High-Performance Computing and Quantum Processinghpc-ua.org/hpc-ua-12/files/proceedings/6.pdfand...

High-Performance Computing and Quantum Processing

Sergey Edward Lyshevski

Department of Electrical and Microelectronic Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA

E-mail: [email protected] URL: http://people.rit.edu/seleee

Abstract. We research novel solutions and apply new transformative findings towards quantum processing engineering

high-performance and enabled-capabilities data processing platforms. In living organisms, various information and

data processing tasks are performed by microscopic atomic and biomolecular fabrics using quantum phenomena and

effects. We examine fundamentals of quantum data processing on measurable, processable and compatible real-valued

physical quantities (variables) in microscopic fabrics. These fabrics comprise and implement devices, modules and

systems. The developed quantum data processing paradigm: (1) Unifies and enables concepts of theoretical computer

science, computer engineering and quantum mechanics; (2) Consistent with the first principles of quantum informatics,

communication and processing; (3) Coherently examines device physics, switching algebra, processing arithmetics and

calculus; (4) Fosters fundamentally-consistent and practical microscopic hardware solutions. We examine two core

problems, e.g., algorithmic and hardware premises. The microscopic processing primitives must exhibit utilizable

quantum-effect state transitions on the measurable physical variables (observables) which result in computable

transforms from viewpoints of devise physics, processing arithmetics, calculus and design. Our new coherent, cohesive

and consistent paradigm promises one to: (i) Ease enormous challenges; (ii) Overcome foremost inconsistencies of

naive algorithmically-centric computing; (iii) Enable new practical inroads, paradigms and solutions; (iv) Guarantee

unprecedented processing capabilities ensuring far-reaching benchmarks; (v) Advance theory and practice of natural

and engineered processing. Our findings support a broad spectrum of transformative research activities and

engineering developments. The results may be used in assessing performance, capabilities and benchmarks of natural

and engineered processing and computing.

Keywords High-performance computing, information theory, microscopic systems, quantum computing, quantum processing

1. Introduction

High-performance computing is very important to solve various computationally-intensive problems. The high-

performance computing is expected to achieve sustainable performance ensuring 1×1015

(petaflops) floating point

operations per second (FLOPS) in practical applications. New reliable and robust hardware and software to ensure

sustained performance and computing are under extensive developments. Petascale supercomputers, in some

applications, can process one quadrillion (1000 trillion) FLOPS. The computers and processors performance and

capabilities can be enabled by:

1. Advanced low-power hardware, self-aware software and cohesive algorithms; 2. Enabling hardware, software and languages which ensure highly-programmable systems; 3. Enabled-functionality low-power nanoscaled microelectronics; etc. Vertebrates and invertebrates exhibit information and data processing. The data processing in living organisms

exceeds exascale performance which by far surpassing the exaflops equivalence and range applied in assessing of

computing. An exaflop is one quintillion (1×1018

) FLOPS. This performance and capabilities cannot be ensured and

sustained by any envisioned super-computer platforms due to fundamental and technological limits.

Advanced digital integrated circuits (ICs) are used to implement various distinct computers designs,

organizations and architectures. Enormous progress was achieved in semiconductor devices and ICs. The

aforementioned astonishing fundamental, applied and technological developments led to mass production of high-

performance ICs and processors with billions of transistors. Enabling materials, processes and tools led to a current

lithography-defined ~32-nm “DRAM Metal 1 Half-Pitch” which is also known as a “technology node” [1]. Various

fundamental limits will emerge within a foreseen scaling towards 20-nm and 10-nm features by 2017 and 2023 [1]. The

technology performance evaluation criteria are scalability, energy efficiency, on/off current, operational reliability,

operational temperature, technology compatibility and architecture compatibility [1]. As the planar solid-state devices

are scaled to only hundreds of nanometer in dimensions, the undesirable quantum phenomena significantly degraded the

overall device and ICs performance and functionality. The focused research activities have being centered on quantum-

effect devices. A significant progress has accomplished in widely deployed resonant-tunneling devices, solid-state,

Міжнародна конференція "Високопродуктивні обчислення" HPC-UA’2012 (Україна, Київ, 8-10 жовтня 2012 року)

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inorganic and organic lasers, etc. [1]. To fully utilize quantum phenomena and enable new features, developments

progressed beyond current macroscopic microelectronic paradigms. We focus on quantum-mechanical microscopic

solutions. The microscopic-centric paradigm ultimately implies new device physics of subatomic/atomic/molecular

devices, novel communication and processing principles, new interfacing and networking schemes, innovative synthesis

and fabrication, etc. Quantum phenomena and effects, exhibited by microscopic systems (subatomic, atomic, molecular

and other), may be utilized to ensure processing tasks. There are enormous challenges and complexities which range

from quantum-mechanical analysis to synthesis, interfacing, testing and characterization of microscopic devices and

systems. Solutions of the aforementioned problem promise one to enable processing with unprecedented performance

and capabilities.

