Shengyu Zhang CSE Dept. @ CUHK. Roadmap Intro to theoretical computer science Intro to quantum computing Export of quantum computing –Formula Evaluation

Shengyu Zhang

CSE Dept. @ CUHK

Roadmap

• Intro to theoretical computer science• Intro to quantum computing• Export of quantum computing

– Formula Evaluation• Solves a classical open question

– N-Representability problem• Addresses the failure of many efforts in quantum chemistry

• Quantum is natural mathematically– Decision tree complexity– Communication complexity

A brief intro to theoretical computer science

• Computation: a sequence of elementary instructions.

• More than knowing the existence, but a step-by-step way to find it.

Efficiency

• Efficient Computation:– Algorithm: design fast algorithms– Computational complexity: classify problems

according to their computational difficulty• Structural

– Measured by resources like time, space, randomness, counting,…

• Interactive• Concrete models: Decision Tree, Communication

Complexity, Circuit

Connections to other sciences

• Import: Use of concepts and techniques from – Math: discrete math, analysis, algebra, topology– Physics

• Export: – Solve TCS questions appearing naturally in

• Statistical Physics, Chemistry, Molecular Biology, Social Science, Economics, Computer & Information Science,

– Concepts such as completeness; – Problems such as P vs. NP

• One of the seven $1M Millennium Problems*1

*1: http://www.claymath.org/millennium/P_vs_NP/

Roadmap

• Intro to theoretical computer science• Intro to quantum computing• Export of quantum computing




Areas in quantum computing

• Quantum algorithms

• Quantum complexity

• Quantum cryptography

• Quantum error correction

• Quantum information theory

• Others: Quantum control / game theory / …

Area 1: Quantum Algorithms

1994 1996 1998 2000 2002 2004 2006 2008

QFT(Quantum Fourier Transform): exponential speedup; slower than expected.

Shor: Factoring & Discrete Log

• Factoring: Given an n-bit number, factor it (into product of two numbers).– The reverse problem of Multiplication, which is very

easy.• Classical (best known) : ~ O(2n^1/3)• Quantum*1: ~ O(n2)

*1: P. Shor. STOC’93, SIAM Journal on Computing, 1997.


1994 1996 1998 2000 2002 2004 2006 2008



• Implication of fast algorithm for Factoring– Theoretical: Church-Turing thesis

– Practical: Breaking RSA-based cryptosystems


1994 1996 1998 2000 2002 2004 2006 2008



• Pell’s Equation: x2 – dy2 = 1.• Problem: Given d, find solutions (x,y) to the above

equation.• Classical (best known):

– ~ 2√log d (assuming the generalized Riemann hypothesis) – ~ d1/4 (no assumptions)

• Quantum*1: poly(log d).

Hallgren: Pell’s Equation

*1: S. Hallgren. STOC’02. Journal of the ACM, 2007.


1994 1996 1998 2000 2002 2004 2006 2008



• Hidden Subgroup Problem (HSP): Given a function f on a group G, which has a hidden subgroup H, s.t. f is– constant on each coset aH,– distinct on different cosets.

Task: find the hidden H.• Factoring, Pell’s Equation both reduce to it.• Efficient quantum algorithms are known for Abelian groups.• Main question: HSP for non-Abelian groups?


Kuperberg: HSP-Dihedral


1994 1996 1998 2000 2002 2004 2006 2008



• Two biggest cases: – HSP for symmetric group Sn: Graph Isomorphism reduce to it.– HSP for dihedral group Dn: Shortest Lattice Vector reduces to it.

• HSP(Dn):– Classical (best known): 2log|G|

– Quantum*1: 2O(√log|G|)



*1: G. Kuperberg. arXiv:quant-ph/0302112, 2003.


1994 1996 1998 2000 2002 2004 2006 2008


QS(Quantum Search): polynomial speedup; most solved.




Grover: Search

• Given n bits x1,…,xn, find an i with xi = 1.– Given n bits x1,…,xn, decide whether ∃i s.t. xi = 1.

• Classical: Θ(n)• Quantum*1: Θ(√n)

*1: L. Grover. Physical Review Letters, 1997.


1994 1996 1998 2000 2002 2004 2006 2008


QS(Quantum Search): polynomial speedup; most solved.




Grover: Search

QW(Quantum Walk): poly and exp speedup; rapidly developed.

AAKV*1: Def

*1: D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani. STOC'01

Many combinatorial/graph problems


1994 1996 1998 2000 2002 2004 2006 2008


AAKV*1: Def

• Classical random walk on graphs: starting from some vertex, repeatedly go to a random neighbor– Many algorithmic applications

• Quantum walk on graphs: even definition is non-trivial.– For instance: classical random walk converges to a stationary

distribution, but quantum walk doesn’t since unitary is reversible.

