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8/14/2019 The Incredible Power of Post Selection (Scott Aaronson)
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A MAX Feature Presentation
P
BQP
PSPACE
=
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Scott Aaronson (IAS)
Scotts Grab Bag o
Cheap Yuks
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Scott Aaronson (IAS)
Dr. Scotts Grab Bago Cheap Yuks
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Scott Aaronson (IAS)
Outlook on the Future of
Quantum Computing
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Scott Aaronson (IAS)
The Remarkable
Ubiquity of Postselection
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Scott Aaronson (IAS)
The Stupendous
Strength ofPostselection
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Scott Aaronson (IAS)
The Hunky, Rippling
Manliness ofPostselection
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Scott Aaronson (IAS)
Lessons Learned in the
Gottesman Institute ofComedy
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Scott Aaronson (IAS)
The Amazing Power of
Postselection
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Learning something about a question by
conditioning on the fact that youre asking it.
What ISPostselection?
BERKELEY CAMBRIDGE
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What about the
quantum case?
Anthropic Computing: A foolproof way to
solve NP-complete problems in polynomial time
(1) Accept the many-worlds interpretation(2) Generate a random truth assignment X
(3) If X doesnt satisfy , kill yourself
Input: A Boolean formula
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In This Talk
This will lead to: An extremely simple proof of the celebrated
Beigel-Reingold-Spielman theorem
Limitations on quantum advice and one-waycommunication
Unrelativized quantum circuit lower bounds
And more!
Ill study what you could do with a quantumcomputer, IF you could measure a qubit andpostselect on its being |1
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PostBQP
Class of languages decidable by a bounded-
error polynomial-time quantum computer, if
at any time you can measure a qubit that
has a nonzero probability of being |1, andassume the outcome will be |1
I hereby define a new
complexity class
(Postselected BQP)
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Another Important Animal: PP
Class of languages decidable by anondeterministic poly-time Turing machine
that accepts iff the majority of its paths do
NP
PP
P#P=PPP
PSPACE
P
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Theorem: PostBQP = PP
Easy half: PostBQP PP
Adleman, DeMarrais, and Huang (1997) showed
BQP PP using Feynman sum-over-histories
Idea: Write acceptance and rejection
probabilities as sums of exponentially many
easy-to-compute terms; then use PP to decide
which sum is greater
For PostBQP, just sum over postselected
outcomes only
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To Show PP PostBQPGiven a Boolean function f:{0,1}n{0,1},
let s=|{x : f(x)=1}|. Need to decide if s>2n-1
From
/ 2
0,1
2n
n
x
x f x
2 2
2 2
2 0 1 1/ 2 2 0 1/ 2 2 2 1,
2 2
n n n
n n
s s sH
s s s s
we can easily prepare
Goal: Decide if these amplitudes have thesame or opposite signs
Prepare |0|+|1H| for some ,.
Then postselect on second qubit being |1
/ 2
2 2 2
0 1/ 2 2 2 1:
/ 2 2 2
n
n
s s
s s
Yields in first qubit
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To Show PP PostBQP
/ 2
2 2 2
0 1/ 2 2 2 1:
/ 2 2 2
n
n
s s
s s
Yields in first qubit
1
0
Suppose s and 2n
-2sare both positive
Then by trying / = 2i
for all i{-n,,n}, welleventually get close to
0 1
2
On the other hand, if2n-2s is negative, then
we wont. QED
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Totally unexpectedly, the PostBQP=PP
theorem turned out to have implicationsforclassical complexity theory
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Beigel, Reingold, Spielman 1990: PP is
closed under intersection
Solved a problem that was open for 18 years
Other Classical Results Proved With
Quantum Techniques:
Kerenidis & de Wolf, A., Aharonov & Regev,
Observation: PostBQP is trivially closed
under intersection PP is too
Given L1,L2PostBQP, to decide if xL1 and x
L2, postselect on both computations
succeeding, and accept iff they both accept
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Other Results that
PostBQP=PP Makes Simpler
(Fortnow and Reingold)
(Fortnow and Rogers)
(Kitaev and Watrous)
PPPPP=
||
PPPPBQP
=
PPQMA
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Quantum Advice
BQP/qpoly:Class of languages decidable by
polynomial-size, bounded-error quantum circuits,
given a polynomial-size quantum advice state |
n that depends only on the input length n
Mike & Ike:We know that manysystems in Nature prefer to sit inhighly entangled states of many
systems; might it be possible to
exploit this preference to obtainextra computational power?
