1. Quantum Computation for Dummies Dan Simon Microsoft Research UW students

2. The Strong Church-Turing Thesis Church-Turing Thesis: Any physically realizable computing machine can be modeled by a Turing Machine (TM) –A statement about the physical world Strong Church-Turing Thesis: Any physically realizable computing machine can be modeled by a polynomial-time probabilistic TM (PPTM) –A physical/economic statement of sorts

3. Consequences of the Thesis Some problems just cannot be efficiently solved by real, physical computing machines Suspected example: NP-complete problems –NP: Class of problems with polynomial-time checkable solutions –NP-complete problems: If these are efficiently solvable, then all NP problems are Many practical examples, esp. in optimization; e.g., TSP

4. Challenges to the Thesis Moore’s Law: Fageddaboudit –It’s just a matter of time…. Parallelism: Only a polynomial factor –Like speed, it eventually hits a wall Analog: Precision is the catch –Precision is (eventually) as costly as speed Chaos: Ditto

5. “You have nothing to do but mention the quantum theory, and people will take your voice for the voice of science, and believe anything.” --George Bernard Shaw, Geneva (1938) Enter Quantum Mechanics…

6. History Benioff (1981): Quantum systems can simulate TM Feynman (1982): Can they do more? It appears possible.... Deutsch (1985): Formalized Quantum TM (QTM) model, constructed an (inefficient) universal QTM (UQTM)

7. More History Deutsch & Jozsa (1992): exponential oracle separation of P (deterministic only) and QP –“promise problem” oracle Bernstein & Vazirani, Yao (1993): –efficient UQTM –Equivalence of quantum circuits and QTMs –Superpolynomial oracle separation of BPP (probabilistic P) and BQP

8. The Breakthroughs Shor (1994): integer factoring, discrete log in BQP Grover (1995): General Search in time

9. Classical Probabilistic Coin flips H H H T TT H 1/2 1/4

10. Probability vs. Amplitude Classical probability is a 1-norm –The probability of an event is just the sum of the probabilities of the paths leading to it –All the probabilities (for all events) must sum to 1 In the quantum world, it becomes a 2-norm –Each path has an amplitude –The amplitude of an event is the sum of the amplitudes of the paths leading to it –Probability = |Amplitude| 2 (for each event) –All the probabilities (for all events) must (still) sum to 1

11. Interference Amplitudes can be negative (even complex!) and still preserve positive probability Different paths can thus “cancel” (negatively interfere with) or “reinforce” (positively interfere with) each other Paths are therefore no longer independent –we must consider the entire parallel collection (superposition) of paths at any given point

12. Quantum Coin Flips H H H T TT H 1/2 -1/2 = 0= 1

13. Another Consequence of Amplitude Probabilistic processes (e.g., computation) can be represented by Markov chains (stochastic matrices--to preserve 1-norm) Quantum processes are represented by unitary matrices (M -1 = M * ) to preserve 2-norm Unitary matrices have unitary inverses –hence quantum processes are always reversible –fortunately, that doesn’t exclude classical computing

14. Stochastic vs. Unitary Stochastic: –Rows, columns, sum to 1 (1-norm) Unitary: –Squared magnitudes in rows, columns sum to 1 (2-norm) –Inverse = Conjugate Transpose (also unitary)

15. Reversible Computation A function is reversibly computable if each step can be computed from the one before it or from the one after it Any computable function can be made reversibly computable (at a constant factor cost) if the input is preserved (i.e., the output on input x is (x,f(x))) –Use reversible gates (e.g., Toffoli gates) –Preserve “work” at each step, then recompute to “clean up”

16. Exploiting Quantum Effects Idea: when searching for needle in haystack…...Just follow all paths by flipping quantum coins, and make the dead ends disappear with negative interference! The catch: you must preserve unitarity… –e.g., use Toffoli gates for all your classical computation, to make it reversible –….but what else can you do?

17. A Simple Trick HT H Tag HHTT 1/2 -1/2 Tag

18. Coherence An “event” can specify the states of multiple objects (coin + tag, multiple coins) Multiple paths interfere only if they lead to exactly the same event Objects must stay “coherent” for this to work –Superposition must be maintained –In particular, observation destroys coherence –That still permits, e.g., (reversible) computation

19. A Simple Trick (2) HT H Tag HHTT 1/2 -1/2 Tag

20. A Slightly Less Simple Trick 0 0... n-1 Tag 0... n-1... 0 n-1 Tag...

21. Shor’s Algorithm for Dummies Events with the same tag interfere negatively (i.e., cancel) unless their value “complements” the periodicity of the tags Seeing such “complementing” event values reveals the tags’ (possibly unknown) period… …Which corresponds to the order of an element in the multiplicative group mod n That’s enough information to factor n

22. Limitations The Church-Turing thesis is unaffected (QM is computable--in PSPACE, even) Some indication that NP may not be in BQP –Algorithm would have to be “non-relativizing” Known methods haven’t (yet) extended to some natural, ostensibly similar problems –Graph isomorphism –Lattice problems

23. Obstacles Getting those funny amplitudes just right –Precision on the quantum scale is required Keeping them just right –Error correcting codes needed ([Shor et al.]) Preventing decoherence –Manipulation and coherence are at cross-purposes – Computing mechanisms themselves may encourage decoherence

24. Implementation? Various proposals –particle spins, energy states to represent bits Best so far: NMR-based implementation of Grover’s search on 4-item “database” –Unlikely to scale well Unknown if any implementation can scale well –Practical limits of coherence are still a mystery