$ \newcommand{\undefined}{} \newcommand{\hfill}{} \newcommand{\qedhere}{\square} \newcommand{\qed}{\square} \newcommand{\ensuremath}[1]{#1} \newcommand{\bit}{\{0,1\}} \newcommand{\Bit}{\{-1,1\}} \newcommand{\Stab}{\mathbf{Stab}} \newcommand{\NS}{\mathbf{NS}} \newcommand{\ba}{\mathbf{a}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bh}{\mathbf{h}} \newcommand{\br}{\mathbf{r}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\Var}{\mathbf{Var}} \newcommand{\dist}{\text{dist}} \newcommand{\norm}[1]{\\|#1\\|} \newcommand{\etal} \newcommand{\ie} \newcommand{\eg} \newcommand{\cf} \newcommand{\rank}{\text{rank}} \newcommand{\tr}{\text{tr}} \newcommand{\mor}{\text{Mor}} \newcommand{\hom}{\text{Hom}} \newcommand{\id}{\text{id}} \newcommand{\obj}{\text{obj}} \newcommand{\pr}{\text{pr}} \newcommand{\ker}{\text{ker}} \newcommand{\coker}{\text{coker}} \newcommand{\im}{\text{im}} \newcommand{\vol}{\text{vol}} \newcommand{\disc}{\text{disc}} \newcommand{\bbA}{\mathbb A} \newcommand{\bbB}{\mathbb B} \newcommand{\bbC}{\mathbb C} \newcommand{\bbD}{\mathbb D} \newcommand{\bbE}{\mathbb E} \newcommand{\bbF}{\mathbb F} \newcommand{\bbG}{\mathbb G} \newcommand{\bbH}{\mathbb H} \newcommand{\bbI}{\mathbb I} \newcommand{\bbJ}{\mathbb J} \newcommand{\bbK}{\mathbb K} \newcommand{\bbL}{\mathbb L} \newcommand{\bbM}{\mathbb M} \newcommand{\bbN}{\mathbb N} \newcommand{\bbO}{\mathbb O} \newcommand{\bbP}{\mathbb P} \newcommand{\bbQ}{\mathbb Q} \newcommand{\bbR}{\mathbb R} \newcommand{\bbS}{\mathbb S} \newcommand{\bbT}{\mathbb T} \newcommand{\bbU}{\mathbb U} \newcommand{\bbV}{\mathbb V} \newcommand{\bbW}{\mathbb W} \newcommand{\bbX}{\mathbb X} \newcommand{\bbY}{\mathbb Y} \newcommand{\bbZ}{\mathbb Z} \newcommand{\sA}{\mathscr A} \newcommand{\sB}{\mathscr B} \newcommand{\sC}{\mathscr C} \newcommand{\sD}{\mathscr D} \newcommand{\sE}{\mathscr E} \newcommand{\sF}{\mathscr F} \newcommand{\sG}{\mathscr G} \newcommand{\sH}{\mathscr H} \newcommand{\sI}{\mathscr I} \newcommand{\sJ}{\mathscr J} \newcommand{\sK}{\mathscr K} \newcommand{\sL}{\mathscr L} \newcommand{\sM}{\mathscr M} \newcommand{\sN}{\mathscr N} \newcommand{\sO}{\mathscr O} \newcommand{\sP}{\mathscr P} \newcommand{\sQ}{\mathscr Q} \newcommand{\sR}{\mathscr R} \newcommand{\sS}{\mathscr S} \newcommand{\sT}{\mathscr T} \newcommand{\sU}{\mathscr U} \newcommand{\sV}{\mathscr V} \newcommand{\sW}{\mathscr W} \newcommand{\sX}{\mathscr X} \newcommand{\sY}{\mathscr Y} \newcommand{\sZ}{\mathscr Z} \newcommand{\sfA}{\mathsf A} \newcommand{\sfB}{\mathsf B} \newcommand{\sfC}{\mathsf C} \newcommand{\sfD}{\mathsf D} \newcommand{\sfE}{\mathsf E} \newcommand{\sfF}{\mathsf F} \newcommand{\sfG}{\mathsf