By the end of this chapter, you will:<\/p>\n\n\n\n
No advanced math. No quantum yet. Just practical learning.<\/p>\n\n\n\n
\nChoose exactly one drink<\/em><\/p>\n\n\n\n
Options:<\/p>\n\n\n\n
Coke ( C )<\/p>\n\n\n\n
Milk Tea (M)<\/p>\n\n\n\n
Orange Juice (O)<\/p>\n<\/blockquote>\n\n\n\n
\n\n\n\nStep 1: Define Binary Variables<\/strong><\/h4>\n\n\n\n
Each choice becomes a binary variable:<\/p>\n\n\n\n
\n
- C = 1 if Coke is chosen, else 0<\/li>\n\n\n\n
- M = 1 if Milk Tea is chosen, else 0<\/li>\n\n\n\n
- O = 1 if Orange Juice is chosen, else 0<\/li>\n<\/ul>\n\n\n\n
Graph view:<\/p>\n\n\n\n
\n
- 3 nodes<\/li>\n\n\n\n
- Each node is ON or OFF<\/li>\n<\/ul>\n\n\n\n
Step 2: Write the Constraint as QUBO<\/h4>\n\n\n\n
Rule:<\/p>\n\n\n\n
\nChoose exactly one drink<\/em><\/p>\n<\/blockquote>\n\n\n\n
QUBO pattern (from Chapter 3):<\/p>\n\n\n\n
(C + M + O \u2212 1)\u00b2<\/code><\/pre>\n\n\n\nWhy this works:<\/p>\n\n\n\n
\n
- = 0 when exactly one is chosen<\/li>\n\n\n\n
- 0 otherwise<\/li>\n<\/ul>\n\n\n\n
This is our energy function<\/strong>.<\/p>\n\n\n\n
Step 3: Expand the Formula<\/h4>\n\n\n\n
Expand:<\/p>\n\n\n\n
C + M + O
+ 2(CM + CO + MO)
\u2212 2(C + M + O)
+ 1<\/code><\/pre>\n\n\n\nIgnore the constant
+1<\/code>.<\/p>\n\n\n\nWhat remains defines:<\/p>\n\n\n\n
\n
- Node weights (linear terms)<\/li>\n\n\n\n
- Edge penalties (pair terms)<\/li>\n<\/ul>\n\n\n\n
This matches the fully connected graph<\/strong> from Chapter 4.<\/p>\n\n\n\n
Step 4: Represent QUBO in Python<\/strong><\/h4>\n\n\n\n
We store QUBO as a dictionary.<\/p>\n\n\n\n
import itertools
# Variable order: C=0, M=1, O=2
Q = {
(0, 0): -1,
(1, 1): -1,
(2, 2): -1,
(0, 1): 2,
(0, 2): 2,
(1, 2): 2,
}<\/code><\/pre>\n\n\n\nEach entry means:<\/p>\n\n\n\n
\n
(i, i)<\/code> \u2192 node weight<\/li>\n\n\n\n(i, j)<\/code> \u2192 edge penalty<\/li>\n<\/ul>\n\n\n\nStep 5: Define Energy Function<\/h4>\n\n\n\n
def energy(x):
e = 0
for (i, j), w in Q.items():
e += w * x[i] * x[j]
return e<\/code><\/pre>\n\n\n\nThis function:<\/p>\n\n\n\n
\n
- Takes a solution like
(1, 0, 0)<\/code><\/li>\n\n\n\n- Calculates total energy<\/li>\n<\/ul>\n\n\n\n
This is exactly how solvers \u201csee\u201d your graph.<\/p>\n\n\n\n
Step 6: Brute Force Solve<\/h4>\n\n\n\n
Because we only have 3 variables, we can try all combinations.<\/p>\n\n\n\n
best = None
for x in itertools.product([0, 1], repeat=3):
e = energy(x)
print(x, e)
if best is None or e < best[1]:
best = (x, e)
print(\"Best solution:\", best)<\/code><\/pre>\n\n\n\nYou will observe:<\/p>\n\n\n\n
\n
- Invalid choices \u2192 higher energy<\/li>\n\n\n\n
- Valid choices \u2192 lowest energy<\/li>\n<\/ul>\n\n\n\n
Step 7: Interpret the Result<\/h4>\n\n\n\n
Valid lowest-energy solutions:<\/p>\n\n\n\n
\n
(1, 0, 0)<\/code> \u2192 Coke<\/li>\n\n\n\n(0, 1, 0)<\/code> \u2192 Milk Tea<\/li>\n\n\n\n(0, 0, 1)<\/code> \u2192 Orange Juice<\/li>\n<\/ul>\n\n\n\nAll are equally good.<\/p>\n\n\n\n
Why?<\/p>\n\n\n\n
Because we added rules<\/strong>, not preferences<\/strong>.<\/p>\n\n\n\n
Step 8: Add Preferences (Objective)<\/h4>\n\n\n\n
Let\u2019s say:<\/p>\n\n\n\n
\n
- Coke is preferred<\/li>\n\n\n\n
- Milk Tea is disliked<\/li>\n<\/ul>\n\n\n\n
Q[(0, 0)] += -2 # reward Coke
Q[(1, 1)] += 3 # penalize Milk Tea<\/code><\/pre>\n\n\n\nRe-run the solver.<\/p>\n\n\n\n
Now the energy landscape tilts.<\/p>\n\n\n\n
One solution becomes the deepest valley.<\/p>\n\n\n\n
\n\n\n\nWhat You Just Learned<\/h3>\n\n\n\n
\n
- QUBO becomes simple Python code<\/li>\n\n\n\n
- Solvers only compare energies<\/li>\n\n\n\n
- Graph intuition directly matches behavior<\/li>\n\n\n\n
- You don\u2019t need special tools to start<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nScaling up Step 9: Simulated Annealing (Conceptual)<\/strong><\/h4>\n\n\n\n
