Quantum Certificate Complexity

Scott Aaronson

UC Berkeley

0-1-NPC - #L - #L/poly - #P - #W[t] - +EXP - +L - +L/poly - +P - +SAC1 - A0PP - AC - AC0 - AC0[m] - ACC0 - AH -

AL – AlgP/poly - AM - AM intersect coAM - AmpMP - AP - AP - APP - APP - APX - AVBPP - AvE - AvP - AW[P] - AWPP - AW[SAT] - AW[*] - AW[t] - βP - BH - BPE - BPEE - BPHSPACE(f(n)) - BPL - BPPKT - BPP-

OBDD - BPPpath - BPQP - BPSPACE(f(n)) - BPTIME(f(n)) - BQNC - BQNP - BQP-OBDD - BQP/log - BQP/qlog - BQTIME(f(n)) - k-BWBP - C=L - C=P - CFL - CLOG - CH - Check - CkP - CNP - coAM - coC=P - cofrIP - Coh -

coMA - coModkP - compIP - compNP -coNE - coNEXP - coNL - coNP - coNP/poly - coRE - coRNC - coRP -

coUCC - CP - CSIZE(f(n)) - CSL - CZK - D#P - Δ2P - δ-BPP - δ-RP - DET - DisNP - DistNP - DP - DSPACE(f(n))

- DTIME(f(n)) - Dyn-FO - Dyn-ThC0 - E - EE - EEE - EESPACE - EEXP - EH - ELEMENTARY - ELkP - EPTAS -

k-EQBP - EQP - EQTIME(f(n)) - ESPACE - EXP - EXP/poly - EXPSPACE - Few - FewP - FNL - FNL/poly - FNP - FO(t(n)) - FOLL – FPNP[log] - FPR - FPRAS - FPT - FPTnu - FPTsu - FPTAS - FQMA - frIP - F-TAPE(f(n)) - F-

TIME(f(n)) - GapL - GapP - GC(s(n),C) - GPCD(r(n),q(n)) - G[t] - HkP - HVSZK - IC[log,poly] - IP - 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 - MIPEXP - (Mk)P - mL - mNC1 - mNL -

mNP - ModkL - ModkP - ModP - ModZkL - mP - MP - MPC - mP/poly - mTC0 - NC - NC0 - NC1 - NC2 - NE - NEE -

NEEE - NEEXP - NEXP - NIQSZK - NISZK - NISZKh - NL - NLIN - NLOG - NL/poly - NPC - NPC - NPI - NP

intersect coNP - (NP intersect coNP)/poly - 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 - P#P[1] - PBP - k-PBP - PC - PCD(r(n),q(n)) - P-close - PCP(r(n),q(n)) - PEXP - PF -

PFCHK(t(n)) - Φ2P - PhP - Π2P - PK - PKC - PL - PL1 - PLinfinity - PLF - PLL - P/log - PNP - PNP[k] - PNP[log] - P-OBDD -

PODN - polyL - PP - PPA - PPAD - PPADS - P/poly - PPP - PQUERY - PR - PR - PrHSPACE(f(n)) - PromiseBPP -

PromiseBQP - PromiseP - PromiseRP - PrSPACE(f(n)) - P-Sel - PSK - PT1 - PTAPE - PTAS - PT/WK(f(n),g(n)) -

PZK - QAC0 - QAC0[m] - QACC0 - QAM - QCFL - QH - QIP - QIP(2) - QMA+ - QMA(2) - QMAlog - QMAM -

QMIP - QMIPle - QMIPne - QNC0 - QNCf0 - QNC1 - QP - QPSPACE - QSZK - R - RE - REG - RevSPACE(f(n)) -

RHL - RL - RNC - RPP - RSPACE(f(n)) - S2P - SAC - SAC0 - SAC1 - SBP - SC - SEH - SFk - Σ2P - SKC - SL -

SLICEWISE PSPACE - SNP - SO-E - SP - span-P - SPARSE - SPL - SPP - SUBEXP - symP - SZK - SZKh - TALLY - TC0 - Θ2P - TREE-REGULAR - UCC - UL - UL/poly - UP - US - VNCk - VNPk - VPk - VQPk - W[1] -

WAPP - W[P] - WPP - W[SAT] - W[*] - W[t] - W*[t] - XP - XPuniform - YACC - ZPE - ZPP

SHAMELESS PLUG

http://w

ww.cs.berkeley.edu/~aaronson/zoo.html

Overview• Most of what’s known about quantum computing

can be cast in the query complexity model

• Despite its simplicity, open problems abound

• We make progress on some of these by studying randomized certificate complexity RC(f) and quantum certificate complexity QC(f)

