🧠 From Qubits to Circuits: Building My Foundations in Quantum Computing

Introduction

Quantum computing is often introduced through spectacle — Schrödinger’s cat, exponential speedups, and claims of breaking classical cryptography. Most introductions stop there.

My journey into quantum computing began differently. I wasn’t trying to “learn quantum computing” as a buzzword. I was trying to understand what computation even means when information itself obeys quantum mechanics — and whether that understanding survives contact with real hardware.

This blog documents the foundational layer of my quantum work: the concepts, mathematics, and circuit intuition I built before touching physical quantum systems. These foundations later became critical when I worked on NMR-based quantum computing and quantum-enhanced neuromorphic sensing.

This is not a tutorial.
It’s a researcher’s reconstruction of how the foundations actually fit together.

1. Classical Information vs Quantum Information

Classical computation is built on certainty.

A classical bit exists as either:

• 0, or

• 1

Quantum computation begins by breaking this assumption.

A qubit exists in a state:

∣ψ⟩=α∣0⟩+β∣1⟩

where:

• α, β ∈ ℂ

• |α|² + |β|² = 1

This equation is not philosophical — it is operational.
The coefficients are probability amplitudes, not probabilities.

The first mental shift I had to make:
Quantum states describe potential measurement outcomes, not hidden classical values.

2. Measurement Is Not Passive

In classical systems, observation does not alter the system.

In quantum systems:

• Measurement projects the state

• Superposition is destroyed

• The measurement basis matters

For a qubit measured in the computational (Z) basis:

• • Probability of 0 = ∣α∣²

    • Probability of 1 = ∣β∣²

      After measurement, the system collapses into the measured eigenstate.

This single fact explains:

• Why copying quantum states is impossible (no-cloning theorem)

• Why quantum algorithms must delay measurement

• Why quantum information behaves fundamentally differently from classical data

3. Mathematical Language: Dirac Notation as a Tool, Not Decoration

Quantum computing cannot be done without mathematical discipline.

I worked with:

• Ket vectors: ∣ψ⟩

• Bra vectors: ⟨ψ∣

• Inner products → probabilities

• Outer products → operators

A single qubit lives in a 2-dimensional complex Hilbert space.
Two qubits live in a tensor product space:

C2⊗C2=C4

This scaling is not cosmetic — it is why quantum systems explode in complexity and why entanglement exists.

4. Quantum Gates as Physical Transformations

Quantum gates are unitary operators.

They must:

• Preserve probability

• Be reversible

• Represent physically realizable transformations

Some gates I worked with extensively:

Single-Qubit Gates

• Pauli-X: Bit flip

• Pauli-Z: Phase flip

• Hadamard (H): Superposition generator

Hadamard is especially important:

H∣0⟩=2​∣0⟩+∣1⟩​

This gate is often presented casually. In reality, it is the entry point to quantum parallelism.

5. Entanglement: Where Classical Intuition Breaks Completely

Entanglement is not “strong correlation”.

An entangled state cannot be factorized into individual qubit states.

Example:

∣Φ+⟩=2​∣00⟩+∣11⟩​

There is no way to express this as:

∣ψ1​⟩⊗∣ψ2​⟩

Key realizations:

• Measurement outcomes are correlated

• Individual qubits do not have independent states

• Information is stored non-locally

This understanding later became critical when I studied ensemble quantum systems like NMR, where individual state access is impossible.

6. Bloch Sphere: Geometry of a Qubit

To develop intuition, I used the Bloch sphere representation extensively.

Any pure qubit state can be written as:
∣ψ⟩=cos(2θ​)∣0⟩+eiϕsin(2θ​)∣1⟩

This maps directly to a point on a sphere:

• Rotations = unitary operations

• Global phase = physically irrelevant

• Relative phase = physically meaningful

This geometric understanding becomes essential when quantum gates are later implemented as physical rotations (e.g., RF pulses in NMR).

7. Quantum Circuits: Thinking in Computation, Not Equations

At the circuit level, I learned to:

• Read quantum circuits gate-by-gate

• Predict final states analytically

• Identify where entanglement is created

• Understand why certain gates must precede others

A crucial insight:

Quantum circuits are constraints on time evolution, not instruction lists.

This mindset is required when moving from ideal simulators to physical implementations.

8. Limits of Pure Theory

At this stage, I understood:

• How quantum algorithms are constructed

• Why quantum speedups are possible

• How information flows in quantum systems

What I did not yet understand:

• How fragile quantum states are

• How difficult control becomes

• How theory degrades under hardware constraints

That gap led directly to my next phase of work:
liquid-state NMR quantum computing, where every abstract concept above had to survive physical reality.

What Comes Next

This foundation was not the destination — it was the filter.
In the next blog, I document how these ideas were translated into:

• Nuclear spins as qubits

• RF pulses as quantum gates

• Hamiltonians instead of circuits

• And why building even a 2-qubit system is brutally hard

👉 Next: Building a 2-Qubit Quantum Computer Using Liquid-State NMR