Computational Intractability As A Law of Physics PSPA

Computational Intractability As A Law of Physics PSPA CE Post BQP NP P Scott Aaronson University of Waterloo

Things we never see… GOLDBACH CONJECTURE: TRUE NEXT QUESTION S E Y Warp drive ES Y Perpetuum mobile Übercomputer Is the absence of these devices something physicists should think about? Goal of talk: Convince you to see the impossibility of übercomputers as a basic principle of physics

Computer Science 101 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 knc steps, for some constants k, c P is the class of all problems that have polynomial-time algorithms

NP: Nondeterministic Polynomial Time Does 37976595177176695379702491479374117272627593 30195046268899636749366507845369942177663592 04092298415904323398509069628960404170720961 97880513650802416494821602885927126968629464 31304735342639520488192047545612916330509384 69681196839122324054336880515678623037853371 49184281196967743805800830815442679903720933 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)?

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

Does P=NP? No. The (literally) $1, 000 question Q: What if P=NP, and the algorithm takes n 10000 steps? A: Then we’d just change the question! Q: Why is it so hard to prove P NP? A: Mostly because algorithms can be so clever!

What about quantum computers? BQP: Bounded-Error Quantum Polynomial-Time Shor 1994: BQP contains integer factoring But factoring isn’t believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? Bennett et al. 1997: “Quantum magic” won’t 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

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

Other Alleged Ways to Solve NP-complete Problems 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) Protein folding: Can also get stuck at local optima (e. g. , Mad Cow Disease) DNA computers: Just massively parallel classical computers!

What would the world actually be like if we could solve NP-complete Proof of Shortest problems efficiently? Riemann hypothesis with 10, 000 symbols? efficient description of stock market data? 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

The NP Hardness Assumption There is no physical means to solve NP complete problems in polynomial time. Rest of talk: Show Alright, what can we say aboutyields this assumption? complexity a new on. P NP linearity of • Implies, but isperspective stronger than, QM, anthropic postselection, • As falsifiable as it getstimelike curves, and closed initial conditions physical theory • Consistent with currently-known • Scientifically fruitful?

1. 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 Can take as an additional argument for why QM is linear 1 solution to NP-complete problem No solutions

2. Anthropic Principle Foolproof way to solve NP-complete problems in polynomial time (at least in the Many-Worlds Interpretation): NP Hardness Assumption First guess ayields random solution. Then, if it’s wrong, kill a nontrivial constraint yourself! on anthropic theorizing: no use of the Anthropic Principle can be valid, if its validity would give us a way to solve Technicality: NP-complete If there are no solutions, problems in you’re out of luck! polynomial time Solution: With tiny probability don’t do anything. Then, if you find yourself in a universe where you didn’t do anything, there probably were no solutions, since otherwise you would’ve 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: Post. BQP (Postselected BQP) Certainly Post. BQP contains NP—but is it even bigger than that?

Some more animals from the complexity zoo… PSPACE: Class of problems solvable with a polynomial amount of memory PP: Class of problems of the form, “out of 2 n possible solutions, are at least half of them correct? ” Adleman, De. Marrais, Huang 1998: BQP PP Proof: Feynman path integral Proof easily extends to show Post. BQP PP

PSPACE In other words, quantum postselection gives exactly the power of PP PP Post. BQP NP P A. 2004: Post. BQP = PP Surprising part: This characterization yields a half-page proof of a celebrated result of Beigel, Reingold, and Spielman, that PP is closed under intersection

3. Time Travel 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

Deutsch’s Model A closed timelike curve (CTC) is a computational resource that, given an efficiently computable function f: {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 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

Let PCTC be the class of problems solvable in polynomial time, if for any function f: {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: PCTC = PSPACE Proof: PCTC PSPACE is easy For PSPACE PCTC: Let sinit, sacc, and srej be the initial, accepting, and rejecting states of a PSPACE machine, and let (s) be the successor state of s. Then set The only fixed point is an infinite loop, with b set to its “true” value

What if we perform a quantum computation around a CTC? Let BQPCTC 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 = PCTC BQPCTC A. , Watrous 2006: BQPCTC = PSPACE If closed timelike curves exist, then quantum computers are no more powerful than classical ones

BQPCTC PSPACE: Proof Sketch Let vec( ) be a “vectorization” of . We can reduce the problem to the following: given a 22 n matrix M, prepare a state such that Solution: Let Then by Taylor expansion, Hence P projects onto the fixed points of M Furthermore, we can compute P exactly in PSPACE, using Csanky’s parallel algorithm for matrix inversion

4. Initial Conditions Normally we assume a quantum computer starts in an “all-0” state, |0… 0. But what if much better initial states were created in the Big Bang, and have been sitting around ever since? Leads to the concept of quantum advice… Useful? |

Let BQP/qpoly be the class of problems solvable in polynomial time by a quantum computer, with help from a polynomial-size “quantum advice state” | n that depends only on the input size n but can otherwise be arbitrary. How big is BQP/qpoly? It’s not obvious why it doesn’t contain every computational problem whatsoever!

Limitations of Quantum Advice A. , 2004 Result #1: BQP/qpoly Post. BQP/poly “Any problem you can solve using short quantum advice, you can also solve using short classical advice, provided you’re willing One to usecan exponentially computation time postulatemore bizarre, to extract what the advice is telling you. ” exponentially-hard-to-prepare initial states in Nature, without Result #2: There existsthe an. NP “oracle” relative to which violating Hardness NP BQP/qpoly Assumption Evidence that NP-complete problems are still hard for quantum computers in the presence of quantum advice

Concluding Remarks PHYSICS E G D I R B S I H T COMPUTATIONAL COMPLEXITY S T S I X E Y D ALREA Prediction: NP Hardness Assumption will eventually be seen as analogous to Second Law of Thermodynamics or impossibility of superluminal signaling Open Question: What is polynomial time in quantum gravity? (First question: What is time in quantum gravity? )

Links to papers, etc. : www. scottaaronson. com