The living organisms provide undisputable evidence of biomolecular sensing, communication and information

processing. The information processing in living organisms has not been comprehended. By proposing, examining and

establishing new premises in high-performance communication and processing, we intent to significantly contribute to:

(i) Devising of enabling engineered processing platforms; (ii) Analysis of revealing and essential aspects of quantum communication and processing.

An exploratory roadmap towards quantum processing is documented in Figure 1 [2, 3].

Size (logarithmic scale)100 1 10 100 1 10 100 1 10

Nanometer Micrometer Millimeter100 1 10

Meter10Picometer

1904US Patent 803684

Molecular and Biomolecular Processing

Solutions:1. Microscopic Devices2. Networked Fabrics3. Quantum Processing4. Processing Calculi 5. Quantum Communication

1946 ENIAC

Natural and Engineered Systems

Processing in Living Organisms

1971Intel 4004

2011AMD 64-bit

940-pin dual-core processor

1930, US Patent 1745175 1934 GB Patent 439457

Fundamental and Technological Limits

Electron (Vacuum)

Tubes

Microelectronics

2008 IBM Blue GeneSupercomputer

Figure 1. Envisioned roadmap: Towards super-high-performance sensing, communication and processing [2, 3]

2. Information Theory With Applications to Communication and Interfacing

Information theory is applied to examine communication. Claude Shannon introduced and applied the entropy in order

to measure the complexity of the set. The sets which have larger entropies require more bits to represent them.

For M objects (symbols) Xi which have probability distribution functions p(Xi), the entropy is given as

∑=

−=M

i

ii XpXpXH1

2 )(log)()(, i=1.2.…, M–1, M. (1)

Example 2. 1. Consider a cubic dice with 6 faces, and, non-cubic dices. The common polyhedron, Zocchihedron and other non-cubic

dices may have a specific number of faces. One may ensure uniform, normal and other distributions. For the

deltohedron, the number of faces is 10. Let

X=[a, b, c, d, e, f] with equal probability 1/6,

and X=[a, b, c, d, e, f, g, h, k, l] with equal probability 1/10.

The entropies H(X) are found to be

61

261

61

261

61

261

61

261

61

261

61

261

1

2 loglogloglogloglog)(log)()( −−−−−−=−= ∑=

M

i

ii XpXpXH=2.585 bit

and ( )101

2101

1

2 log10)(log)()( −=−= ∑=

M

i

ii XpXpXH=3.3219 bit. ■


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Example 2. 2.

Let

−=

p

pX

1y probabilit with 0

y probabilit with 1. The entropy H(X), as a function of p, is given as

)1(log)1(log)(log)()( 221

2 ppppXpXpXHM

i

ii −−−−=−= ∑=

. ■

It is evident that H is positive-definite, H≥0. That is, the number of bits required by the Source Coding

Theorem is positive. In particular, N independent and identically distributed random variables, each with entropy H(X),

can be compressed into more than NH(X) bits with negligible risk of information loss as N→∞; however, if these N random variables are compressed into fewer than NH(X) bits, it is virtually certain that information will be lost.

Examining analog computation and processing on continuous-time signals, a differential entropy can be

applied. For a continuous-time random variable X, the differential entropy is given as

∫−= dxxpxpXH XX )(log)()( 2 , (2) where pX(x) is a one-dimensional probability distribution function of x, 1)( =∫ dxxpX . In general, one has

∫−=− xxx dppXXXXH XXnn )(log)(),,...,,( 2121 . (3) The relative entropy between probability density functions pX(x) and gX(x) is expressed by

∫= xxx

x dg

ppgpH

X

XXXXR

)(

)(log)()( 2 (4)

The differential entropies for various common distribution functions are derived. For Cauchy, exponential,

