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.^[1] Each point on the Bloch sphere corresponds to a possible qubit state, providing a geometric visualization of superposition and phase.^[2] 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.^[3]

Introduction

 

Schematic

Formula and meaning of C

Formula (mode-counting)

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

where:

• ${\displaystyle \alpha \approx 2}$  (polarization factor)^[5]
• ${\displaystyle f_{\mathrm {max} }}$  = maximum frequency cut-off^[6]
• ${\displaystyle c}$  = speed of light^[7]

Definition of C

${\displaystyle C^{2}=\log _{2}(D_{\mathrm {mode} })}$ ^[8]

• Qubit encoding: ${\displaystyle D_{\mathrm {mode} }=2\Rightarrow C^{2}=1}$ ^[9]
• Continuous-variable encoding: ${\displaystyle D_{\mathrm {mode} }=m_{\mathrm {max} }+1}$ , where ${\displaystyle m_{\mathrm {max} }}$  is the maximum photon number.^[10]

 

Parameter

Derivation from mode-counting

1. Volume of a sphere: ${\displaystyle V={\frac {4}{3}}\pi R^{3}}$ ^[11]
2. Number of spatial modes up to ${\displaystyle f_{\mathrm {max} }}$ : ${\displaystyle N_{\mathrm {spatial} }={\frac {8\pi }{3}}V{\frac {f_{\mathrm {max} }^{3}}{c^{3}}}}$ ^[12]
3. Time-slot factor: ${\displaystyle {\text{time slots}}\approx f_{\mathrm {max} }\cdot T}$ ^[13]
4. Total number of orthogonal slots: ${\displaystyle N_{\mathrm {total} }={\frac {8\pi }{3}}V{\frac {f_{\mathrm {max} }^{4}T}{c^{3}}}}$ ^[14]
5. Relation to original notation: with ${\displaystyle X={\frac {4}{3}}\pi R^{3}T}$ : ${\displaystyle N_{\mathrm {total} }=2\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}\cdot X}$ ^[15]

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	${\displaystyle 1.3\times 10^{-3}}$

^[16]

 

Scenarios

Implications and limitations

• Control and addressing: Large ${\displaystyle N}$  does not automatically imply usable capacity; hardware constraints matter.^[17]
• Coherence and dephasing: Limit the effectively usable time ${\displaystyle T}$ .^[18]
• Energy cut-offs: Necessary in CV systems to achieve finite ${\displaystyle D_{\mathrm {mode} }}$ .^[19]

 

Cutoff

Recommended formulation for publication

“The information capacity of a 4D sphere with radius ${\displaystyle R}$  and time window ${\displaystyle T}$  is given by:”^[20]

${\displaystyle N_{\mathrm {eff} }=\alpha \cdot {\frac {4}{3}}\pi R^{3}T\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}}$ ^[21]

where ${\displaystyle \alpha \approx 2}$  and ${\displaystyle C^{2}=\log _{2}(D_{\mathrm {mode} })}$  determines the number of bits per mode.^[22]

References

1. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
2. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
3. Asfaw, Abraham (2022). "Building a quantum engineering undergraduate program". IEEE Transactions on Education. 65 (3): 220–242. doi:10.1109/TE.2022.3144943.
4. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
5. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
6. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
7. Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.
8. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
9. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
10. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
11. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
12. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
13. Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.
14. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
15. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
16. Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.
17. Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77: 174509. doi:10.1103/PhysRevB.77.174509.
18. Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77: 174509. doi:10.1103/PhysRevB.77.174509.
19. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
20. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
21. Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
22. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.