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Title: Quantum Computer Architectures for Physical Simulations Dr. Mike Frank University of Florida CISE Department mpf@cise.ufl.edu

Quantum Computer Architectures for Physical
Simulations Dr. Mike Frank University of
Florida CISE Department mpf_at_cise.ufl.edu
  • Presented at
  • Quantum Computation for Physical Modeling
    Workshop Wed., May 8, 2002

Summary of Current Research
  • One of my two major research projects
  • Reversible Quantum Computing Project
  • Studying developing physical computing theory
  • Technology-independent physical limits of
  • Ultimate models of computing for complexity
  • Nanocomputer systems engineering
  • Optimizing cost-efficiency scalability of future
  • Reversible computing quantum computing
  • Realistic models, CPU architectures, optimal
    scaling advantages

MIT 1995-1999, UF 1999-
RevComp project heritage
  • Grew out of work started by the MIT Information
    Mechanics group in the 1970s.
  • Key members Fredkin, Toffoli, Margolus
  • Occasional collaboration with Feynman, Bennett,
  • This group laid much theoretical groundwork for
    reversible ( eventually quantum) computing.
  • MIT Reversible Computing project (1990s)
  • Group leaders Knight, Margolus
  • Students Younis, DSouza, Becker, Vieri, Frank,
  • Focus Reducing reversible computing to practice
  • CMOS circuit styles, test chips, architectures,
    complexity theory, algorithms, high-level

RevComp group at UF
  • Reversible Quantum Computing group
  • Organized by myself (CISE/ECE depts.)
  • UF collabs. in CISE, ECE, Math, Phys./Chem.
  • Notable graduate DoRon Motter (highest honors)
  • Now doing PhD work in quantum circuits at U.
  • 2 current Ph.D students (all ECE)
  • Current focus
  • Removing remaining barriers to near-term
    practicality of reversible computing
  • Improved circuit styles, efficient power supplies
  • Other applications
  • Long-term study of physical computing theory and
    scaling advantages of reversible/quantum models.

Who We Are
  • Dr. Michael Frank
  • MIT Ph.D. stud. postdoc, 1996-97 1999.
  • Area exam studies on quantum computing.
  • DARPA-funded reversible computing research.
  • 1999-now Head of Reversible Quantum Computing
    group at UFs CISE dept.
  • http//www.cise.ufl.edu/research/revcomp

Some lineage
Who We Are, cont.
  • DoRon Motter
  • Undergrad in UF CISE dept., 1997-2000.
  • Coursework in CS quantum mechanics.
  • Sr. highest honors thesis w. Dr. Frank, 2000.
  • Now a Masters student at U. Mich.
  • Advisor Igor Markov, U. Mich.
  • DARPA-funded project on quantum logic systhesis

Overview of Talk
  • 1. Optimally scalable QC models/architectures
  • Universally maximally scalable (UMS) models
  • Physical limits shape the ultimate model
  • Appropriate programming models
  • 2. Physics sim. algorithms on our QC model
  • Many-particle Schrödinger equation
  • Numerically stable classical reversible sims
  • Quantum versions
  • Quantum field theory
  • 3. QC simulation on classical computers
  • Visualization techniques, various optimizations
  • Polynomial-space techniques

1. Optimally scalable quantum computer
  • Universally maximally scalable (UMS) models
  • Physical limits shape the ultimate model
  • Appropriate programming models

Nanocomputer systems engineering
  • A key goal of my long-term research program
  • Develop key foundations for a new discipline of
    nanocomputer systems engineering suited for
    meeting the challenges of computing at the
  • And convey it to peers, teach it to students.
  • The new field will integrate concerns methods
    from a variety of disciplines
  • Physics of computing ? Algorithm design
  • Systems engineering ? Electrical eng., etc.
  • Computer architecture ? Quantum computing!
  • Complexity theory

Source ITRS 99
½CV2 based on ITRS 99 figures for Vdd and
minimum transistor gate capacitance. T300 K
Physical Computing Theory
  • The study of theoretical models of computation
    that are based on (or closely tied to) physics
  • Make no nonphysical assumptions!
  • Includes the study of
  • Fundamental physical limits of computing
  • Physically-based models of computing
  • Includes reversible and/or quantum models
  • Ultimate (asymptotically optimal) models
  • An asymptotically tight Churchs thesis
  • Model-independent basis for complexity theory
  • Basis for design of future nanocomputer
  • Asymptotic scaling of architectures algorithms
  • Physically optimal algorithms