• Main results I’ll discuss today:

• We’ll need both big quantum lower bound methods (adversary method and polynomial method)

,QC f RC f 2

0 2 0 logR f O Q f Q f n

f:{0,1}n{0,1} is a total Boolean function

D(f) (deterministic query complexity)

R0(f) (zero-error randomized)

R2(f) (bounded-error randomized)

Q2(f) (bounded-error quantum)

Q0(f) (zero-error quantum)

QE(f) (exact quantum)

Background

1, , nf OR x x

Example

0

2

1 2

1

D OR n

R OR n

R OR n

0

2

EQ OR n

Q OR n

Q OR n

Certificate Complexity

CX(f) = min # of queries needed to distinguish X from every Y s.t. f(Y)f(X)

Block Sensitivity

bsX(f) = max # of disjoint blocks B{x1,…,xn} s.t. flipping B changes f(X)

Example: For f=MAJ(x1,x2,x3,x4,x5), letting X=11110,

11110 11110

CX(MAJ)=3 bsX(MAJ)=2

max X

XC f C f

max X

Xbs f bs f

Randomized Certificate Complexity

RCX(f) = min # of randomized queries needed to distinguish X from any Y s.t. f(Y)f(X) with ½ prob.

Quantum Certificate Complexity QC(f)

Example: For f=MAJ(x1,…,xn), letting X=00…0,

RCX(MAJ) = 1

Observations: Anything a prover might provide a verifier besides X, the verifier can compute for itself

One-sided and two-sided error are equivalent

Different notions of nondeterministic quantum query complexity: Watrous 2000, de Wolf 2002

max X

XRC f RC f

Let D0,D1 be distributions over f-1(0), f-1(1) s.t. D0 looks “locally similar” to every 1-input, and D1 looks “locally similar” to every 0-input:

Then

Ambainis’ Adversary Method(special case)

1

0

1

1

2

0 , 1,..., Pr

1 , 1,..., Pr .

1.

i iY D

i iX D

X f i n x y

Y f i n x y

Q f

Claim:

• Any randomized certificate for input X can be made nonadaptive with constant blowup

• By minimax theorem, exists distribution over {Y:f(Y)f(X)} s.t. for all i, xiyi w.p. O(1/RC(f))

• Adversary method then yields

• For upper bound, use “weighted Grover”

QC f RC f

RC f

g

g

1 29

0 12

1 13,14,15,16

0 17

k x x

if k

g k if k

if k

Example where C(f) = (QC(f)2.205)

New Quantum/Classical Relation

0

2

2 0

log

log

R f O RC f ndeg f n

O Q f Q f n

For total f,

where ndeg(f) = min degree of poly p s.t. p(X)0 f(X)=1

Previous: D(f) = O(Q2(f)2Q0(f)2) (de Wolf), D(f) = O(Q2(f)6) (Beals et al.)

Idea (follows Buhrman-de Wolf):

Let p be s.t. p(X)0 f(X)=1

Maxonomials of p are monomials not dominated by other monomials—i.e. maxonomials of x1x2 – x2 + 2x3 are x1x2, 2x3

Nisan-Smolensky: For every 0-input X and maxonomial M of p, X has a sensitive block whose variables are all in M

Consequence: Randomized 0-certificate must intersect each maxonomial w.p. ½

Randomized algorithm: Keep querying a randomized 0-certificate, until either one no longer exists or p=0

Lemma: O(ndeg(f) log n) iterations suffice w.h.p.

Proof: Let S be current set of monomials, and

Initially (S) nndeg(f) ndeg(f)!

We’re done when (S)=0

Claim: Each iteration decreases (S) by expected amount (S)/4e

Reason: 1/e of (S) is concentrated on maxonomials, each of which decreases in degree w.p. ½

deg !M S

S M

Local Proofs• When faced with a hard problem, analyze

limitations of known techniques (Baker-Gill-Solovay, Razborov-Rudich)

• Is• I claim that a ‘yes’ answer would require “global

analysis” of Boolean functions• Given nn lattice of bits X, let f(X)=1 if there’s a

square ‘frame’ of size n1/3n1/3, f(X)=0 otherwise

2?RC f o bs f

0 1/3

0 2/3

n

n

bs f n

RC f n

Open Problems

• Is

• Is

If so we get

deg ?f RC f~ 2

0 ?R f O RC f

4

0 2

2

0 2

,R f O Q f

R f O R f