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## linear algebra for quantum computing

The book explains quantum computation in terms of elementary linear algebra; it assumes the reader will have some familiarity with vectors, matrices, and their basic properties, but offers a review of all the relevant material from linear algebra. We will only use NumPy in this article, and you’ll get an intro at the end to some interactive Jupyter notebooks, so you don’t need to download anything or learn terminal to … 0000076554 00000 n After explaining the development of quantum operations and computations based on linear algebra, the book presents the major quantum algorithms, from seminal algorithms by Deutsch, Jozsa, and Simon through Shor's and Grover's algorithms to recent quantum walks. 0000008381 00000 n A good example of this the GPU or Graphics Processing Unit, or Google's TPU (Tensor Processing Unit) which have enhanced machine learning drastically in recent years. It is also known as a quantum state vector and must meet the requirement that $|a|^2 + |b|^2 = 1$. If you need to refresh your knowledge of these algebra concepts, here are some tutorials that cover the basics: In the topic Understanding quantum computing, you saw that a qubit can be in a state of 1 or 0 or a superposition or both. LinjÃ¤r algebra Ã¤r sprÃ¥ket fÃ¶r kvantberÃ¤kning. trailer << /Info 1176 0 R /ID [ <0a5caf180a95a4d28cbb4470824f9f68> ] /Prev 362481 /Size 1269 /Root 1210 0 R >> startxref 0 %%EOF In this article I will introduce the basic linear algebra you will need to understand quantum computing. LinjÃ¤r algebra Ã¤r standardsprÃ¥ket fÃ¶r att beskriva kvantberÃ¤kning och kvantfysik.Linear algebra is the standard language for describing quantum computing and quantum physics. 0000005361 00000 n The Singularity Research is constantly producing quantum computing notebook which translates textbooks and research papers into interactive notebooks to learn and use quantum computing and quantum machine learning. Quantum computing takes advantage of three different properties of quantum physics to process information. Linear algebra is the standard language for describing quantum computing and quantum physics. 0000030168 00000 n There is a classical computer interface that delegates certain computational tasks to the quantum hardware. The most basic unit of computation in a quantum computer is a qubit, which can be represented as a 2-dimensional complex vector of length one. Quantum computers can have constructive (additive) interference, or they can have destructive interference, where waves are out of sync and cancel. Mathematical tools As a consequence of the Schrödinger equation all quantum gates and quantum evolutions (for a closed system) are unitary similarity transforms of the density matrices representing the states, i.e., 0000029973 00000 n In fact, we tried to avoid quantum circuits and particularities of quantum gates altogether. Using linear algebra, the state of a qubit is described as a vector and is represented by a single column matrix $\begin{bmatrix} a \\ b \end{bmatrix}$. 0000076251 00000 n of basic linear algebra operations is paramount for a wide range of problems. 0000002458 00000 n 0000004639 00000 n Reach out to. Linear algebra is the standard language for describing quantum computing and quantum physics. All you need is a web browser. Once you have downloaded Anaconda, open the Anaconda navigator. So understanding vectors will be foundational and necessary for most of what we will be doing in this book. In the examples above, the state of one qubit was described using a single column matrix $\begin{bmatrix} a \\ b \end{bmatrix}$, and applying an operation to it was described by multiplying the two matrices. Although you don’t need to know it to implement or write quantum programs, it is widely used to describe qubit states, quantum operations, and to predict what a quantum computer will do in response to a sequence of instructions. Quantum operations can also be represented by a matrix. En matris som representerar en kvantÃ¥tgÃ¤rd har ett krav â den mÃ¥ste vara en enhetlig matris.A matrix that represents a quantum operation has one requirement