QUANTUM 01

Bits vs.
qubits

Classical bits, quantum bits, ket notation, and why qubits scale so fast

Sinclair College - Dr. Mike Libassi

TODAY

What we will cover

01Classical bitsWhat n bits can hold

02The qubitKet notation and amplitudes

03MeasurementReading a qubit out

04ScalingWhy 2ⁿ changes everything

Quantum 01 · Bits vs. Qubits2

01 · CLASSICAL BITS

A bit is always exactly 0 or 1

Every classical computer stores information as bits: a voltage that is either low (0) or high (1), never in between.

n bits

Can represent 2ⁿ different values. Four bits give 16 possibilities.

One at a time

Those bits hold only one of the 2ⁿ values at any moment.

n numbers

Describing the state of n bits takes just n numbers, each 0 or 1.

Quantum 01 · Bits vs. Qubits3

02 · THE QUBIT

Writing a qubit down: ket notation

|0⟩=10|1⟩=01

|ψ⟩=α|0⟩ + β|1⟩=αβ

rule: |α|² + |β|² = 1 · P(0) = |α|² · P(1) = |β|²

Amplitudes α and β are the numbers in front of each ket. Square them to get probabilities. Both nonzero means the qubit is in superposition.

Quantum 01 · Bits vs. Qubits4

03 · MEASUREMENT

Reading a qubit gives one ordinary bit

Random

Outcome 0 with probability |α|², outcome 1 with probability |β|².

Collapse

Afterward the qubit is in the state you saw. Measure again and you get the same answer.

Statistics

The amplitudes only show up as percentages across many freshly prepared qubits.

Quantum 01 · Bits vs. Qubits5

04 · SCALING

Each qubit doubles the state

Qubits (n)	Amplitudes (2ⁿ)	Memory to simulate	Fits on
1	2	32 bytes	anything
10	1,024	16 KB	anything
30	≈ 1.07 billion	16 GB	a laptop
37	≈ 137 billion	2 TB	a large server
50	≈ 1.1 quadrillion	16 PB	nothing practical

At roughly 16 bytes per amplitude, classical simulation hits a wall around 45–50 qubits.

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04 · SCALING

Reality check: more qubits isn't infinite power

Readout limit

Measuring n qubits returns only n bits. Algorithms must make the right answer likely.

Decoherence

Noise turns superposition into ordinary 0s and 1s within microseconds to seconds.

Physical vs. logical

Reliable logical qubits are built from many noisy physical qubits.

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WRAP-UP

Key takeaways

A classical bit is always 0 or 1; n bits need n numbers to describe

A qubit is written α|0⟩ + β|1⟩, a two-row column vector of amplitudes

Measuring gives one bit, chosen randomly with probability |amplitude|²

n qubits need 2ⁿ amplitudes, which overwhelms classical simulation past ~50 qubits

Quantum 01 · Bits vs. Qubits8

DISCUSSION

If 50 qubits can't be simulated, why don't today's quantum computers already beat classical ones at everything?

Think about readout limits, noise, and which problems have structure a quantum algorithm can exploit.

Quantum 01 · Bits vs. Qubits & Scaling · Questions? · Open the interactive page

Bits vs.qubits