*1: D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani. STOC'01


1994 1996 1998 2000 2002 2004 2006 2008


AAKV: Def

• Element Distinctness: Given n integers, decide whether they are the all distinct.

• Classical: Θ(n)• Quantum: Θ(n2/3)

– Apply quantum walk on (n,n2/3)-Johnson graph.

*1: A. Ambainis, FOCS’04

Ambainis*1: Ele. Dist.


1994 1996 1998 2000 2002 2004 2006 2008


AAKV: Def

*1: A. Ambainis, A. Childs, B. Reichardt, R. Spalek, S. Zhang. FOCS’07

Ambainis: Ele. Dist.

ACRSZ*1: Formula Evaluation

∧

¬∨

∧

general formulaby {AND-OR-NOT}

∨

Grover’s search: OR function

• Classical: Θ(n)• Quantum: ~ Θ(√n)• apply QW on the formula graph with weight carefully designed for inductions to work.

Area 2: Quantum Complexity

• Quantum complexity– Structural:

• A sample here: BQP in PSPACE

– Interactive: • A sample here: QIP = QIP[3]

– Concrete models: DT and CC• More to come in next section

BQP in PSPACE

• P: problems solvable in polynomial time– One characterization of efficient computation

• BPP: problems solvable in probabilistic polynomial time w/ a small error tolerated– Another characterization of efficient computation

• BQP: problems solvable in polynomial time by a quantum computer w/ a small error tolerated– Yet another characterization of efficient computation,

if you believe large-scale quantum mechanics.

Classical upper bound of BQP

• Central in complexity theory: comparisons of different modes of computations

• How to compare classical and quantum efficient computation?

• An obvious lower bound: BPP ⊆ BQP• An upper bound (of quantum by classical)• [Thm*1] BQP ⊆ PSPACE

– PSPACE: problems solvable in polynomial space.

*1: Bernstein, Vazirani. STOC’93, SIAM J. on Computing, 1997

Where does BQP sit in?

• PH: Polynomial Hierarchy

• Level 3:– Polynomial time

verification V s.t.f(x) = 1 if ∃y1∀y2∃y3

V[x,y1,y2,y3] = 1.

• NP is just level 1.

EXP

PSPACE

PH

NP

P,BPP

BQ

P

Open question: BQP PH?⊆

NPC

Interactive Proof

V

• Interactive Proof: Verifier solves a hard problem with the help of a powerful but untrustworthy Prover.

P …

Computationally unbounded

Probabilistic polynomial time

• If YES: P to convince V.

P, Pr[P convinces V] > 1-δ(δ: completeness error)

• If NO: ∄ P to convince V.

P, Pr[P convinces V] < ε

(ε: soundness error)

Quantum Interactive Proof

• IP: problems solvable by interactive proof system– IP[k]: problems solvable by k-round interactive proof

system

• QIP: problems solvable by quantum interactive proof system– QIP[k]: problems solvable by k-round quantum

interactive proof system

• [Thm*1] QIP = QIP[3]• Classically: IP=IP[3] ⇒ PH collapses to AM *1: Kitaev, Watrous. STOC’00.

Roadmap

• Intro to Theoretical Computer Science• Intro to Quantum Computing• Export of quantum computing




Classical implications of quantum algorithms

• A classical fact on polynomial threshold degree and learnability of a class of functions*1: – thr(f) ≤ r for all fC ⇒ C can be learned in time nO(r)

• Question*2: Any formula f of size n has polynomial threshold function thr(f) = O(n1/2)?

• Recall that we have O(n1/2)-time quantum algorithm for any AND-OR-NOT formula

• Now (roughly): thr(f) ≤ Q(f) ≤ n1/2

• This implies that formulas are learnable in time 2√n. (Matching the known lower bound.)

*1: A. Klivans, R. Servedio, STOC’01; A. Klivans, R. Servedio, R. O’Donnell, FOCS’02 *2: O’Donnell, Servedio, STOC’03

Note that we solved a purely

classical open problem

by giving

a quantum algorithm.

Classical implication of quantum arguments

• It’s not uncommon.

• Quantum computer is not only a potentially more powerful computation machine.

• It’s also a different mathematical model.

• So studies of quantum computing turn out to provide novel perspectives of old (classical) problems

• And some led to complete solutions.

Roadmap





N-Representability problem

• N-Representability problem in quantum chemistry: characterize the allowed set of density operators on N-body fermions satisfying given 2-body correlations.

• An efficient solution would be a breakthrough. • It had attracted a very large of effort, though not

quite successful yet.• [Thm*1] N-Representability is QMA-complete.

