Draft:Bloch sphere representation in mode-counting quantum models

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C-Wave Quantum Computing

C-Wave Quantum Computing is a theoretical model for quantifying the information capacity of a physical system based on mode-counting in space, time, and frequency. The concept introduces the parameter C, which specifies the dimension of the Hilbert space per mode and links it to known results from quantum information theory.

Introduction

The C-Wave 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

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

Problems:

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

Revised formula (mode-counting)

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

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

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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.
• Braunstein, S. L., & van Loock, P. Rev. Mod. Phys. 77, 513 (2005).
• Cywiński, Ł., et al. Phys. Rev. B 77, 174509 (2008).

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