Laplace, Maxwell-Boltzmann, normal and uniform distributions pX(x), the resulting H(X) are reported in [2, 3]. The

differential entropy can be negative. The differential entropy of a Gaussian random variable with

2

2

2

)(

22

1)( σ

πσ

ax

X exp

−−

= , –∞

1×1011 pulse/sec. One has minmax

max

min

maxmin

1

0

rr

r

r

rrer

−−

< . The channel capacity C(r0,rmax) is

−

−=

−

min

00

min

maxmin0 lnln)(

2ln

1 minmaxmax

r

rr

r

rrrC

rr

r

if rmin≤ r≤rmax. Figure 2 document three-dimensional plots for C(r0,rmax)

if rmin=0.5×1011

and rmin=1×1011

pulse/sec. For rmin=0.5×1011

, one finds Cmax=6.106×1012

bits. A very high channel capacity C is achieved.

02

46

810

x 1011

0

5

10

x 1013

0

2

4

6

8

x 1012

r0

rmax

C(r

0, r m

ax)

(a)

02

46

810

x 1011

0

5

10

x 1013

0

1

2

3

4

5

6

x 1012

r0

rmax

C(r

0, r m

ax)

(b)

Figure 2. Three-dimensional plot for the hannel capacity C(r0,rmax):

(a) C(r0,rmax) if rmin=0.5×1011

pulse/sec; (b) C(r0,rmax) if rmin=1×1011

pulses/sec. ■

3. High-Performance Computing: Solutions and Limits

Many enabling solutions were implemented ensuring high-performance computing. The following concepts were

utilized in existing processors:

• parallelism,

• superpipelining,

• multithreading,

• vector processing,

• array processing,

• distributed computing,

• accelerators,

• multi-core architecture,

• multi-level shared memory, etc. These solutions drastically advanced processing performance and capabilities. The advances of computer

engineering and microelectronics enable and support the aforementioned concepts. There are fundamental, hardware and software limits associated with all solutions.

Let us examine the parallelism. We introduce the following ratio

r=tserial/tparallel,

where tserial and tparallel are the processing, communication, interfacing, memory access and retrieval, execution and other

times of the serial (sequential) and parallel (concurrent and distributed) processes, tasks, algorithms, etc. Processes, tasks and algorithms are not scalable, and, many are not parallelizable. Not mentioning memory

hierarchy, even, majority of algorithms are impossible to parallelizable. Inherently serial algorithmic problems are

majority of sequential and combinational logics, conditional statements, numeric problems, etc.

Example 3. 1: Limits of Parallelism There are many processing tasks which must be performed. For example, computing, logics, memory access and retrieval, coding, communication, interfacing, networking, etc. These tasks, many of which can be performed only in series, are hardware-, software-, algorithms-, architecture- and organization-dependent, Many operations and tasks cannot be parallelized. One of the major quantity of parallelism is the speed-up measure Mspeed [2, 3], defined as

( ),

11

1

11

1

seriesseries

parallelseries

series

parallelseries

series

speed

rN

rtt

t

Ntt

tM

PP

−+

=

+−+

+

=

rseries=tseries/(tseries+tparallel), rparallel=1–rseries=1–tseries/(tseries+tparallel),


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where tseries and tparallel are the averaged times to perform series and parallel processing will all related tasks and operations; NP is the number of processors; rseries and rparallel are the series (not parallelizable) and parallelizable ratios. The speed-up measure Mspeed depends on the degree of parallelism which can be achieved. As one may obtain the average estimate for rseries>0 we have rparallel=(1–rseries), rparallelNoptimal, the performance

degrades. Using the derived expression for

I0

optimal

1

nt

tN I= , a three-dimensional plot for Noptomal(nl,tl/t0,) is illustrated in

Figure 4.

0246810

0

0.5

1

0

5

10

15

tI/t

0n

I

No

ptim

al

Figure 4. Plot for Noptomal(nl,tl/t0,) ■


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4. Quantum Processing Fundamentals We initiate transformative knowledge generation and perform fundamental research on [2-5]:

1. Consistent analysis of electron- and photon-induced phenomena which lead to quantum state transitions and utilizable transductions on detectable real-valued measurable physical variables (observables). These measurable variables must be controllable, algorithmically processable and hardware realizable;

2. Quantum- and device-physics consistent fundamentals of communication and processing by microscopic engineered and natural systems;

3. Devising, design, substantiation and demonstration of practical engineering paradigms and technologies on electronic, photonic and photoelectronic sensing, communication and processing by molecular fabrics.

