1. Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson (MIT) Papers & slides at www.scottaaronson.com

2. A quantum superposition involving n particles can require ~2 n complex numbers to specify: Quantum Computing Presents an obvious practical problem when using conventional computers to simulate quantum mechanics Feynman 1981: So then why not turn things around, and build computers that themselves exploit superpositions? 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

3. Applications of Quantum Computing: Proving it’s possible at all! Quantum simulation Breaking public-key cryptography Adiabatic optimization?? Machine learning?? NP NP-complete P Factoring BQP

4. Suppose we just want to demonstrate “quantum supremacy” (i.e., a quantum system that’s hard to simulate classically)— that’s all BosonSampling Our proposal: Identical single photons sent through network of interferometers, then measured at output modes A.-Arkhipov 2011, Bremner-Jozsa-Shepherd 2011: In that case, we can plausibly improve both the hardware requirements and the evidence for classical hardness, compared to Shor’s factoring algorithm We showed: if a fast, classical exact simulation of BosonSampling is possible, then the polynomial hierarchy collapses to the third level. Experimental demonstration has now been achieved with 6 photons (by O’Brien group in Bristol)!

5. Quantum Money 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 A.-Christiano 2012: First quantum money scheme where anyone can verify a bill, and whose security is based on a “conventional” crypto assumption 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

6. A Few Other Things I’ve Worked On The limitations of quantum computers (e.g., for finding collisions in hash functions); the possibility of quantum-secure cryptography What’s the largest possible quantum speedup? (The Forrelation and k-fold Forrelation problems) Quantum computing and the black-hole information loss problem

7. Some Future Directions The need for structure in quantum speedups Quantum copy-protected software Noisy BosonSampling (in dialogue with experimentalists) Rise and fall of complexity in thermodynamic systems