G} \newcommand{\sfH}{\mathsf H} \newcommand{\sfI}{\mathsf I} \newcommand{\sfJ}{\mathsf J} \newcommand{\sfK}{\mathsf K} \newcommand{\sfL}{\mathsf L} \newcommand{\sfM}{\mathsf M} \newcommand{\sfN}{\mathsf N} \newcommand{\sfO}{\mathsf O} \newcommand{\sfP}{\mathsf P} \newcommand{\sfQ}{\mathsf Q} \newcommand{\sfR}{\mathsf R} \newcommand{\sfS}{\mathsf S} \newcommand{\sfT}{\mathsf T} \newcommand{\sfU}{\mathsf U} \newcommand{\sfV}{\mathsf V} \newcommand{\sfW}{\mathsf W} \newcommand{\sfX}{\mathsf X} \newcommand{\sfY}{\mathsf Y} \newcommand{\sfZ}{\mathsf Z} \newcommand{\cA}{\mathcal A} \newcommand{\cB}{\mathcal B} \newcommand{\cC}{\mathcal C} \newcommand{\cD}{\mathcal D} \newcommand{\cE}{\mathcal E} \newcommand{\cF}{\mathcal F} \newcommand{\cG}{\mathcal G} \newcommand{\cH}{\mathcal H} \newcommand{\cI}{\mathcal I} \newcommand{\cJ}{\mathcal J} \newcommand{\cK}{\mathcal K} \newcommand{\cL}{\mathcal L} \newcommand{\cM}{\mathcal M} \newcommand{\cN}{\mathcal N} \newcommand{\cO}{\mathcal O} \newcommand{\cP}{\mathcal P} \newcommand{\cQ}{\mathcal Q} \newcommand{\cR}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cT}{\mathcal T} \newcommand{\cU}{\mathcal U} \newcommand{\cV}{\mathcal V} \newcommand{\cW}{\mathcal W} \newcommand{\cX}{\mathcal X} \newcommand{\cY}{\mathcal Y} \newcommand{\cZ}{\mathcal Z} \newcommand{\bfA}{\mathbf A} \newcommand{\bfB}{\mathbf B} \newcommand{\bfC}{\mathbf C} \newcommand{\bfD}{\mathbf D} \newcommand{\bfE}{\mathbf E} \newcommand{\bfF}{\mathbf F} \newcommand{\bfG}{\mathbf G} \newcommand{\bfH}{\mathbf H} \newcommand{\bfI}{\mathbf I} \newcommand{\bfJ}{\mathbf J} \newcommand{\bfK}{\mathbf K} \newcommand{\bfL}{\mathbf L} \newcommand{\bfM}{\mathbf M} \newcommand{\bfN}{\mathbf N} \newcommand{\bfO}{\mathbf O} \newcommand{\bfP}{\mathbf P} \newcommand{\bfQ}{\mathbf Q} \newcommand{\bfR}{\mathbf R} \newcommand{\bfS}{\mathbf S} \newcommand{\bfT}{\mathbf T} \newcommand{\bfU}{\mathbf U} \newcommand{\bfV}{\mathbf V} \newcommand{\bfW}{\mathbf W} \newcommand{\bfX}{\mathbf X} \newcommand{\bfY}{\mathbf Y} \newcommand{\bfZ}{\mathbf Z} \newcommand{\rmA}{\mathrm A} \newcommand{\rmB}{\mathrm B} \newcommand{\rmC}{\mathrm C} \newcommand{\rmD}{\mathrm D} \newcommand{\rmE}{\mathrm E} \newcommand{\rmF}{\mathrm F} \newcommand{\rmG}{\mathrm G} \newcommand{\rmH}{\mathrm H} \newcommand{\rmI}{\mathrm I} \newcommand{\rmJ}{\mathrm J} \newcommand{\rmK}{\mathrm K} \newcommand{\rmL}{\mathrm L} \newcommand{\rmM}{\mathrm M} \newcommand{\rmN}{\mathrm N} \newcommand{\rmO}{\mathrm O} \newcommand{\rmP}{\mathrm P} \newcommand{\rmQ}{\mathrm Q} \newcommand{\rmR}{\mathrm R} \newcommand{\rmS}{\mathrm S} \newcommand{\rmT}{\mathrm T} \newcommand{\rmU}{\mathrm