– QMA: the quantum analog of NP. – Thus QMA-complete is even harder than NP-hard.

*1: Liu, Christandl, Verstraete. Physical Review Letters, 2007

• This explains the failure of efforts so far.

• And tells researchers to stop trying to solve the generic problem.

Roadmap





decision tree computation

• Task: compute f(x) • The input x can be

accessed by queries in the form of “xi = ?”.

• We only care about the number of queries made

• Query (decision tree) complexity: min # queries needed.

f(x1,x2,x3) = x1∧(x2∨x3)

0

f(x1,x2,x3)=0 x2 = ?

x1 = ?

1

0

f(x1,x2,x3)=1

1

x3 = ?

0 1

f(x1,x2,x3)=0 f(x1,x2,x3)=1

Decision tree complexity

• DTD(f) = the minimum number of queries needed to compute f (on all inputs x)– Superscript D: “deterministic”

• Next we’ll define a natural measure of f and show that it’s a lower bound of DTD(f).

degree

• ∀ f:{0,1}n→{0,1} can be represented by a multi-variate polynomial of deg ≤ n.– f(001) = f(010) = f(111) = 1, and 0 on other x.– f(x1x2x3) = (x1x2x3=001) OR (x1x2x3=010) OR

(x1x2x3=111)= (1-x1)(1-x2)x3 + x1(1-x2)x3 + x1x2x3

– is a deg-3 polynomial.

• [Fact] This polynomial representation is unique.

Decision tree and degree

• [Fact] deg(f) ≤ DT(f)

• Collect all 1-leaves.

• f = OR of all paths to these 1-leaves.

f(x1,x2,x3) = x1∧(x2∨x3)

0

f(x1,x2,x3)=0 x2 = ?

x1 = ?

1

0

f(x1,x2,x3)=1

1

x3 = ?

0 1

f(x1,x2,x3)=0 f(x1,x2,x3)=1

f(x1,x2,x3) = (x1=1,x2=1) OR (x1=1,x2=0,x3=1)

= x1x2 + x1(1-x2)x3

Randomized decision tree

• We can toss coins during the computation.• Or equivalently, we have a random string r and a

collection of decision tree Tr, s.t. for each input xEr[Tr(x)] ≥ 0.99 if f(x) = 1Er[Tr(x)] ≤ 0.01 if f(x) = 0

• Thus a randomized d.t. is a collection S of many deterministic d.t. s.t. for any x, most of the d.t. in S give the correct answer f(x).

• Randomized DT complexity: the max depth of d.t. in S. --- DTR(f)

The error prob 0.01 here can be changed to any ε with an extra

cost about log(1/ε).

Quantum query algorithm

• Instead of coin-tossing, we ask all variables in superposition.

• |i, a, z → |i, axi, z– i: the position we are interested in– a: the register holding the queried variable– z: other part of the work space

i,a,zαi,a,z |i, a, z → i,a,zαi,a,z |i, axi, z

• By def: DTQ(f) ≤ DTR(f) ≤ DTD(f)

• We’ve shown deg(f) ≤ DTD(f)

• Next: We have a similar lower bound for DTR(f).

Approximate degree

• degε(f) = min {deg(f’): |f(x) – f’(x)| ≤ ε}.

• [Fact] degε(f) ≤ DTR(f)

• [proof] d.t.’s in DTR gives a polynomial in degε

– DTR is a collection of d.t. Tr, each of depth d = DTR(f).

– Represent each Tr by a degree≤d polynomial pr.• By the fact in the deterministic case shown just now.

– Now let f’ = Er[pr]; it has degree≤d

– f’(x) = Er[pr(x)] = Er[Tr on x]: ε-approximating f(x).• By the def of DTR(f)

Approximate degree of OR

• degε(f) = min {deg(f’): |f(x) – f’(x)| ≤ ε}.• [Fact] degε(f) ≤ DTR(f)• Question: What’s degε(f) for very simple

functions, such as AND or OR?– Note that deg(AND) = deg(OR) = n.

• AND(x1,…,xn) = x1…xn, • OR(x1,…,xn) = 1-(1-x1)…(1-xn)

• Using the above bound? • It gives nothing!

– DTR(AND) = DTR(OR) = Ω(n).

Because you are still living in the classical world! Mathematically.

Welcome to quantum world

• So we know DTR(f) ≥ degε(f)

• [Theorem*1] DTQ(f) ≥ degε(f)/2

• By this together with Grover’s Search DTQ(OR) = O(√n), we get:

degε(OR) = O(√n)!

