Abstract Representation: Your Ancient Heritage
Learning and TestingQuantum StatesRyan ODonnellJohn
WrightCarnegie MellonCarnegie Mellon
1
Picture byJorge Chama unit vector v in d(qudit)
Any measurement
... can only yieldprobabilistic infoabout the vector.More on
what 3
May rotate space by your favorite unitary Udd.
Measurement outcome is i {1,2,,d}with probability |ei, Uv|2.
... can only yieldprobabilistic infoabout the vector.Details, if
you want to know:Any measurementMore on what 4
Apparatus may itself be probabilisticMore on what 5
Actual output: p1 p2 pd
v1 v2 vd
orthonormalvectors in dMore on what 6An unknown probability
distribution over an unknown set of d orthonormal vectors.And
measuring only gives you someprobabilistic info about the outcome
vector.Thats a triple whammy.Actual output: p1 p2 pd
v1 v2 vd
More on what 7Its expensive, but you can hit the button n
times.
d=3, n=7 example outcome for the particles:v1, v3, v2, v2, v1,
v1, v3with prob. p1 p3 p2 p2 p1 p1 p3
More on what 8Its expensive, but you can hit the button n
times.
d=3, n=7 example outcome for the particles:with prob. p1 p3 p2
p2 p1 p1 p3
v1v3v2v2v1v1v3
(3)7You may measure each particle separately.Or, may do one
giant measurement on (d)n.More on what 9Quantum Problems#1: Learn
v1, , vd, p1, , pd. (Approximately, up to some , w.h.p.)(Quantum
tomography)10Quantum Problems#1: Learn v1, , vd, p1, , pd.
#2: Learn the multiset {p1, , pd}. (Approximately, up to some ,
w.h.p.)(Quantum spectrum estimation)11Quantum Problems#1: Learn v1,
, vd, p1, , pd. #2: Learn the multiset {p1, , pd}.#3: Determine if
{p1, , pd} satisfies
a certain property. E.g., is or has r nonzeros(Quantum spectrum
property testing)
12Quantum Problems#1: Learn v1, , vd, p1, , pd. #2: Learn the
multiset {p1, , pd}.#3: Determine if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisand the associated vis.(Quantum
principal component analysis)13Quantum Problems#1: Learn v1, , vd,
p1, , pd. #2: Learn the multiset {p1, , pd}.#3: Determine if {p1, ,
pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisand the associated vis.Et
cetera14Quantum Problems#1: Learn v1, , vd, p1, , pd. #2: Learn the
multiset {p1, , pd}.#3: Determine if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisand the associated vis.We [OW] have
new tight upper and lower bounds.n O(d2)sufficesO(d2 log d)
shownindependently in[HHJWY15]15Quantum Problems#1: Learn v1, , vd,
p1, , pd. #2: Learn the multiset {p1, , pd}.#3: Determine if {p1, ,
pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisWe [OW] have new tight upper and
lower bounds.n O(d2)sufficesn O(k2)suffices16Quantum Problems#1:
Learn v1, , vd, p1, , pd. #2: Learn the multiset {p1, , pd}.#3:
Determine if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisand the associated vis.We [OW] have
new tight upper and lower bounds.n O(d2)sufficesn
O(kd)suffices17Quantum Problems#1: Learn v1, , vd, p1, , pd. #2:
Learn the multiset {p1, , pd}.#3: Determine if {p1, , pd}
satisfies
a certain property. E.g., is
#4: Learn the k largest pisand the associated vis.We [OW] have
new tight upper and lower bounds.n O(d2)sufficesn = (d)nec. &
suff.n O(kd)suffices18#2: Learn the multiset {p1, , pd}.#3:
Determine if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisFor simplicity, todays focus:
Just the problems depending on {p1, ..., pd}.Whats the
methodology for proving bounds?19An unknown probability
distribution.An unknown set of d orthonormal vectors.Actual output:
p1 p2 pd
v1 v2 vd
More on what 20An unknown probability distribution.Actual
output: p1 p2 pd
v1 v2 vd
If the vectors v1, v2, , vd are known,you can measure the
outcomes exactly.
