PSPACE. PostBQP. BQP. NP. P. Computational Intractability As A Law of Physics. Scott Aaronson University of Waterloo. GOLDBACH CONJECTURE: TRUE NEXT QUESTION. Things we never see…. YES. YES. Warp drive. Ü bercomputer. Perpetuum mobile.
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PostBQP
BQP
NP
P
Scott Aaronson
University of Waterloo
NEXT QUESTION
Things we never see…YES
YES
Warp drive
Übercomputer
Perpetuum mobile
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
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 polynomialtime if it uses at most knc steps, for some constants k,c
P is the class of all problems that have polynomialtime algorithms
Does
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have a prime factor ending in 7?
NPcomplete: NPhard and in NP
Is there a Hamilton cycle (tour that visits each vertex exactly once)?
Hamilton cycleSteiner treeGraph 3coloringSatisfiabilityMaximum clique…
Matrix permanentHalting problem…
FactoringGraph isomorphism…
Graph connectivityPrimality testingMatrix determinantLinear programming…
NPhard
NPcomplete
NP
P
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!
BQP: BoundedError Quantum PolynomialTime
Shor 1994:BQP contains integer factoring
But factoring isn’t believed to be NPcomplete.So the question remains: can quantum computers solve NPcomplete 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
Hi
Hf
Hamiltonian with easilyprepared ground state
Ground state encodes solution to NPcomplete problem
Problem: Eigenvalue gap can be exponentially small
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 NPcomplete problem)
Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease)
DNA computers: Just massively parallel classical computers!
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
What would the world actually be like if we could solve NPcomplete problems efficiently?The NP Hardness AssumptionThere 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?
Can take as an additional argument for why QM is linear
1. Nonlinear variants of the Schrödinger EquationAbrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NPcomplete problems in polynomial time
1 solution to NPcomplete problem
No solutions
Foolproof way to solve NPcomplete problems in polynomial time (at least in the ManyWorlds 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 NPcomplete 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!
I.e. perform a polynomialtime 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?
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: BQPPP
Proof: Feynman path integral
Proof easily extends to show PostBQPPP
PP
PostBQP
NP
BQP
P
In other words, quantum postselection gives exactly the power of PP
Surprising part:
This characterization yields a halfpage proof of a celebrated result of Beigel, Reingold, and Spielman, 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
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
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 polysize circuit, we can immediately get an x{0,1}n such that f(m)(x)=x for some m
Theorem:PCTC = PSPACE
Proof: PCTCPSPACE is easy
For PSPACEPCTC: 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=PCTCBQPCTC
A., Watrous 2006:BQPCTC = PSPACE
If closed timelike curves exist, then quantum computers are no more powerful than classical ones
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,
BQPCTCPSPACE: Proof Sketch
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

Normally we assume a quantum computer starts in an “all0” 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 polynomialsize “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 AdviceA., 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, exponentiallyhardtoprepare initial states in Nature, without violating the NP Hardness Assumption
Result #2:There exists an “oracle” relative to which NP BQP/qpoly
Evidence that NPcomplete problems are still hard for quantum computers in the presence of quantum advice
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?)
Links to papers, etc.:www.scottaaronson.com