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Shengyu Zhang CSE Dept. @ CUHK

Quantum Computing. in. Theoretical Computer Science. 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

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Shengyu Zhang CSE Dept. @ CUHK

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  1. Quantum Computing in Theoretical Computer Science Shengyu Zhang CSE Dept. @ CUHK

  2. 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

  3. 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.

  4. 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

  5. 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/

  6. 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

  7. Areas in quantum computing • Quantum algorithms • Quantum complexity • Quantum cryptography • Quantum error correction • Quantum information theory • Others: Quantum control / game theory / …

  8. Area 1: Quantum Algorithms • 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) 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log QFT (Quantum Fourier Transform): exponential speedup; slower than expected. *1: P. Shor. STOC’93, SIAM Journal on Computing, 1997.

  9. Area 1: Quantum Algorithms • Implication of fast algorithm for Factoring • Theoretical: Church-Turing thesis • Practical: Breaking RSA-based cryptosystems 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log QFT (Quantum Fourier Transform): exponential speedup; slower than expected.

  10. Area 1: Quantum Algorithms • 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). 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log Hallgren: Pell’s Equation QFT (Quantum Fourier Transform): exponential speedup; slower than expected. *1: S. Hallgren. STOC’02. Journal of the ACM, 2007.

  11. Area 1: Quantum Algorithms • 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? 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral QFT (Quantum Fourier Transform): exponential speedup; slower than expected.

  12. Area 1: Quantum Algorithms • 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|) 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral QFT (Quantum Fourier Transform): exponential speedup; slower than expected. *1: G. Kuperberg. arXiv:quant-ph/0302112, 2003.

  13. Area 1: Quantum Algorithms • 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) 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Grover: Search QS (Quantum Search): polynomial speedup; most solved. *1: L. Grover. Physical Review Letters, 1997.

  14. Area 1: Quantum Algorithms 1994 1996 1998 2000 2002 2004 2006 2008 Shor: Factoring & Discrete Log Hallgren: Pell’s Equation Kuperberg: HSP-Dihedral QFT (Quantum Fourier Transform): exponential speedup; slower than expected. Grover: Search Many combinatorial /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

  15. 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

  16. 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

  17. ∧ ¬ ∨ ∨ ∧ Area 1: Quantum Algorithms 1994 1996 1998 2000 2002 2004 2006 2008 • Classical: Θ(n) • Quantum: ~ Θ(√n) • apply QW on the formula graph with weight carefully designed for inductions to work. Grover’s search: OR function general formula by {AND-OR-NOT} AAKV: Def 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

  18. 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

  19. 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.

  20. 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

  21. 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 BQP NP NPC P,BPP Open question: BQP ⊆ PH?

  22. Interactive Proof • Interactive Proof: Verifier solves a hard problem with the help of a powerful but untrustworthy Prover. P V • 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) … Probabilistic polynomial time Computationally unbounded

  23. 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.

  24. 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

  25. Classical implications of quantum algorithms Note that we solved a purely classical open problem by giving a quantum algorithm. • 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 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

  26. 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.

  27. 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

  28. 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. • This explains the failure of efforts so far. • And tells researchers to stop trying to solve the generic problem. *1: Liu, Christandl, Verstraete. Physical Review Letters, 2007

  29. 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

  30. 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) x1 = ? 0 1 f(x1,x2,x3)=0 x2 = ? 0 1 x3 = ? f(x1,x2,x3)=1 0 1 f(x1,x2,x3)=0 f(x1,x2,x3)=1

  31. 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).

  32. 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.

  33. 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 x2 = ? • f(x1,x2,x3) • = (x1=1,x2=1) OR (x1=1,x2=0,x3=1) • = x1x2 + x1(1-x2)x3 0 1 x3 = ? f(x1,x2,x3)=1 0 1 f(x1,x2,x3)=0 f(x1,x2,x3)=1

  34. Randomized decision tree The error prob 0.01 here can be changed to any ε with an extra cost about log(1/ε). • 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 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)

  35. 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)

  36. We’ve shown deg(f) ≤ DTD(f) • Next: We have a similar lower bound for DTR(f).

  37. 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)

  38. 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.

  39. 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

  40. 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

  41. 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) x y Alice Bob F(x,y) F(x,y) *1. A. Yao. STOC’79.

  42. 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 computationalmodels. • Data structures, circuit complexity, streaming algorithms, decision tree complexity, VLSI, algorithmic game theory, optimization, pseudo-randomness…

  43. Rank lower bound • Two-variable function f(x,y) ↔matrix Af = [f(x,y)] • Two-variable Boolean function ↔Booleanmatrix • 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.

  44. 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.

  45. 2 3 1 0 0 ¢ ¢ ¢ 0 1 0 ¢ ¢ ¢ 6 7 I 6 7 = N . . . 6 7 . . . 4 5 . . . 0 0 1 ¢ ¢ ¢ 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?

  46. 2 3 1 0 0 ¢ ¢ ¢ 0 1 0 ¢ ¢ ¢ 6 7 I 6 7 = N . . . 6 7 . . . 4 5 . . . 0 0 1 ¢ ¢ ¢ Approximate rank • Approximate rank: For M = [mij] rankε(M) = min{rank(M’): |Mij – M’ij| ≤ε}. • [Thm*1] CCR(A) ≥ log2rankε(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. Theoretical Computer Science, 1996. *2. M. Rabin, A. Yao. Unpublished.

  47. 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) ≥ log2rankε(M) doesn’t work • [Thm*1] CCR(A) =Ω(n). • So this only gives rankε(A) = 2O(n). • [Thm*2] CCQ(A) ≥ log2rankε(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.

  48. Natural mathematically Algebraic Parameter (degree, rank) 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. Allow perturbation Allow error Approximate Parameter (degε/rankε) Quantum Complexity (DTQ, CCQ) Randomized Complexity (DTR, CCR) ≤ ≤ ≤

  49. Thanks. Questions?

  50. 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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