Quantum Computing With
Closed Timelike Curves
Scott Aaronson (MIT)
Based on joint work with John
Watrous (U. Waterloo)
Motivation
Ordinary 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
EXP
PSPACE
The difference between
space and time in computer
science: BQP
you can reuse
space, but not time
P
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 Model
A 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
Answer
Polynomial
Size Circuit
C
“Closed
Timelike
Curve
Register”
R CTC
R CR
0 0 0
“CausalityRespecting
Register”
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:
x
if f x   1

g x   
x  1 modN if f x   0
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)(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,0
mT,1
mT-1,0
mT-1,1
m2,0
m2,1
m1,0
m1,1
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 Sketch
Let 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) 22n22n matrix M, prepare a state CTC in
BQPSPACE such that
M vecCTC   vecCTC 
Idea: Let
Then
P : lim 1  z I  zM 
1
z 1


 
MP  M lim 1  z  I  zM  z 2 M 2  
z 1

 lim 1  z  M  zM 2  z 2 M 3
Furthermore: z 1
1 z
3
 lim P exactly
zM in
z 2 PSPACE,
M 2  z 3 Mby
using
 small-space
•We can compute
z 1
z
algorithms for matrix inversion discovered in the 1980s (e.g.
1 z
Csanky’s algorithm)
 lim
I  zM  z 2 M 2  z 3 M 3  
z 1
z
• It’s easy to check1 that
z Pv is the
1 vectorization of some

 lim
I  zM 
density matrix
z 1
z
1




lim
1

z
I

zM
So then take (say)
Pvec(I) as the fixed-point CTC





z 1
P
Hence M(Pv)=Pv, so P projects onto the fixed points of M
Coping With Error
Problem: The set of fixed points could be sensitive to
arbitrarily small changes to the superoperator
E.g., consider the two stochastic matrices
1  

,
 0 1  
1  

 
0

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)
Discussion
Three 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 time
equivalent 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
1
x x  y y
2

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
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,
p x x

x
x
Our Response
What Bennett et al. do basically just amounts to
defining CTCs out of existence!
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
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
(In any case, our main result—an upper
bound on BQPCTC and BQPPCTC—is unaffected)
Seth Lloyd’s Response
Bennett 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
p
x
x
x x   p x x x   CTC  x 
x




   p x x x    CTC   p x x x 
 x

 x

Quantum Computing With
Closed Timelike Curves
Scott Aaronson (MIT)
Based on joint work with John
Watrous (U. Waterloo)