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FORCES OF COMPLEXITY

FORCES OF COMPLEXITY. A  MAX Feature Presentation. BQP. =. P. PSPACE. 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).

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FORCES OF COMPLEXITY

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  1. FORCES OF COMPLEXITY A MAX Feature Presentation BQP = P PSPACE

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

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

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

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

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

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

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

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

  10. BERKELEY CAMBRIDGE What ISPostselection? Learning something about a question by conditioning on the fact that you’re asking it.

  11. Input: A Boolean formula  • Accept the many-worlds interpretation • Generate a random truth assignment X • If X doesn’t satisfy , kill yourself What about the quantum case? “Anthropic Computing”: A foolproof way to solve NP-complete problems in polynomial time

  12. In This Talk… I’ll study what you could do with a quantum computer, IF you could measure a qubit and postselect on its being |1 • 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!

  13. PostBQP I hereby define a newcomplexity class… (Postselected BQP) 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

  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 PSPACE P#P=PPP PP NP P

  15. 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 Theorem: PostBQP = PP

  16. Yields in first qubit From we can easily prepare 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 Goal: Decide if these amplitudes have the same or opposite signs Prepare |0|+|1H| for some ,.Then postselect on second qubit being |1

  17. Suppose s and 2n-2s are both positive Then by trying / = 2i for all i{-n,…,n}, we’ll eventually get close to Yields in first qubit To Show PP  PostBQP… On the other hand, if 2n-2s is negative, then we won’t. QED

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

  19. Beigel, Reingold, Spielman 1990: PP is “closed under intersection”Solved a problem that was open for 18 years… Observation: PostBQP is trivially closed under intersection  PP is too Given L1,L2PostBQP, to decide if xL1 and xL2, postselect on both computations succeeding, and accept iff they both accept Other Classical Results Proved With Quantum Techniques:Kerenidis & de Wolf, A., Aharonov & Regev, …

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

  21. Quantum Advice 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?” 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

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

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

  24. Given an input x, clearly lets Bob decide in PostBQP whether xL 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

  25. 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

  26. U: Picks a size-nk quantum circuit uniformly at random and runs it x0 x1 x2 x3 x4 x5 PPP Does Not Have Quantum Circuits of Size nk   Does U accept x0 w.p.  ½?If yes, set x0LIf no, set x0L Conditioned on deciding x0 correctly, does U accept x1 w.p.  ½?If yes, set x1LIf no, set x1L Conditioned on deciding x0 and x1 correctly, does U accept x2 w.p.  ½?If yes, set x2LIf no, set x2L

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

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

  29. 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 - NPcoNP - (NPcoNP)/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)) THE EDWARD H. FARHI CAPITAL LETTER PYROTECHNICS DISPLAY

  30. If PP  BQP/qpoly, then the counting hierarchy—consisting ofetc.—collapses to PP Quantum Karp-Lipton Theorem 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

  31. Even Stronger Separations QMAEXP (a subclass of PEXP) is not in BQP/qpoly QCMAEXP (a subclass of QMAEXP) is not in BQP/poly NONRELATIVIZING A0PP (a subclass of PP) does not have quantum circuits of size nk

  32. Conclusions I started out with a weird philosophical question I ended up with seven or eight results Try it—it works!

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