Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) BQP PSPACE Quantum Computing With Closed Timelike Curves.

Scott Aaronson (MIT)Based on joint work with John

Watrous (U. Waterloo)

BQP

PSPACE

Quantum Computing With Closed Timelike Curves

MotivationOrdinary quantum computing too pedestrian

In the past, CTCs have mostly been studied from the perspective of GR

Studying them from a computer science perspective leads us to ask new questions—like, how hard would Nature have to “work” to ensure causal consistency?

Hopefully, leads to some new insights about causality, linearity of quantum mechanics, space vs. time, ontic vs. epistemic…

Bestiary of Complexity Classes

PSPACE

EXP

BQP

P

The difference between space and time in computer

science: you can reuse space, but not time

Everyone’s first idea for a CTC 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 ModelA closed timelike curve (CTC) is simply a resource that, given an operation f:{0,1}n{0,1}n acting in some region of spacetime, finds a fixed point of f—that is, an x s.t. f(x)=x

Of course, not every f has a fixed point—that’s the Grandfather Paradox!

But since every Markov chain has a stationary distribution, 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

CTC Computation

R CTC R CR

C

0 0 0

Answer

“Causality-Respecting Register”

“Closed Timelike

Curve Register”

Polynomial Size Circuit

You (the “user”) pick a circuit C on two registers, RCR and RCTC, as well as an input x to RCR

Let Cx be the induced operation on RCTC only

Nature is forced to find a distribution DCTC over inputs to RCTC such that Cx(DCTC)=DCTC

(If there’s more than one such DCTC, Nature can choose one “adversarially”)

Then Nature samples a string y from DCTC

Output of the computation: C(x,y)PCTC is the class of decision problems

solvable in this model

How to Use CTCs to Solve Hard Problems: Basic Idea

Given a function f:[N]{0,1} (where N is huge), suppose we want “instantly” to find an input x such that f(x)=1

I claim that we can do so using the following function g:[N][N], acting on a CTC register:

0 ifmod1x

1 if

xfN

xfxxg

What are the fixed points of this evolution?

Theorem: PCTC = PSPACE

Proof: For PCTC PSPACE, just need to find some y such that Cx(m)(m)

(y)=y for some m. Pick any y, then apply Cx 2n times.

For PSPACE PCTC: Have Cx input and output an ordered pair mi,b, where mi is a state of the Turing machine we’re simulating and b is an answer bit, like so:

The only fixed-point distribution is a uniform

distribution over all states of the Turing

machine, with the answer bit set to its “true” value

mT-1,0

mT,0

m1,0

m2,0

mT-1,1

mT,1

m1,1

m2,1

What About The Quantum Case?

You (the “user”) pick a quantum circuit C on two registers, RCR and RCTC, as well as a (classical) input |x to RCR

Let Cx be the induced superoperator acting on RCTC only

Nature is forced to find a mixed state CTC such that Cx(CTC)=CTC

(If there’s more than one such , Nature can choose one “adversarially”)

Output of the computation: C(x,CTC)

Let BQPCTC be the class of problems solvable in this model

Certainly PSPACE = PCTC BQPCTC EXP

Main Result: BQPCTC = PSPACE“If CTCs are possible, then quantum computers are no

more powerful than classical ones”

BQPCTC PSPACE: Proof SketchLet vec() be the “vectorization” of : i.e., a length-22n vector of ’s entries.

We can reduce the problem to the following: given an (implicit) 22n22n matrix M, prepare a state CTC in BQPSPACE such that CTCCTCM vecvec

1

11lim:

zMIzP

zIdea: Let

P

zMIz

zMIz

z

MzMzzMIz

z

MzMzzMz

z

MzzMMz

MzzMIzMMP

z

z

z

z

z

z

1

1

1

1

3322

1

3322

1

322

1

22

1

1lim

1lim

1lim

1lim

1lim

1lim

Then

Hence M(Pv)=Pv, so P projects onto the fixed points of M

Furthermore:

•We can compute P exactly in PSPACE, by using small-space algorithms for matrix inversion discovered in the 1980s (e.g. Csanky’s algorithm)

• It’s easy to check that Pv is the vectorization of some density matrix

So then take (say) Pvec(I) as the fixed-point CTC

Coping With ErrorProblem: The set of fixed points could be sensitive to arbitrarily small changes to the superoperator

E.g., consider the two stochastic matrices

1

01,

10

1

The first has (1,0) as its unique fixed point; the second has (0,1)

However, the particular CTC algorithm used to solve PSPACE problems doesn’t share this property!

Indeed, one can use a CTC to solve PSPACE problems “fault-tolerantly” (building on Bacon 2003)

DiscussionThree ways of interpreting our result:

(1)CTCs exist, so now we know exactly what can be computed in the physical world (PSPACE)!

(2)CTCs don’t exist, and this sort of result helps pinpoint what’s so ridiculous about them

(3)CTCs don’t exist, and we already knew they were ridiculous—but at least we can find fixed points of superoperators in PSPACE!

Our result formally justifies the following intuition:By making time “reusable,” CTCs would make timeequivalent to space as a computational resource.

And Now for the Mudfight!Bennett, Leung, Smith, Smolin 2009: Deutsch’s (and our) model of CTCs is crap

Why? Because if you feed to a CTC computer, the outcome might be different than if you fed x and y separately, then averaged the results

yyxx 2

1

This is a simple consequence of the fact that CTCs induce nonlinearities in quantum mechanics

Bennett et al.’s proposed fix: Force CTC to depend only on the whole distribution over inputs,

xx xxp

Our ResponseWhat Bennett et al. do basically just amounts to defining CTCs out of existence!

That CTCs would strain the normal axioms of physics (like linearity of mixed-state evolution) is obvious … what else did you expect?

At least BQPCTC is a good complexity class, better than their proposed replacement BQPPCTC

Since under their prescription, we might as well treat CTC as a “quantum advice resource” fixed for all time, independent of anything else in the universe

(In any case, our main result—an upper bound on BQPCTC and BQPPCTC—is unaffected)

Seth Lloyd’s ResponseBennett et al.’s fix precludes the possibility that a CTC could form in some “branches of the multiverse” but not others

But quantum gravity theories ought to allow superpositions over different causal structures—so if CTCs can form at all, then why not allow evolutions like

xxCTC

xx

xCTCx

xx

xxpxxp

xxxpxxp

Scott Aaronson (MIT)Based on joint work with John

Watrous (U. Waterloo)

BQP

PSPACE

Quantum Computing With Closed Timelike Curves