Hidden Variables as Fruitful Dead Ends
Scott Aaronson (MIT)
WWJPD?
Ever since I attended his group meetings as a 20-year-old
summer student, John Preskill has been my unbreakable link
between CS and physics—someone whose scientific judgments
I’ve respected above all others’—my lodestar of sanity
Goal of talk: By discussing hidden variables, show how little
of his sanity I’ve learned
“God, Dice, Yadda Yadda”
The “Einsteinian Impulse”: Quantum mechanics is a tool for
calculating probabilities of measurement outcomes. It tells no
clear story about what’s “really there” prior to measurement.
Ergo, one should infer the existence of deeper laws, which tell
the “real story” and from which the probability calculus can
be derived (either exactly or as a limiting approximation)
Don’t you need to be insane
to still believe this in 2013??
The Sentient Quantum Computer
So, what did it feel
like to undergo a
210000-dimensional
Fourier transform?
It’s amazing how
fast you forget
If you believe that a sentient QC would need to have some
definite experience—or distribution over possible experiences—
hidden variables just might be for you
This Talk
Tasting Menu of Hidden-Variable Theories
No-Go Theorems: Bell, Kochen-Specker, and PBR
New Results on -Epistemic Theories [ABCL’13]
Computational Complexity and Hidden Variables
Field Guide to Hidden-Variable Theories
Same predictions as QM Different predictions
Replace wavefunction
“-epistemic theories”
Lots of falsified ideas (Joy
Christian, Stephen
Wolfram…)
Supplement
wavefunction
Bohmian mechanics,
discrete dynamical
theories
Non-equilibrium Bohmian
mechanics (Valentini)
A d-dimensional
-Epistemic Theory is defined by:
A set  of “ontic states” (ontic = philosopher-speak for “real”)
For each pure state |Hd, a probability measure  over
ontic states

Can
trivially satisfy these axioms by setting
pointB=(v
measure
concentrated
on
d,  = the basis
For each=H
orthonormal
,…,v
)
and
i[d],
a
1
d
2
|
itself,
and
R
()=|v
||
i,B
i
“response function” R :[0,1], satisfying
i,B
Gives a completely uninteresting restatement of
quantum mechanics (called the “Beltrametti(Conservation ofBugajski
Probability)
theory”)
(Born Rule)
More Interesting Example: Kochen-Specker Theory
Response functions Ri,B():
deterministically return basis
vector closest to |
Accounts beautifully for one
qubit -epistemically!
(One qutrit: Already a problem…)
Observation: If |=0, then  and  can’t overlap
Call the theory maximally nontrivial if (as above)  and 
overlap whenever | and | are not orthogonal
Discrete Dynamical Theories
 1   u11
Quantum
 
state
  
  N  u1N
u N 1   1 
Unitary
   Quantum
state
matrix
 
u NN   N 
2
  2  s


s


1
11
N1
1
Probability
Probability






distribution
Stochastic
distribution





matrix


2
2


s
s
NN    N 
  N   1N


Such a stochastic matrix S is trivial to find!
 2   2
 1   1




2
2
  N    N
2


1
1 




2
2
 N    N 
2
“Product Dynamics”
(a.k.a. “every Planck
time is a whole new
adventure!”)
Some natural further requirements:
“Indifference”:
“Commutativity”: If UA,UB act only on A,B respectively, then
Robustness to small perturbations in U and |
Bohmian Mechanics
The “actual” particle
positions x are a raft,
floating passively on
the (x,t)
My view: Bohm’s guiding equation
only ocean
looks “inevitable” because he restricted
God only plays
dice at the
Bang!
But space…
then He smashes
attention
to aBig
weird
Hilbert
His dice, and lets x follow the ||2 distribution forever after
Underappreciated Fact: In a finite-dimensional Hilbert space
(like that of quantum gravity), we can’t possibly get Bohm’s kind
of “determinism”
Schrödinger/Nagasawa Theory
(based on iterative matrix scaling; originated in 1931)
7 / 25 3/ 5 4 / 5 3/ 5 
 24 / 25   4 / 5 3/ 5  4 / 5

 