Setup becomes equivalent to:
Learning / testing an unknownprobability distribution on
{1,2,,d}More on what 21Classical Distribution Problems#1: Learn p1,
, pd.(approximately, whp)22Classical Distribution Problems#1: Learn
p1, , pd)#2: Learn the multiset {p1, , pd}.#3: Determine if {p1, ,
pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisEt cetera23Classical Distribution
Problems#1: Learn p1, , pd)#2: Learn the multiset {p1, , pd}.#3:
Determine if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisLast 10 years: some new tight upper
and lower bounds.24#2: Learn the multiset {p1, , pd}.#3: Determine
if {p1, , pd} satisfies
a certain property. E.g., is
#4: Learn the k largest pisLast 10 years: some new tight upper
and lower bounds.Focus: Problems depending on {p1, ..., pd}.
Whats the methodology for proving bounds?
25Typical sample when n=20, d=5 might be
54423131423144554251 Idea 1: Permuting the n positions doesnt
matter. Hence may as well only retain histogram.
Say we care about a property of {p1, , pd}
(e.g., Uniform distribution? Support r?)12345Typical sample when
n=20, d=5 might be
54423131423144554251Say we care about a property of {p1, ,
pd}
(e.g., Uniform distribution? Support r?)Typical sample when
n=20, d=5 might be
12345Idea 2: For properties wecare about, permuting thed symbols
doesnt matter.
Hence may as well sort the histogram.Say we care about a
property of {p1, , pd}
(e.g., Uniform distribution? Support r?)Typical sample when
n=20, d=5 might be
12345Say we care about a property of {p1, , pd}
(e.g., Uniform distribution? Support r?)Typical sample when
n=20, d=5 might be
1st most freq:2nd most freq:3rd most freq:4th most freq:5th most
freq:Say we care about a property of {p1, , pd}
(e.g., Uniform distribution? Support r?)Typical sample when
n=20, d=5 might be
1 := 1st most freq:2 := 2nd most freq:3 := 3rd most freq:4 :=
4th most freq:5 := 5th most freq:(Sorted histogram is AKA a Young
diagram.)Say we care about a property of {p1, , pd}
(e.g., Uniform distribution? Support r?)Classically learning
propertiesof {p1, , pd}, a summary:
The problem has two commuting symmetries: Sn-invariance
(permuting the n outcomes) Sd-invariance (permuting d outcome
names)
Factoring these out, WLOG learner just gets a random Young
diagram (with n boxes, d rows)
Remark: Pr[] = m(p1, , pd),
(some certain symmetric polynomial in p1, , pd)
Quantumly learning properties of {p1, , pd}, a summary:
The problem has two commuting symmetries: Sn-invariance
(permuting the n outcomes) U(d)-invariance (rotating unknown v1, ,
vd)
(some certain symmetric polynomial in p1, , pd)d-dimensional
unitary groupQuantumly learning properties of {p1, , pd}, a
summary:
The problem has two commuting symmetries: Sn-invariance
(permuting the n outcomes) U(d)-invariance (rotating unknown v1, ,
vd)
(some certain symmetric polynomial in p1, , pd)Factoring these
out involves SchurWeyl dualityfrom the representation theory of Sn
and U(d).
This also involves Young diagrams!
