THE QUANTUM COMPLEXITY

OF TIME TRAVEL

Scott Aaronson (MIT)

Things we never see…

Warp drive Perpetuum mobile

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

Übercomputer

Does the absence of these devices tell us anything fundamental about physics?

In the first two cases, the answer is obvious

My view: It’s also obvious in the third case

My Research Interest:What We Can’t Do With Computers We Don’t Have

The Limits of Quantum Computers:

• Could quantum computers solve NP-complete problems in polynomial time?

• Could they break any cryptographic code (not just RSA)?

Evidence strongly suggests no

Most people don’t know this

What about analog computers, or quantum gravity computers, or…

This talk: Closed timelike curve

computers

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

Even in this bizarre setting, still need to quantify computational resources

David Deutsch’s Model

A closed timelike curve (CTC) is a computational resource that, given a function f, immediately finds a fixed point of f—that is, an x such that f(x)=x

Problem: 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

Question: What problems can be solved in this model?

Theorem: Exactly those problems solvable on a classical computer with a polynomial amount of memory, but possibly exponential time (the class PSPACE)

In other words, CTC’s make space and time equivalent as computational resources

You get to specify a polynomial time computation C, mapping n-bit strings to n-bit strings

Then Nature adversarially chooses a fixed point of the computation: a distribution D such C(D)=D

You get a sample from D

The computational model

The Nontrivial Question

What if we can perform a polynomial-time quantum computation inside the closed timelike curve?

Then certainly we can at least do PSPACE, since quantum computers can always simulate classical ones

But can we do more than PSPACE?

Three years ago I raised this as an open problem

Recently John Watrous and I managed to give a negative answer: if closed timelike curves exist, then quantum computers are no more powerful than classical ones

How did we show this?

Furthermore, we can compute P exactly in PSPACE, using Csanky’s NC2 algorithm for matrix inversion

111lim:

zMIzP

zSolution: Let

PMzzMIzMMPz

22

11lim

Then by Taylor expansion,

Hence P projects onto the fixed points of M

Let vec() be a “vectorization” of . We can reduce the problem to the following: given an (implicit) 22n22n matrix M, prepare a state in BQPSPACE such that

vecvec M

Conclusions

If closed timelike curves existed, then besides all the other strange implications, we could efficiently solve PSPACE-complete computational problems

For me, this is just additional evidence that closed timelike curves don’t exist

And yet, even in a world with closed timelike curves, we still wouldn’t have infinite computational power

Also, throwing quantum computing into the mix wouldn’t increase that power any further

THE QUANTUM COMPLEXITY

OF TIME TRAVEL

Scott Aaronson (MIT)