Draft:Bloch sphere representation in mode-counting quantum models

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

 

Schematic

Formula and meaning of C

Formula (mode-counting)

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

where:

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

Definition of C

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

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

 

Parameter

Derivation from mode-counting

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

Numerical examples

<thead> </thead> <tbody> </tbody>

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

 

Scenarios

Implications and limitations

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

 

Cuttoff

Recommended formulation for publication

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

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

where ${\displaystyle \alpha \approx 2}$  and ${\displaystyle C^{2}=\log _{2}(D_{\mathrm {mode} })}$  determines the number of bits per mode.

References

• Nielsen, M. A., & Chuang, I. L. Quantum Computation and Quantum Information EAN 9781107002173.
• Braunstein, S. L., & van Loock, P. Rev. Mod. Phys. 77, 513 (2005) DOI: https://doi.org/10.1103/RevModPhys.77.513.
• Cywiński, Ł., et al. Phys. Rev. B 77, 174509 (2008).DOI: https://doi.org/10.1103/PhysRevB.77.174509
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