The Learnability of Quantum States

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The Learnability of Quantum States
?

• Scott Aaronson
• University of Waterloo

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Outline
A Quantum Occams Razor Theorem - Why you should
want it to be true - Why it is true -
Application to quantum communication -
Application to quantum advice
Sneak Preview Quantum Software
Copy-Protection - What it has to do with
learning - Why it might be possible
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Why do we believe the sun will rise tomorrow?
The hypothesis that it will rise every day until
tomorrow is equally compatible with evidence
In my view, a branch of CS called computational
learning theory has pretty much solved this
Humean Problem of Induction, insofar as it has a
solution
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Occams Razor Theorem(Valiant, Vapnik, Blumer et
al)
If the possible hypotheses have sufficiently
fewer bits than the data youve collected, and if
one of those hypotheses succeeds in explaining
your data, then that hypothesis will probably
also explain most of the data you havent
collected
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Trouble in QuantumLand
Fear not, physicists! Why would he even be
raising this dilemma if he wasnt gonna
demolish it on the very next slide?
To describe a quantum state of n qubits takes 2n
classical bits Indeed, traditional quantum state
tomography requires ?(22n) measurements on copies
of the state Does this mean that a generic
10,000-particle state can never be learned
within the lifetime of the universe? If so, would
call into question the operational status of
many-body quantum states themselves
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The Quantum Occams Razor Theorem
Let ? be an n-qubit mixed state. Let D be a
distribution over two-outcome measurements.
Suppose we draw m measurements E1,,Em
independently from D, and then output a
hypothesis state ? such that
for all i. Then provided ????/10 and
well have
with probability at least 1-? over E1,,Em
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Upshot for Experimentalists
You can do pretty good tomography on an
arbitrary entangled state of n spins, using a
number of measurements that scales only linearly
(!) with n Here pretty good means with respect
to any fixed distribution over observables
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To prove the theorem, we need a notion introduced
by Kearns and Schapire called
Fat-Shattering Dimension
Let C be a class of functions from S to 0,1.
We say a set x1,,xk?S is ?-shattered by C if
there exist reals a1,,ak such that, for all 2k
possible statements of the form f(x1)?a1-? ?
f(x2)?a2? ? ? f(xk)?ak-?, theres some f?C
that satisfies the statement.
Then fatC(?), the ?-fat-shattering dimension of
C, is the size of the largest set ?-shattered by
C.
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Small Fat-Shattering Dimension Implies Small
Sample ComplexityProof uses a 1996 result of
Bartlett and Long
Let C be a class of functions from S to 0,1,
and let f?C. Suppose we draw m elements x1,,xm
independently from some distribution D, and then
output a hypothesis h?C such that h(xi)-f(xi)??
for all i. Then provided ????/7 and
well have
with probability at least 1-? over x1,,xm.
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Upper-Bounding the Fat-Shattering Dimension of
Quantum StatesProof uses Ashwin Nayaks lower
bound for quantum random access codes, which in
turn uses Holevos Theorem on quantum channel
capacity
Let S be the set of two-outcome measurements on n
qubits. Let Cn be the set of functions fS?0,1
defined by f(E)Tr(E?) for some n-qubit mixed
state ?. Then
Quantum Occams Razor Theorem is then just plug
chug
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Simple Application of Quantum Occams Razor
Theorem to Communication Complexity
x
y
f(x,y)
Alice Walker
Bob Dylan

• f Boolean function mapping Alices N-bit string
  x and Bobs M-bit string y to a binary output
• D1(f), R1(f), Q1(f) Deterministic, randomized,
  and quantum one-way communication cost of f
• How much can quantum communication save?
• Its known that D1(f)O(M Q1(f)) for all total f
• In 2004 I showed that for all f, D1(f)O(M
  Q1(f)logQ1(f))