Ultimate Models of Computing
  • We would like models of computing that match the
    real computational power of physics.
  • Not too weak, not too strong.
  • Most traditional models of computing only match
    physics to within polynomial factors.
  • Misleading asymptotic performance of algorithms.
  • Not good enough to form the basis for a real
    systems engineering optimization of
  • Develop models of computing that are
  • As powerful as physically possible on all
  • Realistic within asymptotic constant factors

Unstructured Search Problem
  • Given a set S of N elements and a black-box
    function fS?0,1, find an element x?S such that
    f(x)1, if one exists (or if not, say so).
  • Any NP problem can be cast as an unstructured
    search problem.
  • Not necessarily the optimal approach, however.
  • Bounds on classical run-time
  • ?(N) expected queries in worst case (0 or 1
  • Have to try N/2 elements on average before
    finding soln.
  • Have to try all N if there is no solution.
  • If elements are length-? bit strings,
  • Expected trials is ?(2?) - exponential in ?.

Quantum Unstructured Search
  • Minimum time to solve unstructured search problem
    on a (serial) quantum computer is
  • ?(N1/2) queries (2?/2) (21/2)?
  • Still exponential, but with a smaller base.
  • The minimum of queries can be achieved using
    Grovers algorithm.

Classical Unstructured Search
  • The classical serial algorithm takes ?(N) time.
  • But Suppose we search in parallel!
  • Have MltN processors running in parallel.
  • Each searches a different subset of N/M elements
    of the search space.
  • If processors are ballistic reversible
  • Can cluster them in a dense mesh of diameter
  • Time accounting
  • Computation time ?(N/M)
  • Communication time ?(M1/3) (at lightspeed)
  • Total T ? N/M M1/3 is minimized when M ?
    N3/4 ? N1/4 Faster than Grovers

ClassicalQuantum Parallelism
  • Similar setup to classical parallelism
  • M processors, searching N/M items each.
  • Except, each processor uses Grovers algorithm.
  • Time accounting
  • Computation T ? (N/M)1/2
  • Communication T ? M1/3 (as before)
  • Total T ? (N/M)1/2 M1/3
  • Total is minimized when M?N 3/5
  • Minimized total is T ? N1/5.
  • I.e., quantum unstructured search is really only
    N1/4/N1/5 N1/20 faster than classical!

Scalability Maximal Scalability
  • A multiprocessor architecture accompanying
    performance model is scalable if
  • it can be scaled up to arbitrarily large
    problem sizes, and/or arbitrarily large numbers
    of processors, without the predictions of the
    performance model breaking down.
  • An architecture ( model) is maximally scalable
    for a given problem if
  • it is scalable and if no other scalable
    architecture can claim asymptotically superior
    performance on that problem
  • It is universally maximally scalable (UMS) if it
    is maximally scalable on all problems!
  • I will briefly mention some characteristics of
    architectures that are universally maximally

Universal Maximum Scalability
  • Existence proof for universally maximally
    scalable (UMS) architectures
  • Physics itself is a universal maximally scalable
    architecture because any real computer is
    merely a special case of a physical system.
  • Obviously, no real computer can beat the
    performance of physical systems in general.
  • Unfortunately, physics doesnt give us a very
    simple or convenient programming model.
  • Comprehensive expertise at programming physics
    means mastery of all physical engineering
    disciplines chemical, electrical, mechanical,
    optical, etc.
  • Wed like an easier programming model than this!

Physics Constrains the Ultimate Model
Limits of Quantum Computers
  • Quantum computers remain subject to all the
    fundamental limits previously mentioned!
  • Entropy density limit - only 2n distingable
  • Contrary to press manglings, a quantum computer
    cannot store exponentially large amounts of
    arbitrary data!
  • Information propagation limit - at most c
  • Bells theorem, teleportation, superluminal
    wave velocities do not give gtc information
  • Quantum field theory is explicitly local
  • Computation rate limit - at most 4E/h rate of
    orthogonal transitions, given available energy E.
  • Non-orthogonal ones are faster, but accomplish
    less work
  • Speedups are due to fewer ops needed, not faster

Simple UMS Architectures
  • (I propose) any practical UMS architecture will
    have the following features
  • Processing elements characterized by constant
    parameters (independent of of processors)
  • Mesh-type message-passing interconnection
    network, arbitrarily scalable in 2 dimensions
  • w. limited scalability in 3rd dimension.
  • Processing elements that can be operated in a
    highly reversible mode, at least up to some
  • Enables improved 3-d scalability, in a limited
  • (In long term) Have capability for
    quantum-coherent operation, for extra perf. on
    some probs.

