Quantum Computing using Bloch sphere

Bloch Sphere 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 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:”