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
– 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
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
Shor: Factoring
& Discrete Log
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
• 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.
Area 1: Quantum Algorithms
1994
1996
1998
2000
2002
2004
2006
2008
Shor: Factoring
& Discrete Log
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
• Implication of fast algorithm for Factoring
– Theoretical: Church-Turing thesis
– Practical: Breaking RSA-based cryptosystems
Area 1: Quantum Algorithms
1994
1996
1998
Shor: Factoring
& Discrete Log
2000
2002
2004
2006
2008
Hallgren: Pell’s
Equation
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
• 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).
*1: S. Hallgren. STOC’02. Journal of the ACM, 2007.
Area 1: Quantum Algorithms
1994
1996
1998
Shor: Factoring
& Discrete Log
2000
2002
Hallgren: Pell’s
Equation
2004
2006
2008
Kuperberg:
HSP-Dihedral
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
• 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?
Area 1: Quantum Algorithms
1994
1996
1998
Shor: Factoring
& Discrete Log
2000
2002
Hallgren: Pell’s
Equation
2004
2006
2008
Kuperberg:
HSP-Dihedral
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
• 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.
Area 1: Quantum Algorithms
1994
1996
1998
Shor: Factoring
& Discrete Log
2000
2002
Hallgren: Pell’s
Equation
2004
2006
2008
Kuperberg:
HSP-Dihedral
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
Grover:
Search
QS
(Quantum Search): polynomial speedup; most solved.
• 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.
Area 1: Quantum Algorithms
1994
1996
1998
Shor: Factoring
& Discrete Log
2000
2002
Hallgren: Pell’s
Equation
2004
2006
2008
Kuperberg:
HSP-Dihedral
QFT
(Quantum Fourier Transform): exponential speedup; slower than expected.
Grover:
Many combinatorial
Search
/graph problems
QS
(Quantum Search): polynomial speedup; most solved.
AAKV*1:
Def
QW
(Quantum Walk): poly and exp speedup; rapidly developed.
*1: D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani. STOC'01
Area 1: Quantum Algorithms
1994
1996
1998
2000
2002
2004
2006
2008
• 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.
AAKV*1:
Def
QW
(Quantum Walk): poly and exp speedup; rapidly developed.
*1: D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani. STOC'01
Area 1: Quantum Algorithms
1994
1996
1998
2000
2002
2004
2006
2008
• 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.
AAKV:
Def
Ambainis*1:
Ele. Dist.
QW
(Quantum Walk): poly and exp speedup; rapidly developed.
*1: A. Ambainis, FOCS’04
Area 1: Quantum Algorithms
1994
1996
1998
2000
2002
∧
∨
∨
¬
∧
Grover’s search:
OR function
general formula
by {AND-OR-NOT}
AAKV:
Def
2004
2006
2008
• Classical: Θ(n)
• Quantum: ~ Θ(√n)
• apply QW on the formula
graph with weight carefully
designed for inductions to
work.
Ambainis:
Ele. Dist.
ACRSZ*1: Formula
Evaluation
QW
(Quantum Walk): poly and exp speedup; rapidly developed.
*1: A. Ambainis, A. Childs, B. Reichardt, R. Spalek, S. Zhang. FOCS’07
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?
EXP
PSPACE
PH
NPC
NP
P,BPP
• 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.
Open question: BQP ⊆ PH?
Interactive Proof
• Interactive Proof: Verifier solves a hard problem
with the help of a powerful but untrustworthy
Prover.
• If YES:  P to convince V.
P
…
V
 P, Pr[P convinces V] > 1-δ
(δ: completeness error)
• If NO: ∄ P to convince V.
Computationally
unbounded
 P, Pr[P convinces V] < ε
Probabilistic
polynomial time
(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
– 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
Classical implications of quantum
algorithms
Note that we solved a purely
• A classical fact on polynomial threshold degree
classical open problem
and learnability of a class of functions*1:
– thr(f) ≤ by
r forgiving
all fC ⇒ C can be learned in time nO(r)
• Question*2: Any
formulaalgorithm.
f of size n has
a quantum
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
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
• 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
N-Representability problem
• N-Representability problem in quantum
• This explains the failure of efforts so
chemistry:
characterize
the
allowed
set
of
far.
density operators on N-body fermions satisfying
2-body
correlations.
•given
And tells
researchers
to stop trying to
• solve
An efficient
solution
would be a breakthrough.
the generic
problem.
• 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
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
decision tree computation
f(x1,x2,x3) = x1∧(x2∨x3)
• Task: compute f(x)
• The input x can be
x1 = ?
accessed by queries
0
1
in the form of “xi = ?”.
f(x1,x2,x3)=0
x2 = ?
• We only care about
the number of
0
1
queries made
x3 = ? f(x1,x2,x3)=1
• Query (decision tree)
0
1
complexity: min #
queries needed.
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)
x1 = ?
0
1
f(x1,x2,x3)=0
f(x1,x2,x3)
= (x1=1,x2=1) OR (x1=1,x2=0,x3=1)
= x1x2 + x1(1-x2)x3
x2 = ?
0
1
x3 = ?
0
f(x1,x2,x3)=1
1
f(x1,x2,x3)=0 f(x1,x2,x3)=1
Randomized
decision
tree
The error prob 0.01 here can be
changed to any ε with an extra cost
• We can toss coins during about
the computation.
log(1/ε).
• Or equivalently, we have a random string r and a
collection of decision tree Tr, s.t. for each input x
Er[Tr(x)] ≥ 0.99 if f(x) = 1
Er[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)
Quantum query algorithm
• Instead of coin-tossing, we ask all
variables in superposition.
• |i, a, z → |i, axi, 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, axi, 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!
Because you are still living in the
classical world! Mathematically.
– DTR(AND) = DTR(OR) = Ω(n).
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
• 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
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.
x
y
Alice
Bob
F(x,y)
F(x,y)
• Communication complexity: how many bits are
needed to be exchanged? --- CCD(F)
*1. A. Yao. STOC’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. STOC’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. FOCS’88
*2. N. Nisan and A. Wigderson. Combinatorica, 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 question2as a bait: 3
IN
1 0 ¢¢¢ 0
6 0 1 ¢¢¢ 07
6
7
= 6. .
.
.. 7
4 .. ..
5
0 0 ¢¢¢ 1
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?
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’s2nothing but
3
1 0 ¢¢¢ 0
the Equality problem where
6
7
– f(x,y) = 1 iff x=y.
• [Fact*2] CCR(Eq) = O(1).
• So, quite counterintuitively,
rankε(IN) = O(1)
*1. M. Krause. Theoretical Computer Science, 1996.
*2. M. Rabin, A. Yao. Unpublished.
IN
6 0 1 ¢¢¢
= 6. .
4 .. ..
0 0 ¢¢¢
07
.. 7
.5
1
Not always work
• Another matrix M of dimension 2n2n
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. CCC’01.
Natural mathematically
Algebraic Parameter
(degree, rank)
Allow
perturbation
≤
Complexity Measure
(DTD, CCD)
• The quantum complexity is closer to
the natural math lower bound.
• The tightening gives nontrivial results
randomized complexity can’t yield.
Approximate
Parameter ≤
(degε/rankε)
Quantum
≤
Complexity
(DTQ, CCQ)
≤
Allow
error
Randomized
Complexity
(DTR, CCR)
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