Ideally Scalable Architectures
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
Processing Node
Processing Node
Processing Node
Local memory hierarchy (optimal fixed size)
Local memory hierarchy (optimal fixed size)
Local memory hierarchy (optimal fixed size)
Processing Node
Processing Node
Processing Node
Local memory hierarchy (optimal fixed size)
Local memory hierarchy (optimal fixed size)
Local memory hierarchy (optimal fixed size)
Mesh interconnection network
Some device parameters
  • The following parameters are considered
    fixed (for a given device/node technology)
  • (Maximum) number of bits of state per node
  • (Minimum) node volume
  • (Minimum) transition time (per ? transition)
  • (Minimum) entropy generated per bit erased (?k ln
  • (Minimum) static entropy generation rates
  • For devices even just quiescently maintaining
    state info
  • Related to energy leakage rates, decoherence
  • (Minimum) adiabatic frictional coefficient
  • For devices undergoing reversible transitions
  • (Maximum) quality factor Q of active transitions
  • (Minimum) device cost

System-level parameters
  • The following parameters may be adjusted as the
    problem size increases
  • Number of nodes utilized
  • Arrangement of utilized nodes in x, y, z
  • Spreading nodes out optimizes perf. on some
  • Rate of change of an externally-applied clocking
    signal (time-dependent potential)
  • Allows trading off adiabaticity vs. speed of
    computation, as a function of the number of nodes

Entropy coefficients of some reversible logic
gate operations
  • From Frank, Ultimate theoretical models of
    nanocomputers (Nanotechnology, 1998)
  • SCRL, circa 1997 1 b/Hz
  • Optimistic reversible CMOS 10 b/kHz
  • Merkles quantum FET 1.2 b/GHz
  • Nanomechanical rod logic .07 b/GHz
  • Superconducting PQ gate 25 b/THz
  • Helical logic .01 b/THz

How low can you go? We dont really know!
Thermodynamics Scalability
  • The fastest parallel algorithms for many problems
    ideally require a 3-D mesh topology.
  • Minimizes communication latencies between points
  • But, entropy flux bounds imply entropy generation
    rates can scale only proportionally to systems
    2-D outer (convex hull) surface area.
  • Assuming upper bounds on temperature pressure
  • So, can harness 3rd dimension only to the extent
    that useful operations can be made reversible.
  • Optimizing efficiency requires a careful tradeoff
    between performance, power, cost...

(No Transcript)
Reversible/Adiabatic CMOS
  • Chips designed at MIT, 1996-1999

Minimum Losses w. Leakage
Etot Eadia Eleak
Eleak Pleaktr
Eadia cE / tr
Reversible Emulation - Ben89
k 2 n 3
k 3 n 2
Bennett 89 alg. is not optimal
k 2 n 3
k 3 n 2
Just look at all the spacetime it wastes!!!
Parallel Frank02 algorithm
  • We can move the triangles closer together, to
    eliminate the wasted spacetime.
  • Resulting algorithm is linear time for all n and
    k and dominates Ben89 for time, spacetime,

k3 n2
k2 n3
Emulated time
k4 n1
Real time
Spacetime blowup
Energy saved
Perf. scaling w. of devices
  • If alg. is not limited by communications needs,
  • Use irreversible processors spread in a 2-D
  • Remove entropy along perpendicular dimension.
  • No entropy rate limits,
  • so no speed advantage from reversibility.
  • If alg. requires only local communication, latency
    ? cyc. time, in an NDNDND mesh,
  • Leak-free reversible machine perf. scales better!
  • Irreversible tcyc ?(ND1/3)
  • Reversible tcyc ?(ND1/4) ?(ND1/12) faster!
  • To boost reversibility speedup by 10, one must
    consider 1036-CPU machines (1.7 trillion moles
    of CPUs!)
  • 1.7 trillion moles of H atoms weighs 1.7 million
    metric tons!
  • A 100-m tall hill of H-atom sized CPUs!

