Why Tensors 19 Century Physics Problem and How . . . From Tensors in . . . Computing With Tensors: Modern Algorithm for . . . Modern Algorithm for . . . Potential Applications Quantum Computing . . . of Physics-Motivated New Idea: Tensors to . . . Computing with . . . Mathematics Remaining Open Problem Acknowledgments to Computer Science Title Page ◭◭ ◮◮ Martine Ceberio and Vladik Kreinovich Department of Computer Science ◭ ◮ University of Texas at El Paso Page 1 of 12 El Paso, TX 79968, USA emails mceberio@cs.utep.edu Go Back vladik@utep.edu Full Screen Close Quit
Why Tensors 19 Century Physics 1. Why Tensors Problem and How . . . From Tensors in . . . • Modern computing – main problems include: Modern Algorithm for . . . – large amounts of data; Modern Algorithm for . . . – long time required to process this data. Quantum Computing . . . New Idea: Tensors to . . . • Similar situation – 19 century physics: Computing with . . . – large amounts of data; Remaining Open Problem – long time required to process this data. Acknowledgments Title Page • How the problem was solved then: by using tensors ◭◭ ◮◮ • Natural idea: let us use tensors to solve the problems with modern computing. ◭ ◮ Page 2 of 12 Go Back Full Screen Close Quit
Why Tensors 19 Century Physics 2. 19 Century Physics Problem and How . . . From Tensors in . . . • Physics starts with measuring and describing the val- Modern Algorithm for . . . ues of different physical quantities. Modern Algorithm for . . . • It goes on to equations which enable us to predict the Quantum Computing . . . values of these quantities. New Idea: Tensors to . . . • A measuring instrument usually returns a single nu- Computing with . . . merical value. Remaining Open Problem Acknowledgments • For some physical quantities (like mass m ), the single Title Page measured value is sufficient to describe the quantity. ◭◭ ◮◮ • For other quantities, we need several values. ◭ ◮ • Example: three components E x , E y , and E z describe Page 3 of 12 the electric field. Go Back • Example: to describe the tension inside a solid body, we need values σ ij . Full Screen Close Quit
Why Tensors 19 Century Physics 3. Problem and How Tensors Helped Problem and How . . . From Tensors in . . . • 19 century: a separate equation for each component of Modern Algorithm for . . . the field. Modern Algorithm for . . . • Result: equations cumbersome and difficult to solve. Quantum Computing . . . • Idea: to describe all the components of a physical field New Idea: Tensors to . . . as a single mathematical object: Computing with . . . Remaining Open Problem – a vector a i , Acknowledgments – or, more generally, a tensor a ij , a ijk , . . . Title Page • Result: simplified equations, faster computations. ◭◭ ◮◮ • Originally: mostly vectors (rank-1 tensors) were used. ◭ ◮ • 20 century: Page 4 of 12 – matrices (rank-2 tensors) in quantum physics, Go Back – higher-order tensors such as the rank-4 curvature Full Screen tensor R ijkl in relativity theory. Close Quit
Why Tensors 19 Century Physics 4. From Tensors in Physics to Computing with Ten- Problem and How . . . sors From Tensors in . . . Modern Algorithm for . . . • Reminder: Modern Algorithm for . . . – 19 century physics encountered a problem of too Quantum Computing . . . much data; New Idea: Tensors to . . . – tensors helped. Computing with . . . • Modern computing: suffers from a similar problem. Remaining Open Problem Acknowledgments • Natural idea: tensors can help. Title Page • Two examples justifying our optimism: ◭◭ ◮◮ – modern algorithms for fast multiplication of large ◭ ◮ matrices; Page 5 of 12 – quantum computing. Go Back • Comment: detailed descriptions of these examples fol- Full Screen low. Close Quit
Why Tensors 19 Century Physics 5. Modern Algorithm for Multiplying Large Matrices Problem and How . . . From Tensors in . . . • Definition: Modern Algorithm for . . . a 11 . . . a 1 n b 11 . . . b 1 n c 11 . . . c 1 n Modern Algorithm for . . . = ; . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Computing . . . a n 1 . . . a nn b n 1 . . . b nn c n 1 . . . c nn New Idea: Tensors to . . . c ij = a i 1 · b 1 j + . . . + a ik · b kj + . . . + a in · b nj . Computing with . . . • Problem: for large n , no space for both A and B in the Remaining Open Problem fast (cache) memory. Acknowledgments Title Page • Result: lots of time-consuming data transfers (“cache ◭◭ ◮◮ misses”) between different parts of the memory. ◭ ◮ • Solution: represent each matrix as a matrix of blocks: Page 6 of 12 A 11 . . . A 1 m , A = . . . . . . . . . Go Back A m 1 . . . A mm Full Screen C αβ = A α 1 · B 1 β + . . . + A αγ · B γβ + . . . + A αm · B mβ . Close Quit
