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🇶 Quantum Computing Timeline with Key Contributions and Formulas

About This Repository

This repository is designed to serve as an educational resource, showcasing the progression of quantum computing, key contributions, and foundational formulas. Contributions and discussions are encouraged to expand on these materials and foster collaboration in the field of quantum computing. Feel free to explore, contribute, and share your insights!

Mathematical Foundations for Quantum Mechanics and Quantum Computation

1- Joseph Fourier (1822)
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• Developed the mathematical framework for the Fourier Transform, which is foundational in quantum mechanics and quantum computing.

• Formula for Fourier Transform:

$\huge \color{DeepSkyBlue} \hat{f}(k) = \int_{-\infty}^{\infty} f(x) , e^{-2\pi i k x} , dx$

• Formula for Inverse Fourier Transform:

$\huge \color{DeepSkyBlue} f(x) = \int_{-\infty}^{\infty} \hat{f}(k) , e^{2\pi i k x} , dk$

Where:

• $\large \color{DeepSkyBlue} f(x)$ is the original function in the spatial domain.
• $\large \color{DeepSkyBlue} \hat{f}(k)$ is the transformed function in the frequency domain.
• $\large \color{DeepSkyBlue} x$ represents position, and $k$ represents momentum or frequency.

Relevance in Quantum Mechanics and Computing:

• Quantum Mechanics: Converts wavefunctions between position and momentum spaces.
• Quantum Computing: Basis for the Quantum Fourier Transform (QFT), essential for algorithms like Shor's factoring algorithm.

Early 20th Century – Foundations of Quantum Mechanics

2. Max Planck (1900)
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   • Founder of quantum theory by introducing the concept of energy quantization.

   Formula for quantized energy:

   $\huge \color{DeepSkyBlue} E = h \cdot f$

   Where:

   • $\large \color{DeepSkyBlue} E$ is the energy of the photon.
   • $\large \color{DeepSkyBlue} h$ is Planck's constant ($6.626 \times 10^{-34} , \text{J·s}$).
   • $\large \color{DeepSkyBlue} f$ is the radiation frequency.

3. Albert Einstein (1905)
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   • Explanation of the photoelectric effect, which introduced the concept of photons.

   Formula for the photoelectric effect:

   $\huge \color{DeepSkyBlue} E_{photon} = h \cdot f = W + K$

   Where:

   • $\large \color{DeepSkyBlue} W$ is the work function (minimum energy required to remove an electron).
   • $\large \color{DeepSkyBlue} K$ is the kinetic energy of the ejected electron.

4. Niels Bohr (1913)
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   • Bohr's atomic model with quantized energy levels.

   Formula for the energy levels of the electron in the hydrogen atom:

   $\huge \color{DeepSkyBlue} E_n = -\frac{13.6 , \text{eV}}{n^2}$

   Where:

   • $\large \color{DeepSkyBlue} E_n$ is the energy of level $n$.
   • $\large \color{DeepSkyBlue} n$ is the principal quantum number.

5. Erwin Schrödinger (1926)
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   • Schrödinger's equation, the foundation of wave mechanics.

   Time-dependent form of Schrödinger's equation:

   $\huge \color{DeepSkyBlue} i\hbar \frac{\partial}{\partial t} \psi(r, t) = \hat{H} \psi(r, t)$

   Where:

   • $\large \color{DeepSkyBlue} \psi(r, t)$ is the wave function of the system.
   • $\large \color{DeepSkyBlue} \hat{H}$ is the Hamiltonian operator.
   • $\large \color{DeepSkyBlue} \hbar$ is the reduced Planck constant.

6. Werner Heisenberg (1927)
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   • Uncertainty Principle, central to quantum physics.

   Formula for the Uncertainty Principle:

   $\huge \color{DeepSkyBlue} \Delta x \cdot \Delta p \geq \frac{\hbar}{2}$

   Where:

   • $\large \color{DeepSkyBlue} \Delta x$ is the uncertainty in position.
   • $\large \color{DeepSkyBlue} \Delta p$ is the uncertainty in momentum.

7. Paul Dirac (1928)
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   • Developed the relativistic theory of the electron and contributed to quantum mechanics.

