BQP PSPACE NP P PostBQP Limits on Efficient Computation in the Physical World Scott Aaronson MIT
Mar 26, 2015
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 YES
Goal of talk: Explain why the impossibility of übercomputers is a great question for 21st-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 knc 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
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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 connectivityPrimality testingMatrix determinantLinear programming…
Matrix permanentHalting problem…
Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Factoring
Graph isomorphism…
Does P=NP?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: Because polynomial-time algorithms are so rich
What about quantum computers?
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
BQP: Bounded-Error Quantum Polynomial-Time
Quantum Adiabatic Algorithm (Farhi et al. 2000)
HiHamiltonian with easily-prepared
ground state
HfGround 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 FavoriteDip 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 ~Kn2), 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, PNP
• As falsifiable as it gets
• Consistent with currently-known physical theory
• Scientifically fruitful?
Alright, what can we say about this assumption?
The NP Hardness AssumptionThere 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 (TQFT’s)
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 solutions1 solution to NP-complete problem
Can take as an additional
argument for why QM is linear
4. Anthropic PrincipleFoolproof 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
Technicality: If there are no solutions, you’d seem to be 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
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=PP—and this yields a 1-page proof of the Beigel-Reingold-Spielman theorem, that PP is closed under intersection
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
5. Time Travel
Deutsch’s ModelA 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
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
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 2008: BQPCTC = PSPACE
If closed timelike curves exist, then quantum computers are no more powerful than classical ones
Concluding RemarksAre 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:
www.scottaaronson.com