Lower bound on irrev. time
  • Simulate Nproc ND3 cells for Nsteps ND steps.
  • Consider a sequence of ND update steps.
  • Final cell value depends on ND4 ops in time T.
  • All ops must occur within radius r cT of cell.
  • Surface area A ? T2, rate Rop ? T2 sustainable.
  • Nops ? Rop T ? T3 needs to be at least ND4.
  • ? T must be ?(ND4/3) to do all ND steps.
  • Average time per step must be ?(ND1/3).
  • Any irreversible machine (of any technology or
    architecture) must obey this bound!

Irreversible 3-D Mesh
Reversible 3-D Mesh
Non-local Communication
  • Best computational task for reversibility
  • Each processor must exchange messages with
    another that is ND1/2 nodes away on each cycle
  • Unsure what real-world problem demands this
  • In this case, reversible speedup scales with
    number of CPUs to only the 1/18th power.
  • To boost reversibility speedup by 10, only
    need 1018 (or 1.7 micromoles) of CPUs
  • If each was a 1-nm cluster of 100 C atoms, this
    is only 2 mg mass, volume 1 mm3.
  • Current VLSI Need cost level of 25B before
    you see a speedup.

Open issues for reversible comp.
  • Integrate realistic fundamental models of the
    clocking system into the engineering analysis.
  • There is an open issue about the scalability of
    clock distribution systems.
  • Exist quantum bounds on reusability of timing
  • Not yet clear if reversible clocking is scalable.
  • Fortunately, self-timed reversible computing also
    appears to be a possibility.
  • Not yet clear if this approach works above 1-D
  • Simulation experiments planned to investigate
  • Develop efficient physical realizations of
    nano-scale bit-devices timing systems.

Timing in Adiabatic Systems
  • When multiple adiabatic devices interact, the
    relative timing must be precise, in order to
    ensure that adiabatic rules are met.
  • There are two basic approaches to timing
  • Global (a.k.a. clocked, a.k.a. synchronous)
  • Approach in nearly all conventional irreversible
  • Basis for all practical adiabatic/quantum
    computing mechanisms proposed to date
  • Local (a.k.a. self-timed, a.k.a. asynchronous)
  • Implemented in a few commercial irreversible
  • Feynman 86 showed a self-timed serial reversible
    computation was implementable in QM, in principle
  • Margolus 90 extended this to a 2-D model with
    1-D of parallelism. - Will it work in 3-D?

Global Timing
  • Examples of adiabatic systems designed on the
    basis of global, synchronous timing
  • Adiabatic CMOS with external power/clock rails
  • Superconducting parametric quantron (Likharev)
  • Adiabatic Quantum-Dot Cellular Automaton (Lent)
  • Adiabatic mechanical logics (Merkle, Drexler)
  • All proposed quantum computers
  • But, a problem Synchronous timing may not scale!
  • Work by Janzig others raises issues of possible
    limits due to quantum uncertainty. Unresolved.

Clock/Power Supply Desiderata
  • Requirements for an adiabatic timing signal /
    power supply
  • Generate trapezoidal waveform with very flat
    high/low regions
  • Flatness limits Q of logic.
  • Waveform during transitions is ideally linear,
  • But this does not affect maximum Q, only energy
  • Operate resonantly with logic, with high Q.
  • Power supply Q will limit overall system Q
  • Reasonable cost, compared to logic it powers.
  • If possible, scale Q ? t (cycle time)
  • Required to be considered an adiabatic mechanism.
  • May conflict w. inductor scaling laws!
  • At the least, Q should be high at leakage-limited

(Ideally, independent of t.)
Supply concepts in my research
  • Superpose several sinusoidal signals from
    phase-synchronized oscillators at harmonics of
    fundamental frequency
  • Weight these frequency components as per Fourier
    transform of desired waveform
  • Create relatively high-L integrated inductors via
    vertical, helical metal coils
  • Only thin oxide layers between turns
  • Use mechanically oscillating, capacitive MEMS
    structures in vacuo as high-Q (10k) oscillator
  • Use geometry to get desired wave shape directly

Newer Supply Concepts
  • Transmission-line-based adiabatic resonators.
  • See transparency.
  • MEMS-based resonant power supply
  • See transparency, next slide
  • Ideal adiabatic supplies - Can they exist?
  • Idealized mechanical model See transparency.
  • But, may be quantum limits to reusability/scalabil
    ity of global timing signals.
  • This is a very fundamental issue!

