Quantum Computer Architectures for Quantum Computer Architectures for Physical Simulations Physical Simulations

Dr. Mike FrankDr. Mike FrankUniversity of FloridaUniversity of Florida

CISE DepartmentCISE Departmentmpf@cise.ufl.edumpf@cise.ufl.edu

Presented at:Presented at:Quantum Computation for Quantum Computation for

Physical Modeling WorkshopPhysical Modeling WorkshopWed., May 8, 2002Wed., May 8, 2002

Summary of Current ResearchSummary of Current Research• One of my two major research projects:One of my two major research projects:

– Reversible & Quantum Computing ProjectReversible & Quantum Computing Project• Studying & developing Studying & developing physical computing theory:physical computing theory:

– Technology-independent physical limits of computingTechnology-independent physical limits of computing– ““Ultimate” models of computing for complexity theoryUltimate” models of computing for complexity theory

• Nanocomputer systems engineeringNanocomputer systems engineering– Optimizing Optimizing cost-efficiencycost-efficiency scalability of future computing scalability of future computing

• Reversible computing & quantum computingReversible computing & quantum computing– Realistic models, CPU architectures, optimal scaling advantagesRealistic models, CPU architectures, optimal scaling advantages

MIT 1995-1999,UF 1999-

RevCompRevComp project heritage project heritage• Grew out of work started by the MIT Information Grew out of work started by the MIT Information

Mechanics group in the 1970’s.Mechanics group in the 1970’s.– Key members: Fredkin, Toffoli, MargolusKey members: Fredkin, Toffoli, Margolus

• Occasional collaboration with Feynman, Bennett, …Occasional collaboration with Feynman, Bennett, …– This group laid much theoretical groundwork for This group laid much theoretical groundwork for

reversible (& eventually quantum) computing.reversible (& eventually quantum) computing.

• MIT Reversible Computing project (1990s)MIT Reversible Computing project (1990s)– Group leaders: Knight, MargolusGroup leaders: Knight, Margolus

• Students: Younis, D’Souza, Becker, Vieri, Frank, AmmerStudents: Younis, D’Souza, Becker, Vieri, Frank, Ammer– Focus: Reducing reversible computing to practiceFocus: Reducing reversible computing to practice

• CMOS circuit styles, test chips, architectures, complexity CMOS circuit styles, test chips, architectures, complexity theory, algorithms, high-level languages.theory, algorithms, high-level languages.

RevComp group at UFRevComp group at UF• RevReversible & Quantum ersible & Quantum CompComputing grouputing group

– Organized by myself (CISE/ECE depts.)Organized by myself (CISE/ECE depts.)– UF collabs. in CISE, ECE, Math, Phys./Chem.UF collabs. in CISE, ECE, Math, Phys./Chem.– Notable graduate: DoRon Motter (highest honors)Notable graduate: DoRon Motter (highest honors)

• Now doing PhD work in quantum circuits at U. Mich.Now doing PhD work in quantum circuits at U. Mich.– 2 current Ph.D students (all ECE)2 current Ph.D students (all ECE)

• Current focus:Current focus:– Removing remaining barriers to near-term practicality of Removing remaining barriers to near-term practicality of

reversible computingreversible computing• Improved circuit styles, efficient power suppliesImproved circuit styles, efficient power supplies• Other applicationsOther applications

– Long-term study of Long-term study of physical computing theoryphysical computing theory and scaling and scaling advantages of reversible/quantum models.advantages of reversible/quantum models.

Who We AreWho We Are• Dr. Michael FrankDr. Michael Frank

– MIT Ph.D. stud. & postdoc, 1996-97 & 1999.MIT Ph.D. stud. & postdoc, 1996-97 & 1999.• Area exam studies on quantum computing.Area exam studies on quantum computing.• DARPA-funded reversible computing research. DARPA-funded reversible computing research.

– 1999-now: Head of Reversible & Quantum 1999-now: Head of Reversible & Quantum Computing group at UF’s CISE dept.Computing group at UF’s CISE dept.• http://www.cise.ufl.edu/research/revcomphttp://www.cise.ufl.edu/research/revcomp

Fredkin Toffoli

KnightSussman

MargolusFrank

Minsky

FeynmanSome

lineage:

Who We Are, cont.Who We Are, cont.• DoRon MotterDoRon Motter

– Undergrad in UF CISE dept., 1997-2000.Undergrad in UF CISE dept., 1997-2000.• Coursework in CS + quantum mechanics.Coursework in CS + quantum mechanics.• Sr. highest honors thesis w. Dr. Frank, 2000.Sr. highest honors thesis w. Dr. Frank, 2000.

– Now a Masters student at U. Mich.Now a Masters student at U. Mich.• Advisor: Igor Markov, U. Mich.Advisor: Igor Markov, U. Mich.• DARPA-funded project on quantum logic systhesisDARPA-funded project on quantum logic systhesis

Overview of TalkOverview of Talk• 1. Optimally scalable QC models/architectures1. Optimally scalable QC models/architectures

– Universally maximally scalableUniversally maximally scalable (UMS) models (UMS) models– Physical limits shape the ultimate modelPhysical limits shape the ultimate model– Appropriate programming modelsAppropriate programming models

• 2. Physics sim. algorithms on our QC model2. Physics sim. algorithms on our QC model– Many-particle Schrödinger equationMany-particle Schrödinger equation

• Numerically stable classical reversible simsNumerically stable classical reversible sims• Quantum versionsQuantum versions

– Quantum field theoryQuantum field theory

• 3. QC simulation on classical computers3. QC simulation on classical computers– Visualization techniques, various optimizationsVisualization techniques, various optimizations– Polynomial-space techniquesPolynomial-space techniques

1. Optimally scalable quantum 1. Optimally scalable quantum computer architecturescomputer architectures

–Universally maximally scalableUniversally maximally scalable (UMS) models (UMS) models–Physical limits shape the ultimate modelPhysical limits shape the ultimate model–Appropriate programming modelsAppropriate programming models

Nanocomputer systems engineeringNanocomputer systems engineering• A key goal of my long-term research program: A key goal of my long-term research program:

– Develop key foundations for a new discipline of Develop key foundations for a new discipline of nanocomputer systems engineeringnanocomputer systems engineering suited for meeting suited for meeting the challenges of computing at the nanoscale.the challenges of computing at the nanoscale.

• And convey it to peers, teach it to students.And convey it to peers, teach it to students.

• The new field will integrate concerns & methods The new field will integrate concerns & methods from a variety of disciplines:from a variety of disciplines:– Physics of computingPhysics of computing Algorithm design Algorithm design– Systems engineeringSystems engineering Electrical eng., Electrical eng., etc.etc.– Computer architectureComputer architecture Quantum computing!Quantum computing!– Complexity theoryComplexity theory

ITRS Feature Size Projections

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Physical Computing TheoryPhysical Computing Theory• The study of theoretical models of computation that The study of theoretical models of computation that

are based on (or closely tied to) physicsare based on (or closely tied to) physics– Make no nonphysical assumptions!Make no nonphysical assumptions!

• Includes the study of:Includes the study of:– Fundamental physical limits of computingFundamental physical limits of computing– Physically-based models of computingPhysically-based models of computing

• Includes reversible and/or quantum modelsIncludes reversible and/or quantum models• ““Ultimate” (asymptotically optimal) modelsUltimate” (asymptotically optimal) models

– An asymptotically An asymptotically tighttight “Church’s thesis” “Church’s thesis”– Model-independent basis for complexity theoryModel-independent basis for complexity theory

• Basis for design of future nanocomputer architecturesBasis for design of future nanocomputer architectures– Asymptotic scaling of architectures & algorithmsAsymptotic scaling of architectures & algorithms

• Physically optimal algorithmsPhysically optimal algorithms

Ultimate Models of ComputingUltimate Models of Computing• We would like models of computing that match the We would like models of computing that match the

real computational power of physics.real computational power of physics.– Not too weak, not too strong.Not too weak, not too strong.

• Most traditional models of computing only match Most traditional models of computing only match physics to within physics to within polynomial factorspolynomial factors..– Misleading asymptotic performance of algorithms.Misleading asymptotic performance of algorithms.– Not good enough to form the basis for a real systems Not good enough to form the basis for a real systems

engineering optimization of architectures.engineering optimization of architectures.

• Develop models of computing that are:Develop models of computing that are:– As powerful as physically possible on all problemsAs powerful as physically possible on all problems– Realistic within asymptotic constant factorsRealistic within asymptotic constant factors

Unstructured Search ProblemUnstructured Search Problem• Given a set Given a set SS of of NN elements and a black-box function elements and a black-box function

ff::SS{0,1}, find an element {0,1}, find an element xxSS such that such that ff((xx)=1, if )=1, if one exists (or if not, say so).one exists (or if not, say so).– Any NP problem can be cast as an unstructured search Any NP problem can be cast as an unstructured search

problem.problem.• Not necessarily the optimal approach, however.Not necessarily the optimal approach, however.

• Bounds on classical run-time:Bounds on classical run-time: ((NN) expected queries in worst case (0 or 1 sol’ns):) expected queries in worst case (0 or 1 sol’ns):

• Have to try Have to try NN/2 elements on average before finding sol’n./2 elements on average before finding sol’n.• Have to try all Have to try all NN if there is no solution. if there is no solution.

• If elements are length-If elements are length- bit strings, bit strings,– Expected #trials is Expected #trials is (2(2) - exponential in ) - exponential in . Bad!. Bad!

Quantum Unstructured SearchQuantum Unstructured Search• Minimum time to solve unstructured search Minimum time to solve unstructured search

problem on a (serial) quantum computer is:problem on a (serial) quantum computer is: ((NN1/21/2) queries = (2) queries = (2/2/2) = (2) = (21/21/2))

• Still exponential, but with a smaller base.Still exponential, but with a smaller base.

• The minimum # of queries can be achieved The minimum # of queries can be achieved using using Grover’s algorithmGrover’s algorithm..

Classical Unstructured SearchClassical Unstructured Search• The classical serial algorithm takes The classical serial algorithm takes ((NN) time.) time.• But: Suppose we search in parallel!But: Suppose we search in parallel!

– Have Have M<NM<N processors running in parallel. processors running in parallel.– Each searches a different subset ofEach searches a different subset of

NN//MM elements of the search space. elements of the search space.– If processors are ballistic reversible:If processors are ballistic reversible:

• Can cluster them in a dense mesh of diameter Can cluster them in a dense mesh of diameter ((MM1/31/3).).

• Time accounting:Time accounting:– Computation time: Computation time: ((NN//MM))– Communication time: Communication time: ((MM1/31/3)) (at lightspeed)(at lightspeed)– Total:Total: TT NN//MM + + MM1/31/3 is minimized when is minimized when M M NN3/43/4

NN1/41/4 Faster than Grover’s algorithm! Faster than Grover’s algorithm!

MM1/3

Classical+Quantum ParallelismClassical+Quantum Parallelism• Similar setup to classical parallelism: Similar setup to classical parallelism:

– MM processors, searching processors, searching NN//MM items each. items each.– Except, each processor uses Grover’s algorithm.Except, each processor uses Grover’s algorithm.

• Time accounting:Time accounting:– Computation:Computation: TT ( (NN//MM))1/21/2

– Communication:Communication: TT MM1/31/3 (as before)(as before)– Total:Total: TT ( (NN//MM))1/21/2 + + MM1/31/3

• Total is minimized when Total is minimized when MMN N 3/53/5

– Minimized total is Minimized total is TT NN1/51/5..

• I.e.I.e., quantum unstructured search is really only , quantum unstructured search is really only NN1/41/4//NN1/51/5 = = NN1/201/20 × faster than classical! × faster than classical!

Scalability & Maximal ScalabilityScalability & Maximal Scalability• A multiprocessor architecture & accompanying A multiprocessor architecture & accompanying

performance model is performance model is scalablescalable if: if:– it can be “scaled up” to arbitrarily large problem sizes, it can be “scaled up” to arbitrarily large problem sizes,

and/or arbitrarily large numbers of processors, without the and/or arbitrarily large numbers of processors, without the predictions of the performance model breaking down.predictions of the performance model breaking down.

• An architecture (& model) is An architecture (& model) is maximally scalablemaximally scalable for a for a given problem ifgiven problem if– it is scalable and if no other scalable architecture can claim it is scalable and if no other scalable architecture can claim

asymptotically superior performance on that problemasymptotically superior performance on that problem

• It is It is universally maximally scalableuniversally maximally scalable (UMS) if it is (UMS) if it is maximally scalable on maximally scalable on allall problems! problems!– I will briefly mention some characteristics of architectures I will briefly mention some characteristics of architectures

that are universally maximally scalablethat are universally maximally scalable

Universal Maximum ScalabilityUniversal Maximum Scalability• Existence “proof” for universally maximally scalable Existence “proof” for universally maximally scalable

(UMS) architectures:(UMS) architectures:– Physics itselfPhysics itself is a universal maximally scalable is a universal maximally scalable

“architecture” because “architecture” because anyany real computer real computer isis merely a special merely a special case of a physical system.case of a physical system.

• Obviously, no real computer can beat the performance of physical Obviously, no real computer can beat the performance of physical systems in general.systems in general.

– Unfortunately, physics doesn’t give us a very simple or Unfortunately, physics doesn’t give us a very simple or convenient programming model.convenient programming model.

• Comprehensive expertise at “programming physics” means mastery Comprehensive expertise at “programming physics” means mastery of of allall physical engineering disciplines: chemical, electrical, physical engineering disciplines: chemical, electrical, mechanical, optical, mechanical, optical, etc.etc.

– We’d like an easier programming model than this!We’d like an easier programming model than this!

Physics Constrains the Ultimate ModelPhysics Constrains the Ultimate Model

Limits of Quantum ComputersLimits of Quantum Computers• Quantum computers Quantum computers remain subjectremain subject to all the to all the

fundamental limits previously mentioned!fundamental limits previously mentioned!– Entropy density limit - only 2Entropy density limit - only 2nn disting’able states! disting’able states!

• Contrary to press manglings, a quantum computer Contrary to press manglings, a quantum computer cannotcannot store store exponentially large amounts of arbitrary data!exponentially large amounts of arbitrary data!

– Information propagation limit - at most Information propagation limit - at most cc• Bell’s theorem, teleportation, “superluminal” wave velocities Bell’s theorem, teleportation, “superluminal” wave velocities

do do notnot give > give >cc information propagation information propagation– Quantum field theory is Quantum field theory is explicitlyexplicitly local local

– Computation rate limit - at most 4Computation rate limit - at most 4E/hE/h rate of rate of orthogonalorthogonal transitions, given available energy transitions, given available energy EE..

• Non-orthogonal ones are faster, but accomplish less workNon-orthogonal ones are faster, but accomplish less work• Speedups are due to Speedups are due to fewerfewer ops needed, not ops needed, not fasterfaster ops ops

Simple UMS ArchitecturesSimple UMS Architectures• (I propose) any practical UMS architecture will (I propose) any practical UMS architecture will

have the following features:have the following features:– Processing elements characterized by constant Processing elements characterized by constant

parameters (independent of # of processors)parameters (independent of # of processors)– Mesh-type message-passing interconnection network, Mesh-type message-passing interconnection network,

arbitrarily scalable in 2 dimensionsarbitrarily scalable in 2 dimensions• w. limited scalability in 3rd dimension.w. limited scalability in 3rd dimension.

– Processing elements that can be operated in a highly Processing elements that can be operated in a highly reversible mode, at least up to some limit.reversible mode, at least up to some limit.

• Enables improved 3-d scalability, in a limited regimeEnables improved 3-d scalability, in a limited regime– (In long term) Have capability for quantum-coherent (In long term) Have capability for quantum-coherent

operation, for extra perf. on some probs.operation, for extra perf. on some probs.

Ideally Scalable ArchitecturesIdeally Scalable Architectures

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

CPU

Local memory hierarchy(optimal fixed size)

Processing Node

Conjecture: A 2- or 3-D mesh multiprocessor with a fixed-size memory hierarchy per node is an optimal scalable computer systems design (for any application).

Mesh interconnection network

Some device parametersSome device parameters• The following parameters are considered fixedThe following parameters are considered fixed

(for a given device/node technology):(for a given device/node technology):– (Maximum) number of bits of state per node(Maximum) number of bits of state per node– (Minimum) node volume(Minimum) node volume– (Minimum) transition time (per (Minimum) transition time (per transition) transition)– (Minimum) entropy generated per bit erased ((Minimum) entropy generated per bit erased (k k ln 2)ln 2)– (Minimum) static entropy generation rates(Minimum) static entropy generation rates

• For devices even just quiescently maintaining state infoFor devices even just quiescently maintaining state info• Related to energy leakage rates, decoherence timesRelated to energy leakage rates, decoherence times

– (Minimum) adiabatic frictional coefficient(Minimum) adiabatic frictional coefficient• For devices undergoing reversible transitionsFor devices undergoing reversible transitions

– (Maximum) quality factor (Maximum) quality factor QQ of active transitions of active transitions– (Minimum) device cost(Minimum) device cost

System-level parametersSystem-level parameters• The following parameters may be adjusted as The following parameters may be adjusted as

the problem size increases:the problem size increases:– Number of nodes utilizedNumber of nodes utilized– Arrangement of utilized nodes in Arrangement of utilized nodes in xx, , yy, , zz

• Spreading nodes out optimizes perf. on some problemsSpreading nodes out optimizes perf. on some problems– Rate of change of an externally-applied clocking Rate of change of an externally-applied clocking

signal (time-dependent potential)signal (time-dependent potential)• Allows trading off adiabaticity vs. speed of computation, Allows trading off adiabaticity vs. speed of computation,

as a function of the number of nodesas a function of the number of nodes

Entropy coefficients of some Entropy coefficients of some reversible logic gate operationsreversible logic gate operations

From Frank, “Ultimate theoretical models of From Frank, “Ultimate theoretical models of nanocomputers” (nanocomputers” (NanotechnologyNanotechnology, 1998):, 1998):

• SCRL, circa 1997:SCRL, circa 1997: ~1~1 b/Hzb/Hz• Optimistic reversible CMOS:Optimistic reversible CMOS: ~10~10 b/kHzb/kHz• Merkle’s “quantum FET:”Merkle’s “quantum FET:” ~1.2 ~1.2 b/GHzb/GHz• Nanomechanical rod logic:Nanomechanical rod logic: ~.07~.07 b/GHzb/GHz• Superconducting PQ gate:Superconducting PQ gate: ~25~25 b/THzb/THz• Helical logic:Helical logic: ~.01~.01 b/THzb/THz

How low can you go? We don’t really know!

Thermodynamics & ScalabilityThermodynamics & Scalability• The fastest parallel algorithms for many problems The fastest parallel algorithms for many problems

ideally require a 3-D mesh topology.ideally require a 3-D mesh topology.– Minimizes communication latencies between pointsMinimizes communication latencies between points

• But, entropy flux bounds imply entropy generation rates But, entropy flux bounds imply entropy generation rates can scale only proportionally to system’s 2-D outer can scale only proportionally to system’s 2-D outer (convex hull) surface area.(convex hull) surface area.– Assuming upper bounds on temperature & pressureAssuming upper bounds on temperature & pressure

• So, can harness 3So, can harness 3rdrd dimension only to the extent that dimension only to the extent that useful operations can be made reversible.useful operations can be made reversible.

• Optimizing efficiency requires a careful tradeoff Optimizing efficiency requires a careful tradeoff between performance, power, cost...between performance, power, cost...

Reversible/Adiabatic CMOSReversible/Adiabatic CMOS• Chips designed at MIT, 1996-1999:Chips designed at MIT, 1996-1999:

Minimum Losses w. LeakageMinimum Losses w. Leakage

S

leak

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Etot = Eadia + Eleak

Reversible Emulation - Ben89Reversible Emulation - Ben89

k = 2n = 3

k = 3n = 2

Bennett 89Bennett 89 alg. is alg. is notnot optimal optimal

k = 2n = 3

k = 3n = 2

Just look at all the spacetime it wastes!!!

Parallel “Parallel “Frank02Frank02”” algorithm algorithm• We can move the triangles closer together, to We can move the triangles closer together, to

eliminate the wasted spacetime.eliminate the wasted spacetime.• Resulting algorithm is Resulting algorithm is linear timelinear time for all for all nn and and

kk and dominates and dominates Ben89Ben89 for time, spacetime, & for time, spacetime, & energy!energy!

Real time

Em

ulat

ed ti

me k=2

n=3k=3n=2

k=4n=1

Cost-Efficiency Gains, Modified Ben89

y = 0.3905x0.3896

y = 1.741x0.6198

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Perf. scaling w. # of devicesPerf. scaling w. # of devices• If alg. is If alg. is not not limited by communications needs,limited by communications needs,

– Use Use irreversibleirreversible processors spread in a 2-D layer. processors spread in a 2-D layer.– Remove entropy along perpendicular dimension.Remove entropy along perpendicular dimension.– No entropy rate limits, No entropy rate limits,

• so no speed advantage from so no speed advantage from reversibilityreversibility..

• If alg. requires only If alg. requires only locallocal communication, communication,latency latency cyc. time, in an cyc. time, in an NNDD××NNDD××NNDD mesh, mesh,– Leak-free Leak-free reversiblereversible machine perf. scales better! machine perf. scales better!– Irreversible Irreversible ttcyccyc = = ((NNDD

1/31/3))– Reversible Reversible ttcyccyc = = ((NNDD

1/41/4)… )… ((NNDD1/121/12) × faster!) × faster!

• To boost reversibility speedup by 10×, one must consider ~10To boost reversibility speedup by 10×, one must consider ~103636-CPU -CPU machines (1.7 trillion moles of CPUs!)machines (1.7 trillion moles of CPUs!)

– 1.7 trillion moles of H atoms weighs 1.7 million metric tons!1.7 trillion moles of H atoms weighs 1.7 million metric tons!» A ~100-m tall hill of H-atom sized CPUs!A ~100-m tall hill of H-atom sized CPUs!

Lower bound on irrev. timeLower bound on irrev. time• Simulate Simulate NNprocproc = = NNDD

33 cells for cells for NNstepssteps » » NNDD steps. steps.

• Consider a sequence of Consider a sequence of NNDD update steps. update steps.

• Final cell value depends on Final cell value depends on NNDD44 ops in time ops in time TT..

• All ops must occur within radius All ops must occur within radius rr = = cT cT of cell.of cell.• Surface area Surface area AA TT22, rate , rate RRopop TT22 sustainable. sustainable.

• NNopsops RRop op T T TT33 needs to be at least needs to be at least NNDD44..

T T must be must be ((NNDD4/34/3) to do all ) to do all NNDD steps. steps.

• Average time per step must be Average time per step must be ((NNDD1/31/3).).

• Any irreversible machine (of any technology or Any irreversible machine (of any technology or architecture) must obey this bound!architecture) must obey this bound!

Irreversible 3-D MeshIrreversible 3-D Mesh

Reversible 3-D MeshReversible 3-D Mesh

Non-local CommunicationNon-local Communication• Best computational task for reversibility:Best computational task for reversibility:

– Each processor must exchange messages with Each processor must exchange messages with another that is another that is NNDD

1/21/2 nodes away on each cycle nodes away on each cycle• Unsure what real-world problem demands this pattern!Unsure what real-world problem demands this pattern!

– In this case, reversible speedup scales with number of In this case, reversible speedup scales with number of CPUs to “only” the 1/18CPUs to “only” the 1/18thth power. power.

• To boost reversibility speedup by 10×, “only” need 10To boost reversibility speedup by 10×, “only” need 101818 (or (or 1.7 micromoles) of CPUs1.7 micromoles) of CPUs

• If each was a 1-nm cluster of 100 C atoms, this is only 2 If each was a 1-nm cluster of 100 C atoms, this is only 2 mg mass, volume 1 mmmg mass, volume 1 mm33..

• Current VLSI: Current VLSI: Need cost level of ~$25B before you see a speedup.Need cost level of ~$25B before you see a speedup.

Open issues for reversible comp.Open issues for reversible comp.• Integrate realistic fundamental models of the Integrate realistic fundamental models of the

clocking system into the engineering analysis.clocking system into the engineering analysis.– There is an open issue about the scalability of clock There is an open issue about the scalability of clock

distribution systems.distribution systems.• Exist quantum bounds on reusability of timing signals.Exist quantum bounds on reusability of timing signals.• Not yet clear if reversible clocking is scalable.Not yet clear if reversible clocking is scalable.

– Fortunately, self-timed reversible computing also appears Fortunately, self-timed reversible computing also appears to be a possibility.to be a possibility.

• Not yet clear if this approach works above 1-D models.Not yet clear if this approach works above 1-D models.• Simulation experiments planned to investigate this.Simulation experiments planned to investigate this.

• Develop efficient physical realizations of nano-scale Develop efficient physical realizations of nano-scale bit-devices & timing systems.bit-devices & timing systems.

Timing in Adiabatic SystemsTiming in Adiabatic SystemsWhen multiple adiabatic devices interact, the relative When multiple adiabatic devices interact, the relative timingtiming

must be precise, in order to ensure that adiabatic rules are met.must be precise, in order to ensure that adiabatic rules are met.

• There are two basic approaches to timing:There are two basic approaches to timing:– GlobalGlobal (a.k.a. (a.k.a. clockedclocked, a.k.a. , a.k.a. synchronoussynchronous) timing) timing

• Approach in nearly all conventional irreversible CPUsApproach in nearly all conventional irreversible CPUs• Basis for all practical adiabatic/quantum computing mechanisms Basis for all practical adiabatic/quantum computing mechanisms

proposed to dateproposed to date– LocalLocal (a.k.a. (a.k.a. self-timedself-timed, a.k.a. , a.k.a. asynchronousasynchronous) timing) timing

• Implemented in a few commercial irreversible chips.Implemented in a few commercial irreversible chips.• Feynman ‘86 showed a self-timed Feynman ‘86 showed a self-timed serialserial reversible computation reversible computation

was implementable in QM, in principlewas implementable in QM, in principle• Margolus ‘90 extended this to a 2-D model with 1-D of Margolus ‘90 extended this to a 2-D model with 1-D of

parallelism. - Will it work in 3-D?parallelism. - Will it work in 3-D?

Global TimingGlobal Timing• Examples of adiabatic systems designed on the Examples of adiabatic systems designed on the

basis of global, synchronous timing:basis of global, synchronous timing:– Adiabatic CMOS with external power/clock railsAdiabatic CMOS with external power/clock rails– Superconducting parametric quantron (Likharev)Superconducting parametric quantron (Likharev)– Adiabatic Quantum-Dot Cellular Automaton (Lent)Adiabatic Quantum-Dot Cellular Automaton (Lent)– Adiabatic mechanical logics (Merkle, Drexler)Adiabatic mechanical logics (Merkle, Drexler)– All proposed quantum computersAll proposed quantum computers

• But, a problem: But, a problem: Synchronous timing may not Synchronous timing may not scale!scale!– Work by Janzig & others raises issues of possible Work by Janzig & others raises issues of possible

limits due to quantum uncertainty. Unresolved.limits due to quantum uncertainty. Unresolved.

Clock/Power Supply DesiderataClock/Power Supply Desiderata• Requirements for an adiabatic timing signal / power Requirements for an adiabatic timing signal / power

supply:supply:– Generate trapezoidal waveform with Generate trapezoidal waveform with very flatvery flat high/low regions high/low regions

• Flatness limits Flatness limits QQ of logic. of logic.• Waveform during transitions is Waveform during transitions is ideallyideally linear, linear,

– But this does not affect maximum But this does not affect maximum QQ, only energy coefficient., only energy coefficient.

– Operate resonantly with logic, with high Operate resonantly with logic, with high QQ..• Power supply Power supply QQ will limit overall system will limit overall system QQ

– Reasonable cost, compared to logic it powers.Reasonable cost, compared to logic it powers.– If possible, scale If possible, scale QQ tt (cycle time) (cycle time)

• Required to be considered an Required to be considered an adiabaticadiabatic mechanism. mechanism.• May conflict w. inductor scaling laws!May conflict w. inductor scaling laws!• At the least, At the least, QQ should be high at leakage-limited speed should be high at leakage-limited speed

(Ideally,independentof t.)

Supply concepts in my researchSupply concepts in my research• Superpose several sinusoidal signals from phase-Superpose several sinusoidal signals from phase-

synchronized oscillators at harmonics of fundamental synchronized oscillators at harmonics of fundamental frequencyfrequency– Weight these frequency components as per Fourier Weight these frequency components as per Fourier

transform of desired waveformtransform of desired waveform

• Create relatively high-Create relatively high-LL integrated inductors via integrated inductors via vertical, helical metal coilsvertical, helical metal coils– Only thin oxide layers between turnsOnly thin oxide layers between turns

• Use mechanically oscillating, capacitive MEMS Use mechanically oscillating, capacitive MEMS structures structures in vacuoin vacuo as high- as high-QQ (~10k) oscillator (~10k) oscillator– Use geometry to get desired wave shape directlyUse geometry to get desired wave shape directly

Newer Supply ConceptsNewer Supply Concepts• Transmission-line-based adiabatic resonators.Transmission-line-based adiabatic resonators.

– See transparency.See transparency.

• MEMS-based resonant power supplyMEMS-based resonant power supply– See transparency, & next slideSee transparency, & next slide

• Ideal adiabatic supplies - Can they exist?Ideal adiabatic supplies - Can they exist?– Idealized mechanical model: See transparency.Idealized mechanical model: See transparency.– But, may be quantum limits to But, may be quantum limits to

reusability/scalability of global timing signals.reusability/scalability of global timing signals.• This is a very fundamental issue!This is a very fundamental issue!

A MEMS Supply ConceptA MEMS Supply Concept• Energy storedEnergy stored

mechanically.mechanically.• Variable couplingVariable coupling

strength -> customstrength -> customwave shape.wave shape.

• Can reduce lossesCan reduce lossesthrough balancing,through balancing,filtering.filtering.

• Issue: How toIssue: How toadjust frequency?adjust frequency?

Programming Model DesiderataProgramming Model Desiderata• Should permit optimally efficient quantum Should permit optimally efficient quantum

algorithms (constant-factor slowdowns only).algorithms (constant-factor slowdowns only).• Should have reasonable constant factor overheads.Should have reasonable constant factor overheads.• Unit cell complexity should be kept low for ease Unit cell complexity should be kept low for ease

of design & assembly.of design & assembly.• Should provide a clear separation between Should provide a clear separation between

program and data, where appropriate.program and data, where appropriate.• Should be straightforward to program (and to Should be straightforward to program (and to

write compilers for).write compilers for).

Candidate Programming ModelCandidate Programming Model• Unit-cell capabilities:Unit-cell capabilities:

– A small number of A small number of nn-qubit integer registers.-qubit integer registers.– Perform programmable 2- and 3- qubit ops on selected Perform programmable 2- and 3- qubit ops on selected

data bits (or data bits (or nn-qubit words):-qubit words):• Classical digital ops: CNOT, CCNOT, swaps, Classical digital ops: CNOT, CCNOT, swaps, etc.etc.• 1-bit analog unitary ops1-bit analog unitary ops

– w. precision up to the limit of the qubit device technologyw. precision up to the limit of the qubit device technology– Treat imprecision like decoherence noise, correct it away?Treat imprecision like decoherence noise, correct it away?

– Flow of control:Flow of control:• For reversibility, could have 2 instruction registers, which take For reversibility, could have 2 instruction registers, which take

turns executing & loading each other.turns executing & loading each other.– Data movement:Data movement:

• Streaming between neighboring unit cells.Streaming between neighboring unit cells.

Node Architecture SketchNode Architecture Sketch

Data registers

I/O registers

Execution unit

Instructionregisters

Data pathto/from

neighbornode

(in 2d or 3d)

2. Simulating physical systems 2. Simulating physical systems on our QC modelon our QC model

–Many-particle Schrödinger equationMany-particle Schrödinger equation•Numerically stable classical reversible simsNumerically stable classical reversible sims•Quantum equivalentsQuantum equivalents

–Quantum field theoryQuantum field theory

Simulating Wave MechanicsSimulating Wave Mechanics• The basic problem situation: The basic problem situation:

– Given:Given:• A (possibly complex) initial wavefunctionA (possibly complex) initial wavefunction

in an in an NN-dimensional position basis, and -dimensional position basis, and • a (possibly complex and time-varying) potential energy a (possibly complex and time-varying) potential energy

function ,function ,• a time a time tt after (or before) after (or before) tt00,,

– Compute:Compute:•

• Many practical physics applications...Many practical physics applications...

),( txV

),( 00 tx

),( tx

The Problem with the ProblemThe Problem with the Problem• An efficient technique (when possible): An efficient technique (when possible):

– Convert Convert VV to the corresponding Hamiltonian to the corresponding Hamiltonian HH..– Find the energy eigenstates of Find the energy eigenstates of HH..– Project Project onto eigenstate basis. onto eigenstate basis.– Multiply each component by .Multiply each component by .– Project back onto position basis. Project back onto position basis.

• Problem:Problem:– It may be intractable to find the eigenstates!It may be intractable to find the eigenstates!

• We resort to numerical methods...We resort to numerical methods...

)( 0ttiHe

History of Reversible Schrödinger Sim. History of Reversible Schrödinger Sim.

• Technique discovered by Ed Fredkin and student Technique discovered by Ed Fredkin and student William Barton at MIT in 1975.William Barton at MIT in 1975.

• Subsequently proved by Feynman to exactly Subsequently proved by Feynman to exactly conserve a certain probability measure:conserve a certain probability measure:

PPtt = = RRtt22 + + IItt11··IItt+1+1

• 1-D simulations in C/Xlib written by Frank at MIT 1-D simulations in C/Xlib written by Frank at MIT in 1996. Good behavior observed.in 1996. Good behavior observed.

• 1 & 2-D simulations in Java, and proof of stability 1 & 2-D simulations in Java, and proof of stability by Motter at UF in 2000.by Motter at UF in 2000.

• User-friendly Java GUI by Holz at UF, 2002.User-friendly Java GUI by Holz at UF, 2002.

See http://www.cise.ufl.edu/~mpf/sch

(R=real, I=imag., t=time step index)

Difference EquationsDifference Equations• Consider any system with state Consider any system with state xx that evolves that evolves

according to a diff. eq. that is 1st-order in time:according to a diff. eq. that is 1st-order in time:xx = = ff((xx))

• Discretize time to finite scale Discretize time to finite scale tt, and use a , and use a difference equationdifference equation instead: instead:

xx((t t + + tt) = ) = xx((tt) + ) + t ·ft ·f((xx((tt))))• Problem: Behavior not always numerically Problem: Behavior not always numerically

stable.stable.– Errors can accumulate and grow exponentially.Errors can accumulate and grow exponentially.

Centered Difference EquationsCentered Difference Equations• Discretize derivatives in a Discretize derivatives in a symmetricsymmetric fashion: fashion:

• Leads to update rules like:Leads to update rules like:xx((t t + + tt) = ) = xx((tt tt) + 2) + 2t ·ft ·f((xx((tt))))

• Problem: States at odd- vs. even-Problem: States at odd- vs. even-numbered time steps not constrainednumbered time steps not constrainedto stay close to each other!to stay close to each other!

t

ttxttx

t

x

2

)()(

d

d

+

x1

x2

g

x3 +g

x4+g

2t·f

Centered Schrödinger EquationCentered Schrödinger Equation• Schrödinger’s equation for 1 particle in 1-D:Schrödinger’s equation for 1 particle in 1-D:

• Replace time (& also space) derivatives with centered Replace time (& also space) derivatives with centered differences.differences.

• Centered difference equation has Centered difference equation has realrealpart at odd times that depends only onpart at odd times that depends only onimaginaryimaginary part at even times, & part at even times, &vice-versa.vice-versa.– Drift not an issue - real & imaginaryDrift not an issue - real & imaginary

parts represent different state components! parts represent different state components!

),(),(d

d

2),(

d

d2

22

txtxVxm

txt

i

))(()()( tgitttt

R1

I2

g

R3 +g

I4

g

Proof of StabilityProof of Stability• Technique is proved perfectly numerically stable & Technique is proved perfectly numerically stable &

convergent assuming convergent assuming VV is 0 and is 0 andxx22//tt > > //mm (an angular velocity)(an angular velocity)

• Elements of proof:Elements of proof:– Lax-Richmyer equivalence: convergenceLax-Richmyer equivalence: convergencestability.stability.– Analyze amplitudes of Fourier-transformed basisAnalyze amplitudes of Fourier-transformed basis

• Sufficient due to Parseval’s relationSufficient due to Parseval’s relation– Use theorem (Use theorem (cf.cf. Strikwerda) equating stability to certain Strikwerda) equating stability to certain

conditions on the roots of an conditions on the roots of an amplification amplification polynomial polynomial ((gg,,), ), which are satisfied by our rule.which are satisfied by our rule.

• Empirically, technique looks perfectly stable even for Empirically, technique looks perfectly stable even for more complex potential energy funcs.more complex potential energy funcs.

Phenomena Observed in ModelPhenomena Observed in Model• Perfect reversibilityPerfect reversibility• Wave packet momentumWave packet momentum• Conservation of probability massConservation of probability mass• Harmonic oscillatorHarmonic oscillator• Tunnelling/reflection at potential energy Tunnelling/reflection at potential energy

barriers barriers • Interference fringesInterference fringes• DiffractionDiffraction

Gaussian wave packet moving to the right;Array of small sharp potential-energy barriers

Initial reflection/refraction of wave packet

A little later

Aimed a little higher

A faster-moving particle

Interesting Features of this ModelInteresting Features of this Model• Can be implemented Can be implemented perfectlyperfectly reversibly, with zero reversibly, with zero

asymptotic spacetime overheadasymptotic spacetime overhead– Every last bit is accounted for!Every last bit is accounted for!

• As a result, algorithm can run adiabatically, with As a result, algorithm can run adiabatically, with power dissipation approaching zeropower dissipation approaching zero– Modulo leakage & frictional lossesModulo leakage & frictional losses

• Can map it to a unitary quantum algorithmCan map it to a unitary quantum algorithm– Direct mapping: Direct mapping:

• Classical reversible ops only, no quantum speedupClassical reversible ops only, no quantum speedup– Indirect (implicit) mapping:Indirect (implicit) mapping:

• Simulate Simulate pp particles on particles on kkdd lattice sites using lattice sites using pdpd lg lg kk qubits qubits• Time per update step is order Time per update step is order pdpd lg lg kk instead of instead of kkpdpd

Implicit MappingImplicit Mapping• Use Use pdpd integer registers integer registers xxjj, each lg , each lg kk qubits long qubits long

• Amplitude of joint state of all registers represents Amplitude of joint state of all registers represents amplitude of wavefunction point amplitude of wavefunction point xx

• The difference equation term for dimension The difference equation term for dimension jj amounts to amounts to multiplication of state by matrixmultiplication of state by matrix

– (can be normalized to be) nearly unitary for small (can be normalized to be) nearly unitary for small • Idea: Can approximate Idea: Can approximate DDjj using 1-qubit ops on low-order using 1-qubit ops on low-order

bit of bit of xxii, plus CCNOTs to do carries., plus CCNOTs to do carries.

)21(

)21()21(

iiii

iiiiii

D j

xi

Field Theory SystemsField Theory Systems• Goal:Goal: Simulate field theory for Simulate field theory for pp particle particle typestypes in in dd--

dimensional space over dimensional space over kkdd lattice sites lattice sites• General approach:General approach: At each lattice site, have At each lattice site, have pp integer integer

qubit registers qubit registers nnjj,, denoting the occupancy number of denoting the occupancy number of

particle type particle type jj..– nnjj = 0 or 1 for each type of fermion = 0 or 1 for each type of fermion– nnjj from 0 to from 0 to nnmaxmax for bosons for bosons

• nnmaxmax determined by available total energy determined by available total energy

• Use quantum LGCA model (type I)Use quantum LGCA model (type I)– Interact particles at site using “collision” operatorInteract particles at site using “collision” operator

• Includes particle creation/annihilation operatorsIncludes particle creation/annihilation operators– Stream particles between sites after collision stepStream particles between sites after collision step

3. Quantum computer simulation 3. Quantum computer simulation on classical computerson classical computers

–Visualization techniques, various optimizationsVisualization techniques, various optimizations–Polynomial-space techniquesPolynomial-space techniques

Simulation of QC AlgorithmsSimulation of QC Algorithms• Visualization:Visualization:

– Project states onto 2-D/3-D spacesProject states onto 2-D/3-D spaces• Corresponding to register pairs/triplets.Corresponding to register pairs/triplets.

– Use HSV color space to represent amplitudes.Use HSV color space to represent amplitudes.– Visualize gate ops with continuous color change.Visualize gate ops with continuous color change.

• Simulation Efficiency: Simulation Efficiency: – Optimizations:Optimizations:

• Track only states having non-zero amplitude.Track only states having non-zero amplitude.– Linear-space simulations of Linear-space simulations of nn-qubit systems.-qubit systems.

Visualization TechniqueVisualization Technique

• Illustration: 3 stages of Shor’s algorithm• Register value spatial position of pixel• Phase angle pixel color hue.• Magnitude pixel color saturation.

Initial StateInitial State

After doing Hadamard transform After doing Hadamard transform on all bits of on all bits of aa

After modular exponentiationAfter modular exponentiationbb==xxaa (mod (mod NN))

State After Fourier TransformState After Fourier Transform

Efficient QC SimulationsEfficient QC Simulations• Task: Simulate an Task: Simulate an nn-qubit quantum computer.-qubit quantum computer.• Maximally stupid approach:Maximally stupid approach:

– Store a 2Store a 2nn-element vector-element vector– Multiply it by a full 2Multiply it by a full 2nn××22nn matrix for each gate op matrix for each gate op

• Some obvious optimizations:Some obvious optimizations:– Never store whole matrix (compute dynamically)Never store whole matrix (compute dynamically)– Store only nonzero elements of state vectorStore only nonzero elements of state vector

• Especially helpful when qubits are highly correlatedEspecially helpful when qubits are highly correlated– Do only constant work per nonzero vector elementDo only constant work per nonzero vector element

• Scatter amplitude from each state to 1 or 2 successorsScatter amplitude from each state to 1 or 2 successors– Drop small-probability-mass sets of statesDrop small-probability-mass sets of states

• Linearity of QM implies no chaotic growth of errorsLinearity of QM implies no chaotic growth of errors

Linear-space quantum simulationLinear-space quantum simulation• A popular myth:A popular myth:

– ““Simulating an Simulating an nn-qubit (or -qubit (or nn-particle) quantum system -particle) quantum system takes takes ee((nn)) space (as well as time).” space (as well as time).”

• The usual justification:The usual justification:– It takes It takes ee((nn)) numbers even to numbers even to representrepresent a single a single ((nn)-)-

dimensional state vector, in general.dimensional state vector, in general.

• The hole in that argument:The hole in that argument:– Can simulate the statistical behavior of a quantum system Can simulate the statistical behavior of a quantum system

w/o ever storing a state vector!w/o ever storing a state vector!

• Result Result BQPBQP PSPACEPSPACE known since BV’93... known since BV’93...– But practical poly-space sims are rarely describedBut practical poly-space sims are rarely described

The Basic IdeaThe Basic Idea• Inspiration:Inspiration:

– Feynman’s Feynman’s path integralpath integral formulation of QED. formulation of QED.– Gives the amplitude of a given final configuration Gives the amplitude of a given final configuration

by accumulating amplitude over all paths from by accumulating amplitude over all paths from initial to final configurations.initial to final configurations.

– Each path consists of only a single Each path consists of only a single ((nn)-coordinate )-coordinate configuration at each time, configuration at each time, notnot a full wavefunction a full wavefunction over the configuration space.over the configuration space.

– Can enumerate all paths, Can enumerate all paths, while only ever while only ever representing one path at a time.representing one path at a time.

Simulating Quantum ComputationsSimulating Quantum Computations• Given:Given:

– Any Any nn-qubit quantum computation, expressed as a -qubit quantum computation, expressed as a sequence of 1-qubit gates and CNOT gates.sequence of 1-qubit gates and CNOT gates.

– An initial state An initial state ss00 which is just a basis state in the which is just a basis state in the

classical bitwise basis, classical bitwise basis, e.g.e.g. 0000000000..• Goal:Goal:

– Generate a final basis state stochasically with the Generate a final basis state stochasically with the same probability distribution as the quantum same probability distribution as the quantum computer would do.computer would do.

U1

U3

U4

U2

Matrix RepresentationMatrix Representation• Consider each gate as rank-2Consider each gate as rank-2nn unitary matrix: unitary matrix:

– Each CNOT application is a 0-1 (permutation) Each CNOT application is a 0-1 (permutation) matrix - a classical reversible bit-operation.matrix - a classical reversible bit-operation.

– With appropriate row ordering, each With appropriate row ordering, each UUii gate gate

application is block-diagonal, w. each 2×2 block application is block-diagonal, w. each 2×2 block equal to equal to UUii..

– We need never represent these full matrices!We need never represent these full matrices!– The 1 or 2 nonzero entries in a given row can be The 1 or 2 nonzero entries in a given row can be

located & computed immediately given the row id located & computed immediately given the row id (bit string) and (bit string) and UUii..

The Linear-Space AlgorithmThe Linear-Space Algorithm• Generate a random “coin” Generate a random “coin” cc[0,1]. [0,1]. • Initialize probability accumulator: Initialize probability accumulator: pp0.0.• For each final For each final nn-bit string -bit string yy at time at time tt,,

– Compute its amplitude Compute its amplitude ((yy) as follows:) as follows:• Generate its possible 1 or 2 predecessor stringsGenerate its possible 1 or 2 predecessor strings

xx11 (and maybe (and maybe xx22) given the gate-op preceding ) given the gate-op preceding tt..• For each predecessor, compute its amplitude at time For each predecessor, compute its amplitude at time tt1 1

recursively using this same algorithm,recursively using this same algorithm,– unless unless tt=0, in which case =0, in which case =1 if =1 if xx==ss00, 0 otherwise., 0 otherwise.