Brute force helped us see everything<\/em><\/strong>.<\/p>\n\n\n\n
But it breaks very quickly.<\/p>\n\n\n\n
When variables grow:<\/p>\n\n\n\n
\n
- Combinations explode<\/li>\n\n\n\n
- Brute force becomes impossible<\/li>\n<\/ul>\n\n\n\n
This is where Simulated Annealing (SA)<\/strong> enters.<\/p>\n\n\n\n
\n\n\n\nHow Simulated Annealing Works<\/h4>\n\n\n\n
From Chapter 5 and the graph view:<\/p>\n\n\n\n
\n
- QUBO = energy landscape<\/li>\n\n\n\n
- Valleys = good solutions<\/li>\n\n\n\n
- Hills = bad solutions<\/li>\n<\/ul>\n\n\n\n
Simulated Annealing:<\/p>\n\n\n\n
\n
- Starts at a random configuration<\/li>\n\n\n\n
- Flips one variable (node)<\/li>\n\n\n\n
- Measures energy change<\/li>\n\n\n\n
- Sometimes accepts worse moves<\/li>\n\n\n\n
- Gradually becomes more conservative<\/li>\n<\/ol>\n\n\n\n
Early stage:<\/p>\n\n\n\n
\n
- Explores widely<\/li>\n\n\n\n
- Jumps across hills<\/li>\n<\/ul>\n\n\n\n
Late stage:<\/p>\n\n\n\n
\n
- Settles into a deep valley<\/li>\n<\/ul>\n\n\n\n
This is how solvers escape local minima<\/strong>.<\/p>\n\n\n\n
Why This Matters for Real Problems<\/h4>\n\n\n\n
Most real QUBO problems:<\/p>\n\n\n\n
\n
- Have hundreds or thousands of variables<\/li>\n\n\n\n
- Cannot be solved exactly<\/li>\n\n\n\n
- Need good<\/em> solutions fast<\/li>\n<\/ul>\n\n\n\n
Simulated Annealing:<\/p>\n\n\n\n
\n
- Trades perfection for speed<\/li>\n\n\n\n
- Handles noisy, complex graphs<\/li>\n\n\n\n
- Works well with QUBO structure<\/li>\n<\/ul>\n\n\n\n
This is why it is widely used in practice.<\/p>\n\n\n\n
\n\n\n\nStep 10\u200a\u2014\u200aSimulated Annealing in Python (Hands-On)<\/h4>\n\n\n\n
Now let\u2019s see real code<\/strong>.<\/p>\n\n\n\n
We will use a ready-made solver<\/strong>, just like engineers do in real projects.<\/p>\n\n\n\n
Below is an example using Fixstars Amplify<\/strong> (quantum-inspired, runs on classical hardware).<\/p>\n\n\n\n
Install Library<\/strong><\/h4>\n\n\n\n
pip install amplify<\/code><\/pre>\n\n\n\nDefine the Same QUBO Problem<\/h4>\n\n\n\n
from amplify import BinarySymbolGenerator, solve
from amplify.client import FixstarsClient
# Create binary variables
q = BinarySymbolGenerator()
C, M, O = q.array(3)
# Exactly-one constraint
qubo = (C + M + O - 1) ** 2
# Add preferences
qubo += -2 * C # prefer Coke
qubo += 3 * M # avoid Milk Tea<\/code><\/pre>\n\n\n\nRun Simulated Annealing Solver<\/h4>\n\n\n\n
client = FixstarsClient()
client.parameters.timeout = 1000 # milliseconds
result = solve(qubo, client)
best = result.best
print(\"Solution:\", best.values)
print(\"Energy:\", best.energy)<\/code><\/pre>\n\n\n\nOutput Interpretation<\/h4>\n\n\n\n
Example output:<\/p>\n\n\n\n
Solution: [1, 0, 0]
Energy: -2.0<\/pre>\n\n\n\nMeaning:<\/p>\n\n\n\n
\n
- Coke = ON<\/li>\n\n\n\n
- Milk Tea = OFF<\/li>\n\n\n\n
- Orange Juice = OFF<\/li>\n<\/ul>\n\n\n\n
The solver:<\/p>\n\n\n\n
\n
- Explored many configurations<\/li>\n\n\n\n
- Escaped local minima<\/li>\n\n\n\n
- Found the deepest valley<\/li>\n<\/ul>\n\n\n\n
All without brute force<\/strong>.<\/p>\n\n\n\n
What Just Happened<\/h4>\n\n\n\n
Under the hood, the solver:<\/p>\n\n\n\n
\n
- Flipped nodes ON\/OFF<\/li>\n\n\n\n
- Measured energy change<\/li>\n\n\n\n
- Sometimes accepted worse moves<\/li>\n\n\n\n
- Slowly settled into a low-energy state<\/li>\n<\/ol>\n\n\n\n
Exactly the annealing behavior<\/strong> we discussed conceptually.<\/p>\n\n\n\n
\n\n\n\nWhat You\u2019ve Achieved So Far<\/h3>\n\n\n\n
You can now:<\/p>\n\n\n\n
\n
- Build QUBO models<\/li>\n\n\n\n
- Understand them as graphs<\/li>\n\n\n\n
- Solve small problems exactly<\/li>\n\n\n\n
- Understand how solvers scale<\/li>\n<\/ul>\n\n\n\n
You are no longer \u201creading about QUBO\u201d.<\/p>\n\n\n\n
You are using it<\/strong>.<\/p>\n\n\n\n
