Lesson_9

 Will be working on Sharma lesson 9 today: complex gates. 

The foundational 1995 paper by Barenco et al, 1995:

https://arxiv.org/abs/quant-ph/9503016

A summary of this paper, from CHATGPT_o:

### Summary of the Article

#### [Elementary gates for quantum computation](https://arxiv.org/abs/quant-ph/9503016)

**Authors:** A. Barenco (Oxford), C. H. Bennett (IBM), R. Cleve (Calgary), D. P. DiVincenzo (IBM), N. Margolus (MIT), P. Shor (AT&T), T. Sleator (NYU), J. Smolin (UCLA), H. Weinfurter (Innsbruck)

#### Abstract:

- The paper demonstrates that a set of quantum gates, specifically all one-bit quantum gates (U(2)) and the two-bit exclusive-or (XOR) gate, is universal. This universality means that any unitary operation on n bits (U($2^n$)) can be composed of these gates.

  

- The study also examines the quantity of these elementary gates required to implement other complex gates, such as generalized Deutsch-Toffoli gates. These gates conduct a specific U(2) transformation conditionally based on the logical AND of other input bits.

- The gates studied are pivotal in the design of quantum computational networks, providing foundational insights into quantum computation.

- The authors derive both upper and lower bounds for the number of elementary gates necessary for constructing various two- and three-bit quantum gates, including the asymptotic number for n-bit Deutsch-Toffoli gates. Additionally, the paper makes observations about the requisites for arbitrary n-bit unitary operations.

#### Key Points:

- **Universal Gate Set**: Demonstrates the universality of one-bit quantum gates and the two-bit XOR gate.

- **Implementation**: Studies on implementing complex gates using elementary gates.

- **Bounds and Observations**: Provides upper and lower bounds on the gate count required for various operations, with specific observations for n-bit unitary operations.

- **Quantum Computational Networks**: Importance in constructing networks for quantum computation.

#### Additional Information:

- **Subjects**: Quantum Physics (quant-ph), Condensed Matter (cond-mat), High Energy Physics - Theory (hep-th).

- **Journal Reference**: Phys. Rev. A52 (1995) 3457.

- **DOI**: [10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)

- **Submission Date**: March 23, 1995. 

For more detailed information, you can access the full paper [here](https://arxiv.org/abs/quant-ph/9503016).