraction on a ring) for local classical algorithms.

### Exploring this 2-regular algorithm¶

I did a less "optimized" analysis of the 2-regular algorithm:

1. Take numbers in 6 slots from a list of options or (preferably discrete) distribution.
2. For each of the inner 4, compute a new number.
3. For each of the inner 2, compute a new number.
In [54]:
def step(inps, c, size):
outs = []
for x in inps:
out = []
for i in range(1, size-1):
y = x[i] - c*(x[i-1] + x[i+1])
out.append(y)
outs.append(out)
return outs


I used this to verify the previous result.

In [55]:
opts = [0.85, 0.29, 0.03, 0.01, -0.49, -0.45]

In [56]:
def run_2step_local(poss, c0, c1):
"""Outputs cut fraction given inputs and (c0, c1)"""
l = len(poss)
cases = list(itertools.product(*[poss,]*6))
dones = step(step(cases, c0, 6), c1, 4)
res = sum([(np.sign(d[0]) != np.sign(d[1]) and d[0] != 0 and d[1] != 0) for d in dones]) / len(dones)
return res

In [57]:
out = run_2step_local(opts, 17.96, 0.56)
print(out)
assert out > 5/6, "Should be greater than 5/6"

0.8343192729766804


It's possible there is something more fundamental going on...

### Double-checking my work¶

Here's an un-optimized version of the local linear algorithms. I confirm my previous results here.

In [58]:
@functools.lru_cache(maxsize=int(1e7))
def unoptimized_local_2(c0, c1, v0_1, v0_2, nbrs1, nbrs2, nbr_nbrs1, nbr_nbrs2):
"""Cut fraction (1, 0.5, 0) of a local subgraph"""
v1_1 = v0_1 - c0*sum(nbrs1) - c0*v0_2
v1_2 = v0_2 - c0*sum(nbrs2) - c0*v0_1

v1_nbrs1 = nbrs1 - c0*np.sum(nbr_nbrs1, axis=1) - c0*v0_1
v1_nbrs2 = nbrs2 - c0*np.sum(nbr_nbrs2, axis=1) - c0*v0_2

v2_1 = v1_1 - c1*sum(v1_nbrs1) - c1*v1_2
v2_2 = v1_2 - c1*sum(v1_nbrs2) - c1*v1_1

return 0.5 - 0.5*np.sign(v2_1)*np.sign(v2_2)

In [59]:
def unoptimized_local_2_perf(c0, c1, d, poss):
tot = 0
denom = len(poss)**(2*(d-1)**2 + 2*(d-1) + 2)
print("Number of options:", denom)

nbr_opts = list(itertools.product(*[poss,]*(d-1)))
nbr_nbr_opts = list(itertools.product(*[nbr_opts,]*(d-1)))
assert len(nbr_opts) == len(poss)**(d-1)
assert len(nbr_nbr_opts) == len(poss)**((d-1)**2)

i = 0
for v0_1 in poss:
for v0_2 in poss:
for nbrs1 in nbr_opts:
for nbrs2 in nbr_opts:
for nbr_nbrs1 in nbr_nbr_opts:
for nbr_nbrs2 in nbr_nbr_opts:
i += 1
tot += unoptimized_local_2(c0, c1, v0_1, v0_2, nbrs1, nbrs2, nbr_nbrs1, nbr_nbrs2)

assert i == denom



Here are my previous results:

In [60]:
print("D=2", run(17.96, 0.56, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45]))
print("D=3", run(7.22, 0.58, 3, [-1, 1]) )
print("D=4", run(0.38, 11.20, 4, [-1, 1]) )
print("D=5", run(5.44, 0.33, 5, [-1, 1]) )

D=2 0.3343192729766804
D=3 0.270263671875
D=4 0.23085129261016846
D=5 0.20453225069923064

In [61]:
%%time
unoptimized_local_2_perf(17.96, 0.56, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45])

Number of options: 46656
CPU times: user 1.42 s, sys: 16 ms, total: 1.44 s
Wall time: 1.44 s

Out[61]:
0.3343192729766804
In [62]:
%%time
unoptimized_local_2_perf(7.22, 0.58, 3, [-1, 1])

Number of options: 16384
CPU times: user 588 ms, sys: 36 ms, total: 624 ms
Wall time: 600 ms

Out[62]:
0.270263671875

I have to optimize a little to check $D=4$ and $D=5$:

In [63]:
@functools.lru_cache(maxsize=int(1e7))
def slightly_optimized_local_2(c0, c1, v0_1, v0_2, nbrs1, nbrs2, nbr_nbrsums1, nbr_nbrsums2):
"""Cut fraction (1, 0.5, 0) of a local subgraph"""
v1_1 = v0_1 - c0*sum(nbrs1) - c0*v0_2
v1_2 = v0_2 - c0*sum(nbrs2) - c0*v0_1

v1_nbrs1 = nbrs1 - c0*np.array(nbr_nbrsums1) - c0*v0_1
v1_nbrs2 = nbrs2 - c0*np.array(nbr_nbrsums2) - c0*v0_2

v2_1 = v1_1 - c1*sum(v1_nbrs1) - c1*v1_2
v2_2 = v1_2 - c1*sum(v1_nbrs2) - c1*v1_1

return 0.5 - 0.5*np.sign(v2_1)*np.sign(v2_2)

