A Brief Survey of Quantum Computing

1. A Brief Survey of Quantum Computing Igor Markov http://eecs.umich.edu/~imarkov

2. Outline • A Brief History of Quantum Computing • Background in Mathematics and Quantum Mechanics • What Makes Quantum Computing Work? • read/write ops for quantum storage • potential exp improvement over classical storage • fast computations • handling randomness • Limitations of Quantum Computing • Recent Workshops on Quantum Info Science • Conclusions

3. A Brief History of Q.C. • 1982 – Richard Feynman • could notsimulate quantum effects in poly-time! • tried to use them to perform computation • hope: exp speed-up over classic computation • Late 80s and up to 1992, Deutch and Jozsa • “quantum parallelism” demonstrated • however on somewhat weird problems • 1994 – Peter Shor • can factor integers into primes in quantum poly-time !!! • this immediately brokes RSA encryption

4. A Brief History of Q.C. • 1996 – Lov Grover (“database search”) • needle-in-the-haystack in (haystack)-time !!! • almost immediately: max of N numbers in N time • almost immediately: quantum heuristics • almost immediately: a speed-up for BFS-B&B • 1996 and on: First Quantum Computers built • NMR, ion trap • LANL,IBM,Oxford,MIT,Caltech,Stanford,Berkeley • 3 qbits in 99, 7 qbits in June 2000 (LANL) • rumors: NSA has massive quantum computers • optical and solid-state ideas less promissing • Bad news: need hundreds of qbits at least

5. A Brief History of Q.C. • 1997 or so: Quantum Error Correction • demonstrated by Bell Labs and IBM • tolerable error rate for quantum gates now at 10-5-10-7 • Quantum gates and quantum circuits • any classic computation F can be “quantized” • small penalty for irreversible computations • “controlled F-NOT gate” • Early 90s: developments in Randomized Algorithms • provide techniques to handle Quantum Computation • Quantum Computation is inherently randomized

6. A Brief History of Q.C. • 1994 - Complexity classes for quantum algos • quantum Turing machine described (60+ pages) • QBP similar to BBP, in particular, NPQBP? Unanswered • Lower bounds for quantum complexity • Grover’s search algorithm is optimal • Software emulators of quantum computers available • rely on BDDs to represent “discrete quantum states” • factor 15-bit integers using Shor’s algo on a P-III in 5mins • Quantum Communication/Information Theory • quantum comm. chanels are faster than classical • (communication complexity is well-defined)

7. A Brief History of Q.C. • Quantum Effects immensely useful in crypto • Alice and Bob “share qbits”, “share a Q-RNG” • As of 2000 • still very few useful Quantum Algos faster than classical • Fast Quantum Fourier Transforms • appear behind most(all?) quantum speed-ups • e.g., Shor’s number-factoring algo builds a FT over a cyclic group • general case: FT over a group • Abstract (Non-commutative) Harmonic Analysis (Group Represent.) • constructions available for FQFT over all Abelian groups • construction available for the group of all permutations

8. A Brief History of Q.C. • Two recent Ph.D. dissertations on FQFTs • Markus Puschel (Karlsruhe, Germany), now at CMU • in terms of Quantum Circuits (relatively concrete) • Sean Hallgren (expected from Berkeley), going to MSRI • more abstract (uses algebra and randomized algos) • The Hidden Subgroup Problem (HSP) • FQFTs are typically used to solve HSP (e.g., Shor) • The graph isomorphism reduces to the HSP over Sn • STOC 2000, Hallgren, Russel, Ta-Shma • a general fast algorithm for HSP: works only for normal subgroups

9. A Brief History of Q.C. • Unpublished • the STOC-00 algo always gets thecore of the HS • Open problems in Quantum Algorithms: • FQFTs over all finite groups • classic FFTs are not available for all finite groups! • classic problems in P • e.g., sorting, maxflow • substring matching seems amenable to Q.A. • Graph Auto-/Iso-morphism: attacked in the last 2yrs • NP-complete problems • SAT appears the best candidate • Recent progress in Comp. Group thry relevant to Q.C.

