Chapter 2 – QUBO Building Blocks

(Binary Variables, Penalties, and Objectives)

Goal of this chapter

• Understand the 3 ingredients of every QUBO
• Learn what penalties are and why they work
• Practice with simple, real-life examples

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1. Binary Variables (Yes / No switches)

QUBO only uses binary variables:

• 0 = NO / not chosen / not active
• 1 = YES / chosen / active

Think of them as on/off switches.

Example: 3 drinks

• C = 1 → choose Coke
• M = 1 → choose Milk Tea
• O = 1 → choose Orange Juice

All variables must be:

C, M, O ∈ {0,1}

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2. Objective Function (What you want to optimize)

QUBO always minimizes a formula.

• Smaller value = better
• Bigger value = worse

Example: Drink preference

• Coke is your favorite → 5
• Milk Tea you dislike → +10
• Orange Juice is neutral → 0

Objective:

-5C + 10M + 0O

Lower score = better choice.

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3. Constraints using Penalties (Rules)

Real problems have rules, such as:

• Choose exactly 1 item
• Do not choose A and B together
• Choose at most 2 items

QUBO enforces rules using penalties.

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What is a Penalty?

A penalty is extra cost added when a rule is broken.

• Rule satisfied → penalty = 0
• Rule broken → penalty = positive number

Because QUBO minimizes, it naturally avoids penalties.

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Why penalty = 0 or 1 in simple examples?

Many rules can be written so that:

• Valid solution → penalty = 0
• Invalid solution → penalty = 1

Later, we multiply by a penalty weight to make rules important.

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How to choose penalty weight?

Simple rule:

Penalty weight should be larger than objective values

Objective range	Safe penalty
0 ~ 5	10
0 ~ 10	20
0 ~ 50	100

If penalty is too small → rules may break

If penalty is large → always safe

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4. Super Simple Example: Choose 1 Drink (2 options)

Variables:

C ∈ {0,1}
P ∈ {0,1}

Constraint:

C + P = 1

QUBO penalty form:

(C + P - 1)²

Check it

C	P	Result	Meaning
1	0	0	Valid
0	1	0	Valid
0	0	1	Invalid
1	1	1	Invalid

Valid → penalty = 0

Invalid → penalty = 1

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5. Assessment Problem (3 Drinks)

Problem

You have:

• Coke
• Milk Tea
• Orange Juice

Choose exactly 1.

Variables

C, M, O ∈ {0,1}

Constraint

C + M + O = 1

QUBO formula

(C + M + O - 1)²

Optional penalty weight P:

P(C + M + O - 1)²

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6. Practice (Homework)

Try writing QUBO for:

1. Choose at most 1 drink
2. Choose at least 1 drink
3. Choose exactly 2 drinks
4. Add preferences:
   • Coke = -5
   • Milk Tea = +7
   • Orange Juice = 0

Combine objective + constraint into one QUBO.