BQP PSPACE NP P PostBQP Limits on Efficient Computation in the Physical World Scott Aaronson MIT

Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION Übercomputer But does the absence of these devices have any scientific importance? YES Goal of talk: Explain why the impossibility of übercomputers is a great question for 21 st -century science $3 billion

Problem: Given a graph, is it connected? Each particular graph is an instance The size of the instance, n, is the number of bits needed to specify it An algorithm is polynomial-time if it uses at most kn c steps, for some constants k,c P is the class of all problems that have polynomial-time algorithms CS Theory 101

NP: Nondeterministic Polynomial Time Does have a prime factor ending in 7?

NP-hard: If you can solve it, you can solve everything in NP NP-complete: NP-hard and in NP Is there a Hamilton cycle (tour that visits each vertex exactly once)?

P NP NP- complete NP-hard Graph connectivity Primality testing Matrix determinant Linear programming … Matrix permanent Halting problem … Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … Factoring Graph isomorphism …

Does P=NP? The (literally) $1,000,000 question Q: What if P=NP, and the algorithm takes n steps? A: Then wed just change the question! Q: Why is it so hard to prove P NP? A: Because polynomial-time algorithms are so rich

What about quantum computers? Shor 1994: BQP contains integer factoring But factoring isnt believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? Bennett et al. 1997: Quantum magic wont be enough If we throw away the problem structure, and just consider a landscape of 2 n possible solutions, even a quantum computer needs ~2 n/2 steps to find a correct solution BQP: Bounded-Error Quantum Polynomial-Time

Quantum Adiabatic Algorithm (Farhi et al. 2000) HiHi Hamiltonian with easily-prepared ground state HfHf Ground state encodes solution to NP- complete problem Problem: Eigenvalue gap can be exponentially small

Other Alleged Ways to Solve NP-complete Problems Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: A proposal for massively parallel classical computing The cognitive science approach: Think about it really hard

My Personal Favorite Dip two glass plates with pegs between them into soapy water; let the soap bubbles form a minimum Steiner tree connecting the pegs (thereby solving a known NP-complete problem)

What would the world actually be like if we could solve NP-complete problems efficiently? If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), this would have consequences of the greatest magnitude. Gödel to von Neumann, 1956 Proof of Riemann hypothesis with 10,000,000 symbols? Shortest efficient description of stock market data?

Implies, but is stronger than, P NP As falsifiable as it gets Consistent with currently-known physical theory Scientifically fruitful? Alright, what can we say about this assumption? The NP Hardness Assumption There is no physical means to solve NP complete problems in polynomial time. Rest of talk: Try to give indications that it is

1. Relativity Computing DONE

2. Topological Quantum Field Theories (TQFTs) Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

3. Nonlinear variants of the Schrödinger Equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time No solutions 1 solution to NP-complete problem Can take as an additional argument for why QM is linear

4. Anthropic Principle Foolproof way to solve NP-complete problems in polynomial time (at least in the Many-Worlds Interpretation): First guess a random solution. Then, if its wrong, kill yourself Technicality: If there are no solutions, youd seem to be out of luck! Solution: With tiny probability dont do anything. Then, if you find yourself in a universe where you didnt do anything, there probably were no solutions, since otherwise you wouldve found one

What if we combine quantum computing with the Anthropic Principle? I.e. perform a polynomial-time quantum computation, but where we can measure a qubit and assume the outcome will be |1 Leads to a new complexity class: PostBQP (Postselected BQP) A. 2005: PostBQP=PPand this yields a 1- page proof of the Beigel-Reingold-Spielman theorem, that PP is closed under intersection

Everyones 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 Doesnt take into account the computation youll have to do after getting the answer 5. Time Travel

Deutschs Model A closed timelike curve (CTC) is a computational resource that, given an efficiently computable function f:{0,1} n {0,1} n, immediately finds a fixed point of f that is, an x such that f(x)=x Admittedly, not every f has a fixed point But 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

Let P CTC be the class of problems solvable in polynomial time, if for any function f:{0,1} n {0,1} n described by a poly-size circuit, we can immediately get an x {0,1} n such that f (m) (x)=x for some m Theorem: P CTC = PSPACE

What if we perform a quantum computation around a CTC? Let BQP CTC be the class of problems solvable in quantum polynomial time, if for any superoperator E described by a quantum circuit, we can immediately get a mixed state such that E( ) = Clearly PSPACE = P CTC BQP CTC A., Watrous 2008: BQP CTC = PSPACE If closed timelike curves exist, then quantum computers are no more powerful than classical ones

Concluding Remarks (First question: What is time in quantum gravity?) Are NP-complete problems intractable in the physical universe? I conjecture that they are, but fully understanding why will bring in: Math and computer science (duh): The P vs. NP question Quantum mechanics: The NP vs. BQP question Other physics: Quantum field theory, quantum gravity, closed timelike curves… Biology, cognitive science, economics? Prediction: The NP Hardness Assumption will eventually be seen as analogous to Second Law of Thermodynamics or the impossibility of superluminal signaling Open Question: What is polynomial time in quantum gravity?

Scientific American, March 2008: