A Brief SurveyA Brief Survey of Quantum Computing of Quantum Computing

Igor Markovhttp://eecs.umich.edu/~imarkov

OutlineOutline 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

A Brief History of Q.C.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

A Brief History of Q.C.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

A Brief History of Q.C.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

A Brief History of Q.C.A Brief History of Q.C.

1994 - Complexity classes for quantum algos– quantum Turing machine described (60+ pages)– QBP similar to BBP, in particular, NPQBP? 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)

A Brief History of Q.C.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

A Brief History of Q.C.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

A Brief History of Q.C.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.

Need to Delve into Math!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

Need to Delve into Math!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

Background in MathematicsBackground 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…

Tensor ProductsTensor Products

x00+ x01+ x10 +x11– “product” of two copies of x0+ x1– isn’t that the Cartesian (direct, VW) product ?

x000+ x001+x010+x011 +x100+ x101+x110+x111– “product” of three copies of x0+ x1– is not a Cartesian product (has dim=8)– is a tensor product of vector spaces: VW

VW adds dimensions, VW multiplies them

Terminology and NotationTerminology 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 VVV…V

Can compose registers from existing ones– is associative

MeasurementMeasurement 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

ReversibilityReversibility 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!

Orthogonal and Unitary MatricesOrthogonal 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

What Makes What Makes Quantum Computing Work?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”

Writing into Quantum StorageWriting 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

Specifying Quantum Specifying Quantum ComputationsComputations

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

Quantum ParallelismQuantum 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!

Quantum Algo DevelopmentQuantum 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

Handling RandomnessHandling 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

Handling RandomnessHandling 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

LimitationsLimitations

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 useful

during quantum computations

Lower bounds for quantum computations– OR, AND: (N), PARITY: N/2, MAJORITY: (N)

Recent WorkshopsRecent 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

Recent WorkshopsRecent 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

Strong Groups on Q. AlgorithmsStrong 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

ConclusionsConclusions

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

ReferencesReferences

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