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Computational Intractability As A Law of Physics
BQP PSPACE NP P PostBQP Scott Aaronson University of Waterloo
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Things we never see… YES YES Warp drive Perpetuum mobile Übercomputer
GOLDBACH CONJECTURE: TRUE NEXT QUESTION YES YES Warp drive 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
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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
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NP: Nondeterministic Polynomial Time
Does have a prime factor ending in 7?
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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)?
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NP P NP-hard NP-complete
Matrix permanent Halting problem … Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … NP-complete NP Factoring Graph isomorphism … Graph connectivity Primality testing Matrix determinant Linear programming … P
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The (literally) $1,000,000 question
Does P=NP? No. The (literally) $1,000,000 question Q: What if P=NP, and the algorithm takes n10000 steps? A: Then we’d just change the question! Q: Why is it so hard to prove PNP? A: Mostly because algorithms can be so clever!
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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 2n possible solutions, even a quantum computer needs ~2n/2 steps to find a correct solution
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Quantum Adiabatic Algorithm (Farhi et al. 2000)
Hf Hamiltonian with easily-prepared ground state Ground state encodes solution to NP-complete problem Problem: Eigenvalue gap can be exponentially small
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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!
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What would the world actually be like if we could solve NP-complete problems efficiently?
Proof of Riemann hypothesis with 10,000,000 symbols? Shortest efficient description of stock market data? If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude. —Gödel to von Neumann, 1956
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The NP Hardness Assumption There is no physical means to solve NP complete problems in polynomial time. Rest of talk: Show how complexity yields a new perspective on linearity of QM, anthropic postselection, closed timelike curves, and initial conditions Alright, what can we say about this assumption? Implies, but is stronger than, PNP As falsifiable as it gets Consistent with currently-known physical theory Scientifically fruitful?
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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
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2. 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 it’s wrong, kill yourself! NP Hardness Assumption yields a nontrivial constraint on anthropic theorizing: no use of the Anthropic Principle can be valid, if its validity would give us a way to solve NP-complete problems in polynomial time Technicality: If there are no solutions, you’re out of luck! 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!
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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) Certainly PostBQP contains NP—but is it even bigger than that?
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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 2n possible solutions, are at least half of them correct?” Adleman, DeMarrais, Huang 1998: BQP PP Proof: Feynman path integral Proof easily extends to show PostBQP PP
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BQP PP NP P PostBQP PSPACE
2004: PostBQP = PP BQP PP NP P PostBQP PSPACE In other words, quantum postselection gives exactly the power of 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
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THIS DOES NOT WORK 3. Time Travel Why not?
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
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Deutsch’s 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 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
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Let PCTC 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: 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
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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
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BQPCTC PSPACE: Proof Sketch
Let vec() be a “vectorization” of . We can reduce the problem to the following: given a 22n22n 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
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4. Initial Conditions | Useful?
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? |
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Limitations of Quantum Advice A., 2004
Result #1: BQP/qpoly PostBQP/poly “Any problem you can solve using short quantum advice, you can also solve using short classical advice, provided you’re willing to use exponentially more computation time to extract what the advice is telling you.” One can postulate bizarre, exponentially-hard-to-prepare initial states in Nature, without violating the NP Hardness Assumption Result #2: There exists an “oracle” relative to which NP BQP/qpoly Evidence that NP-complete problems are still hard for quantum computers in the presence of quantum advice
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COMPUTATIONAL COMPLEXITY
Concluding Remarks COMPUTATIONAL COMPLEXITY THIS BRIDGE ALREADY EXISTS PHYSICS 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?)
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Links to papers, etc.: www.scottaaronson.com
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