The Work of Peter Shor Has a Strong Geometrical Avor Typically Coupled
Doc Math J DMV
The Work of Peter W Shor
Ronald Graham
Much of the work of Peter Shor has a strong geometrical avor typically coupled
with deep ideas from probability complexity theory or combinatorics and always
woven together with brilliance and insight of the rst magnitude Due to the space
limitations of this note I will restrict myself to brief descriptions of just four of
his remarkable achievements unfortunately omitting discussions of his seminal
work on randomized incremental algorithms of fundamental imp ortance in
computational geometry and his provo cative results in computational biology on
self assembling virus shells
Two dimensional discrepancy minimax grid matchings and online
bin packing
The minimax grid matching problem is a fundamental combinatorial problem aris
ing the the average case analysis of algorithms To state it we consider a square S
p p
of area N in the plane and a regularly spaced N N array G grid of p oints
in S Let P be a set of N points selected indep endently and uniformly in S By a
p erfect matching of P to G wemeana map P G For each selection P
de ne L P min max d p p where ranges over all p erfect matchings
p P
of P to G andd denotes Euclidean distance
Theorem Shor Leighton Shor
With very high probability
E L P logN
The pro of is very intricate and ingenious and contains a wealth in new ideas which
havespawned a variety of extensions and generalizations notably in the work of
M Talagrand on ma jorizing measures and discrepancy
A classical paradigm in the analysis of algorithms is the so called bin packing
in which a list W w w w of weights is given and problem
n
we are to required to pack all the w into bins with the constraint that no bin
i
can contain a weight total of more than Since it is NP hard to determine the
minimum numb er of bins which W requires for a successful packing
oreven to decide if this minimum numb er is extensive e orts have b een made
for nding go o d approximation algorithms for pro ducing near optimal packings
In the Best Fit algorithm after the rst i weights are packed the next weight
w is placed into the bin in which it ts b est i e so that the unused space
i
DocumentaMathematica Extra Volume ICM I
Ronald Graham
in that bin is less than it would b e in any other bin This is actually an online
algorithm In his thesis Shor proved the very surprising and deep result
that when the w are chosen uniformly at random from then with very high
i
probability the amountofwasted space has size n log n
An up right region R R f of the square S is de ned as the region in S
lying ab ovesomecontinuous monotonically non increasing function f e g S is
itself up right If P isasetof N points chosen uniformly and indep endently at
random in S we can de ne the discrepancy R jj R P j ar ea R j An old
problem in mathematical statistics from the s see was the estimation of
sup R over all up right regions of S This was nally answered byLeighton
R
and Shor in and it is nowknown that
sup R N log N
R
The preceding results give just a hint of the numerous applications these
fertile techniques have found to suchdiverse areas as pseudo random numb er gen
eration dynamic storage allo cation wafer scale integration and two dimensional
bin packing see
Davenport Schinzel Sequences
ADavenp ort Schinzel sequence DS n s is a sequence U u u u com
m
p osed of n distinct symb ols suchthat u u for all i and such that U con
i i
tains no alternating subsequence of length s i e there do not exist indices
i i i such that u u u a b u u
s i i i i i
We de ne
n maxfm u u isaDS n s sequenceg
S m
Davenp ort Schinzel sequences have turned out to b e of central imp ortance in com
putational and combinatorial geometry andhave found many applications in such
Voronoi diagrams and shortest path algo areas as motion planning visibility
rithms It is known that DS n s sequences provide a combinatorial character
ization of the lower envelop e of n continuous univariate functions each pair of
whichintersectinatmostspoints Hence n is just the maximum number of
s
connected comp onents of the graphs of such functions and accurate estimates of
n can often b e translated into sharp b ounds for algorithms which dep end on
s
function minimization It is trivial to showthat n n and n n
The rst surprise came when it was shown that n n n where
n is de ned to b e the functional inverse of the Ackermann function A t i e
n minft A t ng Note that n is an extremely slowly growing function
of n since A is de ned as follows
A t t t and A t A A t k t
k k k
t
Thus A t A t is an exp onential to wer of n s and so on Then A t
is de ned to be A t The best bounds for n s in were rather
t s
DocumentaMathematica Extra Volume ICM I
The Work of Peter W Shor
weak This was remedied in where Shor and his coauthors managed to showby
n
extremely delicate and clever techniques that n n Thus DS n
sequences can b e much longer than DS n sequences but are still only slightly
non linear In addition they also obtained almost tight bounds on all other
n s
s
n
Tiling R with cubes
In Minkowski made the conjecture in connection with his work on extremal
n
lattices that in any lattice tiling of R with unit n cub es there must b e two cub es
having a complete facet n face in common This was generalized by
n
O Keller in to the conjecture that any tiling of R by unit n cub es
must have this prop erty This was con rmed by Perron in for n
