1. Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) BQP PSPACE Quantum Computing With Closed Timelike Curves

2. Motivation Ordinary quantum computing too pedestrian In the past, CTCs have mostly been studied from the perspective of GR Studying them from a computer science perspective leads us to ask new questionslike, how hard would Nature have to work to ensure causal consistency? Hopefully, leads to some new insights about causality, linearity of quantum mechanics, space vs. time, ontic vs. epistemic…

3. Bestiary of Complexity Classes PSPACE EXP BQP P The difference between space and time in computer science: you can reuse space, but not time

4. Everyones first idea for a CTC 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 Doesnt take into account the computation youll have to do after getting the answer

5. Deutschs Model A closed timelike curve (CTC) is simply a resource that, given an operation f:{0,1} n {0,1} n acting in some region of spacetime, finds a fixed point of fthat is, an x s.t. f(x)=x Of course, not every f has a fixed pointthats the Grandfather Paradox! But since every Markov chain has a stationary distribution, theres always a distribution D such that f(D)=D Probabilistic Resolution of the Grandfather Paradox - Youre born with ½ probability - If youre born, you back and kill your grandfather - Hence youre born with ½ probability

6. CTC Computation R CTC R CR C 000 Answer Causality- Respecting Register Closed Timelike Curve Register Polynomial Size Circuit

7. You (the user) pick a circuit C on two registers, R CR and R CTC, as well as an input x to R CR Let C x be the induced operation on R CTC only Nature is forced to find a distribution D CTC over inputs to R CTC such that C x (D CTC )=D CTC (If theres more than one such D CTC, Nature can choose one adversarially) Then Nature samples a string y from D CTC Output of the computation: C(x,y) P CTC is the class of decision problems solvable in this model

8. How to Use CTCs to Solve Hard Problems: Basic Idea Given a function f:[N] {0,1} (where N is huge), suppose we want instantly to find an input x such that f(x)=1 I claim that we can do so using the following function g:[N] [N], acting on a CTC register: What are the fixed points of this evolution?

9. Theorem: P CTC = PSPACE (m) Proof: For P CTC PSPACE, just need to find some y such that C x (m) (y)=y for some m. Pick any y, then apply C x 2 n times. For PSPACE P CTC : Have C x input and output an ordered pair m i,b, where m i is a state of the Turing machine were simulating and b is an answer bit, like so: The only fixed-point distribution is a uniform distribution over all states of the Turing machine, with the answer bit set to its true value m T-1,0 m T,0 m 1,0 m 2,0 m T-1,1 m T,1 m 1,1 m 2,1

10. What About The Quantum Case? You (the user) pick a quantum circuit C on two registers, R CR and R CTC, as well as a (classical) input |x to R CR Let C x be the induced superoperator acting on R CTC only Nature is forced to find a mixed state CTC such that C x ( CTC )= CTC (If theres more than one such, Nature can choose one adversarially) Output of the computation: C(x, CTC )

11. Let BQP CTC be the class of problems solvable in this model Certainly PSPACE = P CTC BQP CTC EXP Main Result: BQP CTC = PSPACE If CTCs are possible, then quantum computers are no more powerful than classical ones

12. BQP CTC PSPACE: Proof Sketch Let vec( ) be the vectorization of : i.e., a length-2 2n vector of s entries. We can reduce the problem to the following: given an (implicit) 2 2n 2 2n matrix M, prepare a state CTC in BQPSPACE such that

13. Idea: Let Then Hence M(Pv)=Pv, so P projects onto the fixed points of M Furthermore: We can compute P exactly in PSPACE, by using small-space algorithms for matrix inversion discovered in the 1980s (e.g. Csankys algorithm) Its easy to check that Pv is the vectorization of some density matrix So then take (say) Pvec(I) as the fixed-point CTC

14. Coping With Error Problem: The set of fixed points could be sensitive to arbitrarily small changes to the superoperator E.g., consider the two stochastic matrices The first has (1,0) as its unique fixed point; the second has (0,1) However, the particular CTC algorithm used to solve PSPACE problems doesnt share this property! Indeed, one can use a CTC to solve PSPACE problems fault-tolerantly (building on Bacon 2003)

15. Discussion Three ways of interpreting our result: (1)CTCs exist, so now we know exactly what can be computed in the physical world (PSPACE)! (2)CTCs dont exist, and this sort of result helps pinpoint whats so ridiculous about them (3)CTCs dont exist, and we already knew they were ridiculousbut at least we can find fixed points of superoperators in PSPACE! Our result formally justifies the following intuition: By making time reusable, CTCs would make time equivalent to space as a computational resource.

16. And Now for the Mudfight! Bennett, Leung, Smith, Smolin 2009: Deutschs (and our) model of CTCs is crap Why? Because if you feed to a CTC computer, the outcome might be different than if you fed x and y separately, then averaged the results This is a simple consequence of the fact that CTCs induce nonlinearities in quantum mechanics Bennett et al.s proposed fix: Force CTC to depend only on the whole distribution over inputs,

17. Our Response What Bennett et al. do basically just amounts to defining CTCs out of existence! That CTCs would strain the normal axioms of physics (like linearity of mixed-state evolution) is obvious … what else did you expect? At least BQP CTC is a good complexity class, better than their proposed replacement BQPP CTC Since under their prescription, we might as well treat CTC as a quantum advice resource fixed for all time, independent of anything else in the universe (In any case, our main resultan upper bound on BQP CTC and BQPP CTC is unaffected)

18. Seth Lloyds Response Bennett et al.s fix precludes the possibility that a CTC could form in some branches of the multiverse but not others But quantum gravity theories ought to allow superpositions over different causal structuresso if CTCs can form at all, then why not allow evolutions like

19. Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) BQP PSPACE Quantum Computing With Closed Timelike Curves