It is important to progress from theory, to its substantiation, engineering solutions and technologies by: 1. Studying and evaluating quantum utilizable transductions on detectable real-valued measurable physical

variables; 2. Verifying principles and mechanisms of energy conversion, sensing, communication and processing in

microscopic systems as applied to practical engineered solutions. Under some hypotheses and conjectures, microscopic system can be mathematically modeled (mapped) by using wave functions in different spaces. For example, spatial, momentum and other spaces are used. The

spatiotemporal wave function Ψ(r,t)=ψ(r)ϕ(t) is found by solving the time-dependent Schrödinger equation

t

titH

∂

Ψ∂=Ψ

),(),(ˆ

rr h , ti

n

nn

tE

i

n

nn

n

nnn

n

ececttctωψψϕψ −

∞

=

−∞

=

∞

=

∑∑∑ ===Ψ=Ψ111

)()()()(),(),( rrrrr h , Ĥ∈÷, Ψ∈÷, c∈÷, (7)

where Ĥ is the total Hamiltonian operator, Ĥ=Ĥ0+ĤE+ĤP; Ĥ0 is the unperturbed Hamiltonian in the absence of external

excitations, and, for an unperturbed microscopic system Ĥ0ψn=Enψn, Π+∇−=22

21

0ˆ h

mH ; ĤE and ĤP are the excitation

and perturbation Hamiltonians; Π is the potential function; cn(t) are the complex probability amplitudes, and, *cn*2 is the

probability that a microscopic system at any given time t is in a state with En (the probability that a measurement of the

energy at t would yield En), 11

2=∑

∞

=n nc .

From (7), using Ψ(r,t)=ψ(r)ϕ(t), one has

Ĥ0ψn=Enψn, Et

i =∂

∂ϕ

ϕ

1h ,

tE

i

et h−

=)(ϕ . (8)

Wave functions Ψ, derived using various spaces, may yield mathematically-consistent estimates on various quantities, such as probabilities, allowed states, expectation values, etc. The expectation value of a quantum canonical

variable C∈ú, with an associated operator Ĉ∈÷, is

dVtCtC ),(ˆ),(* rr ΨΨ=∫ , C∈ú. (9)

The governing equation for the operator Ĉ∈÷ is given as

]ˆ,ˆ[1ˆˆ HCit

CC

dt

d

h+

∂

∂= , [Ĉ, Ĥ]=ĈĤ – ĤĈ, Ĉ∈÷. (10)

Our quantum-mechanically-consistent modeling and analysis result in a set of equations (7)-(10) with the resulting model mapping of microscopic systems as

MMMM(k(Ψ),Ĉ,C)∈KKKK×Ĉ×C, Ĉ∈÷, C∈ú. (11) Various use of mathematical operations, manipulations on operators, superposition and other premises of all-

algorithmic quantum communication and processing were outlined in [6-11]. There is a need to depart from abstract

quantum computing which assumes the practicality of [6-11]:

• Naive all-algorithmic computing on not detectable, not observable and not measurable mathematical operators;

• Algorithmic schemes of “quantum logic gates” aggregated within postulated computing structures by means of elusive quantum interconnects and circuits;

• Macroscopic microelectronic devices including the so-called “single-electron-transistor”, “single-photon-transistor”, “quantum dots”; etc.

By applying the aforementioned solutions, it is unclear how one may ensure a practical computing. It is

unlikely that any processing tasks can be accomplished by directly or indirectly abstractly using quantum

indeterminacy, incompleteness, mathematical operators (wave functions, probability density and others), hidden

variables, superposition of states, etc.

