Quantum Computation for Dummies UW students Dan Simon

Quantum Computation for Dummies. UW students Dan Simon Microsoft Research

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

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

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

Enter Quantum Mechanics… “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)

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)

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

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

Classical Probabilistic Coin flips H H T 1/2 H T 1/4 1/4

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

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

Quantum Coin Flips H H T H T 1/2 1/2 -1/2 =1 =0

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

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

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”

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?

A Simple Trick H H H 1/2 Tag T 1/2 Tag H 1/2 Tag T -1/2 Tag

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

A Simple Trick (2) H H H 1/2 Tag T 1/2 Tag H 1/2 Tag T -1/2 Tag

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

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

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

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

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