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Quantum Versus Classical Proofs and Advice n x{0,1} | Scott Aaronson Waterloo MIT Greg Kuperberg UC Davis Can “quantum proofs” let us verify certain theorems exponentially faster than classical proofs? Yes (we think!) But to argue for the power of quantum proofs, we’ll have to introduce a new kind of evidence: “Quantum Oracle Separations” (It’s not just that we failed to find the old kind of evidence—we can tell you exactly why we failed) Schrödinger’s Zoo QMA: Quantum Merlin-Arthur Class of problems for which a “yes” answer can be verified in quantum polynomial-time, with help from a polynomial-size quantum witness state QCMA: Quantum Classical Merlin-Arthur Same, except now the witness has to be classical Closely related to quantum proofs is quantum advice… BQP/qpoly: Class of problems solvable in quantum polynomial time, with help from a “quantum advice state” |n that depends only on the input length n BQP/poly: Same, except now advice has to be classical Surely it should at least be easy to separate these classes by oracles… PP/poly PP Dream on! QMA BQP/qpoly QCMA BQP/poly MA BQP P/poly This Talk: Quantum Oracle Separations Theorem: There exist “quantum oracles” U and V such that QMAU QCMAU and BQPV/qpoly BQPV/poly Quantum oracle: A sequence of unitary transformations {Un} that a quantum algorithm can apply in a black-box fashion Models subroutines that take quantum input and produce quantum output A new kind of evidence that two complexity classes are different Idea has already found other applications in quantum computing [A07] [MS07] The Oracle Problem We’ll Use Choose an n-qubit state | uniformly at random Let U be the unitary that maps ||0 to ||1, and ||0 to ||0 whenever |=0 Problem: Given oracle access to U, decide whether • (YES) U=U for some , or • (NO) U=I is the identity transformation Clearly this problem is in QMAU (The witness: | itself) Claim: The problem is not in QCMAU Underlying Question: How much does an nk-bit classical hint help in searching for an unknown 2n-dimensional unit vector? Intuition: Not much! 2n-dimensional unit sphere 2 n k “advice regions” To prove the intuition, we need a geometric lemma… Let be a probability measure over N-dimensional unit vectors Call p-uniform if it can be obtained by starting from the uniform measure, and then conditioning on an event that occurs with probability p Lemma: If is p-uniform, then for every fixed quantum state |, EX | 2 1 log 1 / p O N Intuition: Best you can do is let be the uniform measure over the fraction p of states that are closest to | | Lower Bound Theorem: Suppose we’re given oracle access to an n-qubit unitary U, and want to decide whether (i) U=I is the identity operator, or Quantum oracles relative to (ii) U=U for some secret which QMAU “marked QCMAUstate” and |. U/qpoly Then even BQP if we’re given an m-bitU/poly classical witness BQP in support of case (ii), we need now follow by still standard arguments n 2 m 1 queries to U to verify the witness. Proof uses BBBV hybrid argument Almost-Matching Upper Bound Theorem: We can find an n-qubit “marked state” | using an m-bit classical hint, together with 2n O 1 m queries to the quantum oracle U. (Provided m2n) Idea: A “mesh” of 2m states. Merlin tells Arthur the state closest to |, which Arthur then uses as a starting point for Grover’s algorithm But What About A Classical Oracle Separation Between QMA and QCMA? We’ve had essentially one candidate problem for this: Group Non-Membership (Babai) Problem: Given a group G, a subgroup HG, and an element xG, is xH? Here G and H are specified as black-box groups I.e. every xG is labeled by a meaningless string s(x), and we’re given an oracle that maps s(x) and s(y) to s(xy) and s(x-1) Group Non-Membership (as an oracle problem) is known to be in AM but outside MA Watrous (2000) showed how to solve GNM in QMA, using the state 1 The Group Conclusion: H h Non-Membership H hHproblem cannot, alas, lead to an as a witness oracle separation between QMA and QCMA. Our result: Arthur can verify xH using (1) a polynomial-size classical witness from Merlin, and (2) polynomially many quantum queries to the group oracle (but possibly an exponential amount of computation) Idea: “Pull the group out of the black box” Isomorphism claimed by Merlin Explicit group Black-box group Merlin gives Arthur an explicit group , together with a claimed isomorphism f:G (defined by its action on generators) Arthur checks that f is a homomorphism using the BCLR tester He checks that f is one-to-one by solving an instance of the Hidden Subgroup Problem (f is one-to-one kernel of f is trivial) Ettinger-Høyer-Knill: Hidden Subgroup Problem has polynomial quantum query complexity Once we’ve replaced G by an explicit group, no more queries to the group oracle are needed Open Problems Can we prove a classical oracle separation between QMA and QCMA? Bigger question: Whenever we prove a quantum oracle separation, can we also prove a classical one? Is Group Non-Membership in QCMA? (I.e. is the computational complexity polynomial, in addition to the query complexity?) Other quantum oracle separations? QMA vs. QMA(2)