Quantum Parallelism

The ability of quantum computers to process multiple inputs simultaneously through superposition.

Quantum parallelism refers to a quantum computer’s ability to evaluate a function on many inputs at once by exploiting superposition.

The Basic Idea

Classically, to evaluate $f(x)$ on $N$ inputs, you need $N$ evaluations.

Quantumly, you can create a superposition of all inputs and evaluate once:

$$|x⟩|0⟩\stackrel{U_f}{\to }|x⟩|f(x)⟩$$

Apply to superposition: $$\frac{1}{\sqrt{N}}\sum _x|x⟩|0⟩\stackrel{U_f}{\to }\frac{1}{\sqrt{N}}\sum _x|x⟩|f(x)⟩$$

One quantum operation, $N$ function evaluations encoded in the state!

Example: 3 Qubits

With 3 input qubits in superposition:

$$H^{\otimes 3}|000⟩=\frac{1}{\sqrt{8}}(|000⟩+|001⟩+\dots +|111⟩)$$

All 8 inputs ($0$ through $7$) exist simultaneously.

The Catch

Here’s the critical limitation: you can’t read all the answers.

Measurement gives you ONE random result. The superposition collapses.

Before measurement: |f(0)⟩ + |f(1)⟩ + |f(2)⟩ + ... (all answers)
After measurement:  |f(k)⟩ (one random answer)

How Quantum Algorithms Use It

Quantum algorithms don’t just compute all answers. They use interference to extract useful information:

Deutsch-Josza

Determine if function is constant or balanced in ONE query (classically needs ~N/2).

Grover’s Search

Find the marked item in $O(\sqrt{N})$ queries instead of $O(N)$.

Shor’s Factoring

Use parallelism + QFT to find periodicity, enabling factoring.

The Pattern

1. Create superposition over all inputs
2. Evaluate function (quantum parallelism)
3. Apply interference (constructive for answers, destructive for non-answers)
4. Measure to extract the useful result

Common Misconception

Wrong: “Quantum computers try all solutions and pick the right one”

Right: Quantum computers use interference to make correct answers more probable. Designing this interference is the hard part, which is why we have only a handful of useful quantum algorithms.

Exponential State Space

With $n$ qubits:

• $2^n$ basis states can be in superposition
• But only $n$ qubits of information can be extracted (Holevo bound)

The challenge is designing algorithms that concentrate useful information into measurable outcomes.

See also: Superposition, Quantum Algorithm, Grover’s Algorithm