Which Problems Are . . . Is Speed Up Possible? Quantum Computing Computation Quantum Computing: . . . Potential Use of . . . in Quantum Space-Time Quantum Space-Time . . . Geometry of Quantum . . . Can Lead to Using Quantum . . . a Super-Polynomial Title Page Speedup ◭◭ ◮◮ ◭ ◮ Michael Zakharevich Page 1 of 13 Department of Mathematics Stanford University Go Back ymzakharevich@yahoo.com Full Screen Vladik Kreinovich Close Department of Computer Science University of Texas at El Paso Quit vladik@utep.edu
Which Problems Are . . . 1. Which Problems Are Feasible: Brief Reminder Is Speed Up Possible? • In theoretical computer science: researchers usually dis- Quantum Computing tinguish between Quantum Computing: . . . Potential Use of . . . – problems that can be solved in polynomial time, Quantum Space-Time . . . i.e., in time ≤ P ( n ) where n is input length, and Geometry of Quantum . . . – problems that require more computation time. Using Quantum . . . • Terminology: Title Page – problems solvable in polynomial time are usually ◭◭ ◮◮ called feasible , ◭ ◮ – while others are called intractable . Page 2 of 13 • Warning: this association is not perfect. Go Back • Example: an algorithm that requires 10 100 · n steps is Full Screen – polynomial time, but Close – not practiclaly feasible. Quit
Which Problems Are . . . 2. Is Speed Up Possible? Is Speed Up Possible? • Problem: some problems are intractable – i.e., require Quantum Computing algorithms which are too slow (intractable). Quantum Computing: . . . Potential Use of . . . • Clarification: they are slow when we use the physical Quantum Space-Time . . . processes which are currently used in computers. Geometry of Quantum . . . • Natural idea: use new physical processes, processes Using Quantum . . . that have not been used in modern computers. Title Page • Question: is it possible to make computations drasti- ◭◭ ◮◮ cally faster? ◭ ◮ • Reformulation: is it possible to make intractable prob- Page 3 of 13 lems feasible? Go Back • This may happen: if a physical process provides a super- polynomial (= faster than polynomial) speed-up. Full Screen Close Quit
Which Problems Are . . . 3. Quantum Computing Is Speed Up Possible? • Question (reminder): find physical processes that would Quantum Computing make computations drastically faster. Quantum Computing: . . . Potential Use of . . . • Most active research in this direction – quantum com- Quantum Space-Time . . . puting. Geometry of Quantum . . . • Fact: quantum processes can speed up computations. Using Quantum . . . • Example: Grover’s algorithm searches in an un-sorted Title Page √ list of size N in time N . ◭◭ ◮◮ • Application: to problems that can be solved by N = 2 n ◭ ◮ time exhaustive search. Page 4 of 13 • Example: SAT – given a propositional formula F ( x ), Go Back find x = ( x 1 , . . . , x n ) s.t. F ( x ) holds. Full Screen • Exhaustive search: try all 2 n possible combinations of Close x i ∈ { false, true } . Quit
Which Problems Are . . . 4. Quantum Computing: Limitations Is Speed Up Possible? • Reminder: SAT under quantum computing. Quantum Computing Quantum Computing: . . . • Grover’s algorithm: reduces the computation time from N = 2 n to Potential Use of . . . √ √ 2 n = 2 n/ 2 . Quantum Space-Time . . . N = Geometry of Quantum . . . • Limitation: this is still a polynomial-time speed-up: Using Quantum . . . – let T c ( n ) be non-quantum time, then quantum time Title Page � is T q ( n ) = T c ( n ); ◭◭ ◮◮ – when T q ( n ) is polynomial, so is T c ( n ) = T 2 q ( n ) :-) ◭ ◮ • Fact: some known quantum algorithms are exponen- tially faster than the best known non-quantum ones. Page 5 of 13 • Example: Shor’s algorithm for factoring large integers. Go Back • Limitation: it is not clear whether a similar fast non- Full Screen quantum algorithm is possible. Close • The only proven quantum speed-ups are polynomial. Quit
Which Problems Are . . . 5. Potential Use of Curved Space-Time Is Speed Up Possible? • Parallelization – a natural source of speed-up. Quantum Computing Quantum Computing: . . . • Claim: in Euclidean space-time, parallelization only Potential Use of . . . leads to a polynomial speed-up. Quantum Space-Time . . . • Fact: the speed of all the physical processes is bounded Geometry of Quantum . . . by the speed of light c . Using Quantum . . . • Conclusion: in time T , we can only reach computa- Title Page tional units at a distance ≤ R = c · T . ◭◭ ◮◮ • The volume V ( R ) of this area (inside of the sphere of radius R = c · T ) is proportional to R 3 ∼ T 3 . ◭ ◮ • So, we can use ≤ V/ ∆ V ∼ T 3 computational elements. Page 6 of 13 Go Back • Interesting: in Lobachevsky space-time, Full Screen V ( R ) ∼ exp( R ) ≫ Polynomial( R ) . Close • Hence, we can fit more processors – and thus get a Quit drastic speed-up.
