THE QUANTUM COMPLEXITY OF TIME TRAVEL

1. THE QUANTUM COMPLEXITY OF TIME TRAVEL Scott Aaronson (MIT)

2. GOLDBACH CONJECTURE: TRUE NEXT QUESTION Things we never see… Warp drive Übercomputer Perpetuum mobile Does the absence of these devices tell us anything fundamental about physics? In the first two cases, the answer is obvious My view: It’s also obvious in the third case

3. My Research Interest:What We Can’t Do With Computers We Don’t Have • The Limits of Quantum Computers: • Could quantum computers solve NP-complete problems in polynomial time? • Could they break any cryptographic code (not just RSA)? This talk: Closed timelike curve computers Evidence strongly suggests no Most people don’t know this What about analog computers, or quantum gravity computers, or…

4. Everyone’s first idea for a time travel computer: Do an arbitrarily long computation, then send the answer back in time to before you started THIS DOES NOT WORK • Why not? • Ignores the Grandfather Paradox • Doesn’t take into account the computation you’ll have to do after getting the answer Even in this bizarre setting, still need to quantify computational resources

5. David Deutsch’s Model A closed timelike curve (CTC) is a computational resource that, given a function f, immediately finds a fixed point of f—that is, an x such that f(x)=x Problem: Not every f has a fixed point! But there’s always a distribution D such that f(D)=D Probabilistic Resolution of the Grandfather Paradox- You’re born with ½ probability- If you’re born, you back and kill your grandfather- Hence you’re born with ½ probability

6. The computational model You get to specify a polynomial time computation C, mapping n-bit strings to n-bit strings Then Nature adversarially chooses a fixed point of the computation: a distribution D such C(D)=D You get a sample from D Question: What problems can be solved in this model? Theorem: Exactly those problems solvable on a classical computer with a polynomial amount of memory, but possibly exponential time (the class PSPACE) In other words, CTC’s make space and time equivalent as computational resources

7. The Nontrivial Question What if we can perform a polynomial-time quantum computation inside the closed timelike curve? Then certainly we can at least do PSPACE, since quantum computers can always simulate classical ones But can we do more than PSPACE? Three years ago I raised this as an open problem Recently John Watrous and I managed to give a negative answer: if closed timelike curves exist, then quantum computers are no more powerful than classical ones

8. Let vec() be a “vectorization” of . We can reduce the problem to the following: given an (implicit) 22n22n matrix M, prepare a state  in BQPSPACE such that Solution: Let Then by Taylor expansion, How did we show this? Hence P projects onto the fixed points of M Furthermore, we can compute P exactly in PSPACE, using Csanky’s NC2 algorithm for matrix inversion

9. Conclusions If closed timelike curves existed, then besides all the other strange implications, we could efficiently solve PSPACE-complete computational problems For me, this is just additional evidence that closed timelike curves don’t exist And yet, even in a world with closed timelike curves, we still wouldn’t have infinite computational power Also, throwing quantum computing into the mix wouldn’t increase that power any further

10. THE QUANTUM COMPLEXITY OF TIME TRAVEL Scott Aaronson (MIT)