â it must be a unitary matrix. $$\begin{bmatrix} a \\ b \end{bmatrix} \otimes \begin{bmatrix} c \\ d \end{bmatrix} =\begin{bmatrix} a \begin{bmatrix} c \\ d \end{bmatrix} \\ b \begin{bmatrix}c \\ d \end{bmatrix} \end{bmatrix} = \begin{bmatrix} ac \\ ad \\ bc \\ bd \end{bmatrix}. Du behÃ¶ver inte kunna det fÃ¶r att implementera eller skriva kvantprogram, men det anvÃ¤nds ofta fÃ¶r att beskriva kvantbitstillstÃ¥nd och kvantÃ¥tgÃ¤rder samt fÃ¶r att fÃ¶rutsÃ¤ga vad en kvantdator kommer att gÃ¶ra som svar pÃ¥ en sekvens med instruktioner. Linear algebra, the language of matrices and vectors, is the fundamental language of quantum computing and quantum information. Kom ihÃ¥g att varje kvantbit Ã¤r ett vektorrum och dÃ¤rfÃ¶r inte kan multipliceras rakt av.Remember that each qubit is a vector space, so they can't just be multiplied. These were all designed to solve specific kinds of problems, such as enhancing machine learning tasks. If you are interested in discovering the inner workings of quantum algorithms and trying to understand where this quantum supremacy or advantage comes from, check out these lectures. In fact, any computational task a classical computer can solve, a quantum computer can also solve. 0000061923 00000 n Write a paragraph or two about how you would like to contribute. 0000001560 00000 n 0000027871 00000 n If you want you can download the notebooks and run them locally. 0000007180 00000 n For the entire linear algebra series of notebooks check out this page. Quantum Algorithms via Linear Algebra provides a great alternative introduction to the fascinating area of quantum computing. The following matrices all represent valid quantum state vectors:$$\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}, \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix}, \text{ and }\begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{i}{\sqrt{2}} \end{bmatrix}.$$. 0000077598 00000 n A matrix is unitary if the inverse of the matrix is equal to the conjugate transpose of the matrix. It succeeds in giving a mathematically precise, and complete, exposition that invokes only elementary linear algebra. It was used for rendering many of the quantum circuit diagrams in the following lectures. Om du behÃ¶ver uppdatera dina kunskaper om dessa algebrabegrepp finns hÃ¤r nÃ¥gra sjÃ¤lvstudier som beskriver grunderna: If you need to refresh your knowledge of these algebra concepts, here are some tutorials that cover the basics: Jupyter Notebook-sjÃ¤lvstudie om linjÃ¤r algebra, Jupyter notebook tutorial on linear algebra, Jupyter Notebook-sjÃ¤lvstudie om komplex aritmetik, Jupyter notebook tutorial on complex arithmetic, Vectors and matrices in quantum computing, Med linjÃ¤r algebra betecknas statusen fÃ¶r en kvantbit som en vektor och representeras av en, Using linear algebra, the state of a qubit is described as a vector and is represented by a single column. In this case, the vectors have n complex entries. 0000076975 00000 n This style of presentation strips away unnecessary notation and abstraction and brings the beautiful ideas underlying these algorithms into a sharp focus.$$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},$$, och anvÃ¤nds fÃ¶r att Ã¤ndra status fÃ¶r en kvantbit frÃ¥n 0 till 1 (eller vice versa), till exempel, and is used to flip the state of a qubit from 0 to 1 (or vice-versa), for example,$$\begin{bmatrix}0 &1\\ 1 &0\end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$,$$H = \dfrac{1}{\sqrt{2}}\begin{bmatrix}1 &1\\ 1 &-1\end{bmatrix},$$, och placerar en kvantbit i ett superpositionstillstÃ¥nd dÃ¤r den har lika stor sannolikhet att kollapsa Ã¥t ettdera hÃ¥llet sÃ¥som det visas hÃ¤r, and puts a qubit into a superposition state where it has an even probability of collapsing either way, as shown here,$$\frac{1}{\sqrt{2}}\begin{bmatrix}1 &1\\ 1 &-1\end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix}=\left(\frac{1}{\sqrt{2}}\right)^2=\frac{1}{2}., En matris som representerar en kvantÃ¥tgÃ¤rd har ett krav â den mÃ¥ste vara en, A matrix that represents a quantum operation has one requirement â it must be a, Representera tillstÃ¥nd fÃ¶r tvÃ¥ kvantbitar.

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