Scott Aaronson (MIT)

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Quantum Computing With Closed Timelike Curves
BQP
PSPACE

• Scott Aaronson (MIT)
• Based on joint work with John Watrous (U.
  Waterloo)

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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 questionslike, 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
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Bestiary of Complexity Classes
EXP
PSPACE
The difference between space and time in computer
science you can reuse space, but not time
BQP
P
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Everyones 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
• Doesnt take into account the computation youll
  have to do after getting the answer

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Deutschs Model
A closed timelike curve (CTC) is simply a
resource that, given an operation f0,1n?0,1n
acting in some region of spacetime, finds a fixed
point of fthat is, an x s.t. f(x)x Of course,
not every f has a fixed pointthats the
Grandfather Paradox! But since every Markov chain
has a stationary distribution, theres always a
distribution D such that f(D)D
Probabilistic Resolution of the Grandfather
Paradox- Youre born with ½ probability- If
youre born, you back and kill your grandfather-
Hence youre born with ½ probability
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CTC Computation
Polynomial Size Circuit
Closed Timelike Curve Register
Causality-Respecting Register
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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
theres 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
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How to Use CTCs to Solve Hard Problems Basic Idea
Given a function fN?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 gN?N, acting on a CTC register
What are the fixed points of this evolution?
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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 were 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
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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 theres more than one such ?,
Nature can choose one adversarially) Output of
the computation C(x,?CTC)
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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
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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
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Idea Let
Then

• Furthermore
• We can compute P exactly in PSPACE, by using
  small-space algorithms for matrix inversion
  discovered in the 1980s (e.g. Csankys algorithm)
• Its easy to check that Pv is the vectorization
  of some density matrix
• So then take (say) Pvec(I) as the fixed-point
  ?CTC

Hence M(Pv)Pv, so P projects onto the fixed
points of M
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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
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 doesnt share this
property! Indeed, one can use a CTC to solve
PSPACE problems fault-tolerantly (building on
Bacon 2003)
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Discussion

• Three ways of interpreting our result
• CTCs exist, so now we know exactly what can be
  computed in the physical world (PSPACE)!
• CTCs dont exist, and this sort of result helps
  pinpoint whats so ridiculous about them
• CTCs dont exist, and we already knew they were
  ridiculousbut 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.
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And Now for the Mudfight!
Bennett, Leung, Smith, Smolin 2009 Deutschs
(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
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,
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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 resultan upper bound on
BQPCTC and BQPPCTCis unaffected)
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Seth Lloyds 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
structuresso if CTCs can form at all, then why
not allow evolutions like
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Quantum Computing With Closed Timelike Curves
BQP
PSPACE

• Scott Aaronson (MIT)
• Based on joint work with John Watrous (U.
  Waterloo)