Scott Aaronson (MIT) Papers and slides at scottaaronson

1. Exploring the Limits of the Efficiently ComputableResearch Directions I Like In Complexity and Physics Scott Aaronson (MIT) Papers and slides at www.scottaaronson.com

2. Quantum Mechanics in One Slide Probability Theory: Quantum Mechanics: Linear transformations that conserve 1-norm of probability vectors:Stochastic matrices Linear transformations that conserve 2-norm of amplitude vectors:Unitary matrices

3. Quantum Computing A general entangled state of n qubits requires ~2n amplitudes to specify: Presents an obvious practical problem when using conventional computers to simulatequantum mechanics Feynman 1981: So then why not turn things around, and build computers that themselvesexploit superposition? Could such a machine get any advantage over a classical computer with a random number generator? If so, it would have to come from interference between amplitudes

4. Interesting BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993 Shor 1994: Factoring integers is in BQP NP-complete NP Factoring BQP P

5. This Talk: Three Recent Directions 1. Meeting Experimentalists Halfway Using complexity theory to find quantum advantage in systems of current experimental interest (e.g. linear-optical networks), which fall short of universal quantum computers 2. Publicly-Verifiable Quantum Money First scheme based on a “standard” crypto assumption 3. Rise and Fall of Complexity in Thermodynamic Systems Resource-bounded sophistication and coffee cups

6. 1. Meeting Experimentalists Halfway

7. BosonSampling (A.-Arkhipov 2011) Classical counterpart: Galton’s Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip A rudimentary type of quantum computing, involving only non-interacting photons

8. n identical photons enter, one per input mode Assume for simplicity they all leave in different modes—there are possibilities The beamsplitter network defines a column-orthonormal matrix ACmn, such that In general, we consider a network of beamsplitters, with n input “modes” (locations) and m>>n output modes where nnsubmatrix of A corresponding to S Amazing connection between permanents and physics, which even leads to a simpler proof of Valiant’s famous result that the permanent is #P-complete [A. 2011]

9. So, Can We Use Quantum Optics to Solve a #P-Complete Problem? That sounds way too good to be true… Explanation: If X is sub-unitary, then |Per(X)|2 will usually be exponentially small. So to get a reasonable estimate of |Per(X)|2 for a given X, we’d generally need to repeat the optical experiment exponentially many times

10. Better idea: Given ACmn as input, let BosonSampling be the problem of merely sampling from the same distribution DA that the beamsplitter network samples from—the one defined by Pr[S]=|Per(AS)|2 Theorem (A.-Arkhipov 2011): Suppose BosonSampling is solvable in classical polynomial time. Then P#P=BPPNP Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence that a weaker system can do something classically hard Better Theorem: Suppose we can sample DA even approximately in classical polynomial time. Then in BPPNP, it’s possible to estimate Per(X), with high probability over a Gaussian random matrix We conjecture that the above problem is already #P-complete. If it is, then a fast classical algorithm for approximateBosonSampling would already have the consequence thatP#P=BPPNP

11. Prove #P-completeness for natural average-case approximation problems (like permanents of Gaussians)—thereby yielding hardness for approximate BosonSampling As a first step, understand the distribution of Per(X), X Gaussian Early experimental implementations have been done (Rome, Brisbane, Vienna, Oxford)! But so far with just 3-4 photons. For scaling, will be crucial to understand the complexity of BosonSampling when a constant fraction of photons are lost Can the BosonSampling model solve hard “conventional” problems? How do we verify that a BosonSampling device is working correctly? [A.-Arkhipov 2014, A.-Nguyen 2014] BosonSampling with thermal states: fast classical algorithm to approximate Per(X) when X is positive semidefinite? Challenges

12. Can a quantum computer solve problems for which a classical computer can’t even efficiently verify the answers? Or better yet: that are still classically hard even if P=NP? BQP vs. the Polynomial Hierarchy Boils down to: are there problems in BQP but not in PH? BosonSampling: A candidate for such a problem. If it’s solvable anywhere in BPPPH, then PH collapses. A. 2009: Unconditionally, there’s a black-box sampling problem (Fourier Sampling) solvable in BQP but not in BPPPH Given a Boolean function output z{0,1}n with probability

13. “Forrelation”: Given two Boolean functions f,g:{0,1}n{-1,1}, estimate how correlated g is with the Fourier transform of f: What’s the largest possible quantum speedup? A.-Ambainis 2014: This problem is solvable using only 1 quantum query, but requires at least ~2n/2/n queries classically Furthermore, this separation is essentially the largest possible! Any N-bit problem that’s solvable with k quantum queries, is also solvable with ~N1-1/2k classical queries Conjecture (A. 2009): Forrelation  Polynomial Hierarchy

14. 2. Publicly-Verifiable Quantum Money

15. Idea: Quantum states that can be created by a bank, traded as currency, and verified as legitimate, but can’t be cloned by counterfeiters, because of quantum mechanics’ No-Cloning Theorem Quantum Money Wiesner ca. 1970: First quantum money scheme, but only the bank could verify the bills. If anyone can verify a bill, then computational assumptions clearly needed, in addition to QM A.-Christiano 2012: First quantum money scheme where anyone can verify a bill, and whose security is based on a “conventional” crypto assumption

16. Our Hidden Subspace Scheme Quantum money state: Mint can easily choose a random A and prepare |A Corresponding “serial number” s: Somehow describes how to check membership in A and in A (the dual subspace of A), yet doesn’t reveal A or A Our proposal: Random low-degree polynomials p1,…,pm and q1,…,qm that vanish on A and A respectively

17. Procedure to Verify Money State(assuming ability to decide membership in A and A) • Project onto A elements (reject if this fails) • Hadamard all n qubits to map |A to |A • Project onto A elements (reject if this fails) • Hadamard all n qubits to return state to |A A A Theorem: The above just implements a projection onto |A—i.e., it accepts | with probability ||A|2

18. Security Theorem: There’s no efficient counterfeiting procedure, assuming there’s no an efficient quantum algorithm to learn a basis for A with 2-O(n) probability, given p1,…,pm and q1,…,qm. [Recently: Attack on noiseless version of scheme] Theorem: If the A and A membership tests are black boxes, then any counterfeiting procedure requires Ω(2n/2) queries to them.

19. 3. Rise and Fall of Complexity in Thermodynamic Systems

20. How to Measure Interesting Structure? Can define structure and in many other ways One simpleminded measure: the Kolmogorov complexity of a coarse-grained description of our cellular automaton or other system Sean Carroll’s example:

21. The Coffee Automaton A., Carroll, Mohan, Ouellette, Werness 2015: A probabilistic nn reversible system that starts half “coffee” and half “cream.” At each time step, we randomly “shear” half the coffee cup horizontally or vertically (assuming a toroidal cup) We prove that the apparent complexity of this image has a rising-falling pattern, with a maximum of at least ~n1/6

22. Other Work Quantum lower bound for the collision problem [A. 2002] Quantum (+classical!) lower bound for local search [A. 2004] First direct product theorem for quantum search [A. 2004] PostBQP = PP[A. 2004] BQP/qpolyQMA/poly, learnability of quantum states [A.-Drucker 2010, A. 2004, A. 2006] Algebrization [A.-Wigderson 2008]

23. Some Future Directions Quantum copy-protected software Complexity theory of quantum states and unitary transformations Classification of quantum gate sets Noisy BosonSampling The power of quantum proofs See also my talk at Perimeter on Wednesday at 10:30, for complexity and quantum gravity!