Quantum Computing using Bloch sphere

Bloch Sphere Quantum Computing Relation to the Bloch Sphere The C-Wave model can be described using the Bloch sphere, a standard representation of single-qubit states in quantum computing. Each point on the Bloch sphere corresponds to a possible qubit state, providing a geometric visualization of superposition and phase. By mapping C-Wave’s mode-counting approach onto the Bloch sphere, the model can be related to conventional qubit-based quantum computing, allowing comparison with established quantum information frameworks.

Introduction

The Bloch sphere model was developed to revise an earlier, inconsistent formulation. The main objectives are:

To make the mathematical formulation physically consistent.
To define the parameter C unambiguously.
To connect the concept to quantum information theory.
To establish testable relationships and provide numerical estimates.

Schematic

New formula and meaning of C

Original formula

$N=\left({\frac {4}{3}}\pi R^{3}T\right)^{C^{2}}$

Problems:

Units are not consistent.
C undefined.
Exponential scaling without physical justification.

Revised formula (mode-counting)

$N_{\mathrm {eff} }=\alpha \cdot {\frac {4}{3}}\pi R^{3}T\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}$

where:

$\alpha \approx 2$ (polarization factor)
$f_{\mathrm {max} }$ = maximum frequency cut-off
$c$ = speed of light

Definition of C

$C^{2}=\log _{2}(D_{\mathrm {mode} })$

Qubit encoding: $D_{\mathrm {mode} }=2\Rightarrow C^{2}=1$
Continuous-variable encoding: $D_{\mathrm {mode} }=m_{\mathrm {max} }+1$ , where $m_{\mathrm {max} }$ is the maximum photon number.

Parameter

Derivation from mode-counting

Volume of a sphere: $V={\frac {4}{3}}\pi R^{3}$
Number of spatial modes up to $f_{\mathrm {max} }$ : $N_{\mathrm {spatial} }={\frac {8\pi }{3}}V{\frac {f_{\mathrm {max} }^{3}}{c^{3}}}$
Time-slot factor: ${\text{time slots}}\approx f_{\mathrm {max} }\cdot T$
Total number of orthogonal slots: $N_{\mathrm {total} }={\frac {8\pi }{3}}V{\frac {f_{\mathrm {max} }^{4}T}{c^{3}}}$
Relation to original notation: with $X={\frac {4}{3}}\pi R^{3}T$ : $N_{\mathrm {total} }=2\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}\cdot X$

Numerical examples

Scenario	$R$	$f_{\mathrm {max} }$	$T$	$N_{\mathrm {total} }$
A — Superconducting chip	5 mm	10 GHz	300 μs	$4.88\times 10^{5}$
B — Optical scale	1 cm	200 THz	1 μs	$2.08\times 10^{21}$
C — Microscale	1 μm	1 THz	1 ns	$1.3\times 10^{-3}$

Scenarios

Implications and limitations

Control and addressing: Large $N$ does not automatically imply usable capacity; hardware constraints matter.
Coherence and dephasing: Limit the effectively usable time $T$ .
Energy cut-offs: Necessary in CV systems to achieve finite $D_{\mathrm {mode} }$ .

Cuttoff

Recommended formulation for publication

“The information capacity of a 4D sphere with radius $R$ and time window $T$ is given by:”