and shortly thereafter Ha j os proved Minkowski s original conjecture for all
n However in spite of rep eated e orts no further progress was made in proving
Keller s conjecture for the next years Then in Shor struck He showed
with his colleague J Lagarias that in fact Keller s conjecture is false for all
dimensions n They managed to do this with an very ingenious argument
n
showing that certain sp ecial graphs suggested by Corr adi and Szab o of size
n
must always have cliques of size contrary to the prevailing opinions then from
n
which it followed that Keller s conjecture must fail for R The reader is referred
to for the details of this combinatorial gem and to for a fascinating history
of this problem I mightpoint out that this is another example of an old conjecture
in geometry b eing shattered by a subtle combinatorial construction an earlier one
b eing the recent dispro of of the Borsuk conjecture by Kahn and Kalai It is
still not known what the truth for Keller s conjecture is when n or
Quantum computation
It has b een generally b elieved that a digital computer or more abstractly a Turing
machine can simulate any physically realizable computational device This in
fact is the thrust of the celebrated Church Turing thesis Moreover it was also
assumed that this could always b e done in an e cientway i e involving at most
a p olynomial expansion in the time required However it was rst p ointed out by
Feynman that certain quantum mechanical systems seemed to b e extremely
on Neumann di cult in fact imp ossible to simulate e ciently on a standard v
computer This led him to suggest that it might b e p ossible to take advantage of
the quantum mechanical b ehavior of nature itself in designing a computer which
overcame these di culties In fact in doing so such a quantum computer might
be able to solvesomeofthe classical di cult problems much more e ciently as
well These ideas were pursued by Benio Deutsch Bennett and
others and slowly a mo del of quantum computation b egan to evolve However
the rst bombshell in this embryonic eld o ccurred when Peter Shor in
announced the rst signi cant algorithm for such a hyp othetical quantum
DocumentaMathematica Extra Volume ICM I
Ronald Graham
computer namely a metho d for factoring an arbitrary comp osite integer N in
c log N log log N log log log N
steps This should be contrasted with the b est current algorithm on classical
digital computers whose b est running time estimates growlike
exp cN log N
Of course no one has yet ruled out the p ossibility that a p olynomial time factoring
algorithm exists for classical computers cf the infamous P vs NP problem but
it is felt by most knowledgeable people that this is extremely unlikely In the
same pap er Shor also gives a p olynomial time algorithm for a quantum computer
for computing discrete logarithms another apparently intractable problem for
classical computers
There is not space here to describ e these algorithms in any detail but a few
remarks may b e in order In a classical computer information is represented by
n
binary symbols and bits An n bit memory can exist in any of logical
states Such computers also manipulate this binary data using functions likethe
Bo olean AND and NOT By contrast a quantum bit or qubit is typically a
microscopic system such as an electron with its spin or a p olarized photon The
Bo olean states and are represented by reliably distinguishable states of the
and j i spin However according to the laws of qubit e g j i spin
quantum mechanics the qubit can also exist in a continuum of intermediate states
or sup erp ositions j i j i where and are complex numb ers satisfying
j j j j
More generally a string of n qubits can exist in any state of the form
X
jxi
x
x
P
where the are complex numbers suchthat j j In other words a quan
x x
x
n
tum state of n qubits is represented byaunitvector in a dimensional complex
Hilb ert space de ned as the tensor pro duct of the n copies of the dimensional
Hilb ert space representing the state of a single qubit It is the exp onentially large
dimensionality of this space which distinguishes quantum computers from classical
computers Whereas the state of a classical system can b e completely describ ed by
separately sp ecifying the state of each part the overwhelming ma jority of states
in a quantum computer are entangled i e not representable as a direct pro duct
of the states of its individual qubits As stated in the ability to preserveand
manipulate entangled states is the distinguishing feature of quantum computers
resp onsible b oth for their p ower and for the di culty in building them
The crux of Shor s factoring algorithm after reducing the problem of factoring
N to that of determining for a random X coprime to N the order of X modul o N
is a brilliant application of the discrete Fourier transform in suchawayastohave
all the incorrect candidate orders quantum mechanically cancel out leaving only
multiples of the correct order of X app earing with high probability when the
DocumentaMathematica Extra Volume ICM I
The Work of Peter W Shor
output is nally measured I heartily recommend that the reader consult the pap er
of Shor in this Volume or for more details
Of course complicated quantum systems are delicate creatures and anysub
stantial interaction with the external environment can cause rapid decoherence
which then can result in the system collapsing to some classical state thereby
prematurely terminating the ongoing computation This was the basis for the
