|
8 | 8 | } |
9 | 9 | }, |
10 | 10 | "source": [ |
11 | | - "[<img src=\"images/quantum_algorithms_tutorials.png\" alt=\"drawing\" width=\"100\" align=\"left\"/>](https://github.com/DavitKhach/quantum-algorithms-tutorials)\n", |
| 11 | + "[<img src=\"images/quantum_algorithms_tutorials.png\" alt=\"drawing\" width=\"100\" align=\"left\"/>][5]\n", |
12 | 12 | "\n", |
13 | 13 | "<h2 align=\"center\">\n", |
14 | 14 | "Quantum parallelism and Deutsch–Jozsa algorithm\n", |
15 | 15 | "</h2>\n", |
16 | 16 | "\n", |
17 | | - "[Open with the nbviewer.](https://nbviewer.jupyter.org/github/DavitKhach/quantum-algorithms-tutorials/blob/master/quantum_parallelism_Deutsch_Jozsa.ipynb)\n", |
| 17 | + "**[[Homepage][5]]**\\\n", |
| 18 | + "**[[Open with the nbviewer.][6]]**\n", |
18 | 19 | "\n", |
19 | 20 | "<h3 align=\"left\">\n", |
20 | | - "Quantum parallelism\n", |
| 21 | + "1. Quantum parallelism\n", |
21 | 22 | "</h3>\n", |
22 | 23 | "\n", |
23 | 24 | "In this tutorial, we are going to introduce 3 separate problems and we will mainly follow the [[1]] book's approach of describing them. After giving descriptions we will implement them with Qiskit. So, let's start with quantum parallelism, which is one of the powerful tools/properties that is used in the quantum computers. Quantum parallelism enables us to simultaneously compute the outputs of a function for given different inputs. Let's say we have two registers one for input the other one for output. The initial state is given by $\\left| x \\right\\rangle \\left| 0 \\right\\rangle$ wavefunction, where $\\left| x \\right\\rangle$ is the register that takes $x$ input and $\\left| 0 \\right\\rangle$ is the output register where the value of the $f(x)$ function will be stored. After applying the circuit that corresponds to the given function the wavefunction becomes $\\left| x \\right\\rangle \\left| f(x) \\right\\rangle$. Here is the circuit:\n", |
|
103 | 104 | " [1]: https://www.cambridge.org/am/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-computation-and-quantum-information-10th-anniversary-edition?format=HB\n", |
104 | 105 | " [2]: https://quantumcomputing.stackexchange.com/questions/9566/implementing-four-oracles-for-the-deutsch-algorithm-preferably-on-ibm-q-experie\n", |
105 | 106 | " [3]: https://www.youtube.com/watch?v=awpnsGl08bc&list=LLEPoMdv-DZViRaaObznY_-A&index=113&t=0s\n", |
106 | | - " [4]: https://courses.edx.org/courses/course-v1:MITx+8.370.2x+1T2018/course/" |
| 107 | + " [4]: https://courses.edx.org/courses/course-v1:MITx+8.370.2x+1T2018/course/\n", |
| 108 | + " [5]: https://github.com/DavitKhach/quantum-algorithms-tutorials\n", |
| 109 | + " [6]: https://nbviewer.jupyter.org/github/DavitKhach/quantum-algorithms-tutorials/blob/master/quantum_parallelism_Deutsch_Jozsa.ipynb" |
107 | 110 | ] |
108 | 111 | }, |
109 | 112 | { |
|
212 | 215 | "metadata": {}, |
213 | 216 | "source": [ |
214 | 217 | "<h3 align=\"left\">\n", |
215 | | - "Deutsch's algorithm\n", |
| 218 | + "2. Deutsch's algorithm\n", |
216 | 219 | "</h3>\n", |
217 | 220 | "\n", |
218 | 221 | "In the Deutsch algorithm, we are solving a problem with two qubits. The problem can be described this way [[1]]: Alice and Bob live in different cities and they are playing a \"funny\" game. Alice prepares her qubits and sends them to Bob. Bob applies a function on Alice's qubits and sends them back to Alice. Bob is allowed to choose only one of the 4 possible functions, that are either constant ($f(0) = f(1)$) or balanced ($f(0) \\ne f(1)$), and input and output of the function is one bit (0ドル$ or 1ドル$). Alice should find out from the measurment result was the chosen function balanced or constant. In the course of this algorithm, we will use the concept of quantum parallelism. Here is the circuit for the algorithm (a modified version of the Fig. 1.19 from [[1]]):\n", |
|
541 | 544 | "metadata": {}, |
542 | 545 | "source": [ |
543 | 546 | "<h3 align=\"left\">\n", |
544 | | - "The Deutsch-Jozsa algorithm\n", |
| 547 | + "3. The Deutsch-Jozsa algorithm\n", |
545 | 548 | "</h3>\n", |
546 | 549 | "\n", |
547 | 550 | "Deutsch-Jozsa algorithm is the $n$ qubit version of the Deutsch algorithm. The functions now take $n$ bit strings (e.g. \"0101011\") as an input and return one bit output (0ドル$ or 1ドル$). Bob still is allowed only to use either balanced or constant functions and Alice should find out the property of that function (is it constant or balanced). Bob has two options for constant functions (for all imput bit strings $x$ the $f(x) = 0$ or $f(x) = 1$) and $\\begin{pmatrix} 2^n \\\\ 2^{n - 1} \\end{pmatrix}$ options to choose a balanced function. The main steps of the algorithm are presented in the following circuit (a modified version of the Fig.1.20 from the [[1]]):\n", |
|
796 | 799 | "cell_type": "markdown", |
797 | 800 | "metadata": {}, |
798 | 801 | "source": [ |
| 802 | + "**[[Homepage][5]]**\n", |
| 803 | + "\n", |
| 804 | + "<h3 align=\"left\">\n", |
| 805 | + "References\n", |
| 806 | + "</h3>\n", |
| 807 | + "\n", |
799 | 808 | "[[1]] [M.A. Nielsen, I.L. Chuang, Cambridge University Press New York, \"Quantum Computation and Quantum Information: 10th Anniversary Edition\n", |
800 | 809 | "10th\" (2011)][1]\n", |
801 | 810 | "\n", |
|
805 | 814 | "\n", |
806 | 815 | "[[4]] [edx course: Quantum Information Science I, Part 2][4]\n", |
807 | 816 | "\n", |
| 817 | + "\n", |
808 | 818 | " [1]: https://www.cambridge.org/am/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-computation-and-quantum-information-10th-anniversary-edition?format=HB\n", |
809 | 819 | " [2]: https://quantumcomputing.stackexchange.com/questions/9566/implementing-four-oracles-for-the-deutsch-algorithm-preferably-on-ibm-q-experie\n", |
810 | 820 | " [3]: https://www.youtube.com/watch?v=awpnsGl08bc&list=LLEPoMdv-DZViRaaObznY_-A&index=113&t=0s\n", |
811 | | - " [4]: https://courses.edx.org/courses/course-v1:MITx+8.370.2x+1T2018/course/" |
| 821 | + " [4]: https://courses.edx.org/courses/course-v1:MITx+8.370.2x+1T2018/course/\n", |
| 822 | + " [5]: https://github.com/DavitKhach/quantum-algorithms-tutorials\n", |
| 823 | + " [6]: https://nbviewer.jupyter.org/github/DavitKhach/quantum-algorithms-tutorials/blob/master/quantum_parallelism_Deutsch_Jozsa.ipynb" |
812 | 824 | ] |
813 | 825 | } |
814 | 826 | ], |
|
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