Lecture Notes on Quantum Mechanics & Quantum Annealing

24 Pages·2017·0.38 MB·English

Lecture Notes on Quantum Mechanics & Quantum
Annealing – Some Background for the D-Wave Quantum
Computer and Adiabatic Quantum Computation
SUMMER2016–VERSIONPRL-QMN-2016.v1.2b–Copyright(cid:13)c 2016and2017
Ken Kreutz–Delgado
Electrical&ComputerEngineering,JacobsSchoolofEngineering
QualcommInstitutePatternRecognitionLaboratory,Calit2
UniversityofCalifornia,SanDiego
August10,2017
1 Preface
These lecture notes were provided in the summer of 2016 to a “D-Wave Interest Group” (DWIG) pre-
dominantly comprised of first-year MS students in the Electrical and Computer Engineering (ECE) de-
partment at the University of California, San Diego. Participation was entirely voluntary and no course
credit was given. The DWIG, which met weekly, was interested in participating in a course of directed
readingaimedatunderstandingAdiabaticQuantumComputation(AQC)andQuantumAnnealing(QA).
The purpose of the lecture notes given below was to supplement and expand on some of the quantum
mechanics (QM) discussion provided in the NASA/JPL/D-Wave/USC report [24], the D-Wave research
memorandum[22],andthemonograph[18]. Specifically,becausetheparticipantsintheDWIGhadlittle
or no prior exposure to quantum mechanics, the notes attempt to “fill in the blanks” of some of the dis-
cussion given in references [24, 22, 18]. In particular, other than some aspects of its pedagogic content,
thereisnoclaimtooriginalityofthesenotesoverandabovethematerialpresentedinstandardreferences
suchas[24,22,18],amongothers.
The background of the approximately ten participants during Summer 2016 was typical for contem-
porary BS degree graduates of EE and CS departments who have had additional courses in machine
learning and pattern recognition roughly corresponding to the level of [3]. In particular, the participants
had a level of proficiency in probability and random variables roughly corresponding to the level of the
textbook [21], and in linear algebra roughly corresponding to the level of the textbook [20]. Typical
for EE/CS graduates, they were also knowledgeable about the theory of ordinary differential equations.
Otherthanthecursoryintroductiongiveninstandardlowerdivisionphysicscourses,theparticipantshad
hadnopriorexposuretoquantummechanics.
Fortunately, the prior exposure to linear algebra at the relatively sophisticated and geometrically-
motivatedlevelof[20]meantthattheparticipantseffectivelyhadalreadybeenintroducetothetheoryof
finite-dimensional Hilbert spaces (except for the concept of a dual space, which was straightforward to
1
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 2
present and motivate). With this background, and some required translation of notation and terminology
from linear algebra to the Dirac notation-based finite-dimensional Hilbert space formalism, the DWIG
could be quickly exposed to the (non-relativistic) finite-dimensional quantum mechanics of systems of
distinguishable (non-identical) spin-1 particles (qubits) needed to understand AQC. This was accom-
2
plished by assigning as mandatory reading the first seven chapters of the very accessible textbook by
Susskind and Friedman [26], which has a primary focus on spin-1 systems and provides an introduction
2
to the density matrix. Suggested additional sources for the mathematics of qubit systems were [14] and
[12],theformerreferenceprovidingadetaileddiscussionofmanipulatingthePaulispinmatricesandthe
latterprovidingadditionaldiscussionofthedensitymatrix.1
As relevant background to these notes, prior lectures were given on the classical (non-quantum me-
chanical) stochastic Ising spin-glass model used both for modeling neural networks and for optimization
of boolean objective functions via simulated annealing. The discussion of stochastic neural networks
was based on material drawn from the classic textbooks of Hertz, Krogh, and Palmer [9] and Amit [1],
while the discussion of simulated annealing (SA) was drawn from the foundational paper [15] and the
presentation in [9].2 Because, as discussed in [15, 9], classical equilibrium statistical mechanics plays
a fundamental role in simulated annealing-based optimization, prior lectures based on material drawn
from [4, 19] were given to provide an overview of the classical canonical ensemble, Boltzmann-Gibbs
distribution,andHelmholtzfreeenergy. Further,AsnotedintheD-Wavememorandum[22],equilibrium
statisticalmechanicsisequallyimportantforunderstandingkeyaspectsofquantumannealing,motivating
thediscussionofquantumandclassicalstatisticalmechanicsgiveninthenotesbelow.3
In the lectures given to the DWIG, it is emphasized that the D-Wave computer is an extraordinarily
sophisticatedandadvancedappliedphysics,engineering,andmathematicalsystemthatcanbeunderstood
andengagedwithatseverallevels:
1. Most fundamental is the device physics and engineering level. This requires a deep understanding
of experimental and applied low-temperature physics and devices, such as SQUIDs, which are
basedonlow-temperaturephenomenaandstructuressuchastheJosephsonjunction.