Upshot: again, WLOG the learner gets a randomYoung diagram, but
with a weird distributionQuantumly learning properties of {p1, ,
pd}, a summary:
The problem has two commuting symmetries: Sn-invariance
(permuting the n outcomes) U(d)-invariance (rotating unknown v1, ,
vd)
(some certain symmetric polynomial in p1, , pd)
Factoring these out, WLOG learner just gets a random Young
diagram (n boxes, d rows), with:
Pr[] = f s(p1, , pd)
(some certain symmetric polynomial in p1, , pd)Quantumly
learning properties of {p1, , pd}, a summary:
The problem has two commuting symmetries: Sn-invariance
(permuting the n outcomes) U(d)-invariance (rotating unknown v1, ,
vd)
(some certain symmetric polynomial in p1, , pd) Factoring these
out, WLOG learner just gets a random Young diagram (n boxes, d
rows), with:
Pr[] = f s(p1, , pd)
(some certain symmetric polynomial in p1, , pd)Schur polynomial#
SYTs of shape Factoring these out, WLOG learner just gets a random
Young diagram (d rows, n boxes), with:
Pr[] = f s(p1, , pd)
There is a combinatorial interpretation of this! can be
generated as follows:
Pick a random word w of length n, each letterrandomly drawn from
[d] according to the pisApply the RSK algorithm to w Let be the
shape of the resulting tableauRSK Algorithm
Gilbert de B.RobinsonCraige (Ea Ea)SchenstedDonald E.KnuthRSK
Algorithm On input w = 54423131423144554251
RSK Algorithm On input w = 54423131423144554251RSK applet byTom
Roby40RSK Algorithm On input w = 544231314231445542511 = longest
incr. subseq
1 + 2 = longest union of2 incr. subseqs
1 + 2 + 3 = longest union of3 incr. subseqs
1 + 2 + 3 + + d = nAlternativecharacterization:RSK Algorithm On
input w =
54423131423144554251Alternativecharacterization:5442313142314455425154423131423144554251544231314231445542515442313142314455425154423131423144554251Cor:
Height = longeststrictly decreasingsubsequenceSummaryp1, , pd are
unknown probabilities.Learner specifies n.w ~ [d]n drawn with
i.i.d. letters according to pis.Classical case: get to see sorted
histogram, .Quantum case: get to see L.I.S. information, .Learner
tries to infer things about {p1, , pd}.Remark:
MajorizationClassical case: get to see sorted histogram, .Quantum
case: get to see L.I.S. information, .When w =
54423131423144554251partial sums of partial sums of Example 1:
Estimating pMaxClassical case: get to see sorted histogram,
.Quantum case: get to see L.I.S. information, .Classical case:
Output 1/n.Easy analysis: -accurate if n = O(1/2).Quantum case:
Output 1/n. (?)Does the L.I.S. concentrate around pMaxn ?Not too
hard: -accurate if n = (d/2).Harder (?) [OW15b]: if n = O(1/2).a
universal constant,independent of d,of pMaxp2ndMax, etc.p1 = = pd =
1/d pMax as small as possible 1 as small as possible as rectangular
as possible Variance{1, , d} small small
Example 2: Testing uniformity
Intuition:Can we determine ?
Example 2: Testing uniformityNo, but we can determine a similar
quantity:Can we determine ?
, where Why? Goes back to representation theory of Sn,Fourier
analysis. Up to scaling, is
Methodological summaryHalf the time its probabilistic
combinatoricsof increasing subsequences in random words.
Half the time its representation theory of Snand theory of
(shifted) symmetric polynomials.Conclusion
Suppose p1 = = pd = 1/d.Suppose d n.In fact, let d = .All
letters unique w is a random permutation.Distribution on is the
Plancherel Distribution.Now as n , there are many beautiful
theoremsabout the limiting shape of .
n=1000example
(pic by Dan Romik)LIS = LDS ~
where 1 ~ TracyWidom Suppose p1 = = pd = 1/d,with d = .As n
,
As n ,scaled curve
Suppose p1 = = pd = 1/d,with d = .Suppose p1 = = pd = 1/d,with d
= .The case of finite d is less studied.The case of pis not all
equal is still less studied.The case when n is not assumed to
besufficiently large as a function of dis still less studied.But
these cases are highly motivated byquantum mechanics, and I think
manybeautiful theorems are waiting to be discovered.Thanks!Gluttons
for punishment:I will give a 3-hour blackboard versionof this talk
next Friday, Oct. 9 at 1:30pmin Harvards Maxwell-Dworkin 119.
Gluttons for pizza:Apparently there will be pizza at 1:15pm.