U} \newcommand{\rmV}{\mathrm V} \newcommand{\rmW}{\mathrm W} \newcommand{\rmX}{\mathrm X} \newcommand{\rmY}{\mathrm Y} \newcommand{\rmZ}{\mathrm Z} \newcommand{\bb}{\mathbf{b}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bw}{\mathbf{w}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\paren}[1]{( #1 )} \newcommand{\Paren}[1]{\left( #1 \right)} \newcommand{\bigparen}[1]{\bigl( #1 \bigr)} \newcommand{\Bigparen}[1]{\Bigl( #1 \Bigr)} \newcommand{\biggparen}[1]{\biggl( #1 \biggr)} \newcommand{\Biggparen}[1]{\Biggl( #1 \Biggr)} \newcommand{\abs}[1]{\lvert #1 \rvert} \newcommand{\Abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\bigabs}[1]{\bigl\lvert #1 \bigr\rvert} \newcommand{\Bigabs}[1]{\Bigl\lvert #1 \Bigr\rvert} \newcommand{\biggabs}[1]{\biggl\lvert #1 \biggr\rvert} \newcommand{\Biggabs}[1]{\Biggl\lvert #1 \Biggr\rvert} \newcommand{\card}[1]{\left| #1 \right|} \newcommand{\Card}[1]{\left\lvert #1 \right\rvert} \newcommand{\bigcard}[1]{\bigl\lvert #1 \bigr\rvert} \newcommand{\Bigcard}[1]{\Bigl\lvert #1 \Bigr\rvert} \newcommand{\biggcard}[1]{\biggl\lvert #1 \biggr\rvert} \newcommand{\Biggcard}[1]{\Biggl\lvert #1 \Biggr\rvert} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\Norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\bignorm}[1]{\bigl\lVert #1 \bigr\rVert} \newcommand{\Bignorm}[1]{\Bigl\lVert #1 \Bigr\rVert} \newcommand{\biggnorm}[1]{\biggl\lVert #1 \biggr\rVert} \newcommand{\Biggnorm}[1]{\Biggl\lVert #1 \Biggr\rVert} \newcommand{\iprod}[1]{\langle #1 \rangle} \newcommand{\Iprod}[1]{\left\langle #1 \right\rangle} \newcommand{\bigiprod}[1]{\bigl\langle #1 \bigr\rangle} \newcommand{\Bigiprod}[1]{\Bigl\langle #1 \Bigr\rangle} \newcommand{\biggiprod}[1]{\biggl\langle #1 \biggr\rangle} \newcommand{\Biggiprod}[1]{\Biggl\langle #1 \Biggr\rangle} \newcommand{\set}[1]{\lbrace #1 \rbrace} \newcommand{\Set}[1]{\left\lbrace #1 \right\rbrace} \newcommand{\bigset}[1]{\bigl\lbrace #1 \bigr\rbrace} \newcommand{\Bigset}[1]{\Bigl\lbrace #1 \Bigr\rbrace} \newcommand{\biggset}[1]{\biggl\lbrace #1 \biggr\rbrace} \newcommand{\Biggset}[1]{\Biggl\lbrace #1 \Biggr\rbrace} \newcommand{\bracket}[1]{\lbrack #1 \rbrack} \newcommand{\Bracket}[1]{\left\lbrack #1 \right\rbrack} \newcommand{\bigbracket}[1]{\bigl\lbrack #1 \bigr\rbrack} \newcommand{\Bigbracket}[1]{\Bigl\lbrack #1 \Bigr\rbrack} \newcommand{\biggbracket}[1]{\biggl\lbrack #1 \biggr\rbrack} \newcommand{\Biggbracket}[1]{\Biggl\lbrack #1 \Biggr\rbrack} \newcommand{\ucorner}[1]{\ulcorner #1 \urcorner} \newcommand{\Ucorner}[1]{\left\ulcorner #1 \right\urcorner} \newcommand{\bigucorner}[1]{\bigl\ulcorner #1 \bigr\urcorner} \newcommand{\Bigucorner}[1]{\Bigl\ulcorner #1 \Bigr\urcorner} \newcommand{\biggucorner}[1]{\biggl\ulcorner #1 \biggr\urcorner} \newcommand{\Biggucorner}[1]{\Biggl\ulcorner #1 \Biggr\urcorner} \newcommand{\ceil}[1]{\lceil #1 \rceil} \newcommand{\Ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\bigceil}[1]{\bigl\lceil #1 \bigr\rceil} \newcommand{\Bigceil}[1]{\Bigl\lceil #1 \Bigr\rceil} \newcommand{\biggceil}[1]{\biggl\lceil #1 \biggr\rceil} \newcommand{\Biggceil}[1]{\Biggl\lceil #1 \Biggr\rceil} \newcommand{\floor}[1]{\lfloor #1 \rfloor} \newcommand{\Floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\bigfloor}[1]{\bigl\lfloor #1 \bigr\rfloor} \newcommand{\Bigfloor}[1]{\Bigl\lfloor #1 \Bigr\rfloor} \newcommand{\biggfloor}[1]{\biggl\lfloor #1 \biggr\rfloor} \newcommand{\Biggfloor}[1]{\Biggl\lfloor #1 \Biggr\rfloor} \newcommand{\lcorner}[1]{\llcorner #1 \lrcorner} \newcommand{\Lcorner}[1]{\left\llcorner #1 \right\lrcorner} \newcommand{\biglcorner}[1]{\bigl\llcorner #1 \bigr\lrcorner} \newcommand{\Biglcorner}[1]{\Bigl\llcorner #1 \Bigr\lrcorner} \newcommand{\bigglcorner}[1]{\biggl\llcorner #1 \biggr\lrcorner} \newcommand{\Bigglcorner}[1]{\Biggl\llcorner #1 \Biggr\lrcorner} \newcommand{\ket}[1]{| #1 \rangle} \newcommand{\bra}[1]{\langle #1 |} \newcommand{\braket}[2]{\langle #1 | #2 \rangle} \newcommand{\ketbra}[1]{| #1 \rangle\langle #1 |} \newcommand{\e}{\varepsilon} \newcommand{\eps}{\varepsilon} \newcommand{\from}{\colon} \newcommand{\super}[2]{#1^{(#2)}} \newcommand{\varsuper}[2]{#1^{\scriptscriptstyle (#2)}} \newcommand{\tensor}{\otimes} \newcommand{\eset}{\emptyset} \newcommand{\sse}{\subseteq} \newcommand{\sst}{\substack} \newcommand{\ot}{\otimes} \newcommand{\Esst}[1]{\bbE_{\substack{#1}}} \newcommand{\vbig}{\vphantom{\bigoplus}} \newcommand{\seteq}{\mathrel{\mathop:}=} \newcommand{\defeq}{\stackrel{\mathrm{def}}=} \newcommand{\Mid}{\mathrel{}\middle|\mathrel{}} \newcommand{\Ind}{\mathbf 1} \newcommand{\bits}{\{0,1\}} \newcommand{\sbits}{\{\pm 1\}} \newcommand{\R}{\mathbb R} \newcommand{\Rnn}{\R_{\ge 0}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\C}{\mathbb C} \newcommand{\A}{\mathbb A} \newcommand{\Real}{\mathbb R} \newcommand{\mper}{\,.} \newcommand{\mcom}{\,,} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\vol}{vol} \DeclareMathOperator{\val}{val} \DeclareMathOperator{\opt}{opt} \DeclareMathOperator{\Opt}{Opt} \DeclareMathOperator{\Val}{Val} \DeclareMathOperator{\LP}{LP} \DeclareMathOperator{\SDP}{SDP} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Inf}{Inf} \DeclareMathOperator{\size}{size} \DeclareMathOperator{\poly}{poly} \DeclareMathOperator{\polylog}{polylog} \DeclareMathOperator{\min}{min} \DeclareMathOperator{\max}{max} \DeclareMathOperator{\argmax}{arg\,max} \DeclareMathOperator{\argmin}{arg\,min} \DeclareMathOperator{\qpoly}{qpoly} \DeclareMathOperator{\qqpoly}{qqpoly} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\Conv}{Conv} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\perm}{perm} \DeclareMathOperator{\mspan}{span} \DeclareMathOperator{\mrank}{rank} \DeclareMathOperator{\E}{\mathbb