• Add predecessor amplitudes, weighted by entries.Add predecessor amplitudes, weighted by entries.– Maybe output Maybe output yy, using roulette wheel algorithm:, using roulette wheel algorithm:

• Accumlate Pr[Accumlate Pr[yy] into total: ] into total: p p p +||p +||((yy)||)||22

• Output Output yy and halt if and halt if pp>>cc..

A Further OptimizationA Further Optimization• Don’t even have to enumerate all final states!Don’t even have to enumerate all final states!

– Instead: Stochasically follow a trajectory.Instead: Stochasically follow a trajectory.

• Basic idea:Basic idea:– Keep track of 1 “current” state & its amplitude Keep track of 1 “current” state & its amplitude 00..– For CNOTs: Deterministically transform state.For CNOTs: Deterministically transform state.– For For UUs:s:

• Calculate amplitude Calculate amplitude 11 of “neighbor” state w. path-integral of “neighbor” state w. path-integral• Calculate amplitudes Calculate amplitudes 00 and and 11 after qubit op after qubit op• Choose 1 successor as new current state, using |Choose 1 successor as new current state, using |||22 distrib. distrib.

0

1

u00

u10 u01

u11

’0

’1

Possiblesuccessors

Current state

“Neighbor” state

Complexity ComparisonComplexity Comparison• To simulate To simulate tt gate ops ( gate ops (cc CNOTs + CNOTs + uu 1-bit unitary 1-bit unitary

ops) of an ops) of an nn-qubit quantum computer:-qubit quantum computer:SpaceSpace TimeTime

Traditional method:Traditional method: 2 2nn tt·2·2nn

Path-integral method:Path-integral method: tt··nn n n·2·2tt

– (Actually, only the (Actually, only the uu unitary ops, not all unitary ops, not all tt ops ops or all or all nn qubits, contribute to any of the exponents here.)qubits, contribute to any of the exponents here.)

• Upshot:Upshot:– Lower space usage can allow larger systems to be Lower space usage can allow larger systems to be

simulated, for short periods.simulated, for short periods.– Run time is competitive for case when Run time is competitive for case when tt < < nn

ConclusionConclusion• A grab-bag of tricks and techniquesA grab-bag of tricks and techniques• Outline of a research program is taking shapeOutline of a research program is taking shape• Quantum computing is really interesting...Quantum computing is really interesting...

– Now if only I can get someone to pay me to devote Now if only I can get someone to pay me to devote my full time to studying it!my full time to studying it!

Slides left over from USC talkSlides left over from USC talk

To import as neededTo import as needed

Reversibility of PhysicsReversibility of Physics• The universe is (apparently) a closed systemThe universe is (apparently) a closed system• Closed systems evolve via unitary transformsClosed systems evolve via unitary transforms

– ApparentApparent wavefunction collapse wavefunction collapse doesn’tdoesn’t contradict this contradict this (confirmed by work of Everett, Zurek, (confirmed by work of Everett, Zurek, etc.etc.))

• Time-evolution of concrete state of universe (or Time-evolution of concrete state of universe (or closed subsystems) is closed subsystems) is reversiblereversible::– Invertible (bijective)Invertible (bijective)– Deterministic looking backwards in timeDeterministic looking backwards in time– Total info. (log # of poss. states) doesn’t decreaseTotal info. (log # of poss. states) doesn’t decrease

• Can increase, though, if volume is increasingCan increase, though, if volume is increasing

• Information cannot be destroyed!Information cannot be destroyed!

Illustrating Landauer’s principleIllustrating Landauer’s principle

0

0

1

1

……

Nstates

Nstates

0s0

sN-1

s’0

s’N-1

0…

0

0

…

s0’’

sN-1’’

sN’’

s2N-1’’

……

2Nstates

Unitary(1-1)

evolution

Before bit erasure After bit erasure

Benefits of Reversible ComputingBenefits of Reversible Computing• Reduces energy/cooling costs of computingReduces energy/cooling costs of computing

– Improves performance per unit power consumedImproves performance per unit power consumed

• Given heat flux limits in the cooling system,Given heat flux limits in the cooling system,– Improves performance per unit convex hull areaImproves performance per unit convex hull area

• A faster machine in a given size box.A faster machine in a given size box.

• For communication-intensive parallel algorithms,For communication-intensive parallel algorithms,– Improves performance, period!Improves performance, period!

All these benefits are by small polynomial factors in All these benefits are by small polynomial factors in the integration scale & the device properties.the integration scale & the device properties.

Quantum Computing pros/consQuantum Computing pros/cons• Pros:Pros:

– Removes an unnecessary restriction on the types of quantum Removes an unnecessary restriction on the types of quantum states & ops usable for computation.states & ops usable for computation.

– Opens up exponentially shorterOpens up exponentially shorter paths paths to solving to solving somesome types of types of problems (problems (e.g.e.g., factoring, simulation), factoring, simulation)

• Cons:Cons:– Sensitive, requires overhead for error correction.Sensitive, requires overhead for error correction.– Also, still remains subject to fundamental physical bounds on Also, still remains subject to fundamental physical bounds on

info. density, & rate of state change!info. density, & rate of state change!• Myth: Myth: “A quantum memory can store an exponentially large amount “A quantum memory can store an exponentially large amount

of data.”of data.”• Myth:Myth: “A quantum computer can perform operations at an “A quantum computer can perform operations at an

exponentially faster rate than a classical one.”exponentially faster rate than a classical one.”

Some goals of my QC workSome goals of my QC work• Develop a UMS model of computation that Develop a UMS model of computation that

incorporates quantum computing.incorporates quantum computing.• Design & simulate quantum computer Design & simulate quantum computer

architectures, programming languages, architectures, programming languages, etc.etc. • Describe how to do the systems-engineering Describe how to do the systems-engineering

optimization of quantum computers for various optimization of quantum computers for various problems of interest.problems of interest.

ConclusionConclusion• As we near the physical limits of computing, As we near the physical limits of computing,

– Further improvements will require an increasingly Further improvements will require an increasingly sophisticated interdisciplinary integration of concerns across sophisticated interdisciplinary integration of concerns across many levels of engineering.many levels of engineering.

• I am developing a principled I am developing a principled nanocomputer systems nanocomputer systems engineeringengineering methodology methodology– And applying it to the problem of determining the real cost-And applying it to the problem of determining the real cost-

efficiency of new models of computing:efficiency of new models of computing:• Reversible computingReversible computing• Quantum computingQuantum computing

• Building the foundations of a new discipline that will be Building the foundations of a new discipline that will be critical in coming decades.critical in coming decades.