A MEMS Supply Concept
  • Energy stored mechanically.
  • Variable coupling strength -gt custom wave shape.
  • Can reduce losses through balancing, filtering.
  • Issue How to adjust frequency?

Programming Model Desiderata
  • Should permit optimally efficient quantum
    algorithms (constant-factor slowdowns only).
  • Should have reasonable constant factor overheads.
  • Unit cell complexity should be kept low for ease
    of design assembly.
  • Should provide a clear separation between program
    and data, where appropriate.
  • Should be straightforward to program (and to
    write compilers for).

Candidate Programming Model
  • Unit-cell capabilities
  • A small number of n-qubit integer registers.
  • Perform programmable 2- and 3- qubit ops on
    selected data bits (or n-qubit words)
  • Classical digital ops CNOT, CCNOT, swaps, etc.
  • 1-bit analog unitary ops
  • w. precision up to the limit of the qubit device
  • Treat imprecision like decoherence noise, correct
    it away?
  • Flow of control
  • For reversibility, could have 2 instruction
    registers, which take turns executing loading
    each other.
  • Data movement
  • Streaming between neighboring unit cells.

Node Architecture Sketch
Instruction registers
Data path to/from neighbor node (in 2d or 3d)
I/O registers
Execution unit
Data registers
2. Simulating physical systems on our QC model
  • Many-particle Schrödinger equation
  • Numerically stable classical reversible sims
  • Quantum equivalents
  • Quantum field theory

Simulating Wave Mechanics
  • The basic problem situation
  • Given
  • A (possibly complex) initial wavefunction
    in an N-dimensional position basis,
  • a (possibly complex and time-varying) potential
    energy function ,
  • a time t after (or before) t0,
  • Compute
  • Many practical physics applications...

The Problem with the Problem
  • An efficient technique (when possible)
  • Convert V to the corresponding Hamiltonian H.
  • Find the energy eigenstates of H.
  • Project ? onto eigenstate basis.
  • Multiply each component by .
  • Project back onto position basis.
  • Problem
  • It may be intractable to find the eigenstates!
  • We resort to numerical methods...

History of Reversible Schrödinger Sim.
See http//www.cise.ufl.edu/mpf/sch
  • Technique discovered by Ed Fredkin and student
    William Barton at MIT in 1975.
  • Subsequently proved by Feynman to exactly
    conserve a certain probability measure
  • Pt Rt2 It?1It1
  • 1-D simulations in C/Xlib written by Frank at MIT
    in 1996. Good behavior observed.
  • 1 2-D simulations in Java, and proof of
    stability by Motter at UF in 2000.
  • User-friendly Java GUI by Holz at UF, 2002.

(Rreal, Iimag., ttime step index)
Difference Equations
  • Consider any system with state x that evolves
    according to a diff. eq. that is 1st-order in
    time x f(x)
  • Discretize time to finite scale ?t, and use a
    difference equation instead x(t ?t) x(t)
    ?t f(x(t))
  • Problem Behavior not always numerically stable.
  • Errors can accumulate and grow exponentially.

Centered Difference Equations
  • Discretize derivatives in a symmetric fashion
  • Leads to update rules like x(t ?t) x(t ?
    ?t) 2?t f(x(t))
  • Problem States at odd- vs. even- numbered time
    steps not constrained to stay close to each other!




Centered Schrödinger Equation
  • Schrödingers equation for 1 particle in 1-D
  • Replace time ( also space) derivatives with
    centered differences.
  • Centered difference equation has real part at odd
    times that depends only on imaginary part at even
    times, vice-versa.
  • Drift not an issue - real imaginary parts
    represent different state components!


Proof of Stability
  • Technique is proved perfectly numerically stable
    convergent assuming V is 0 and ?x2/?t gt ?/m
    (an angular velocity)
  • Elements of proof
  • Lax-Richmyer equivalence convergence?stability.
  • Analyze amplitudes of Fourier-transformed basis
  • Sufficient due to Parsevals relation
  • Use theorem (cf. Strikwerda) equating stability
    to certain conditions on the roots of an
    amplification polynomial ?(g,?), which are
    satisfied by our rule.
  • Empirically, technique looks perfectly stable
    even for more complex potential energy funcs.

