Chapter 3: How to Think in QUBO

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(From Daily Life to Math, Step by Step)

QUBO is not about math first.
QUBO is about thinking clearly.

Goal of this chapter

• Learn a repeatable method to convert real problems into QUBO
• Understand how to think in variables, objectives, and constraints
• Practice with everyday, non-technical examples

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1. The Universal QUBO Thinking Process

No matter how complex a problem looks, QUBO modeling always follows the same steps:

1. List choices (Yes / No decisions)
2. Create binary variables for each choice
3. Decide what you want to optimize (objective)
4. List rules you must follow (constraints)
5. Convert rules into penalties
6. Combine everything into one formula

Now let’s apply every step explicitly.

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Example 1: Choose Exactly One Drink

You have 3 drinks: Coke, Milk Tea, Orange Juice

You want to choose exactly one.

Step 1: List the Choices

Ask yourself:

What are the Yes / No decisions?

Choices:

• Choose Coke?
• Choose Milk Tea?
• Choose Orange Juice?

That’s it. No math yet.

Step 2: Create Binary Variables

Turn each choice into a variable:

C, M, O ∈ {0,1}

Meaning:

• 1 → chosen
• 0 → not chosen

So:

• C = 1 means Coke is chosen
• M = 1 means Milk Tea is chosen
• O = 1 means Orange Juice is chosen

Step 3: Decide What You Want to Optimize

In this example:

• You don’t care which drink
• You just want one drink

So the objective can be zero:

 Objective = 0

This is normal. Not every problem needs rewards.

Step 4: List the Rules

Rule in human language:

You must choose exactly one drink

Translate to math:

C + M + O = 1

Still not QUBO yet.

Step 5: Convert the Rule into a Penalty

QUBO cannot say “forbidden”. It only says:

Break the rule → pay a cost

We do this by squaring:

(C + M + O — 1)²

Why this works:

• Rule satisfied → value = 0 (no penalty)
• Rule broken → value > 0 (penalty)

We multiply by a penalty weight P:

P(C + M + O - 1)²

Step 6: Combine Everything into One Formula

Final QUBO:

Q = P(C + M + O — 1)²

This single formula fully describes the problem.

A solver will try all combinations and pick:

• exactly one variable = 1
• others = 0

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Example 2: Prefer One Option

Now let’s add preference.

You like Coke more than the others.

Step 1 and Step 2 are same as Example 1 so let’s jump to Step 3.

Step 3: Objective (Preference)

You like Coke, so reward it:

Objective = -3C

Negative = good.

Step 4: Rule

Choose exactly one:

C + M + O = 1

Step 5: Penalty

P(C + M + O — 1)²

Step 6: Final QUBO

Q = -3C + P(C + M + O — 1)²

Now the solver:

• Must choose one drink
• Will prefer Coke if possible

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Example 3: Multiple Rules

• You preference Coke -> Orange juice -> Milk Tea
• You can choose at most 2 drinks
• You cannot choose Coke and Orange juice together

Step 1 and Step 2 are same as Example 1 & 2 so let’s jump to Step 3.

Step 3: Objective (Multiple rules)

Objective = -5C -3M -4O

Step 4: Rules

1. At most 2 drinks:
   C + M+ O ≤ 2
2. Coke and Orange juice cannot both be chosen:
   C + O ≤ 1

Step 5: Penalties

Convert rules:

P₁ for Rule 1:

(C + M + O — 2)²

P₂ for Rule 2:

(C + O — 1)²

Step 6: Final QUBO

Q = -5C -3M -4O + P₁(C + M + O- 2)² + P₂(C + O - 1)²

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Mental Shortcut (Very Important)

Whenever you face a new problem, ask:

1. What are my Yes / No choices?
2. What do I want more or less of?
3. What must never happen?
4. Which option do I prefer more than the others?

If you can answer these three questions, you are already 80% done.

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Final Thought

You’re no longer memorizing formulas. You’re thinking in patterns.

In the next chapter, we’ll:

• Visualize QUBO as graphs
• Understand interactions intuitively
• Prepare for real solvers

Keep going 👍