1. A MAX Feature Presentation P BQP PSPACE =

2. Scott Aaronson (IAS) Scotts Grab Bag o Cheap Yuks

3. Scott Aaronson (IAS) Dr. Scotts Grab Bag o Cheap Yuks

4. Scott Aaronson (IAS) Outlook on the Future of Quantum Computing

5. Scott Aaronson (IAS) The Remarkable Ubiquity of Postselection

6. Scott Aaronson (IAS) The Stupendous Strength of Postselection

7. Scott Aaronson (IAS) The Hunky, Rippling Manliness of Postselection

8. Scott Aaronson (IAS) Lessons Learned in the Gottesman Institute of Comedy

9. Scott Aaronson (IAS) The Amazing Power of Postselection

10. Learning something about a question by conditioning on the fact that youre asking it. What IS Postselection? BERKELEYCAMBRIDGE

11. 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

12. 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! Ill study what you could do with a quantum computer, IF you could measure a qubit and postselect on its being |1

13. 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 new complexity class… (Postselected BQP)

14. 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 =P PP PSPACE P

15. 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

16. 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>2 n-1 Fromwe can easily prepare Goal: Decide if these amplitudes have the same or opposite signs Prepare |0 | + |1 H| for some,. Then postselect on second qubit being |1 Yieldsin first qubit

17. To Show PP PostBQP… Yieldsin first qubit Suppose s and 2 n -2s are both positive Then by trying / = 2 i for all i {-n,…,n}, well eventually get close to On the other hand, if 2 n -2s is negative, then we wont. QED

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

20. 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 L 1,L 2 PostBQP, to decide if x L 1 and x L 2, postselect on both computations succeeding, and accept iff they both accept

21. Other Results that PostBQP=PP Makes Simpler (Fortnow and Reingold) (Fortnow and Rogers) (Kitaev and Watrous)

22. 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?

23. How powerful is quantum advice? Could it let us solve problems that are not even recursively enumerable given classical advice of similar size?!

24. Limitations of Quantum Advice NP BQP/qpoly relative to an oracle ( Uses direct product theorem for quantum search) BQP/qpoly PostBQP/poly ( = PP/poly) Closely related: for all (partial or total) Boolean functions f : {0,1} n {0,1} m {0,1},

25. Alices Classical Advice Bob, suppose you used the maximally mixed state in place of your quantum advice. Then x 1 is the lexicographically first input for which youd output the right answer with probability less than ½. But suppose you succeeded on x 1, and used the resulting reduced state as your advice. Then x 2 is the lexicographically first input after x 1 for which youd output the right answer with probability less than ½... x1x1 x2x2 Given an input x, clearly lets Bob decide in PostBQP whether x L

26. 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 2 p(n) -1 orthogonal states Alice needs to specify at most p(n) inputs x 1,x 2,…, since each one cuts Bobs total success probability by least half, but the probability must be (2 -p(n) ) by the end

27. P PP Does Not Have Quantum Circuits of Size n k Does U accept x 0 w.p. ½? If yes, set x 0 L If no, set x 0 L U : Picks a size-n k quantum circuit uniformly at random and runs it x0x0 x1x1 x2x2 x3x3 x4x4 x5x5 Conditioned on deciding x 0 correctly, does U accept x 1 w.p. ½? If yes, set x 1 L If no, set x 1 L Conditioned on deciding x 0 and x 1 correctly, does U accept x 2 w.p. ½? If yes, set x 2 L If no, set x 2 L

28. For any k, defines a language L that does not have quantum circuits of size n k Why? Intuitively, each iteration cuts the number of potential circuits in half, but there were at most circuits to begin with On the other hand, clearly L P PP Even works for quantum circuits with quantum advice!

29. 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 x t and quantum circuits C t. Repeatedly use postselection to find An input x t on which C t fails A circuit C t+1 that succeeds on x 1,…,x t Even works for quantum circuits with quantum advice!