Why Tensors 19 Century Physics 6. Modern Algorithm for Multiplying Large Matrices: Problem and How . . . Tensor Interpretation From Tensors in . . . Modern Algorithm for . . . • Main idea: Modern Algorithm for . . . – we start with a large matrix A of elements a ij ; Quantum Computing . . . – we represent it as a matrix consisting of block sub- New Idea: Tensors to . . . matrices A αβ . Computing with . . . • Tensor interpretation: Remaining Open Problem Acknowledgments – each element of the original matrix is now repre- Title Page sented as ◭◭ ◮◮ – an ( x, y )-th element of a block A αβ , ◭ ◮ – i.e., as an element of a rank-4 tensor ( A αβ ) xy . Page 7 of 12 • Fact: an increase in rank improves efficiency. Go Back • Analogy: a representation of a rank-1 vector as a rank- Full Screen 2 spinor works in relativistic quantum physics. Close Quit
Why Tensors 19 Century Physics 7. Quantum Computing as Computing with Tensors Problem and How . . . From Tensors in . . . • Classical bit: a system with two states 0 and 1. Modern Algorithm for . . . • Quantum bit (qubit): superposition principle – we can Modern Algorithm for . . . have states c 0 · | 0 � + c 1 · | 1 � . Quantum Computing . . . • Probabilities: Prob(0) = | c 0 | 2 and Prob(1) = | c 1 | 2 , New Idea: Tensors to . . . hence | c 0 | 2 + | c 1 | 2 = 1 . Computing with . . . Remaining Open Problem • n -(qu)bit system: a general state is Acknowledgments c 0 ... 00 ·| 0 . . . 00 � + c 0 ... 01 ·| 0 . . . 01 � + . . . + c 1 ... 11 ·| 1 . . . 11 � . Title Page • Conclusion: each state is a tensor c i 1 ...i n of rank n . ◭◭ ◮◮ • Advantage: store the entire tensor in only n (qu)bits. ◭ ◮ • Resulting efficiency of quantum computing: Page 8 of 12 – search in an unsorted array of size n in √ n time Go Back (Grover); Full Screen – factoring large integers in polynomial time (Shor). Close Quit
Why Tensors 19 Century Physics 8. New Idea: Tensors to Describe Constraints Problem and How . . . From Tensors in . . . • A general constraint between n real-valued quantities Modern Algorithm for . . . is a subset S ⊆ R n . Modern Algorithm for . . . • A natural idea: represent this subset block-by-block – Quantum Computing . . . by enumerating sub-blocks that contain elements of S . New Idea: Tensors to . . . • Fact: each block bi 1 . . . i n can be described by n indices Computing with . . . i 1 , . . . , i n . Remaining Open Problem Acknowledgments • Result: we can describe a constraint by a boolean- Title Page valued tensor t i 1 ...i n for which: ◭◭ ◮◮ • t i 1 ...i n =“true” if b i 1 ...,i n ∩ S � = ∅ ; and ◭ ◮ • t i 1 ...i n =“false” if b i 1 ...,i n ∩ S = ∅ . Page 9 of 12 • Fact: processing such constraint-related sets can also Go Back be naturally described in tensor terms. Full Screen • Fact: this speeds up computations. Close Quit
Why Tensors 19 Century Physics 9. Computing with Tensors Can Also Help Physics Problem and How . . . From Tensors in . . . • So far: we have shown that tensors can help comput- Modern Algorithm for . . . ing. Modern Algorithm for . . . • New idea: relation between tensors and computing can Quantum Computing . . . also help physics. New Idea: Tensors to . . . • Example: Kaluza-Klein-type high-dimensional space- Computing with . . . time models of modern physics. Remaining Open Problem • Einstein’s idea: use “tensors” with integer or circular Acknowledgments Title Page values. ◭◭ ◮◮ • From the mathematical viewpoint: such “tensors” are unusual. ◭ ◮ • In computer terms: integer or circular data types are Page 10 of 12 very natural. Go Back • Fact: integers and circular data are even more efficient Full Screen to process than standard real numbers. Close Quit
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