E} \DeclareMathOperator{\pE}{\tilde{\mathbb E}} \DeclareMathOperator{\Pr}{\mathbb P} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\Cone}{Cone} \DeclareMathOperator{\junta}{junta} \DeclareMathOperator{\NSS}{NSS} \DeclareMathOperator{\SA}{SA} \DeclareMathOperator{\SOS}{SOS} \DeclareMathOperator{\Stab}{\mathbf Stab} \DeclareMathOperator{\Det}{\textbf{Det}} \DeclareMathOperator{\Perm}{\textbf{Perm}} \DeclareMathOperator{\Sym}{\textbf{Sym}} \DeclareMathOperator{\Pow}{\textbf{Pow}} \DeclareMathOperator{\Gal}{\textbf{Gal}} \DeclareMathOperator{\Aut}{\textbf{Aut}} \newcommand{\iprod}[1]{\langle #1 \rangle} \newcommand{\cE}{\mathcal{E}} \newcommand{\E}{\mathbb{E}} \newcommand{\pE}{\tilde{\mathbb{E}}} \newcommand{\N}{\mathbb{N}} \renewcommand{\P}{\mathcal{P}} \notag $
$ \newcommand{\sleq}{\ensuremath{\preceq}} \newcommand{\sgeq}{\ensuremath{\succeq}} \newcommand{\diag}{\ensuremath{\mathrm{diag}}} \newcommand{\support}{\ensuremath{\mathrm{support}}} \newcommand{\zo}{\ensuremath{\{0,1\}}} \newcommand{\pmo}{\ensuremath{\{\pm 1\}}} \newcommand{\uppersos}{\ensuremath{\overline{\mathrm{sos}}}} \newcommand{\lambdamax}{\ensuremath{\lambda_{\mathrm{max}}}} \newcommand{\rank}{\ensuremath{\mathrm{rank}}} \newcommand{\Mslow}{\ensuremath{M_{\mathrm{slow}}}} \newcommand{\Mfast}{\ensuremath{M_{\mathrm{fast}}}} \newcommand{\Mdiag}{\ensuremath{M_{\mathrm{diag}}}} \newcommand{\Mcross}{\ensuremath{M_{\mathrm{cross}}}} \newcommand{\eqdef}{\ensuremath{ =^{def}}} \newcommand{\threshold}{\ensuremath{\mathrm{threshold}}} \newcommand{\vbls}{\ensuremath{\mathrm{vbls}}} \newcommand{\cons}{\ensuremath{\mathrm{cons}}} \newcommand{\edges}{\ensuremath{\mathrm{edges}}} \newcommand{\cl}{\ensuremath{\mathrm{cl}}} \newcommand{\xor}{\ensuremath{\oplus}} \newcommand{\1}{\ensuremath{\mathrm{1}}} \notag $
$ \newcommand{\transpose}[1]{\ensuremath{#1{}^{\mkern-2mu\intercal}}} \newcommand{\dyad}[1]{\ensuremath{#1#1{}^{\mkern-2mu\intercal}}} \newcommand{\nchoose}[1]{\ensuremath} \newcommand{\generated}[1]{\ensuremath{\langle #1 \rangle}} \notag $
