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An Invitation to Quantum Complexity Theory The Study of What We Cant Do With Computers We Dont Have Scott Aaronson (MIT) QIP08, New Delhi BQP NP- complete SZK

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So then why cant we just ignore quantum computing, and get back to real work?

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Because the universe isnt classical My picture of reality, as an 11-year-old messing around with BASIC programming: + details Fancier version: Extended Church-Turing Thesis (Also some peoples current picture of reality)

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Shors factoring algorithm presents us with a choice 1. the Extended Church-Turing Thesis is false, 2. textbook quantum mechanics is false, or 3. theres an efficient classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true! Either In my view, this is why everyone should care about quantum computing, whether or not quantum factoring machines are ever built

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Outline of Talk What is quantum complexity theory? The black-box model Three examples of what we know Five examples of what we dont

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Quantum Complexity Theory Today, we know fast quantum algorithms to factor integers, compute discrete logarithms, solve certain Diophantine equations, simulate quantum systems … but not to solve NP-complete problems. Quantum complexity theory is the field where we step back and ask: How much of the classical theory of computation is actually overturned by quantum mechanics? And how much of it can be salvaged (even if in a strange new quantum form)? But first, what is the classical theory of computation?

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Classical Complexity Theory A polytheistic religion with many local gods: EXP PSPACE IP MIP BPP RP ZPP SL NC AC 0 TC 0 MA AM SZK But also some gods everyone prays to: P: Class of problems solvable efficiently on a deterministic classical computer NP: Class of problems for which a yes answer has a short, efficiently-checkable proof Major Goal: Disprove the heresy that the P and NP gods are equal

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In both classical and (especially) quantum complexity theory, much of what we know today can be stated in the black-box model This is a model where we count only the number of questions to some black box or oracle f: f x f(x) and ignore all other computational steps The Black-Box Model

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Algorithms state has the form A query maps each basis state |x,w to |x,w f(x) (f(x) gets reversibly written to the workspace) Between two query steps, can apply an arbitrary unitary operation that doesnt depend on f Query complexity = minimum number of steps needed to achieve for all f Quantum Black-Box Algorithms

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Given a function f:[N] {0,1}, suppose we want to know whether theres an x such that f(x)=1. How many queries to f are needed? Example Of Something We Can Prove In The Black-Box Model Classically, its obvious the answer is ~N On the other hand, Grover gave a quantum algorithm that needs only ~ N queries Bennett, Bernstein, Brassard, and Vazirani proved that no quantum algorithm can do better

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Given a periodic function f:[N] [N], how many queries to f are needed to determine its period? Example #2 Classically, one can show ~N queries are needed by any deterministic algorithm, and ~ N by any randomized algorithm On the other hand, Shor (building on Simon) gave a quantum algorithm that needs only O(log N) queries. Indeed, this is the core of his factoring algorithm So quantum query complexity can be exponentially smaller than classical! Beals, Buhrman, Cleve, Mosca, de Wolf: But only if theres some promise on f, like that its periodic

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Given a function f:[N] [N], how many queries to f are needed to determine whether f is one-to-one or two-to-one? (Promised that its one or the other) Example #3 Classically, ~ N (by the Birthday Paradox) A., Shi: This is the best possible Quantum algorithms cant always exploit structure to get exponential speedups! By combining the Birthday Paradox with Grovers algorithm, Brassard, Høyer, and Tapp gave a quantum algorithm that needs only ~N 1/3 queries

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More formally, does BPP=BQP? BPP (Bounded-Error Probabilistic Polynomial- Time): Class of problems solvable efficiently with use of randomness Note: Its generally believed that BPP=P BQP (Bounded-Error Quantum Polynomial-Time): Class of problems solvable efficiently by a quantum computer Open Problem #1: Are quantum computers more powerful than classical computers? (In the real, non-black-box world?)

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Most of us believe (hope?) that BPP BQP among other things, because factoring is in BQP! On the other hand, Bernstein and Vazirani showed that BPP BQP PSPACE Therefore, you cant prove BPP BQP without also proving P PSPACE. And that would be almost as spectacular as proving P NP!

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Contrary to almost every popular article ever written on the subject, most of us think the answer is no For generic combinatorial optimization problems, the situation seems similar to that of black-box modelwhere you only get the quadratic speedup of Grovers algorithm, not an exponential speedup Open Problem #2: Can Quantum Computers Solve NP-complete Problems In Polynomial Time? As for proving this … dude, we cant even prove classical computers cant solve NP-complete problems in polynomial time! More formally, is NP BQP? (Conditional result?)

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Most of us dont believe NP BQP … but what about BQP NP? If a quantum computer solves a problem, is there always a short proof of the solution that would convince a skeptic? (As in the case of factoring?) My own opinion: Not enough evidence even to conjecture either way Open Problem #3: Can Quantum Computers Be Simulated In NP?

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Is BQP PH (where PH is the Polynomial-Time Hierarchy, a generalization of NP to any constant number of quantifiers)? Gottesmans Question: If a quantum computer solves a problem, can it itself interactively prove the answer to a skeptic (who doesnt even believe quantum mechanics)? The latter question carries a $25 prize! See Related Problems

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That is, does QMA=QCMA? QMA (Quantum Merlin-Arthur): A quantum generalization of NP. Class of problems for which a yes answer can be proved by giving a polynomial-size quantum state |, which is then checked by a BQP algorithm. QCMA: A hybrid between QMA and NP. The proof is classical, but the algorithm verifying it can be quantum Known: QMA-complete problems [Kitaev et al.], quantum oracle separation between QMA and QCMA [A.-Kuperberg] Open Problem #4: Are Quantum Proofs More Powerful Than Classical Proofs?

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Does QMA(2)=QMA? QMA(2): Same as QMA, except now the verifier is given two quantum proofs | and |, which are guaranteed to be unentangled with each other Liu, Christandl, and Verstraete gave a problem called pure state N-representability, which is in QMA(2) but not known to be in QMA Recently A., Beigi, Fefferman, and Shor showed that, if a 3SAT instance of size n is satisfiable, this can be proved using two unentangled proofs of n polylog n qubits each Open Problem #5: Are Two Quantum Proofs More Powerful Than One?

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