.360 .640
.078 .130
.019
.360
.013 .059
.640
.410
.065
.922 .347
.640
.230
.461 .360
.230
.575
.461
Set (i,j) entry of joint
probabilities matrix to
|uij|2, as a first guess
Normalize the columns
Normalize the rows
Can prove this process converges for every U,|! Beautiful
math involved: KL divergence, Max-Flow/Min-Cut Theorem…
Bell/CHSH No-Go Theorem
Implication for dynamical theories: Impossible to satisfy
both indifference and commutativity
Implication for -epistemic theories: Can’t reproduce
QM using =AliceJohn and “local” response functions
Kochen-Specker No-Go Theorem
There exist unit vectors v1,…,v31R3 that can’t be colored red
or blue so that in every orthonormal basis, exactly one vi is red
Implication for dynamical theories: Can’t have
dynamics in all bases that “mesh” with each other
Implication for -epistemic theories: If theory is
deterministic (Ri,B(){0,1}), then Ri,B() must depend on
all vectors in B, not just on vi
PBR (Pusey-Barrett-Rudolph 2011) No-Go Theorem
Suppose we assume =
(“-epistemic theories must behave well under tensor product”)
Then there’s a 2-qubit entangled measurement M, such
that the only way to explain M’s behavior on the 4 states
is using a “trivial” theory that doesn’t mix 0 and +.
(Can be generalized to any pair of states, not just |0 and |+)
Bell’s Theorem: Can’t “locally” simulate all separable
measurements on a fixed entangled state
PBR Theorem: Can’t “locally” simulate a fixed entangled
measurement on all separable states (at least nontrivially so)
But suppose we drop PBR’s tensor assumption. Then:
Theorem (A.-Bouland-Chua-Lowther ‘13): There’s a maximallynontrivial -epistemic theory in any finite dimension d
Albeit an extremely weird one!
Solves the main open problem of Lewis et al. ‘12
Ideas of the construction:
Cover Hd with -nets, for all =1/n
Mix the states in pairs of small balls
(B,B), where |,| both belong
to some -net
(“Mix” = make their ontic distributions overlap)
To mix all non-orthogonal states,
take a “convex combination” of
countably many such theories
On the other hand, suppose we want our theory to be
symmetric—meaning that
and
Theorem (ABCL’13): There’s
no symmetric, maximallynontrivial -epistemic theory
in dimensions d3
To prove, easiest to start with
“strongly symmetric”
theories—special case where
has the same form for every 
“Speedo
Region”
Proof Sketch
To generalize to the “merely” symmetric case
(()=f(|||)), we use some measure
theory and differential geometry, to show that
the ’s can’t possibly “evade” |
And strangely, our current proof works only for
complex
spaces,
real
Hilbert
spaces
Measuring
| inHilbert
the basis
B={|not
,|
,|
} must
yield some
1
2
3
outcome with
nonzero
probability—suppose
|1to
 a
Trying
to adapt
to the real case leads
Kakeya-like problem
By sliding from |2 to |3, we can find a state | orthogonal
to |1 such that | is nevertheless in the support of . Then
applying B to | yields |1 with nonzero probability,
contradicting the Born rule
Hidden Variables and Quantum Computing
Some people believe scalable QC is fundamentally impossible
I’ve never understood how such people could be right, unless
Nature were describable by a “classical polynomial-time hidden
variable theory” (some of the skeptics admit this, others don’t)
Well-known problem: It’s incredibly hard to construct such a
theory that doesn’t contradict QM on existing experiments!
Needed: A “Sure/Shor separator” (A. 2004),
between the many-particle quantum states
we’re sure we can create and those that suffice
for things like Shor’s algorithm
PRINCIPLED LINE
Scalable Quantum Computing:
“The Bell inequality violation of the 21st century”
Admittedly, quantum computers seem to differ from Bell
violation in being directly useful for something
But in a recent advance, [A.-Arkhipov 2011] solved that
problem!
BosonSampling
Recently demonstrated with
3-4 photons [Broome et al., Tillmann
et al., Walmsley et al., Crespi et al.]
Ironically, dynamical hidden-variable theories
could also increase the power of QC even further
Yes, these theories reproduce standard QM at each
individual time. But they also define a distribution over
trajectories. And because of correlations, sampling a whole
trajectory might be hard even for a quantum computer!
Concrete evidence comes from the Collision Problem:
Given a list of N numbers where every number appears
twice, find any collision pair
13 10 4 1 8 7 12 9 11 5 6 4 2 13 10 3 2 7 9 11 5 1 6 12 3 8
Models graph isomorphism, breaking crypto hash functions
Any quantum algorithm to solve the collision problem
needs at least ~N1/3 steps [A.-Shi 2002] (and this is tight)
How to solve the collision problem
super-fast by sampling a trajectory
[A. 2005]
1
N
N
i
i 1
xi
“Measurement”
of 2nd register
1
i  j

2
x
1
i  j

2
x
i
Two bitwise Fourier
transforms
i
GOAL: When we inspect the hidden-variable
trajectory, see both |i and |j with high probability
By sampling a trajectory, you can also do
Grover search in ~N1/3 steps instead of ~N1/2 (!)
N1/3
iterations
of Grover’s
quantum
search
algorithm
Probability of observing
the marked item after T
iterations is ~T2/N
Hidden variable
Conjectured
World Map
NP
Satisfiability, Traveling
Salesman, etc.
DQP
Dynamical
Quantum
Polynomial
Time
BQP
Quantum
Polynomial
Time
Graph Isomorphism
Approximate
Shortest Vector
Factoring
P
Polynomial
Time
Upshot: If, at your death, your whole life flashed before
you in an instant, then you could solve Graph
Isomorphism in polynomial time
(Assuming you’d prepared beforehand by putting your brain in
appropriate quantum states, and a dynamical hidden-variable
theory satisfying certain reasonable axioms was true)
But probably still not NP-complete problems!
DQP is basically the only example I know of a
computational model that generalizes quantum
computing, but only “slightly”
(Contrast with nonlinear quantum mechanics, postselection, closed
timelike curves…)
Concluding Thought
Hidden-variable
theories are like
mathematical
sandcastles on the
shores of QM
But it’s hard not to wonder:
just how convincing a castle
can one build, before the
sand reasserts its sandiness?
Yes, they tend to topple
over when pushed
(by mathematical demands if
they match QM’s predictions, or
by experiments if they don’t)
And yes, people who
think they can live in one
are almost certainly
deluding themselves
80+ years after it was first asked, the answers to this question
(both positive and negative) continue to offer surprises, making us
wonder how well we really know sand and water…
Open Problems in Hiddenvariableology
In the Schrödinger/Nagasawa theory, are the probabilities
obtained by matrix scaling robust to small perturbations of U
and |?
Can we upper-bound the complexity of sampling hiddenvariable histories? (Best upper bound I know is EXP)
What’s the computational complexity of simulating Bohmian
mechanics?
Are there symmetric -epistemic theories in dimensions d3
that mix some ontic distributions (not necessarily all of them)?
In -epistemic theories, what’s the largest possible amount of
overlap between two ontic distributions  and , in terms of
|||?