Day: March 28, 2026

Quantum Computing: Inversion About the Mean

Quantum Computing: Grover’s Algorithm

The Mathematics of Inversion about the Mean

1. Initial State & The “Root 8” Factor

In a 3-qubit system, there are 23 = 8 possible states. To maintain a total probability of 1, the starting amplitude for each state must be:

Amplitude (a) = 1 / √8 ≈ 0.353

When we apply the HZH Diffusion Operator, the Hadamard gates redistribute this “0.353” energy through constructive and destructive interference.

2. Step 1.2: Hadamard Ket Expansion

Each state in the superposition expands into 8 terms. The Oracle has already marked |101> by making its coefficient negative:

(1/8)[+|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>] ← H|000> (1/8)[+|000>-|001>+|010>-|011>+|100>-|101>+|110>-|111>] ← H|001> (1/8)[+|000>+|001>-|010>-|011>+|100>+|101>-|110>-|111>] ← H|010> (1/8)[+|000>-|001>-|010>+|011>+|100>-|101>-|110>+|111>] ← H|011> (1/8)[+|000>+|001>+|010>+|011>-|100>-|101>-|110>-|111>] ← H|100> -(1/8)[+|000>-|001>+|010>-|011>-|100>+|101>-|110>+|111>] ← Marked H|101> (1/8)[+|000>+|001>-|010>-|011>-|100>-|101>+|110>+|111>] ← H|110> (1/8)[+|000>-|001>-|010>+|011>-|100>+|101>+|110>-|111>] ← H|111>

3. The Full Interference Map

Source Row 000001010011100101110111
H|000>++++++++
H|001>+–+–+–+–
H|010>++––++––
H|011>+––++––+
H|100>++++––––
-H|101>*–+–++–+–
H|110>++––––++
H|111>+––+–++–
Net Signs 6+, 2-5+, 3-2+, 6-5+, 3-5+, 3-2+, 6-5+, 3-2+, 6-

4. Final Amplitude Breakdown

The Net Count from the map is scaled by the original “1/√8” amplitude to give the final result:

Output State Net Count Calculation (Net/8 × 0.353) Final Amplitude
|000> (and others) 6+, 2- (6 – 2)/8 × 0.353 +0.177
|101> (Target) 2+, 6- (2 – 6)/8 × -0.353 +0.884

The “Stopping Rule”

Grover’s algorithm is periodic. For 3 qubits, 2 rounds is the “sweet spot.” A 3rd round would actually decrease the success probability as the vector rotates past the target. Stopping at the peak amplitude of π/4 √N rounds ensures near-certainty!