The Incredible Power of Post Selection (Scott Aaronson)

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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Dr. Scotts Grab Bago Cheap Yuks


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Outlook on the Future of

Quantum Computing


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The Remarkable

Ubiquity of Postselection


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The Stupendous

Strength ofPostselection


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The Hunky, Rippling

Manliness ofPostselection


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Lessons Learned in the

Gottesman Institute ofComedy


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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

The Incredible Power of Post Selection (Scott Aaronson)

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