Which Problems Are . . . 6. Quantum Space-Time Models Is Speed Up Possible? • So far: we had two separate approaches: Quantum Computing Quantum Computing: . . . – use of quantum effects, and Potential Use of . . . – use of curved space-time Quantum Space-Time . . . • In physics: quantum and space-time effects are related: Geometry of Quantum . . . via quantization of space-time. Using Quantum . . . • Natural idea: combine the two approaches. Title Page ◭◭ ◮◮ • Specifics: how quantum effects affect space-time: ◭ ◮ – Heisenberg’s Uncertainty Principle: in regions of size ε , energy uncertainty is ∆ E ∼ � · ε − 1 ; Page 7 of 13 – so, when size is ε , a lot of energy enters the region; Go Back – this energy curves space-time; Full Screen – hence, on a small scale, space-time is very curved Close (“foam”-like). Quit
Which Problems Are . . . 7. Quantum Space-Time Models (cont-d) Is Speed Up Possible? • Reminder: all fluctuations in area of size ε have energy Quantum Computing Quantum Computing: . . . E ∼ � · ε − 1 . Potential Use of . . . • Energy ∆ E of a single fluctuation: Quantum Space-Time . . . Geometry of Quantum . . . – according to Einstein’s General Relativity, action � Using Quantum . . . is L = R dV dt ; Title Page – action is energy times time, hence � ◭◭ ◮◮ ∆ E ∼ R dV ≈ R · V ; ◭ ◮ – for a fluctuation of size c · ε ( c ≈ 1), volume is Page 8 of 13 V ∼ ε 3 and curvature is R ∼ ε − 2 ; Go Back – hence, ∆ E ∼ ε . Full Screen • Conclusion: the total number of such fluctuations is Close E/ ∆ E ∼ � · ε − 2 . Quit
Which Problems Are . . . 8. Geometry of Quantum Space-Time Models Is Speed Up Possible? • Reminder: in each ε -size area, there are n ∼ � · ε − 2 of Quantum Computing ( c · ε )-fluctuations. Quantum Computing: . . . Potential Use of . . . • Question: how many processors N ( ε ) of size ε can we Quantum Space-Time . . . fit in a given region? Geometry of Quantum . . . • We can have one proc. on each of these fluctuations: Using Quantum . . . N ( c · ε ) ≈ � · ε − 2 · N ( ε ) . Title Page • N (1) ≈ 1; ◭◭ ◮◮ • N ( c ) ≈ � · c − 2 ; ◭ ◮ • N ( c 2 ) ≈ � · c − 4 · N ( c ) = � 2 · c − (2+4) ; • N ( c 3 ) ≈ � · c − 6 · N ( c 2 ) = � 3 · c − (2+4+6) ; Page 9 of 13 • . . . Go Back • N ( c k ) ≈ � k · c − (2+4+ ... +2 k ) . Full Screen • Here, 2 + 4 + . . . + 2 k = 2 · (1 + 2 + . . . + k ) = Close 2 · k · ( k + 1) ≈ k 2 , so N ( c k ) ≈ � k · c − k 2 . Quit 2
Which Problems Are . . . 9. Using Quantum Space-Time (ST) Models in Com- Is Speed Up Possible? putations Quantum Computing • For the same technological level ε , we compare: Quantum Computing: . . . Potential Use of . . . – parallel computations in non-quantum ST, and Quantum Space-Time . . . – parallel computations in quantum ST. Geometry of Quantum . . . • Non-quantum ST: N n ( ε ) ∼ V 0 ε 3 ∼ ε − 3 , so ε ∼ N − 1 / 3 Using Quantum . . . . n Title Page • Quantum ST: we know that N q ( c k ) ≈ � k · c − k 2 . ◭◭ ◮◮ • To get N q ( ε ), take k s.t. c k = ε , i.e., k ∼ ln( ε ), then ◭ ◮ N q ( ε ) ∼ exp( α · ln 2 ( ε )) . Page 10 of 13 • Substituting ε ∼ N − 1 / 3 Go Back and ln( ε ) ∼ ln( N n ), we get n Full Screen N q ∼ exp( β · ln 2 ( N n )) = N β · ln( N n ) . n Close • Fact: this expression grows faster than any polynomial. Quit
Which Problems Are . . . 10. Acknowledgments Is Speed Up Possible? • This work was supported in part by NSF grants: Quantum Computing Quantum Computing: . . . – Cyber-ShARE Center of Excellence (HRD-0734825), Potential Use of . . . – Computing Alliance of Hispanic-Serving Institutions Quantum Space-Time . . . CAHSI (CNS-0540592), Geometry of Quantum . . . and by NIH Grant 1 T36 GM078000-01. Using Quantum . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 13 Go Back Full Screen Close Quit
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