Quantum Algorithms

Reference pages for every algorithm in the Qubibyte simulator. Each page has the default static Qubi, a circuit diagram, and the expected state for that circuit.

Computational kets are written |qn-1 ... q0⟩ with q0 the least significant bit (rightmost), matching the simulator. Qubi rotation and CP/P angles are multiples of π.

Entanglement

Entanglement

Bell State

Two-qubit circuit that prepares a maximally entangled Bell pair. Measuring one qubit determines the other.

2 qubits · O(1) Entanglement

GHZ State

Equal superposition of all-zero and all-one. The n-qubit form of a Bell pair.

3 qubits · O(n) Entanglement

W State

Equal superposition of every single-excitation basis state. Entanglement survives the loss of one qubit.

3 qubits · O(n)

Building blocks

Concept

State Preparation

Prepares arbitrary superpositions and probability distributions over computational basis states with exact amplitudes.

3 qubits · O(k · n) Concept

Qubit Preparation

Prepares independent single-qubit probabilities of being in the |1⟩ state using single-qubit rotations.

3 qubits · O(n) Concept

Phase Kickback

A controlled gate can move a target eigenvalue onto the control. This is the engine inside DJ, Grover, and QPE.

2 qubits · O(1) Concept

Phase Oracle

Marks one computational basis state with a −1 phase using X-conjugated multi-controlled Z.

3 qubits · O(1) Concept

SWAP Test

Estimates |⟨ψ|φ⟩|² with a controlled-SWAP and one ancilla. Does not measure the data states in Z.

3 qubits · O(1)

Communication

Communication

Superdense Coding

Alice sends two classical bits by transmitting one qubit of a shared Bell pair.

2 qubits · O(1) Communication

Quantum Teleportation

Moves an unknown qubit state from q0 to q2 using a Bell pair and two classical bits. The original is destroyed.

3 qubits · O(1) Communication

Quantum Steganography

Hides a secret bit in the phase of a Bell pair. Bob decodes with the cover key. The reduced state Eve sees does not reveal s.

2 qubits · O(1)

Core algorithms

Algorithm

Deutsch-Jozsa

One oracle query decides whether a Boolean function is constant or balanced.

3 qubits · O(1) vs O(2ⁿ) Algorithm

Bernstein-Vazirani

Recovers a hidden n-bit string s from f(x) = s·x (mod 2) in one query.

4 qubits · O(1) vs O(n) Algorithm

Grover's Algorithm

Unstructured search by amplitude amplification. Optimal iteration count is round(π/4 · √(N/M) − 1/2).

3 qubits · O(√N) Algorithm

Quantum Fourier Transform

Maps the computational basis to the Fourier (frequency) basis. Subroutine of QPE and Shor.

3 qubits · O(n²) Algorithm

Inverse QFT

Exact adjoint of the QFT. Readout step of phase estimation.

3 qubits · O(n²) Algorithm

Quantum Phase Estimation

Estimates φ in U|ψ⟩ = e^{2πiφ}|ψ⟩. Counting q0 is the LSB of the binary readout.

4 qubits · O(n²) + cost of U Algorithm

Shor's Algorithm

Period finding with a real modular multiplier. The counting register shows all r peaks, and continued fractions plus a gcd give the factors.

8 qubits · O((log N)³)

Error correction

Error Correction

Bit-Flip Error Correction

Three-qubit repetition code for X errors. Encode, optional fault, decode, Toffoli correction.

3 qubits · O(1)

Parameter playground

Change defaults (Bell type, Grover target, QPE unitary) and copy Qubi or open the simulator. Dedicated pages keep a fixed default circuit so the write-up stays exact.