Phenomena Observed in Model
  • Perfect reversibility
  • Wave packet momentum
  • Conservation of probability mass
  • Harmonic oscillator
  • Tunnelling/reflection at potential energy
  • Interference fringes
  • Diffraction

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 Model
  • Can be implemented perfectly reversibly, with
    zero asymptotic spacetime overhead
  • Every last bit is accounted for!
  • As a result, algorithm can run adiabatically,
    with power dissipation approaching zero
  • Modulo leakage frictional losses
  • Can map it to a unitary quantum algorithm
  • Direct mapping
  • Classical reversible ops only, no quantum speedup
  • Indirect (implicit) mapping
  • Simulate p particles on kd lattice sites using pd
    lg k qubits
  • Time per update step is order pd lg k instead of

Implicit Mapping
  • Use pd integer registers xj, each lg k qubits
  • Amplitude of joint state of all registers
    represents amplitude of wavefunction point x
  • The difference equation term for dimension j
    amounts to multiplication of state by matrix
  • (can be normalized to be) nearly unitary for
    small ?
  • Idea Can approximate Dj using 1-qubit ops on
    low-order bit of xi, plus CCNOTs to do carries.

Field Theory Systems
  • Goal Simulate field theory for p particle types
    in d-dimensional space over kd lattice sites
  • General approach At each lattice site, have p
    integer qubit registers nj, denoting the
    occupancy number of particle type j.
  • nj 0 or 1 for each type of fermion
  • nj from 0 to nmax for bosons
  • nmax determined by available total energy
  • Use quantum LGCA model (type I)
  • Interact particles at site using collision
  • Includes particle creation/annihilation operators
  • Stream particles between sites after collision

3. Quantum computer simulation on classical
  • Visualization techniques, various optimizations
  • Polynomial-space techniques

Simulation of QC Algorithms
  • Visualization
  • Project states onto 2-D/3-D spaces
  • Corresponding to register pairs/triplets.
  • Use HSV color space to represent amplitudes.
  • Visualize gate ops with continuous color change.
  • Simulation Efficiency
  • Optimizations
  • Track only states having non-zero amplitude.
  • Linear-space simulations of n-qubit systems.

Visualization Technique
  • Illustration 3 stages of Shors algorithm
  • Register value ? spatial position of pixel
  • Phase angle ? pixel color hue.
  • Magnitude ? pixel color saturation.

Initial State
After doing Hadamard transform on all bits of a
After modular exponentiation bxa (mod N)
State After Fourier Transform
Efficient QC Simulations
  • Task Simulate an n-qubit quantum computer.
  • Maximally stupid approach
  • Store a 2n-element vector
  • Multiply it by a full 2n2n matrix for each gate
  • Some obvious optimizations
  • Never store whole matrix (compute dynamically)
  • Store only nonzero elements of state vector
  • Especially helpful when qubits are highly
  • Do only constant work per nonzero vector element
  • Scatter amplitude from each state to 1 or 2
  • Drop small-probability-mass sets of states
  • Linearity of QM implies no chaotic growth of

Linear-space quantum simulation
  • A popular myth
  • Simulating an n-qubit (or n-particle) quantum
    system takes e?(n) space (as well as time).
  • The usual justification
  • It takes e?(n) numbers even to represent a single
    ?(n)-dimensional state vector, in general.
  • The hole in that argument
  • Can simulate the statistical behavior of a
    quantum system w/o ever storing a state vector!
  • Result BQP ? PSPACE known since BV93...
  • But practical poly-space sims are rarely described

The Basic Idea
  • Inspiration
  • Feynmans path integral formulation of QED.
  • Gives the amplitude of a given final
    configuration by accumulating amplitude over all
    paths from initial to final configurations.
  • Each path consists of only a single
    ?(n)-coordinate configuration at each time, not a
    full wavefunction over the configuration space.
  • Can enumerate all paths, while only ever
    representing one path at a time.

Simulating Quantum Computations
  • Given
  • Any n-qubit quantum computation, expressed as a
    sequence of 1-qubit gates and CNOT gates.
  • An initial state s0 which is just a basis state
    in the classical bitwise basis, e.g. ?00000?.
  • Goal
  • Generate a final basis state stochasically with
    the same probability distribution as the quantum
    computer would do.