We outline the following Three Core Principles on which our results are centered:

Principle 1: Engineering Quantum Mechanics – Coherent quantum physics and information science as applied

to information and data processing;


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Principle 2: Design and Algorithmic Solutions – Processing on detectable real-valued measurable physical

variables using distinguishable, quantifiable and computable transforms and schemes with corresponding data

processing arithmetics (logics and calculi), organizations and architectures;

Principle 3: Physical Implementation on Device and System Levels – Utilize quantum phenomena which lead

to state transitions and utilizable transductions on measurable physical variables (observables) in molecular

fabrics. These variables should be quantum-mechanically achievable, algorithmically processsable and

hardware realizable to perform processing. It is important that for quantum processing, the time-dependent Shannon entropy H(t,p) is given as

H(t,p)=i

N

ii

pp ln1

∑=

− . (12)

5. High-Performance Quantum Processing Using three principles, we research quantum-mechanically consistent, algorithmically cohesive and hardware coherent quantum data processing. We use the detectable real-valued measurable physical variables (quantities) during

controllable quantum transductions [2-5]. Only measurable and algorithmically processable variables vvvv∈VVVV lead to quantum-mechanically, device- and algorithmically (arithmetically) consistent processing. For example, vvvv=[E, ω, λ, T,…]T. It is unlikely that computing can be accomplished by using wave functions, kets, eigenkets or other mathematical quantities, operators, etc. Consider controlled physical microscopic devices. We utilize the distinguishable and computable transforms T which are mappings of the utilizable initial, intermediate and final state transductions (SI, ST and SF) on vvvv=[v1,…,vk]

T.

The microscopic devices may accomplish the following irreversible and reversible utilizable transductions

S I:vvvvIYS T:vvvvTYSF:vvvvF and S I:vvvvI]ST:vvvvT]SF:vvvvF. (13) Consider a physical microscopic processing fabrics with processing primitives P1,…,Pk. Each Pj exhibits transductions Sj(vvvv) on detectable, measurable and processable variables vvvvj yielding distinguishable and computable transforms Tj(S,vvvv). Using Tj(S,v), consistent with device physics and admissible arithmetic operand Aj, one has T Σ=T1B …BTk. (14) The processing can be accomplished by using the infinite- and finite-valued logics. Analog, digital and hybrid processing schemes may be supported by microscopic devices. Considering multiple-valued logic, the switching

function on r-valued vj is f:{0,…,r–1}n→

FFFF{0,…,r–1}m with a truth vector FFFF. Any f can be represented as f=AAAA(FFFF,TTTT). (15) The evolutions of quantum transductions are mapped as

vj,l²QQQQvj,l+1=Qj(vj,l) (16) which defines the evolution of physically-realizable computable transforms Tj(S,vvvv). The transductions Sj(vvvv) on vvvv in physical microscopic systems (devices) can be controlled by using device-specific control schemes by varying systems

energetics, potential or other quantities denoted as H [20-22]. The controllable evolutions on measurable vj, are mapped as

vj,l²QQQQvj,l+1=Qj(vj,l,Hj,l), (17) where Qj denotes transductions Sj on vj consistent with AAAA(FFFF,TTTT). The binary switching function is f:{0,1}

n→

F{0,1}m.

An n-variable r-valued function f, with n

rr different combinations, is defined as a mapping of a finite set

{0,…,r–1}n into a finite set {0,…,r–1}

m, e.g.,

f:{0,…,r–1}n→

F{0,…,r–1}

m. (18)

Truth vectors on n binary and r-valued variables [x1,…,xn] are

FFFF=[f(0),f(1),…,f(2n–1)]T and FFFF=[f(0),f(1),…,f(rn–1)]T. (19) Digital computing and digital design use algebraic maps, Boolean algebra, digital (binary and multiple-valued)

logics, decision diagrams, data structures, fundamental expansions, polynomial expressions, sequential networks,

probabilistic concepts, stochastic schemes, and other approaches [12, 13]. If quantum-mechanical consistency,

algorithmically cohesiveness and hardware coherency are satisfied, some aforementioned concepts may be applied for

quantum processing. Any arithmetically- and algorithmically-defined computable function must be:

1. Definable, realizable and implementable as derived by using the distinguishable and computable transforms;

2. Implementable using the utilizable quantum transductions on measurable and algorithmically processable variables.

The calculi and arithmetics of quantum processing are reported in [2-5, 12, 13].


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6. Conclusions We examined quantum phenomena which are exhibited and utilized to ensure high-performance quantum processing. The following three-fold objectives were achieved:

1. The microscopic systems which may enable quantum communication and processing were devices and examined;

2. Phenomena and mechanisms, possibly utilized by natural systems to accomplish high-performance communication and processing, were studied;

3. A novel paradigms of high-performance quantum processing was developed. We enabled a knowledge base and discovered new solutions. Our transformative findings are substantiated by means of basic, applied and numerical studies which are consistent with experiments, biophysics, quantum mechanics, information theory, computer science and computer engineering.

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