10. Need to Delve into Math! • What for ? • Quantum Mechanics is Mathematics • except for the bra/ket notation from Physics • very counter-intuitive, paradoxes abound • Mathematics does not change with technology • e.g., ion-trap versus NMR, electrons versus photons • Clear and expressive terms • improve the learning curve • “suggest” new applications and techniques • e.g., connections to Optical Computing

11. Need to Delve into Math! • Finite-dimensional Linear Algebra • vectors, unitary matrices, tensor products, etc. • Quantum Mechanics uses Linear Algebra • Probability • Quantum Mech. says: measurement is randomized • Abstract Algebra • Finite Groups and Group Representation Theory • generalize most of Linear Algebra and Spectral Theory • in use by quantum theorists since 1930s • e.g., to explain the [Mendeleev’s] Periodic Table of Elements

12. Background in Mathematics • Finite-dimensional Vector Spaces • made of (x1,x2,x3,x4,…xn) • xk are complex numbers ! • a 2-dimensional (“0-1”) space has dim 4 over reals • geometric intuition for dim=5+ very limited • “free” vector spaces on symbols • basis represented by a set of symbols, e.g.,  and  • e.g., x0+ x1 or x0+x1+x2+ x3 • tensor products of vector spaces…

13. Tensor Products • x00+ x01+ x10+x11 • “product” of two copies of x0+ x1 • isn’t that the Cartesian (direct, VW) product ? • x000+ x001+x010+x011 +x100+ x101+x110+x111 • “product” of three copies of x0+ x1 • is nota Cartesian product (has dim=8) • is a tensor product of vector spaces: VW • VW adds dimensions, VW multiplies them

14. Terminology and Notation • Qbit • a linear combination of (1) and(0) • i.e., an element of [a copy of] the 0-1 space V • in practice: one can only distinguish  • “qbit” may be used as “placeholder” (variable) for above • Quantum register/variable • one or more qbits (that make one value) • i.e., an element of VVV…V • Can compose registers from existing ones •  is associative

15. Measurement • Bad news • any measurement changes its subject • however, this is good news for cryptography • quantum states cannot be cloned (theorem) • after we “measure” qbit x0+ x1 • we can only detect or  • the bit changes to or  depending on the observation • Good news • x0 andx1 show up as probabilities of the outcomes • can measure many qbits at once, in many ways • can detect “pure states” • or, more generally, “orthogonal subspaces of states” • probabilities expressed via scalar products

16. Reversibility • Reversible ordinary computations • are permutations of bit-strings (not necess. … of bits) • Quantum computations • map quantum registers to quantum registers • must be linear and preserve scalar products • must be matrices of a certain type • must be reversible (can’t lose information) • must generalize permutations • e.g., matrices that permute basis vectors • but there are more • Bad news: cannot measure during computation!

17. Orthogonal and Unitary Matrices • Forget about complex numbers for a while • real-valued matrix A is orthogonal iff ATA=E • property:preserves scalar product •  preserves lengths and angles • Now back to complex numbers • complex-valued A is unitary iff A*A=E • properties: similar to orthogonal matrices

18. What Makes Quantum Computing Work? • Quantum storage • size = the number of qbits • N qbits can represent more info that N classical bits: • there are 2N “pure states” of the form … • a generic quantum state is a linear combination of pure states • it’s practical to measure the sign () of each pure state • “dense coding”: sending 2 classical bits through 1 qbit • Need to • read/write quantum storage • compute with it, handle randomness • Cannot copy, can only exchange • “quantum communication”

19. Writing into Quantum Storage • Need to set input registers (difficult) • main problem: cannot create quantum info • details depend on technology • with NMR, registers are in near-Bernoulli states • each qbit is in the state (1+) +(1-) • need special computations to get any other state! • can manufacture the state , but that’s useless • recent non-trivial result by Vazirani (Berkeley): • having  and any qbit is as good as having one qbit