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How powerful is quantum advice?
Could it let us solve problems that are not
even recursively enumerable givenclassical advice of similar size?!
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Limitations of Quantum Advice
NP BQP/qpoly relative to an oracle(Uses direct product theorem for quantum search)
BQP/qpoly PostBQP/poly( = PP/poly)
( ) ( ) ( )( ).log 111 fQfmQOfD
=
Closely related: for all (partial or total) Booleanfunctions f : {0,1}n {0,1}m {0,1},
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Alices Classical Advice
Bob, suppose you used themaximally mixed state in place of yourquantum advice. Then x1 is the
lexicographically first input for which
youd output the right answer withprobability less than .
But suppose you succeeded on x1,
and used the resulting reduced state
as your advice. Then x2 is the
lexicographically first input after x1 for
which youd output the right answer
with probability less than ...
x1
x2Given an input x,
clearly lets Bob
decide in PostBQP
whether xL
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But how many inputs must Alice specify?
We can boost a quantum advice state so
that the error probability on any input is atmost (say) 2-100n; then Bob can reuse the
advice on as many inputs as he likes
We can decompose the maximally mixedstate on p(n) qubits as the boosted advice
plus 2p(n)-1 orthogonal states
Alice needs to specify at most p(n) inputs
x1,x2,, since each one cuts Bobs total
success probability by least half, but the
probability must be (2-p(n)) by the end
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PPP Does Not Have Quantum
Circuits of Size nk
Does U accept x0 w.p. ?If yes, set x0L
If no, set x0L
U: Picks a size-nk quantumcircuit uniformly at random
and runs it
x0
x1
x2
x3
x4
x5
Conditioned on deciding x0
correctly, does U accept x1
w.p. ?If yes, set x1L
If no, set x1L
Conditioned on deciding x0
and x1 correctly, does U
accept x2 w.p. ?
If yes, set x2L
If no, set x L
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For any k, defines a language L that does nothave quantum circuits of size nk
Why? Intuitively, each iteration cuts the
number of potential circuits in half, but therewere at most circuits to begin with
kn2~
On the other hand, clearly L PPP
Even works for
quantum circuits
with quantum
advice!
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And now for a grand finale0-1-NPC - #AC0 - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC1 - A 0PP - AC - AC0 - AC
0[m] - ACC0 - AH - AL
- AlgP/poly - AM - AM-EXP - AM intersect coAM - AM[polylog] - AmpMP - AmpP-BQP - AP - AP - APP - APP - APX - AUC-
SPACE(f(n)) - AVBPP - AvE - AvP - AW[P] - AWPP - AW[SAT] - AW[*] - AW[t] - P - BH - BPE - BPEE - BPHSPACE(f(n)) -
BPL - BPNP - BPP - BPPcc - BPPKT - BPP-OBDD - BPPpath - BPQP - BPSPACE(f(n)) - BPTIME(f(n)) - BQNC - BQNP -
BQP - BQP/log - BQP/poly - BQP/qlog - BQP/qpoly - BQP-OBDD - BQPtt/poly - BQTIME(f(n)) - k-BWBP - C=AC0 - C=L -
C=P - CFL - CLOG - CH - Check - CkP - CNP - coAM - coC=P - cofrIP - Coh - coMA - coModkP - compIP - compNP - coNE