In [64]:
def slightly_optimized_local_2_perf(c0, c1, d, poss):
tot = 0
denom = len(poss)**(2*(d-1)**2 + 2*(d-1) + 2)
print("Number of options:", denom)

nbr_opts = list(itertools.product(*[poss,]*(d-1)))
nbr_nbr_opts = list(itertools.product(*[nbr_opts,]*(d-1)))
nbr_nbr_sums = [tuple(i) for i in list(np.sum(nbr_nbr_opts, axis=2))]

assert len(nbr_opts) == len(poss)**(d-1)
assert len(nbr_nbr_sums) == len(poss)**((d-1)**2)

i = 0
for v0_1 in poss:
for v0_2 in poss:
for nbrs1 in nbr_opts:
for nbrs2 in nbr_opts:
for nbr_nbrs1 in nbr_nbr_sums:
for nbr_nbrs2 in nbr_nbr_sums:
i += 1
if i % 10000000 == 0:
print(i)
tot += slightly_optimized_local_2(c0, c1, v0_1, v0_2, nbrs1, nbrs2, nbr_nbrs1, nbr_nbrs2)

assert i == denom


In [65]:
%%time
slightly_optimized_local_2_perf(17.96, 0.56, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45])

Number of options: 46656
CPU times: user 632 ms, sys: 8.02 ms, total: 640 ms
Wall time: 639 ms

Out[65]:
0.3343192729766804
In [66]:
%%time
slightly_optimized_local_2_perf(7.22, 0.58, 3, [-1, 1])

Number of options: 16384
CPU times: user 119 ms, sys: 8.02 ms, total: 127 ms
Wall time: 110 ms

Out[66]:
0.270263671875
In [67]:
%%time
# this will take awhile
slightly_optimized_local_2_perf(0.38, 11.20, 4, [-1, 1])

Number of options: 67108864
10000000
20000000
30000000
40000000
50000000
60000000
CPU times: user 1min 9s, sys: 292 ms, total: 1min 9s
Wall time: 1min 9s

Out[67]:
0.23085129261016846

I have to optimize further just to get $D=5$ to run in a reasonable amount of time.

In [68]:
@functools.lru_cache(maxsize=int(1e7))
def more_optimized_local_2(d, c0, c1, v0_1, v0_2, s1, s2, diff1, diff2):
"""Cut fraction (1, 0.5, 0) of a local subgraph"""
v1_1 = v0_1 - c0*s1 - c0*v0_2
v1_2 = v0_2 - c0*s2 - c0*v0_1

v2_1 = v1_1 - c1*(s1 - c0*diff1 - c0*v0_1*(d-1)) - c1*v1_2
v2_2 = v1_2 - c1*(s2 - c0*diff2 - c0*v0_2*(d-1)) - c1*v1_1

return 0.5 - 0.5*np.sign(v2_1)*np.sign(v2_2)

In [69]:
def more_optimized_local_2_perf(c0, c1, d, poss):
tot = 0
denom = len(poss)**(2*(d-1)**2 + 2*(d-1) + 2)
print("Number of options:", denom)

nbr_opts = list(itertools.product(*[poss,]*(d-1)))
nbr_nbr_opts = list(itertools.product(*[nbr_opts,]*(d-1)))
nbr_nbr_sums = [tuple(i) for i in list(np.sum(nbr_nbr_opts, axis=2))]

nbr_sums = [i for i in list(np.sum(nbr_opts, axis=1))]
diffs = [i for i in list(np.sum(nbr_nbr_sums, axis=1))]

assert len(nbr_sums) == len(poss)**(d-1)
assert len(diffs) == len(poss)**((d-1)**2)

poss_u = np.unique(poss, return_counts=True)
nbrs_u = np.unique(nbr_sums, return_counts=True)
diff_u = np.unique(diffs, return_counts=True)

i = 0
for v0_1, m1 in zip(*poss_u):
for v0_2, m2 in zip(*poss_u):
for nbrs1, m3 in zip(*nbrs_u):
for nbrs2, m4 in zip(*nbrs_u):
for nbr_nbrs1, m5 in zip(*diff_u):
for nbr_nbrs2, m6 in zip(*diff_u):
i += m1*m2*m3*m4*m5*m6
tot += m1*m2*m3*m4*m5*m6*more_optimized_local_2(d, c0, c1, v0_1, v0_2, nbrs1, nbrs2, nbr_nbrs1, nbr_nbrs2)

assert i == denom


In [70]:
%%time
more_optimized_local_2_perf(17.96, 0.56, 2, [0.85, 0.29, 0.03, 0.01, -0.49, -0.45])

Number of options: 46656
CPU times: user 445 ms, sys: 3 µs, total: 445 ms
Wall time: 443 ms

Out[70]:
0.3343192729766804
In [71]:
%%time
more_optimized_local_2_perf(7.22, 0.58, 3, [-1, 1])

Number of options: 16384
CPU times: user 46 ms, sys: 5 µs, total: 46 ms
Wall time: 44.3 ms

Out[71]:
0.270263671875
In [72]:
%%time
more_optimized_local_2_perf(0.38, 11.20, 4, [-1, 1])

Number of options: 67108864
CPU times: user 220 ms, sys: 15 µs, total: 220 ms
Wall time: 217 ms

Out[72]:
0.23085129261016846
In [73]:
%%time
more_optimized_local_2_perf(5.44, 0.33, 5, [-1, 1])

Number of options: 4398046511104
CPU times: user 1.09 s, sys: 8.01 ms, total: 1.1 s
Wall time: 1.1 s

Out[73]:
0.20453225069923064

Looks like the numbers match! Thanks for reading!

In [74]:
print("Ending time:", time.ctime())

Ending time: Tue Dec 29 04:35:18 2020