\n\n\n\nWhat\u2019s Next<\/h3>\n\n\n\n
In the next chapter, we will:<\/p>\n\n\n\n
\n
- Introduce the Ising model<\/strong><\/li>\n\n\n\n
- See how QUBO maps to Ising<\/li>\n<\/ul>\n\n\n\n
This connects modeling to real solvers and machines.<\/p>\n","protected":false},"excerpt":{"rendered":"
Goal of this chapter By the end of this chapter, you will: No advanced math. No quantum yet. Just practical learning. Problem We Will Solve (Again) Choose exactly one drink Options: Coke ( C ) Milk Tea (M) Orange Juice (O) Step 1: Define Binary Variables Each choice becomes a binary variable: Graph view: Step 2: Write the Constraint as QUBO Rule: Choose exactly one drink QUBO pattern (from Chapter 3): Why this works: This is our energy function. Step 3: Expand the Formula Expand: Ignore the constant +1. What remains defines: This matches the fully connected graph from Chapter 4. Step 4: Represent QUBO in Python We store QUBO as a dictionary. Each entry means: Step 5: Define Energy Function This function: This is exactly how solvers \u201csee\u201d your graph. Step 6: Brute Force Solve Because we only have 3 variables, we can try all combinations. You will observe: Step 7: Interpret the Result Valid lowest-energy solutions: All are equally good. Why? Because we added rules, not preferences. Step 8: Add Preferences (Objective) Let\u2019s say: Re-run the solver. Now the energy landscape tilts. One solution becomes the deepest valley. What You Just Learned Scaling up Step 9: Simulated Annealing (Conceptual) Brute force helped us see everything. But it breaks very quickly. When variables grow: This is where Simulated Annealing (SA) enters. How Simulated Annealing Works From Chapter 5 and the graph view: Simulated Annealing: Early stage: Late stage: This is how solvers escape local minima. Why This Matters for Real Problems Most real QUBO problems: Simulated Annealing: This is why it is widely used in practice. Step 10\u200a\u2014\u200aSimulated Annealing in Python (Hands-On) Now let\u2019s see real code. We will use a ready-made solver, just like engineers do in real projects. Below is an example using Fixstars Amplify (quantum-inspired, runs on classical hardware). Install Library Define the Same QUBO Problem Run Simulated Annealing Solver Output Interpretation Example output: Solution: [1, 0, 0]Energy: -2.0 Meaning: The solver: All without brute force. What Just Happened Under the hood, the solver: Exactly the annealing behavior we discussed conceptually. What You\u2019ve Achieved So Far You can now: You are no longer \u201creading about QUBO\u201d. You are using it. What\u2019s Next In the next chapter, we will: This connects modeling to real solvers and machines.<\/p>\n","protected":false},"author":1,"featured_media":60,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-59","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"jetpack_featured_media_url":"https:\/\/qc.thelazydreamer.com\/wp-content\/uploads\/2025\/12\/Gemini_Generated_Image_hyw350hyw350hyw3.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/posts\/59","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=59"}],"version-history":[{"count":1,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/posts\/59\/revisions"}],"predecessor-version":[{"id":61,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/posts\/59\/revisions\/61"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=\/wp\/v2\/media\/60"}],"wp:attachment":[{"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=59"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=59"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qc.thelazydreamer.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=59"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}