Hilbert Space (ℋ) | The Vector Space of Quantum Mechanics
Quick Technical Answer:
Hilbert Space (ℋ) is a complete, complex inner product vector space that contains all possible quantum states of a physical system. For an n-qubit quantum register, the Hilbert space dimension scales exponentially as dim(ℋ) = 2^n.
Formula / Unitary:
\mathcal{H}_n = \mathbb{C}^{2^n}, \quad \dim(\mathcal{H}_n) = 2^n, \quad \langle \phi | \psi \rangle = \sum_{i} \phi_i^* \psi_i
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Frequently Asked Questions
How many complex numbers are needed to describe a 50-qubit Hilbert space?
A 50-qubit register possesses 2^50 ≈ 1.125 × 10^15 complex amplitudes. At 16 bytes per double-precision complex number, storing this statevector requires 18 Petabytes of RAM.