- coNEXP - coNL - coNP - coNPcc - coNP/poly - coNQP - coRE - coRNC - coRP - coSL - coUCC - coUP - CP - CSIZE(f(n))
- CSL - CZK - D#P - 2P - -BPP - -RP - DET - DiffAC0 - DisNP - DistNP - DP - DQP - DSPACE(f(n)) - DTIME(f(n)) -
DTISP(t(n),s(n)) - Dyn-FO - Dyn-ThC0 - E - EE - EEE - EESPACE - EEXP - EH - ELEMENTARY - ELkP - EPTAS - k-EQBP
- EQP - EQTIME(f(n)) - ESPACE - BPP - NISZK - EXP - EXP/poly - EXPSPACE - FBQP - Few - FewP - FH - FNL -FNL/poly - FNP - FO(t(n)) - FOLL - FP - FPNP[log] - FPR - FPRAS - FPT - FPTnu - FPTsu - FPTAS - FQMA - frIP - F-
TAPE(f(n)) - F-TIME(f(n)) - GA - GAN-SPACE(f(n)) - GapAC0 - GapL - GapP - GC(s(n),C) - GI - GPCD(r(n),q(n)) - G[t] -HeurBPP - HeurBPTIME(f(n)) - HkP - HVSZK - IC[log,poly] - IP - IPP - L - LIN - LkP - LOGCFL - LogFew - LogFewNL -
LOGNP - LOGSNP - L/poly - LWPP - MA - MA' - MAC 0 - MA-E - MA-EXP - mAL - MaxNP - MaxPB - MaxSNP - MaxSNP0 -
mcoNL - MinPB - MIP - MIP*[2,1] - MIPEXP - (Mk)P - mL - mNC1 - mNL - mNP - ModkL - ModkP - ModP - ModZkL - mP -
MP - MPC - mP/poly - mTC0 - NC - NC0 - NC1 - NC2 - NE - NE/poly - NEE - NEEE - NEEXP - NEXP - NEXP/poly - NIQSZK
- NISZK - NISZKh - NL - NL/poly - NLIN - NLOG - NP - NPC - NP cc - NPC - NPI - NPcoNP - (NPcoNP)/poly - NP/log -
NPMV - NPMV-sel - NPMVt - NPMVt-sel - NPO - NPOPB - NP/poly - (NP,P-samplable) - NPR - NPSPACE - NPSV -
NPSV-sel - NPSVt
- NPSVt
-sel - NQP - NSPACE(f(n)) - NT - NTIME(f(n)) - OCQ - OptP - P - P/log - P/poly - P#P - P#P[1] -
PAC0 - PBP - k-PBP - PC - Pcc - PCD(r(n),q(n)) - P-close - PCP(r(n),q(n)) - PermUP - PEXP - PF - PFCHK(t(n)) - PH - PHcc- 2P - PhP - 2P - PINC - PIO - PK - PKC - PL - PL
1 - PLinfinity - PLF - PLL - PLS - PNP - PNP[k] - PNP[log] - PNP[log^2] - P-OBDD -
PODN - polyL - PostBQP - PP - PP/poly - PPA - PPAD - PPADS - PPP - PPP - PPSPACE - PQUERY - PR - PR -
PrHSPACE(f(n)) - PromiseBPP - PromiseBQP - PromiseP - PromiseRP - PrSPACE(f(n)) - P-Sel - PSK - PSPACE - PT 1 -
PTAPE - PTAS - PT/WK(f(n),g(n)) - PZK - QAC0 - QAC0[m] - QACC0 - QAM - QCFL - QCMA - QH - QIP - QIP(2) - QMA -QMA+ - QMA(2) - QMAlog - QMAM - QMIP - QMIPle - QMIPne - QNC
0 - QNCf0 - QNC1 - QP - QPLIN - QPSPACE - QSZK - R
- RE - REG - RevSPACE(f(n)) - RHL - RL - RNC - RP - RPP - RSPACE(f(n)) - S2P - S2-EXPPNP - SAC - SAC0 - SAC1 -
SAPTIME - SBP - SC - SEH - SelfNP - SF k - 2P - SKC - SL - SLICEWISE PSPACE - SNP - SO-E - SP - SP - span-P -SPARSE - SPL - SPP - SUBEXP - symP - SZK - SZK h - TALLY - TC
0 - TFNP - 2P - TreeBQP - TREE-REGULAR - UAP -
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Quantum Karp-Lipton Theorem
If PP BQP/qpoly, then the countinghierarchyconsisting of
etc.collapses to PP
,,,PPPPPP
PPPPPP
But theres more: With no assumptions, PP
does not have quantum circuits of size nk
And more: PEXP requires quantum circuits
of size f(n), where f(f(n))2n
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Even Stronger Separations
QMAEXP (a subclass of PEXP) is not in
BQP/qpoly
QCMAEXP (a subclass of QMAEXP) is not in
BQP/poly
A0PP (a subclass of PP) does not havequantum circuits of size nk
NONRELATI
VIZING
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Conclusions
I started out with a weird philosophical question
Try itit works!
I ended up with seven or eight results