Quantum Computer Architectures for Physical Simulations Dr. Mike Frank University of Florida CISE Department [e-mail]

1
Quantum Computer Architectures for Physical
Simulations Dr. Mike FrankUniversity of
FloridaCISE Departmentmpf_at_cise.ufl.edu

• Presented at
• Quantum Computation for Physical Modeling
  WorkshopWed., May 8, 2002

2
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
  computing
• Ultimate models of computing for complexity
  theory
• Nanocomputer systems engineering
• Optimizing cost-efficiency scalability of future
  computing
• Reversible computing quantum computing
• Realistic models, CPU architectures, optimal
  scaling advantages

MIT 1995-1999,UF 1999-
3
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,
  Ammer
• Focus Reducing reversible computing to practice
• CMOS circuit styles, test chips, architectures,
  complexity theory, algorithms, high-level
  languages.

4
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.
  Mich.
• 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.

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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

Feynman
Some lineage
Knight
Sussman
Minsky
Frank
Fredkin
Toffoli
Margolus
6
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

7
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

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1. Optimally scalable quantum computer
architectures

• Universally maximally scalable (UMS) models
• Physical limits shape the ultimate model
• Appropriate programming models

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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
  nanoscale.
• 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

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Source ITRS 99
11
½CV2 based on ITRS 99 figures for Vdd and
minimum transistor gate capacitance. T300 K
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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
  architectures
• Asymptotic scaling of architectures algorithms
• Physically optimal algorithms

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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
  architectures.
• Develop models of computing that are
• As powerful as physically possible on all
  problems
• Realistic within asymptotic constant factors

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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
  solns)
• 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 ?.
  Bad!

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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.

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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
  ?(M1/3).
• 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
  algorithm!

M1/3
M
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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!

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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
  scalable

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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!

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Physics Constrains the Ultimate Model
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Limits of Quantum Computers

• Quantum computers remain subject to all the
  fundamental limits previously mentioned!
• Entropy density limit - only 2n distingable
  states!
• 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
  propagation
• 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
  ops

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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
  limit.
• Enables improved 3-d scalability, in a limited
  regime
• (In long term) Have capability for
  quantum-coherent operation, for extra perf. on
  some probs.

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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
application).
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
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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
  2)
• (Minimum) static entropy generation rates
• For devices even just quiescently maintaining
  state info
• Related to energy leakage rates, decoherence
  times
• (Minimum) adiabatic frictional coefficient
• For devices undergoing reversible transitions
• (Maximum) quality factor Q of active transitions
• (Minimum) device cost

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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
  problems
• 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

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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!
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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...

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Reversible/Adiabatic CMOS

• Chips designed at MIT, 1996-1999

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Minimum Losses w. Leakage
Etot Eadia Eleak
Eleak Pleaktr
Eadia cE / tr
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Reversible Emulation - Ben89
k 2n 3
k 3n 2
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Bennett 89 alg. is not optimal
k 2n 3
k 3n 2
Just look at all the spacetime it wastes!!!
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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,
  energy!

k3n2
k2n3
Emulated time
k4n1
Real time
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Spacetime blowup
Energy saved
k
n
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Perf. scaling w. of devices

• If alg. is not limited by communications needs,
• Use irreversible processors spread in a 2-D
  layer.
• 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!

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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!

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Irreversible 3-D Mesh
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Reversible 3-D Mesh
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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
  pattern!
• 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.

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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
  signals.
• 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
  models.
• Simulation experiments planned to investigate
  this.
• Develop efficient physical realizations of
  nano-scale bit-devices timing systems.

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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)
  timing
• Approach in nearly all conventional irreversible
  CPUs
• Basis for all practical adiabatic/quantum
  computing mechanisms proposed to date
• Local (a.k.a. self-timed, a.k.a. asynchronous)
  timing
• Implemented in a few commercial irreversible
  chips.
• 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?

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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.

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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
  coefficient.
• 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
  speed

(Ideally,independentof t.)
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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

45
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!

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A MEMS Supply Concept

• Energy storedmechanically.
• Variable couplingstrength -gt customwave shape.
• Can reduce lossesthrough balancing,filtering.
• Issue How toadjust frequency?

47
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).

48
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
  technology
• 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.

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Node Architecture Sketch
Instructionregisters
Data pathto/fromneighbornode(in 2d or 3d)
I/O registers
Execution unit
Data registers
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2. Simulating physical systems on our QC model

• Many-particle Schrödinger equation
• Numerically stable classical reversible sims
• Quantum equivalents
• Quantum field theory

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Simulating Wave Mechanics

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

52
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...

53
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)
54
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.

55
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 constrainedto stay close to each other!

2?tf
x1
g
x2

g
x3

g
x4

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Centered Schrödinger Equation

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

R1
g
?
I2
g
R3

g
I4
?
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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.

58
Phenomena Observed in Model

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

59
Gaussian wave packet moving to the rightArray
of small sharp potential-energy barriers
60
Initial reflection/refraction of wave packet
61
A little later
62
Aimed a little higher
63
A faster-moving particle
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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
  kpd

65
Implicit Mapping

• Use pd integer registers xj, each lg k qubits
  long
• 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.

xi
66
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
  operator
• Includes particle creation/annihilation operators
• Stream particles between sites after collision
  step

67
3. Quantum computer simulation on classical
computers

• Visualization techniques, various optimizations
• Polynomial-space techniques

68
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.

69
Visualization Technique

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

70
Initial State
71
After doing Hadamard transform on all bits of a
72
After modular exponentiationbxa (mod N)
73
State After Fourier Transform
74
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
  op
• Some obvious optimizations
• Never store whole matrix (compute dynamically)
• Store only nonzero elements of state vector
• Especially helpful when qubits are highly
  correlated
• Do only constant work per nonzero vector element
• Scatter amplitude from each state to 1 or 2
  successors
• Drop small-probability-mass sets of states
• Linearity of QM implies no chaotic growth of
  errors

75
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

76
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.

77
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.

U2
U3
U4
U1
78
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.

79
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
  otherwise.
• 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.

80
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
  ?0.
• For CNOTs Deterministically transform state.
• For Us
• Calculate amplitude ?1 of neighbor state w.
  path-integral
• Calculate amplitudes ?0 and ?1 after qubit op
• Choose 1 successor as new current state, using
  ?2 distrib.

u00
?0
?0
Current state
u10
Possiblesuccessors
u01
?1
?1
u11
Neighbor state
81
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

82
Conclusion

• 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!

83
Slides left over from USC talk

• To import as needed

84
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
  decrease
• Can increase, though, if volume is increasing
• Information cannot be destroyed!

85
Illustrating Landauers principle
Before bit erasure
After bit erasure
s0
0
0
s0
Nstates



sN-1
0
0
sN-1
Unitary(1-1)evolution
2Nstates
s0
sN
1
0
Nstates



0
sN-1
s2N-1
1
86
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
  properties.

87
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,
  simulation)
• Cons
• Sensitive, requires overhead for error
  correction.
• Also, still remains subject to fundamental
  physical bounds on info. density, rate of state
  change!
• 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
  one.

88
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.

89
Conclusion

• 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.