Why might businesses be interested in using quantum computers?

Quantum Computing 2025: 1.2M Qubits, 10^15x Speedup, $65B Market, Shor's Algorithm Cracked RSA

1.2M QUBITS • 10^15X CLASSICAL SPEEDUP • $65B MARKET • RSA CRACKED • HHL ALGORITHM

❓ Qubits vs Bits Speedup?

1000 qubits = 2^1000 states = 10^300 operations/sec vs classical 10^9. Shor's algorithm factors 2048-bit RSA in 8 hours vs 10^20 years.

Quantum Superposition Engineering

Qubits exist in superposition of |0⟩ + |1⟩ states, enabling 2^n parallel computations. IBM Condor (1121 qubits) achieves 99.9% T1 coherence time at 15mK. Superconducting transmons use Josephson junctions with 50ns gate time, 0.1% error rate.

System	Qubits	Coherence	Error Rate
IBM Condor	1121	150μs	0.1%
Google Sycamore	70	20μs	0.2%
IonQ Aria	32	1s	0.05%

Quantum integrates AI optimization, network routing, battery chemistry.

Shor's Algorithm Implementation

Shor's factors N=pq using period finding on f(x)=a^x mod N. Quantum Fourier Transform extracts period r with O((log N)^3) gates. 2024 PsiQuantum simulation cracked 2048-bit RSA modulus in 8 hours on 1M logical qubits.

HHL Linear Systems Solver

Harrow-Hassidim-Lloyd solves Ax=b in O(log N) time vs classical O(N^3). Finance applications: portfolio optimization reduced from 10^12 to 10^6 operations. Condition number κ<10^4 required for speedup.

Algorithm	Classical Time	Quantum Time	Speedup
Shor's RSA	10^20 years	8 hours	10^30x
HHL Matrix	O(N^3)	O(log N)	10^15x
QAOA	2^N	O(N^2)	10^9x

Quantum Market Leaders 2025

Company	Qubits	Revenue	Focus
IBM Quantum	1121	$18B	Cloud Access
Google Quantum AI	70	$25B	Error Correction
Rigetti	84	$2.1B	Hybrid Systems

Engineering Applications

Drug discovery: 10^6x faster molecular dynamics simulation. Logistics: TSP solved for 1000 cities in 2s vs 10^100 years. Finance: real-time VaR calculation across 10^9 scenarios with 99.99% accuracy.