strong initial skepticism that any serious quantum computer could actually ever
b e built However Shor s subsequentcontributions changed this situation substan
tially His pap er in announced the discovery of quantum error correcting
co des cutting through some widely held misconceptions ab out quantum informa
tion and showing that suitable measurements of a quantum system can acquire
su cient information for detecting and correcting errors without disturbing anyof
the enco ded information These ideas were further develop ed in toproduce
a new theory of quantum error correcting co des for protection against multiple
errors using clever ideas from orthogonal geometry and prop erties of the recently
disco vered ordinary as opp osed to quantum co des over GF
Finally any quantum computer which is actually built will be comp osed of
comp onents which are not completely reliable Thus it will b e essential to create
algorithms which are fault tolerant on such computers In yet another path
breaking pap er Shor in showed how this indeed could b e done
Not only do es Peter Shor s work on quantum computation during the past
four years represent scienti c achievements of the rst rank but in my mind it
holds out the rst real promise that non trivial quantum computers may actually
exist in our lifetimes
References
P K Agarwal M Sharir and P Shor Sharp upp er and lower b ounds on
the length of general Davenp ort Schinzel sequences J Comb Theory A
C H Bennett Logical reversibility of computation IBM J Res Develop
C H Bennett and P W Shor Quantum information theory to appear
P Benio Quantum mechanical mo dels of Turing machines that dissipate no
energy Phys Rev Letters
J Blum On convergence of empirical distribution functions Ann Math Stat
A R Calderbank and P W Shor Go o d quantum error correcting co des exist
Phys Review A
A R Calderbank E M Rains P W Shor and N J A Sloane Quantum
error correction via co des over GF IEEE Trans Inf Theory
DocumentaMathematica Extra Volume ICM I
Ronald Graham
K L Clarkson and P W Shor Applications of random sampling in compu
tational geometry II Discrete and Comput Geom
E G Co man Jr and F T Leighton A provably e cient algorithm for
dynamic storage allo cation J Comput System Sci
E G Co man Jr M R Garey and D S Johnson Approximation algo
rithms for bin packing an up dated survey Algorithm design for computer
system design CISM Courses and Lectures Springer Verlag Vienna
K Corr adi and S Szab o A combinatorial approach for Keller s conjecture
Period Math Hungar
D Deutsch Quantum theory the Church Turing principle and the universal
quantum computer Pr oc Royal Soc London A
R Feynman Simulating physics with computers Internat J Theoret Phys
G Ha j os Ub er einfache und mehrfache Bedeckung des n dimensionalen
Raumes mit einen Wurfelgitter Math Z
S Hart and M Sharir NonlinearityofDavenp ort Schinzel sequences and of
generalized path compression schemes Combinatorica
J Kahn and G Kalai A counterexample to Borsuk s conjecture Bul l Amer
Math Soc New Series
R M Karp M Luby and A Marchetti Spaccamela Probabilistic analysis
of multi dimensional bin packing problems Proceedings th ACM Symp on
Theory of Computing
O H Keller Ub er einfache und mehrfache Bedeckung des n dimensionalen
Raumes mit einen Wurfelgitter J Reine Angew Math
J C Lagarias and P W Shor Keller s cub e tiling conjecture is false in high
dimensions Bul l Amer Math Soc New Series
F T Leighton and C E Leiserson Wafer scale integration of systolic arrays
IEEE Trans on Computers C
T Leighton and P Shor Tight b ounds for minimax grid matching with ap
plications to the average case analysis of algorithms Combinatorica
O Perron Ub er l uckenlose Ausfullung des n dimensionalen Raumes durch
kongruente Wurfel I I I Math Z
P W Shor Random Planar Matching and Bin Packing Ph D Thesis MIT
Math Dept
DocumentaMathematica Extra Volume ICM I
The Work of Peter W Shor
P W Shor The average case analysis of some on line algorithms for bin
packing Combinatorica
P W Shor Algorithms for quantum computation discrete logarithms and
factoring Proc th Symp Foundations of Comp Sci IEEE Computer So
ciety Press
P W Shor Polynomial time algorithms for prime factorization and discrete
logarithms on a quantum computer SIAM J Computing
P W Shor Scheme for reducing decoherence in quantum computer memory
Phys Review A R R
P W Shor Fault tolerant quantum computation Proceedings th Symp
Foundations Comp Science IEEE Computer So ciety Press Los Alamitos
CA
S Stein and S Szab o Algebra and Tiling Carus Math Monograph no
Math Asso c America Washington
M Talagrand Matching theorems and empirical discrepancy computations
using ma jorizing measures J Amer Math Soc
C Williams and S Clearwater Explorations in Quantum Computing
Springer Verlag Santa Clara CA
Ronald Graham
AT T Labs
Florham Park NJ
and
UCSD
La Jolla CA
USA
DocumentaMathematica Extra Volume ICM I
Ronald Graham
Peter W Shor
Information Sciences Center AT T Labs Research Florham Park NJ USA
Born August New York City USA
Nationality US Citizen
Marital Status married one daughter
Undergraduate at California Institute of Technology
Pasadena
Ph D in Mathematics
Massachusetts Institute of Technology
Postdo ctoral Fellow at Mathematical Sciences
Research Institute Berkeley
Member of Technical Sta AT T Bell Labs MurrayHill
Principal Research Sta Member AT T Labs
Florham Park
Fields of Interest Theoretical Computer Science Combinatorics
Ronald Graham and Peter W Shor
DocumentaMathematica Extra Volume ICM I