Quantum circuit model

Qubits as the working unit

A qubit is a two-level quantum system. Its state is a normalized complex vector in a two-dimensional space. The computational basis states, written |0⟩ and |1⟩, are the labels a later measurement can return. Any pure single-qubit state is a weighted sum of those labels, with weights called amplitudes.

Unlike a classical bit, the qubit is not secretly 0 or 1 before measurement. The amplitudes are the description of the state. Measurement forces one basis label and discards the rest of the superposition. That distinction is why a circuit is specified as a unitary evolution first and a readout second.

|ψ⟩ = α|0⟩ + β|1⟩,    |α|² + |β|² = 1

Gates are reversible linear maps

A gate is a unitary matrix: it preserves length and is invertible by its conjugate transpose. Hadamard creates an equal superposition from |0⟩. Pauli X, Y, and Z are the three axis flips and phase flips. A controlled-NOT correlates two qubits so that the target flips when the control is |1⟩. Parameterized rotations, R_z(θ) and its cousins, carry the continuous angles that variational algorithms learn.

A circuit is the composition of these gates in a declared order. Composition is matrix multiplication, but research code almost never builds the full matrix. It stores a gate list: which qubits, which angle, which moment in the schedule. That list is the artifact Kryptur papers archive, because a gate list can be replayed on a simulator or on a device.

Width, depth, and entanglement

Width is the number of qubits the circuit touches. Depth is the number of time steps after gates that do not share qubits are allowed to run together. Width dominates memory for exact state-vector work, because the state holds one amplitude per basis string. Depth dominates runtime and, on real devices, exposure to noise.

Entanglement is correlation that cannot be written as a product of single-qubit states. It appears when a superposition on one qubit is copied into a correlation with another, typically by Hadamard followed by a controlled gate. Product states stay cheap to store. Entangled states do not. That is the practical fork between exact state-vector methods and compressed tensor methods later in this series.

• A product state of n qubits needs only 2n amplitudes if each qubit is stored alone.
• A general pure state needs 2^n amplitudes. At n = 30 that is about a billion complex numbers.
• Circuit depth does not change the number of amplitudes. It changes how expensive it is to update them and how much noise a device accumulates.

How Kryptur uses the model

Variational circuits in the Kryptur programme, including QAOA layers for combinatorial problems, are angle-bearing gate lists. A classical optimizer proposes angles. The circuit is executed to estimate a cost. The angles move. The circuit model is the contract between those two sides: the classical side never edits amplitudes directly, and the quantum side never sees the optimizer’s line search.

Writing the circuit down independently of any vendor library keeps the research claim portable. The same gate list can be reasoned about on paper, sampled in a small-n check, or handed to a hardware backend. The concept page that follows this one treats what it costs to apply that gate list to a full state vector.