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
H H T 1/2 1/2 H T H T 1/4 1/4 1/4 1/4 Classical Probabilistic Coin flips
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
H H T H T H T 1/2 1/2 1/2 -1/2 = 1 = 0 Quantum Coin Flips
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 alwaysreversible • fortunately, that doesn’t exclude classical computing
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)
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?
Tag Tag H T H T Tag Tag Tag Tag 1/2 1/2 1/2 -1/2 A Simple Trick H H T
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
Tag Tag H T H T Tag Tag Tag Tag 1/2 1/2 1/2 -1/2 A Simple Trick (2) H H T
Tag Tag 0 n-1 0 n-1 Tag ... Tag ... Tag ... Tag ... Tag A Slightly Less Simple Trick 0 n-1 0 ... ... ...
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