$ \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\R} % real numbers \newcommand{\N}} % natural numbers \newcommand{\Z} % integers \newcommand{\F} % a field \newcommand{\Q} % the rationals \newcommand{\C}{\mathbb{C}} % the complexes \newcommand{\poly}} \newcommand{\polylog}} \newcommand{\loglog}}} \newcommand{\zo}{\{0,1\}} \newcommand{\suchthat} \newcommand{\pr}[1]{\Pr\left[#1\right]} \newcommand{\deffont}{\em} \newcommand{\getsr}{\mathbin{\stackrel{\mbox{\tiny R}}{\gets}}} \newcommand{\Exp}{\mathop{\mathrm E}\displaylimits} % expectation \newcommand{\Var}{\mathop{\mathrm Var}\displaylimits} % variance \newcommand{\xor}{\oplus} \newcommand{\GF}{\mathrm{GF}} \newcommand{\eps}{\varepsilon} \notag $
$ \newcommand{\class}[1]{\mathbf{#1}} \newcommand{\coclass}[1]{\mathbf{co\mbox{-}#1}} % and their complements \newcommand{\BPP}{\class{BPP}} \newcommand{\NP}{\class{NP}} \newcommand{\RP}{\class{RP}} \newcommand{\coRP}{\coclass{RP}} \newcommand{\ZPP}{\class{ZPP}} \newcommand{\BQP}{\class{BQP}} \newcommand{\FP}{\class{FP}} \newcommand{\QP}{\class{QuasiP}} \newcommand{\VF}{\class{VF}} \newcommand{\VBP}{\class{VBP}} \newcommand{\VP}{\class{VP}} \newcommand{\VNP}{\class{VNP}} \newcommand{\RNC}{\class{RNC}} \newcommand{\RL}{\class{RL}} \newcommand{\BPL}{\class{BPL}} \newcommand{\coRL}{\coclass{RL}} \newcommand{\IP}{\class{IP}} \newcommand{\AM}{\class{AM}} \newcommand{\MA}{\class{MA}} \newcommand{\QMA}{\class{QMA}} \newcommand{\SBP}{\class{SBP}} \newcommand{\coAM}{\class{coAM}} \newcommand{\coMA}{\class{coMA}} \renewcommand{\P}{\class{P}} \newcommand\prBPP{\class{prBPP}} \newcommand\prRP{\class{prRP}} \newcommand\prP{\class{prP}} \newcommand{\Ppoly}{\class{P/poly}} \newcommand{\NPpoly}{\class{NP/poly}} \newcommand{\coNPpoly}{\class{coNP/poly}} \newcommand{\DTIME}{\class{DTIME}} \newcommand{\TIME}{\class{TIME}} \newcommand{\SIZE}{\class{SIZE}} \newcommand{\SPACE}{\class{SPACE}} \newcommand{\ETIME}{\class{E}} \newcommand{\BPTIME}{\class{BPTIME}} \newcommand{\RPTIME}{\class{RPTIME}} \newcommand{\ZPTIME}{\class{ZPTIME}} \newcommand{\EXP}{\class{EXP}} \newcommand{\ZPEXP}{\class{ZPEXP}} \newcommand{\RPEXP}{\class{RPEXP}} \newcommand{\BPEXP}{\class{BPEXP}} \newcommand{\SUBEXP}{\class{SUBEXP}} \newcommand{\NTIME}{\class{NTIME}} \newcommand{\NL}{\class{NL}} \renewcommand{\L}{\class{L}} \newcommand{\NQP}{\class{NQP}} \newcommand{\NEXP}{\class{NEXP}} \newcommand{\coNEXP}{\coclass{NEXP}} \newcommand{\NPSPACE}{\class{NPSPACE}} \newcommand{\PSPACE}{\class{PSPACE}} \newcommand{\NSPACE}{\class{NSPACE}} \newcommand{\coNSPACE}{\coclass{NSPACE}} \newcommand{\coL}{\coclass{L}} \newcommand{\coP}{\coclass{P}} \newcommand{\coNP}{\coclass{NP}} \newcommand{\coNL}{\coclass{NL}} \newcommand{\coNPSPACE}{\coclass{NPSPACE}} \newcommand{\APSPACE}{\class{APSPACE}} \newcommand{\LINSPACE}{\class{LINSPACE}} \newcommand{\qP}{\class{\tilde{P}}} \newcommand{\PH}{\class{PH}} \newcommand{\EXPSPACE}{\class{EXPSPACE}} \newcommand{\SigmaTIME}[1]{\class{\Sigma_{#1}TIME}} \newcommand{\PiTIME}[1]{\class{\Pi_{#1}TIME}} \newcommand{\SigmaP}[1]{\class{\Sigma_{#1}P}} \newcommand{\PiP}[1]{\class{\Pi_{#1}P}} \newcommand{\DeltaP}[1]{\class{\Delta_{#1}P}} \newcommand{\ATIME}{\class{ATIME}} \newcommand{\ASPACE}{\class{ASPACE}} \newcommand{\AP}{\class{AP}} \newcommand{\AL}{\class{AL}} \newcommand{\APSPACE}{\class{APSPACE}} \newcommand{\VNC}[1]{\class{VNC^{#1}}} \newcommand{\NC}[1]{\class{NC^{#1}}} \newcommand{\AC}[1]{\class{AC^{#1}}} \newcommand{\ACC}[1]{\class{ACC^{#1}}} \newcommand{\TC}[1]{\class{TC^{#1}}} \newcommand{\ShP}{\class{\# P}} \newcommand{\PaP}{\class{\oplus P}} \newcommand{\PCP}{\class{PCP}} \newcommand{\kMIP}[1]{\class{#1\mbox{-}MIP}} \newcommand{\MIP}{\class{MIP}} $
$ \newcommand{\textprob}[1]{\text{#1}} \newcommand{\mathprob}[1]{\textbf{#1}} \newcommand{\Satisfiability}{\textprob{Satisfiability}} \newcommand{\SAT}{\textprob{SAT}} \newcommand{\TSAT}{\textprob{3SAT}} \newcommand{\USAT}{\textprob{USAT}} \newcommand{\UNSAT}{\textprob{UNSAT}} \newcommand{\QPSAT}{\textprob{QPSAT}} \newcommand{\TQBF}{\textprob{TQBF}} \newcommand{\LinProg}{\textprob{Linear Programming}} \newcommand{\LP}{\mathprob{LP}} \newcommand{\Factor}{\textprob{Factoring}} \newcommand{\CircVal}{\textprob{Circuit Value}} \newcommand{\CVAL}{\mathprob{CVAL}} \newcommand{\CircSat}{\textprob{Circuit Satisfiability}} \newcommand{\CSAT}{\textprob{CSAT}} \newcommand{\CycleCovers}{\textprob{Cycle Covers}} \newcommand{\MonCircVal}{\textprob{Monotone Circuit Value}} \newcommand{\Reachability}{\textprob{Reachability}} \newcommand{\Unreachability}{\textprob{Unreachability}} \newcommand{\RCH}{\mathprob{RCH}} \newcommand{\BddHalt}{\textprob{Bounded Halting}} \newcommand{\BH}{\mathprob{BH}} \newcommand{\DiscreteLog}{\textprob{Discrete Log}} \newcommand{\REE}{\mathprob{REE}} \newcommand{\QBF}{\mathprob{QBF}} \newcommand{\MCSP}{\mathprob{MCSP}} \newcommand{\GGEO}{\mathprob{GGEO}} \newcommand{\CKTMIN}{\mathprob{CKT-MIN}} \newcommand{\MINCKT}{\mathprob{MIN-CKT}} \newcommand{\IdentityTest}{\textprob{Identity Testing}} \newcommand{\Majority}{\textprob{Majority}} \newcommand{\CountIndSets}{\textprob{\#Independent Sets}} \newcommand{\Parity}{\textprob{Parity}} \newcommand{\Clique}{\textprob{Clique}} \newcommand{\CountCycles}{\textprob{#Cycles}} \newcommand{\CountPerfMatchings}{\textprob{\#Perfect Matchings}} \newcommand{\CountMatchings}{\textprob{\#Matchings}} \newcommand{\CountMatch}{\mathprob{\#Matchings}} \newcommand{\ECSAT}{\mathprob{E#SAT}} \newcommand{\ShSAT}{\mathprob{#SAT}} \newcommand{\ShTSAT}{\mathprob{#3SAT}} \newcommand{\HamCycle}{\textprob{Hamiltonian