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Theorem R1(f)O(M Q1(f)) for all f, partial or
total
Proof By Yaos minimax principle, Alice can
consider a worst-case distribution Dx over Bobs
input y Alices classical message will consist of
y1,,yT drawn from Dx, together with
f(x,y1),,f(x,yT) Here T?(Q1(f)) Bob searches
for a quantum message ? that yields the right
answers on y1,,yT (certainly such a ? exists) By
the Quantum Occams Razor Theorem, with high
probability such a ? yields the right answers on
most y drawn from Dx
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Computational Complexity of Learning Quantum
States
I showed that, if you find a state ? that
explains O(n) measurements drawn from D, with
high probability that ? will correctly explain
most future measurements drawn from D. This says
nothing about the computational problem of
finding ?!
Indeed, if ? can always be prepared by a
polynomial-time quantum algorithm, then no
one-way function is secure against quantum attack.
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To say more, we need to visit the bestiary
YQP Yaroslav Quantum Polynomial-Time
Class of problems solvable efficiently on a
quantum computer, with the help of
polynomial-size untrusted quantum advice
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Theorem AvgBQP/qpoly AvgYQP/poly Or in
English We can use trusted classical advice to
verify that untrusted quantum advice will work on
most inputs.
Proof Idea The classical advice will consist of
training inputs x1,,xm, as well as whether
xi?L for all 1?i?m Given a purported advice state
??, first check that ?? yields the right
answers on x1,,xm, and only then use it on the x
you care about By Quantum Occams Razor Theorem,
mO(poly(n)) is enough to ensure ?? will work on
most inputs w.h.p. The technical part is to do
the verification without damaging ?? too badly
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Quantum Copy-Protection
We say a program P is copy-protected if theres
no efficient algorithm that, given Ps source
code, outputs two programs with the same
input/output behavior as P Classically,
copy-protection is trivially impossible (tell
that to Sony/BMG) Quantumly well, its called
the No-Cloning Theorem for a reason Connection
to learning If P can be learned from
input/output behavior, then it cant be
copy-protected
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A Weird Example
Let G be a finite group, such that we can
efficiently prepare G? (a uniform superposition
over g?G) Let H?G be a subgroup with H ?
G/polylogG Let f(g)1 if g?H and f(g)0
otherwise Given H? (a uniform superposition over
H), Watrous showed that we can efficiently
compute f Test whether H? and gH? are equal or
orthogonal Conversely, given a black box that
computes f, we can efficiently prepare H? First
prepare G?, then postselect on f(g)1 So any
program for f can be piratedbut (apparently)
only in an indirect, quantum way
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The Pirates Nightmare
In the quantum world, can any program that cant
be learned be copy-protected? Main Result There
exists a quantum oracle relative to which the
answer is yes Upshot Even if the answer is no,
we cant prove it without using quantumly
nonrelativizing techniques
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Handwaving Proof Idea
For each circuit C, choose a meaningless quantum
label ?C? according to the Haar measure The
quantum oracle will map ?C?x?0? to
?C?x?C(x)?, as well as C?0? to
C??C? Intuitively, then, being given ?C? is
no better than being given a black box for C To
prove this, we need to simulate an algorithm that
prepares ?C? given another copy of ?C?, by an
algorithm that prepares ?C? given only black-box
access to C Strategy Mimic the copying
algorithm, by mocking up a random pure state
?? that plays the same role as ?C?
Problem Mocking up a random pure state takes
exponential time
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Solution Pseudorandom States
where p is a degree-d univariate polynomial over
GF(2n) for some dpoly(n), and p0(x) is the
leading bit of p(x)
Clearly the ?p?s can be prepared in polynomial
time Lemma If p is chosen uniformly at random,
then ?p? looks like it was chosen under the
Haar measure- Even if we get polynomially many
copies of ?p?- Even if we query the quantum
oracle, which depends on ?p? So the simulator
can use ?p?s in place of ?C?s
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Open Problems
Can we tighten the Quantum Occams Razor
Theorem?The best lower bounds I can prove go
like ?(n/??2), or ?(n/??4) in the case where each
measurement is applied only once Does BQP/qpoly
YQP/poly?I.e., can we use classical advice to
verify quantum advice in the worst-case
setting? Is D1(f) O(M Q1(f))? Or even
O(MQ1(f))?Even more ambitiously, could learning
theory techniques help us show that
R1(f)O(Q1(f)) for all total f? In the real
world, are there nontrivial programs that can be
quantumly copy-protected?What about point
functions (f(x)1 if x equals a secret password
s otherwise f(x)0)?