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 not simulate 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-BB
• 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
• Grovers search algorithm is optimal
• Software emulators of quantum computers available
• rely on BDDs to represent discrete quantum
  states
• factor 15-bit integers using Shors 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., Shors 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 the core 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 Mendeleevs 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 dim5 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?
• isnt that the Cartesian (direct, V?W) product
  ?
• x000??? x001???x010???x011??? x100???
  x101???x110???x111???
• product of three copies of x0? x1?
• is not a Cartesian product (has dim8)
• 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 and x1 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 (cant 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 ATAE
• property preserves scalar product
• ? preserves lengths and angles
• Now back to complex numbers
• complex-valued A is unitary iff AAE
• 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
• its 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 thats 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 dont 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 Moores 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 software is 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