*1: Beals, Buhrman, Cleve, Mosca, de Wolf, STOC’98, J. of the ACM, 2001

Roadmap





Communication complexity*1

• Two parties, Alice and Bob, jointly compute a function F(x,y) with x known only to Alice and y only to Bob.

• Communication complexity: how many bits are needed to be exchanged? --- CCD(F)

Alice Bob

F(x,y) F(x,y)

x y

*1. A. Yao. A. Yao. STOCSTOC’79.’79.

Why CC is interesting?

• Reason 1: Mathematically interesting and challenging.

• Reason 2: Rich connections to other areas in TCS

• Though defined in an information theoretical setting, it turned out to provide lower bounds to many computational models.– Data structures, circuit complexity, streaming

algorithms, decision tree complexity, VLSI, algorithmic game theory, optimization, pseudo-randomness…

Rank lower bound

• Two-variable function f(x,y) ↔ matrix Af = [f(x,y)]– Two-variable Boolean function ↔ Boolean matrix

• Rank lower bound*1:

CCD(f) ≥ log2 rank(Mf),

where Mf = [f(x,y)]x,y

• [proof] – Decompose A into monochromatic combinatorial rectangles.

• CCD(f) ≥ log2 # monochromatic combinatorial rectangles

– Each rectangle has rank 1.– Rank is subadditive.

*1. K. Melhorn and E. Schmidt. K. Melhorn and E. Schmidt. STOCSTOC’82.’82.

Log Rank Conjecture

• Big open problem:

• Log Rank Conjecture*1: ∀ total Boolean f,

CCD(f) = poly(log2 rank(Mf))

– Largest known gap*2: CCD(f) = (log2 rank(Mf))1.63…

*1. L. Lovász and M. Saks. 1. L. Lovász and M. Saks. FOCSFOCS’88’88*2. N. Nisan and A. Wigderson. 2. N. Nisan and A. Wigderson. CombinatoricaCombinatorica, 1995., 1995.

Variant of rank

• Next: we’ll introduce a natural variant of rank and show that it’s a lower bound of CCR(f)

• One cute question as a bait:

the N-dim identity matrix has rank N.• Question: If you can perturb each entry by 0.01, how

much can you decrease the rank?

IN =

2

6664

1 0 ¢¢¢ 00 1 ¢¢¢ 0...

......

0 0 ¢¢¢ 1

3

7775

Approximate rank

• Approximate rank: For M = [mij]rankε(M) = min{rank(M’): |Mij – M’ij| ≤ ε}.

• [Thm*1] CCR(A) ≥ log2 rankε(A)

• Back to our question of rankε(IN): It’s nothing but the Equality problem where – f(x,y) = 1 iff x=y.

• [Fact*2] CCR(Eq) = O(1).• So, quite counterintuitively,

rankε(IN) = O(1)*1. M. Krause. 1. M. Krause. Theoretical Computer ScienceTheoretical Computer Science, 1996., 1996.*2. M. Rabin, A. Yao. *2. M. Rabin, A. Yao. UnpublishedUnpublished..

IN =

2

6664

1 0 ¢¢¢ 00 1 ¢¢¢ 0...

......

0 0 ¢¢¢ 1

3

7775

Not always work

• Another matrix M of dimension 2n2n

M[x,y] = 1 iff ∃i s.t. xi = yi = 1.– An important matrix in TCS.

• CCR(M) ≥ log2 rankε(M) doesn’t work – [Thm*1] CCR(A) = Ω(n).– So this only gives rankε(A) = 2O(n).

• [Thm*2] CCQ(A) ≥ log2 rankε(A) / 2• Thus rankε(A) = 2O(√n).*1. Kalyanasundaram and Schintger, SIAM Journal on Discrete Mathematics, 1992. Razborov. Theoretical Computer Science, 1992.*2. H. Buhrman and R. de Wolf. 2. H. Buhrman and R. de Wolf. CCCCCC’01.’01.

Natural mathematically

Complexity Measure(DTD, CCD)

Algebraic Parameter(degree, rank)

Allow error

Randomized Complexity(DTR, CCR)

ApproximateParameter(degε/rankε)

Allow perturbation

QuantumComplexity(DTQ, CCQ)

≤

≤≤ ≤

• The quantum complexity is closer to the natural math lower bound. • The tightening gives nontrivial results randomized complexity can’t yield.

A brief intro to quantum computing

• Feymann’82: Idea

• Deutsch’85,’89: quantum Turing machine and quantum circuit

• Bernstein-Vazirani’93, Yao’93: ground of quantum complexity theory

• Shor’94: fast quantum algorithm for Factoring and Discrete Log

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Shengyu Zhang CSE Dept. @ CUHK. Roadmap Intro to theoretical computer science Intro to quantum computing Export of quantum computing –Formula Evaluation