20. Specifying Quantum Computations • Need to mathematically describe a computation (in particular, show existence) • note: a q. computation is a unitary operator U of exp size that is followed by a measurement projection P • need to show an “efficient algorithm” or argue existence • Need to express it in terms of quantum gates • i.e., Quantum Logic Synthesis • e.g., Markus Puschel did this in his Ph.D. dissertation for FQFTs

21. Quantum Parallelism • Consider a reversible classical computation F • maps N bits into N bits • Can construct a quantum computation that • maps N qbits into N qbits • maps an arbitrary linear combination of classical N –bit stringsinto a linear combination of classical N –bit strings • agrees with F on pure states • takes the same time to compute as F • This looks like a “cheap” exponential speed-up! • is not • we cannot measure arbitrary linear combinations!

22. Quantum Algo Development • “Q.C.” can mean “a Q.C. that does not exist yet” • Bottom-up Quantum Algorithm development • what can be done, given existing quantum gates? • Top-down Quantum Algorithm development • reduce a problem to seemingly easier problems • choose sub-problems with hope of being solvable • proceed recursively • The two have not converged yet in many cases

23. Handling Randomness • Measurement (reading quantum storage) • inherently randomized • The Quantum Oracle model • computation + measurement considered black-box • the input is classical, therefore • oracle calls can be repeated many times • Complexity estimates are products of • the complexity of the quantum oracle • the number of oracle calls

24. Handling Randomness • After all, the answer may be wrong!!! • the probability of getting a correct answer, as function of # of oracle calls, is part of the game • good news: if we can get the right answer with probability ½+, the rest is trivial • typically, it suffices to be correct with arbitrarily small but bounded (from below) probability: BPP versus QBP • If we apply another quantum algorithm to a wrong answer, the error may be magnified! • need error-correction • classic approaches don’t work because of the “no-cloning” theorem • completely new techniques were demonstrated by IBM

25. Limitations • Classic decidability same as quantum • the only difference between classic and quantum computing is the cost • Classic computation can simulate quantumin poly-space (but exp-time) • exp. quantum storage is only usefulduring quantum computations • Lower bounds for quantum computations • OR, AND: (N), PARITY: N/2, MAJORITY: (N)

26. Recent Workshops • October `99 NSF workshop (see handout) • over 100 participants • celebrities (Freedman, U. Vazirani, Yao etc) • NSF, NIST, Los Alamos N.L., DOD, DOE, NSA, DARPA, Naval Res. Lab., Army Res. Office • IBM/Watson, Bellcore, MSFT, NEC R.I., Litton, Mitre • Berkeley, Caltech, MIT, Princeton, Stanford, UIUC U. Maryland, U.Michigan, U. Texas, 10+ more • Oxford, Innsbruck, European Commission

27. Recent Workshops • October `99 NSF workshop • catalogized existing knowledge • outlined challenges and new opportunities • suggested Q.C. may help maintaining Moore’s law • Princeton `97 • Los Alamos `98 • MSRI / Berkeley 2000 • STOC and FOCS have Q.C. papers every year

28. Strong Groups on Q. Algorithms • UC Berkeley • Los Alamos, U. of New Mexico, U. of Arizona • IBM, AT & T • CalTech • Montreal, Canada • Copenhagen, Denmark • Karlsruhe, Germany • Oxford, UK

29. Conclusions • Technological promise of Quantum Computers • not clear, but many people are hopeful • Research on Quantum Computing • achieved great progress in the last 6 years • is overall popular, both in software and in hardware • requires a solid background in Mathematics • quantum softwareis a high-risk area until hardware exists • research on lower complexity bounds – less risky, but overly popular • Need more “Killer Apps” (assuming hardware comes) • 2 killer apps available now: search and number-factoring

30. References • Eleanor Rieffel and Wolfgang Polak,“An Introduction to Quantum Computing for Non-Physicists”, • http://xxx.lanl.gov/quant-ph/980916 v2 • also in ACM Computing Surveys • Dorit Aharonov, “Quantum Computation”, • http://xxx.lanl.gov/quant-ph/9812037 • also in Annual Reviews of Computational Physics B