A MAX Feature Presentation

P

BQP

PSPACE=

Scott Aaronson (IAS)

Scott’s Grab Bag o’ Cheap Yuks

Scott Aaronson (IAS)

Dr. Scott’s Grab Bag o’ Cheap Yuks

Scott Aaronson (IAS)

Outlook on the Future of Quantum Computing

Scott Aaronson (IAS)

The Remarkable Ubiquity of

Postselection

Scott Aaronson (IAS)

The Stupendous Strength of

Postselection

Scott Aaronson (IAS)

The Hunky, Rippling Manliness of Postselection

Scott Aaronson (IAS)

Lessons Learned in the Gottesman

Institute of Comedy

Scott Aaronson (IAS)

The Amazing Power of Postselection

Learning something about a question by conditioning on the fact that you’re asking it.

What IS Postselection?

BERKELEY CAMBRIDGE

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 doesn’t satisfy , kill yourself

Input: A Boolean formula

In This Talk…

This will lead to:

• An extremely simple proof of the celebrated Beigel-Reingold-Spielman theorem

• Limitations on quantum advice and one-way communication

• Unrelativized quantum circuit lower bounds

• And more!

I’ll study what you could do with a quantum

computer, IF you could measure a qubit and postselect on its being |1

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, and assume the outcome will be |1

I hereby define a newcomplexity class…

(Postselected BQP)

Another Important Animal: PP

Class of languages decidable by a nondeterministic poly-time Turing machine that accepts iff the majority of its paths do

NP

PP

P#P=PPP

PSPACE

P

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

To Show PP PostBQP…Given 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 22 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 the same or opposite signs

Prepare |0|+|1H| for some ,.Then postselect on second qubit being |1

/ 22 2 2

0 1/ 2 2 2 1:

/ 2 2 2

n

n

s s

s s

Yields in first qubit

To Show PP PostBQP…

/ 22 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-2s are both positive

Then by trying / = 2i for all i{-n,…,n}, we’ll eventually get close to

0 1

2

On the other hand, if 2n-2s is negative, then we won’t. QED

Totally unexpectedly, the PostBQP=PP theorem turned out to have implications for classical complexity theory…

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 xL2, postselect on both computations succeeding, and accept iff they both accept

Other Results that PostBQP=PP Makes Simpler

(Fortnow and Reingold)

(Fortnow and Rogers)

(Kitaev and Watrous)

PPPPP ||

PPPPBQP PPQMA

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 many systems in Nature ‘prefer’ to sit in highly entangled states of many systems; might it be possible to exploit this preference to obtain extra computational power?”

How powerful is quantum advice?

Could it let us solve problems that are not even recursively enumerable given classical advice of similar size?!

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) Boolean functions f : {0,1}n {0,1}m {0,1},

Alice’s Classical Advice

Bob, suppose you used the maximally mixed state in place of your

quantum advice. Then x1 is the lexicographically first input for which you’d output the right answer with

probability less than ½.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 you’d output the right answer

with probability less than ½...

x1

x2

Given an input x, clearly lets Bob

decide in PostBQP whether xL

But how many inputs must Alice specify?

We can boost a quantum advice state so that the error probability on any input is at most (say) 2-100n; then Bob can reuse the advice on as many inputs as he likes

We can decompose the maximally mixed state 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 Bob’s total success probability by least half, but the probability must be (2-p(n)) by the end

PPP Does Not Have Quantum Circuits of Size nk

Does U accept x0 w.p. ½?If yes, set x0LIf no, set x0L

U: Picks a size-nk quantum circuit uniformly at random

and runs it

x0

x1

x2

x3

x4

x5

Conditioned on deciding x0 correctly, does U accept x1 w.p. ½?If yes, set x1LIf no, set x1L

Conditioned on deciding x0 and x1 correctly, does U accept x2 w.p. ½?If yes, set x2LIf no, set x2L

For any k, defines a language L that does not have quantum circuits of size nk

Why? Intuitively, each iteration cuts the number of potential circuits in half, but there were at most circuits to begin with

kn2~

On the other hand, clearly L PPP

Even works for quantum circuits

with quantum advice!

Also…If a function f:{0,1}n{0,1} has a polynomial-size quantum circuit, then a PostBQP machine can find such a circuit using queries to f

Reminiscent of a classical learning theory result of Bshouty, Cleve, et al.

Intuition: Guess random inputs xt and quantum circuits Ct. Repeatedly use postselection to find

• An input xt on which Ct fails

• A circuit Ct+1 that succeeds on x1,…,xt

Even works for quantum circuits

with quantum advice!

And now for a grand finale…0-1-NPC - #AC0 - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC1 - A0PP - AC - AC0 - AC0[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] - βP - BH - BPE - BPEE - BPHSPACE(f(n)) - BPL - BP•NP - 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 - Δ2P - δ-BPP - δ-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' - MAC0 - 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 - NPcc - 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 - Φ2P - PhP - Π2P - PINC - PIO - PK - PKC - PL - PL1 - 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 - PT1 - 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 - QNC0 - QNCf0 - QNC1 - QP - QPLIN - QPSPACE - QSZK - R - RE - REG - RevSPACE(f(n)) - RHL - RL - RNC - RP - RPP - RSPACE(f(n)) - S2P - S2-EXP•PNP - SAC - SAC0 - SAC1 - SAPTIME - SBP - SC - SEH - SelfNP - SFk - Σ2P - SKC - SL - SLICEWISE PSPACE - SNP - SO-E - SP - SP - span-P - SPARSE - SPL - SPP - SUBEXP - symP - SZK - SZKh - TALLY - TC0 - TFNP - Θ2P - TreeBQP - TREE-REGULAR - UAP - UCC - UE - UL - UL/poly - UP - US - VNCk - VNPk - VPk - VQPk - W[1] - WAPP - W[P] - WPP - W[SAT] - W[*] - W[t] - W*[t] - XOR-MIP*[2,1] - XP - XPuniform - YACC - ZPE - ZPP - ZPTIME(f(n))

Quantum Karp-Lipton Theorem

If PP BQP/qpoly, then the counting hierarchy—consisting ofetc.—collapses to PP

,,,PPPPPP PPPPPP

But there’s 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

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 have quantum circuits of size nk

NONRELATIVIZING

Conclusions

I started out with a weird philosophical question

Try it—it works!

I ended up with seven or eight results