Shengyu Zhang CSE 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:

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 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(2 n^1/3 ) Quantum* 1 : ~ O(n 2 ) *1: P. Shor. STOC’93, SIAM Journal on Computing, 1997.

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Shor: Factoring & Discrete Log Implication of fast algorithm for Factoring –Theoretical: Church-Turing thesis –Practical: Breaking RSA-based cryptosystems

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Shor: Factoring & Discrete Log Pell’s Equation: x 2 – dy 2 = 1. Problem: Given d, find solutions (x,y) to the above equation. Classical (best known): –~ 2 √log d (assuming the generalized Riemann hypothesis) –~ d 1/4 (no assumptions) Quantum* 1 : poly(log d). Hallgren: Pell’s Equation *1: S. Hallgren. STOC’02. Journal of the ACM, 2007.

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Shor: Factoring & Discrete Log 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? Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Shor: Factoring & Discrete Log Two biggest cases: –HSP for symmetric group S n : Graph Isomorphism reduce to it. –HSP for dihedral group D n : Shortest Lattice Vector reduces to it. HSP(D n ): –Classical (best known): 2 log|G| –Quantum* 1 : 2 O(√log|G|) Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral *1: G. Kuperberg. arXiv:quant-ph/ , 2003.

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. QS (Quantum Search): polynomial speedup; most solved. Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral Grover: Search Given n bits x 1,…,x n, find an i with x i = 1. –Given n bits x 1,…,x n, decide whether ∃i s.t. x i = 1. Classical: Θ(n) Quantum* 1 : Θ(√n) *1: L. Grover. Physical Review Letters, 1997.

Area 1: Quantum Algorithms QFT (Quantum Fourier Transform): exponential speedup; slower than expected. QS (Quantum Search): polynomial speedup; most solved. Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral 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

Area 1: Quantum Algorithms QW (Quantum Walk): poly and exp speedup; rapidly developed. 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

Area 1: Quantum Algorithms QW (Quantum Walk): poly and exp speedup; rapidly developed. AAKV: Def Element Distinctness: Given n integers, decide whether they are the all distinct. Classical: Θ(n) Quantum: Θ(n 2/3 ) –Apply quantum walk on (n,n 2/3 )-Johnson graph. *1: A. Ambainis, FOCS’04 Ambainis* 1 : Ele. Dist.

Area 1: Quantum Algorithms QW (Quantum Walk): poly and exp speedup; rapidly developed. AAKV: Def *1: A. Ambainis, A. Childs, B. Reichardt, R. Spalek, S. Zhang. FOCS’07 Ambainis: Ele. Dist. ACRSZ* 1 : Formula Evaluation ∧ ¬ ∨ ∧ general formula by {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 BQP 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 –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 A classical fact on polynomial threshold degree and learnability of a class of functions* 1 : –thr(f) ≤ r for all f  C ⇒ C can be learned in time n O(r) Question* 2 : Any formula f of size n has polynomial threshold function thr(f) = O(n 1/2 )? Recall that we have O(n 1/2 )-time quantum algorithm for any AND-OR-NOT formula Now (roughly): thr(f) ≤ Q(f) ≤ n 1/2 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 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 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 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 Task: compute f(x) The input x can be accessed by queries in the form of “x i = ?”. We only care about the number of queries made Query (decision tree) complexity: min # queries needed. f(x 1,x 2,x 3 ) = x 1 ∧ (x 2 ∨ x 3 ) 0 f(x 1,x 2,x 3 )=0 x 2 = ? x 1 = ? 1 0 f(x 1,x 2,x 3 )=1 1 x 3 = ? 01 f(x 1,x 2,x 3 )=0f(x 1,x 2,x 3 )=1

Decision tree complexity DT D (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 DT D (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(x 1 x 2 x 3 ) = (x 1 x 2 x 3 =001) OR (x 1 x 2 x 3 =010) OR (x 1 x 2 x 3 =111) = (1-x 1 )(1-x 2 )x 3 + x 1 (1-x 2 )x 3 + x 1 x 2 x 3 –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(x 1,x 2,x 3 ) = x 1 ∧ (x 2 ∨ x 3 ) 0 f(x 1,x 2,x 3 )=0 x 2 = ? x 1 = ? 1 0 f(x 1,x 2,x 3 )=1 1 x 3 = ? 01 f(x 1,x 2,x 3 )=0f(x 1,x 2,x 3 )=1 f(x 1,x 2,x 3 ) = (x 1 =1,x 2 =1) OR (x 1 =1,x 2 =0,x 3 =1) = x 1 x 2 + x 1 (1-x 2 )x 3

Randomized decision tree We can toss coins during the computation. Or equivalently, we have a random string r and a collection of decision tree T r, s.t. for each input x E r [T r (x)] ≥ 0.99 if f(x) = 1 E r [T r (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. --- DT R (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, a  x i, z  –i: the position we are interested in –a: the register holding the queried variable –z: other part of the work space  i,a,z α i,a,z |i, a, z  →  i,a,z α i,a,z |i, a  x i, z  By def: DT Q (f) ≤ DT R (f) ≤ DT D (f)

We’ve shown deg(f) ≤ DT D (f) Next: We have a similar lower bound for DT R (f).