Matrix Representation
  • Consider each gate as rank-2n unitary matrix
  • Each CNOT application is a 0-1 (permutation)
    matrix - a classical reversible bit-operation.
  • With appropriate row ordering, each Ui gate
    application is block-diagonal, w. each 22 block
    equal to Ui.
  • We need never represent these full matrices!
  • The 1 or 2 nonzero entries in a given row can be
    located computed immediately given the row id
    (bit string) and Ui.

The Linear-Space Algorithm
  • Generate a random coin c?0,1.
  • Initialize probability accumulator p?0.
  • For each final n-bit string y at time t,
  • Compute its amplitude ?(y) as follows
  • Generate its possible 1 or 2 predecessor strings
    x1 (and maybe x2) given the gate-op preceding t.
  • For each predecessor, compute its amplitude at
    time t?1 recursively using this same algorithm,
  • unless t0, in which case ?1 if ?x?s0, 0
  • Add predecessor amplitudes, weighted by entries.
  • Maybe output y, using roulette wheel algorithm
  • Accumlate Pry into total p ? p ?(y)2
  • Output y and halt if pgtc.

A Further Optimization
  • Dont even have to enumerate all final states!
  • Instead Stochasically follow a trajectory.
  • Basic idea
  • Keep track of 1 current state its amplitude
  • For CNOTs Deterministically transform state.
  • For Us
  • Calculate amplitude ?1 of neighbor state w.
  • Calculate amplitudes ?0 and ?1 after qubit op
  • Choose 1 successor as new current state, using
    ?2 distrib.

Current state
Possible successors
Neighbor state
Complexity Comparison
  • To simulate t gate ops (c CNOTs u 1-bit unitary
    ops) of an n-qubit quantum computer
  • Space Time
  • Traditional method 2n t2n
  • Path-integral method tn n2t
  • (Actually, only the u unitary ops, not all t ops
    or all n qubits, contribute to any of the
    exponents here.)
  • Upshot
  • Lower space usage can allow larger systems to be
    simulated, for short periods.
  • Run time is competitive for case when t lt n

  • A grab-bag of tricks and techniques
  • Outline of a research program is taking shape
  • Quantum computing is really interesting...
  • Now if only I can get someone to pay me to devote
    my full time to studying it!

Slides left over from USC talk
  • To import as needed

Reversibility of Physics
  • The universe is (apparently) a closed system
  • Closed systems evolve via unitary transforms
  • Apparent wavefunction collapse doesnt contradict
    this (confirmed by work of Everett, Zurek, etc.)
  • Time-evolution of concrete state of universe (or
    closed subsystems) is reversible
  • Invertible (bijective)
  • Deterministic looking backwards in time
  • Total info. (log of poss. states) doesnt
  • Can increase, though, if volume is increasing
  • Information cannot be destroyed!

Illustrating Landauers principle
Before bit erasure
After bit erasure
N states

Unitary (1-1) evolution
2N states
N states

Benefits of Reversible Computing
  • Reduces energy/cooling costs of computing
  • Improves performance per unit power consumed
  • Given heat flux limits in the cooling system,
  • Improves performance per unit convex hull area
  • A faster machine in a given size box.
  • For communication-intensive parallel algorithms,
  • Improves performance, period!
  • All these benefits are by small polynomial
    factors in the integration scale the device

Quantum Computing pros/cons
  • Pros
  • Removes an unnecessary restriction on the types
    of quantum states ops usable for computation.
  • Opens up exponentially shorter paths to solving
    some types of problems (e.g., factoring,
  • Cons
  • Sensitive, requires overhead for error
  • Also, still remains subject to fundamental
    physical bounds on info. density, rate of state
  • Myth A quantum memory can store an
    exponentially large amount of data.
  • Myth A quantum computer can perform operations
    at an exponentially faster rate than a classical

Some goals of my QC work
  • Develop a UMS model of computation that
    incorporates quantum computing.
  • Design simulate quantum computer architectures,
    programming languages, etc.
  • Describe how to do the systems-engineering
    optimization of quantum computers for various
    problems of interest.

  • As we near the physical limits of computing,
  • Further improvements will require an increasingly
    sophisticated interdisciplinary integration of
    concerns across many levels of engineering.
  • I am developing a principled nanocomputer systems
    engineering methodology
  • And applying it to the problem of determining the
    real cost-efficiency of new models of computing
  • Reversible computing
  • Quantum computing
  • Building the foundations of a new discipline that
    will be critical in coming decades.
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