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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 Bag o 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 of Postselection

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

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

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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 IS Postselection? BERKELEYCAMBRIDGE

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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-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

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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, and assume 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 a nondeterministic poly-time Turing machine that accepts iff the majority of its paths do NP PP P #P =P PP 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 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

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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

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

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

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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 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?

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How powerful is quantum advice? Could it let us solve problems that are not even recursively enumerable given classical 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) Closely related: for all (partial or total) Boolean functions f : {0,1} n {0,1} m {0,1},

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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

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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 at most (say) n ; 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

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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

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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!

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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!

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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))

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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

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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

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

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