2. Closely tied to the device and engineering physics level are the detailed mathematical models that
guidethedesign,construction,andassemblyofthecomponentsoftheD-Wavecomputertoensure
that the system behaves like a controllable quantum mechanical spin-glass system of intercon-
necteddistinguishablequbits(viatheuseof“manufacturedspins”[13]). Foradiscussionofissues
pertainingtoLevels1and2,theDWIGparticipantswerereferredtoreferences[8,13,24].
3. Thenexthigherlevelofabstractionisprovidedbythequantummechanicalspin-glassbehaviorthat
is enforced by Levels 1 and 2. This behavior is used to analyze, design, and implement quantum
annealing algorithms capable of combinatorial optimization (specifically quadratic unconstrained
binaryoptimization(QUBO)problems)subjecttodeviceimposedconstraints.
4. This level of abstraction assumes that the D-Wave computer functions as a black box capable of
solving a certain family of QUBO problems. The onus is on the user to translate a combinatorial
optimizationproblemofinteresttoQUBOform.
1MoreadvancedQMreferencesusedinthedraftingofthesenotesare[16,6,2,5,23].
2Inthediscussionsandnotesbelow,following[18],weoftenrefertoSAas“ClassicalAnnealing”(CA)todistinguishSA
fromthe“QuantumAnnealing”(QA)describedin[24,18].
3Usefulreferencesforquantumstatisticalmechanicsare[10,17,19,11,25]. TheextensionoftheIsingspin-glassmodel
tothequantummechanicalHeisenbergmodelofferromagnetismisbrieflydiscussedin[11].
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 3
5. The highest level of abstraction exists when a user merely specifies a combinatorial optimization
problem and a “compiler” exists that is capable of translating it to QUBO form that can be solved
bytheD-Wavecomputer.
2 Finite Dimensional Hilbert Space and Hermitian Operators
2.1 General Background
Let H be a finite-dimensional Hilbert space of dimension n over the field of complex numbers. It is an
axiomofquantummechanicsthatacomplexHilbertspacecanserveasthestate-spaceof(thepurestates
of)aquantummechanicalsystemofinterest.4
Let M denote an observable quantity in physical space that can take experimentally measured real
values m ,··· ,m . It is a second axiom of quantum mechanics that the physical observable M is
0 n−1
ˆ
represented by M, a self-adjoint (hermitian) linear operator in the Hilbert space H whose eigenvalues
are precisely equal to the experimentally measurable values m . The situation is as follows, when a
i
ˆ
measurement of M is taken, one of the real eigenvalues of M, m = m , is observed and the quantum
i
state“jumps”tothecorrespondingeigenvector,|φ(cid:105) = |φ (cid:105),oftheoperatorMˆ.5
i
Theeigenvectorsandeigenvaluesoftheself-adjointoperatorMˆ aregivenby,6
ˆ
M|φ (cid:105) = m |φ (cid:105), j = 0,··· ,n−1
j j j
with m ∈ R. The information provided by the eigenvectors and eigenvalues are encoded in the spectral
j
representation(orspectraldecomposition),7
n−1 n−1
(cid:88) (cid:88)
Mˆ = m |φ (cid:105)(cid:104)φ | = m Pˆ(φj), (1)
j j j j
j=0 j=0
where
Pˆ(φj) = |φ (cid:105)(cid:104)φ |.
j j
is the orthogonal projection operator onto the one-dimensional subspace spanned by |φ (cid:105).8 The identity
j
ˆ
operatorI canberesolvedas,
n−1 n−1
(cid:88) (cid:88)
Iˆ= |φ (cid:105)(cid:104)φ | = Pˆ(φj).
j j
j=0 j=0
ˆ
Byinductionontheeigenvalue/vectorequation,M|φ (cid:105) = m |φ (cid:105),wehave
j j j
αMˆk|φ (cid:105) = αmk|φ (cid:105) for j = 0,··· ,n−1 and k = 0,1,···
j j j
4We use the standard “Copenhagen interpretation” axiomatic approach articulated by von Neumann; see the discussion
in [6]. The derivations given in this note are (for the most part) valid and straightforward because we are working in a
finite-dimensional Hilbert space; things are tricker in infinite-dimensional Hilbert spaces. For example, linear operators on
a finite-dimensional Hilbert space H, n = dim(H) < ∞, are all bounded, have point spectra (n eigenvalues only in the
spectrum),andadomainofdefinitionequaltoH[7].
5Thisoccursprobabilistically,aswediscusswhenwepresent“Axiom3”inthenextsection.
6Weassumethatthe(orthogonal)eigenvectorshavebeennormalizedsothat(cid:104)φ |φ (cid:105)=δ .
i j ij
7Allself-adjointlinearoperatorsonafinitedimensionalHilbertspaceofdimensionnhaveacompletesetofnorthonormal
eigevectors that can be used as a basis for the space, and n associated real eigenvalues. This means that every self-adjoint
operatoronafinitedimensionalHilbertspacehasadiscretespectrum.