30. And now for a grand finale… 0-1-NPC - #AC 0 - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC1 - A 0 PP - AC - AC0 - AC 0 [m] - ACC 0 - 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 - BP H SPACE(f(n)) - BPL - BPNP - BPP - BPP cc - BPP KT - BPP-OBDD - BPP path - BPQP - BPSPACE(f(n)) - BPTIME(f(n)) - BQNC - BQNP - BQP - BQP/log - BQP/poly - BQP/qlog - BQP/qpoly - BQP-OBDD - BQP tt /poly - BQTIME(f(n)) - k-BWBP - C = AC 0 - C = L - C = P - CFL - CLOG - CH - Check - C k P - CNP - coAM - coC = P - cofrIP - Coh - coMA - coModkP - compIP - compNP - coNE - coNEXP - coNL - coNP - coNP cc - coNP/poly - coNQP - coRE - coRNC - coRP - coSL - coUCC - coUP - CP - CSIZE(f(n)) - CSL - CZK - D#P - Δ 2 P - δ-BPP - δ-RP - DET - DiffAC 0 - DisNP - DistNP - DP - DQP - DSPACE(f(n)) - DTIME(f(n)) - DTISP(t(n),s(n)) - Dyn-FO - Dyn-ThC 0 - E - EE - EEE - EESPACE - EEXP - EH - ELEMENTARY - EL k P - 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 - FP NP [log] - FPR - FPRAS - FPT - FPT nu - FPT su - FPTAS - FQMA - frIP - F- TAPE(f(n)) - F-TIME(f(n)) - GA - GAN-SPACE(f(n)) - GapAC 0 - GapL - GapP - GC(s(n),C) - GI - GPCD(r(n),q(n)) - G[t] - HeurBPP - HeurBPTIME(f(n)) - H k P - HVSZK - IC[log,poly] - IP - IPP - L - LIN - L k P - LOGCFL - LogFew - LogFewNL - LOGNP - LOGSNP - L/poly - LWPP - MA - MA' - MAC 0 - MA-E - MA-EXP - mAL - MaxNP - MaxPB - MaxSNP - MaxSNP 0 - mcoNL - MinPB - MIP - MIP*[2,1] - MIPEXP - (M k )P - mL - mNC 1 - mNL - mNP - Mod k L - Mod k P - ModP - ModZ k L - mP - MP - MPC - mP/poly - mTC 0 - NC - NC 0 - NC 1 - NC 2 - NE - NE/poly - NEE - NEEE - NEEXP - NEXP - NEXP/poly - NIQSZK - NISZK - NISZK h - NL - NL/poly - NLIN - NLOG - NP - NPC - NP cc - NPC - NPI - NP coNP - (NP coNP)/poly - NP/log - NPMV - NPMV-sel - NPMV t - NPMV t -sel - NPO - NPOPB - NP/poly - (NP,P-samplable) - NPR - NPSPACE - NPSV - NPSV-sel - NPSV t - NPSV t -sel - NQP - NSPACE(f(n)) - NT - NTIME(f(n)) - OCQ - OptP - P - P/log - P/poly - P #P - P #P [1] - PAC 0 - PBP - k-PBP - PC - P cc - PCD(r(n),q(n)) - P-close - PCP(r(n),q(n)) - PermUP - PEXP - PF - PFCHK(t(n)) - PH - PH cc - Φ 2 P - PhP - Π 2 P - PINC - PIO - PK - PKC - PL - PL 1 - PL infinity - PLF - PLL - PLS - P NP - P NP[k] - P NP[log] - P NP[log^2] - P-OBDD - PODN - polyL - PostBQP - PP - PP/poly - PPA - PPAD - PPADS - PPP - P PP - PPSPACE - PQUERY - PR - PR - Pr H SPACE(f(n)) - PromiseBPP - PromiseBQP - PromiseP - PromiseRP - PrSPACE(f(n)) - P-Sel - PSK - PSPACE - PT 1 - PTAPE - PTAS - PT/WK(f(n),g(n)) - PZK - QAC 0 - QAC 0 [m] - QACC 0 - QAM - QCFL - QCMA - QH - QIP - QIP(2) - QMA - QMA+ - QMA(2) - QMA log - QMAM - QMIP - QMIP le - QMIP ne - QNC 0 - QNCf 0 - QNC 1 - QP - QPLIN - QPSPACE - QSZK - R - RE - REG - RevSPACE(f(n)) - RHL - RL - RNC - RP - RPP - RSPACE(f(n)) - S 2 P - S 2 -EXPP NP - SAC - SAC 0 - SAC 1 - SAPTIME - SBP - SC - SEH - SelfNP - SF k - Σ 2 P - SKC - SL - SLICEWISE PSPACE - SNP - SO-E - SP - SP - span-P - SPARSE - SPL - SPP - SUBEXP - symP - SZK - SZK h - TALLY - TC 0 - TFNP - Θ 2 P - TreeBQP - TREE-REGULAR - UAP - UCC - UE - UL - UL/poly - UP - US - VNC k - VNP k - VP k - VQP k - W[1] - WAPP - W[P] - WPP - W[SAT] - W[*] - W[t] - W*[t] - XOR-MIP*[2,1] - XP - XP uniform - YACC - ZPE - ZPP - ZPTIME(f(n))

31. Quantum Karp-Lipton Theorem If PP BQP/qpoly, then the counting hierarchyconsisting of etc.collapses to PP But theres more: With no assumptions, PP does not have quantum circuits of size n k And more: PEXP requires quantum circuits of size f(n), where f(f(n)) 2 n

32. Even Stronger Separations QMA EXP (a subclass of PEXP) is not in BQP/qpoly QCMA EXP (a subclass of QMA EXP ) is not in BQP/poly A 0 PP (a subclass of PP) does not have quantum circuits of size n k NONRELATIVIZING

33. Conclusions I started out with a weird philosophical question Try itit works! I ended up with seven or eight results