   Dirac equation for relativistic particles:

   $\huge \color{DeepSkyBlue} (i\hbar \gamma^\mu \partial_\mu - mc)\psi = 0$

   Where:

   • $\large \color{DeepSkyBlue} \gamma^\mu$ are the Dirac matrices.
   • $\large \color{DeepSkyBlue} m$ is the mass of the particle.

Mid-20th Century – Foundation for Quantum Information

8. John von Neumann (1932)
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   • Formalized the mathematics of quantum mechanics and introduced operator theory.

   Formula for the density matrix in Hilbert space:

   $\huge \color{DeepSkyBlue} \rho = \sum_i p_i |\psi_i\rangle \langle\psi_i|$

   Where:

   • $\large \color{DeepSkyBlue} \rho$ is the density matrix.
   • $\large \color{DeepSkyBlue} p_i$ are the probabilities of the quantum states.
   • $\large \color{DeepSkyBlue} |\psi_i\rangle$ are the individual quantum states.

9. [()Claude Shannon] (1948)
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   • Although Shannon is primarily known for classical information theory, his definition of entropy plays a crucial role in both quantum computing and quantum information theory. Shannon's entropy measures the uncertainty of a random variable, and this concept extends to quantum systems, forming the foundation for quantum information theory.

   Formula for Shannon Entropy (used in quantum information theory):

   $\huge \color{DeepSkyBlue} H = -\sum p_i \log p_i$

   Where:

   • $\large \color{DeepSkyBlue} H$ is the entropy of the system (quantifies uncertainty or information).
   • $\large \color{DeepSkyBlue} p_i$ represents the probability of the $\large \color{DeepSkyBlue} i^{th}$ event or outcome.

10. Stephen Wiesner (1970)

• Introduced the concepts of quantum cryptography and quantum money, making significant contributions to the development of secure communication systems. Wiesner's work laid the groundwork for the field of quantum cryptography.

• Quantum Money Quantum Currency

  • Wiesner's idea of "quantum money" suggests using quantum states to encode currency, making it practically impossible to forge due to the properties of quantum states.
  • The exact mathematical formulation varies depending on the implementation, but the general idea relies on quantum superposition and entanglement to ensure security and authenticity.

11. Richard Feynman (1981)

• Introduced the idea of quantum computers as simulators for physical systems.

Simplified formula for simulating quantum systems:
$\huge \color{DeepSkyBlue} U(t) = e^{-iHt/\hbar}$

Where:

• $\large \color{DeepSkyBlue} U(t)$ is the time evolution operator.
• $\large \color{DeepSkyBlue} H$ is the Hamiltonian of the system.

12. Gilles Brassard (1984)

• Co-founder of the BB84 protocol, the first functional quantum cryptography system. The BB84 protocol is a [quantum key distribution] (QKD) protocol that allows two parties to securely exchange cryptographic keys over a potentially insecure channel. The security of BB84 relies on the principles of quantum mechanics, particularly quantum superposition and the no-cloning theorem.

• BB84 Protocol Formula:
  The Quantum Bit Error Rate (QBER) is used to measure the efficiency and security of the BB84 protocol by determining the rate of errors that occur during the transmission of quantum bits (qubits). It is calculated as follows:

  $\huge \color{DeepSkyBlue} QBER = \frac{\text{observable error}}{\text{total bits sent}}$

  Where:

  • Observable error refers to the number of bits where the transmitted and received values differ due to noise or eavesdropping.
  • Total bits sent refers to the total number of qubits transmitted during the key distribution process.

  This formula is essential for determining the level of interference and security in quantum communication systems. The lower the QBER, the higher the security of the quantum key distribution process.

13. David Deutsch (1985)

• Proposed the concept of the quantum Turing machine and formulated the first quantum algorithm.

Formulation of Deutsch's Algorithm:
$\huge \color{DeepSkyBlue} |q\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$

Where:

• $\large \color{DeepSkyBlue} |q\rangle$ is the superposed state of a qubit.

Contributions and References

This repository is a tribute to these great thinkers who have shaped physics and quantum computing. Their ideas and theories continue to inspire new generations of scientists and innovators.

How to Contribute

Feel free to add information or corrections. This repository encourages contributions from everyone interested in Quantum Computing! -->

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