Approximate degree deg ε (f) = min {deg(f’): |f(x) – f’(x)| ≤ ε}. [Fact] deg ε (f) ≤ DT R (f) [proof] d.t.’s in DT R gives a polynomial in deg ε –DT R is a collection of d.t. T r, each of depth d = DT R (f). –Represent each T r by a degree≤d polynomial p r. By the fact in the deterministic case shown just now. –Now let f’ = E r [p r ]; it has degree≤d –f’(x) = E r [p r (x)] = E r [T r on x]: ε-approximating f(x). By the def of DT R (f)

Approximate degree of OR deg ε (f) = min {deg(f’): |f(x) – f’(x)| ≤ ε}. [Fact] deg ε (f) ≤ DT R (f) Question: What’s deg ε (f) for very simple functions, such as AND or OR? –Note that deg(AND) = deg(OR) = n. AND(x 1,…,x n ) = x 1 …x n, OR(x 1,…,x n ) = 1-(1-x 1 )…(1-x n ) Using the above bound? It gives nothing! –DT R (AND) = DT R (OR) = Ω(n). Because you are still living in the classical world! Mathematically.

Welcome to quantum world So we know DT R (f) ≥ deg ε (f) [Theorem* 1 ] DT Q (f) ≥ deg ε (f)/2 By this together with Grover’s Search DT Q (OR) = O(√n), we get: deg ε (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. Communication complexity: how many bits are needed to be exchanged? --- CC D (F) AliceBob F(x,y) xy A. Yao. STOC’79. *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 A f = [f(x,y)] –Two-variable Boolean function ↔ Boolean matrix Rank lower bound* 1 : CC D (f) ≥ log 2 rank(M f ), where M f = [f(x,y)] x,y [proof] –Decompose A into monochromatic combinatorial rectangles. CC D (f) ≥ log 2 # monochromatic combinatorial rectangles –Each rectangle has rank 1. –Rank is subadditive. K. Melhorn and E. Schmidt. STOC’82. *1. K. Melhorn and E. Schmidt. STOC’82.

Log Rank Conjecture Big open problem: Log Rank Conjecture* 1 : ∀ total Boolean f, CC D (f) = poly(log 2 rank(M f )) –Largest known gap* 2 : CC D (f) = (log 2 rank(M f )) 1.63… 1. L. Lovász and M. Saks. FOCS’88 *1. L. Lovász and M. Saks. FOCS’88 2. N. Nisan and A. Wigderson. Combinatorica, *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 CC R (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? I N = ¢¢¢ 0 01 ¢¢¢ ¢¢¢

Approximate rank Approximate rank: For M = [m ij ] rank ε (M) = min{rank(M’): |M ij – M’ ij | ≤ ε}. [Thm* 1 ] CC R (A) ≥ log 2 rank ε (A) Back to our question of rank ε (I N ): It’s nothing but the Equality problem where –f(x,y) = 1 iff x=y. [Fact* 2 ] CC R (Eq) = O(1). So, quite counterintuitively, rank ε (I N ) = O(1) 1. M. Krause. Theoretical Computer Science, *1. M. Krause. Theoretical Computer Science, *2. M. Rabin, A. Yao. Unpublished. I N = ¢¢¢ 0 01 ¢¢¢ ¢¢¢

Not always work Another matrix M of dimension 2 n  2 n M[x,y] = 1 iff ∃i s.t. x i = y i = 1. –An important matrix in TCS. CC R (M) ≥ log 2 rank ε (M) doesn’t work –[Thm* 1 ] CC R (A) = Ω(n). –So this only gives rank ε (A) = 2 O(n). [Thm* 2 ] CC Q (A) ≥ log 2 rank ε (A) / 2 Thus rank ε (A) = 2 O(√n). *1. Kalyanasundaram and Schintger, SIAM Journal on Discrete Mathematics, Razborov. Theoretical Computer Science, H. Buhrman and R. de Wolf. CCC’01. *2. H. Buhrman and R. de Wolf. CCC’01.

Natural mathematically Complexity Measure (DT D, CC D ) Algebraic Parameter (degree, rank) Allow error Randomized Complexity (DT R, CC R ) Approximate Parameter (deg ε /rank ε ) Allow perturbation Quantum Complexity (DT Q, CC Q ) ≤ ≤ ≤≤ 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