Cycle}} \newcommand{\Permanent}{\textprob{Permanent}} \newcommand{\ModPermanent}{\textprob{Modular Permanent}} \newcommand{\GraphNoniso}{\textprob{Graph Nonisomorphism}} \newcommand{\GI}{\mathprob{GI}} \newcommand{\GNI}{\mathprob{GNI}} \newcommand{\GraphIso}{\textprob{Graph Isomorphism}} \newcommand{\QuantBoolForm}{\textprob{Quantified Boolean Formulae}} \newcommand{\GenGeography}{\textprob{Generalized Geography}} \newcommand{\MAXTSAT}{\mathprob{Max3SAT}} \newcommand{\GapMaxTSAT}{\mathprob{GapMax3SAT}} \newcommand{\ELIN}{\mathprob{E3LIN2}} \newcommand{\CSP}{\mathprob{CSP}} \newcommand{\Lin}{\mathprob{Lin}} \newcommand{\ONE}{\mathbf{ONE}} \newcommand{\ZERO}{\mathbf{ZERO}} \newcommand{\yes} \newcommand{\no} $


〈致謝〉





First and foremost, I would like to express my gratitude to my Ph.D. advisor, Professor Boaz Barak. His expansive vision in computer science and its intersection with other fields has had a profound impact on me. I am also grateful to Boaz for introducing me to many physics topics covered in this book. Without his unconditional support and trust in my abilities, this book would not have been possible.

I would also like to thank all of my collaborators who joined me on this exciting journey, exploring various research topics and diverse fields. Special thanks go to the following individuals:

To Xun Gao: Thank you for your endless patience in teaching me physics. I also enjoyed our open-ended discussions on science and research.

To Wei-Ping Chan: Thank you for your guidance as I navigated my initial struggles with biology. You opened my eyes to the beauty of the natural world.

To Brabeeba Wang: Thank you for inspiring me with our conversations on mathematics, neuroscience, music, and food.

To Lisa Yang: Thank you for exploring physics from a computer science perspective with me, particularly in the areas of quantum and gravity.

To Kai-Min Chung (my former advisor): Thank you for showing me how to think deeply and ask questions. I look forward to hearing your thoughts on the neuroscience part of this book.

To SueYeon Chung (my future advisor): Thank you for introducing me to the interplay between statistical physics and neuroscience.

I also want to thank the six guest speakers (Lijie Chen, Xun Gao, Khadijeh Sona Najafi, Angel Hsing-Chi Hwang, Zhiqian Wang, and Brabeeba Wang) and the twelve teaching assistants (Avantika Agarwal, Kartikeya Arora, Salvador Buse, Aaron K. Clark, Erick Galinkin, Simone Maria Giancola, Aman Goel, Reijo Jaakkola, Pawan Mishra, K Prahlad Narasimhan, Tirukkovalluri Sri Sowmya, and Mostafa Touny) in my mini-course.

最後，我想感謝本書撰寫過程中曾經提供許多寶貴建議的朋友：Teresa Huang、余天呈、侯宗誠、周咸正、詹偉平，以及本書的封面及編排設計師楊涵婷。