8ItisstraightforwardtoshowthatPˆ(φj)isself-adjointandidempotent.
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 4
foranyscalarα ∈ C. Asaconsequence,
ˆ
Poly(M)|φ (cid:105) = Poly(m )|φ (cid:105) for j = 0,··· ,n−1
j j j
foranypolynomialfunctionPoly(z)and,moregenerally,
ˆ
F(M)|φ (cid:105) = F(m )|φ (cid:105) for j = 0,··· ,n−1 (2)
j j j
for any analytic function F(z). Note that F(m ) and |φ (cid:105), j = 0,···n − 1, are the eigenvalues and
j j
ˆ ˆ
orthonormaleigenvectorsofF(M)andthereforetheoperatorF(M)hasaspectraldecomposition,
n−1 n−1
(cid:88) (cid:88)
F(Mˆ) = F(m )|φ (cid:105)(cid:104)φ | = F(m )Pˆ(φj). (3)
j j j j
j=0 j=0
2.2 Energy Eigenstates and Energy Representation
ˆ
Consider the very important observable represented by the operator H, the Hamiltonian operator. It
representsrepresentstheobservableenergyE whichtakesrealvaluesE ,··· ,E . Thepossibleenergy
0 n−1
valuesareassumedtobeorderedas,
E ≤ E ≤ E ≤ ··· ≤ E ,
0 1 2 n−1
where E(0) = E is the ground state and E(1) = min{E |E > E } is the first excited state.9 The
0 j j 0
eigenvectorsandeigenvalues(energyvalues)ofHˆ are,ofcourse,relatedby,10
ˆ
H|ψ (cid:105) = E |ψ (cid:105), j = 0,··· ,n−1
j j j
n−1 n−1
(cid:88) (cid:88)
Hˆ = E |ψ (cid:105)(cid:104)ψ | = E Pˆ(ψj),
j j j j
j=0 j=0
and
n−1 n−1
(cid:88) (cid:88)
Iˆ= |ψ (cid:105)(cid:104)ψ | = Pˆ(ψj)
j j
j=0 j=0
withE ∈ R.
j
ˆ
Whenageneralstatevector|φ(cid:105)isrepresentedintermsoftheeigenvectorsofH,
(cid:32) (cid:33)
n−1 n−1
(cid:88) (cid:88)
ˆ
|φ(cid:105) = I|φ(cid:105) = |ψ (cid:105)(cid:104)ψ | |φ(cid:105) = c |ψ (cid:105), c = (cid:104)ψ |φ(cid:105),
j j j j j j
j=0 j=0
we call this the energy representation of |φ(cid:105).11 As a preliminary application of the very useful Equations
(2)and(3),notethattheyimply
e−βHˆ|ψ (cid:105) = e−βEj|ψ (cid:105).
j j
and
n−1
e−βHˆ = (cid:88)e−βEj|ψ (cid:105)(cid:104)ψ |.
j j
j=0
9MoregenerallyweindexdistinctenergylevelsbyE((cid:96))inorderofincreasingmagnitude,E((cid:96)+1) >E((cid:96)). Notethatifthe
groundstateisnondegeneratethenE(1) =E .
1
10Againassumethattheeigenvectorshavebeennormalized,(cid:104)ψ |ψ (cid:105)=δ .
i j ij
11Notethattheenergyrepresentationisequivalenttothecolumnvectorrepresentationc=(c ,··· ,c )T ∈Cn.
0 n−1
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 5
3 Probabilities and the Density Matrix
Given that the system is in state |ψ(cid:105) ∈ H, it is a third axiom of quantum mechanics that the probability
thatameasurementwillobservethesystemtobeinstate|φ(cid:105)isgivenby,12
p (φ) = |(cid:104)ψ|φ(cid:105)|2.
ψ
Notethat,13
(cid:16) (cid:17)
|(cid:104)ψ|φ(cid:105)|2 = (cid:104)ψ|φ(cid:105)(cid:104)φ|ψ(cid:105) = tr |ψ(cid:105)(cid:104)ψ||φ(cid:105)(cid:104)φ| = trPˆ(ψ)Pˆ(φ).
Suppose,then,thatthequantummechanicalsystemis(hasbeenpreparedtobe)instateψ. Wedefine
thedensitymatrix (akadensityoperator)by
ρˆ= Pˆ(ψ) = |ψ(cid:105)(cid:104)ψ|.
Note that in this case ρˆ ⇐⇒ ψ, so that p (φ) = p (φ), which begins to explain why in quantum
ψ ρˆ
mechanicsthedensitymatrixρˆitselfisalsocalledthestateofthesystem. Withthisdefinitionwehave,
p (φ) = trρˆPˆ(φ) with ρˆ= Pˆ(ψ) = |ψ(cid:105)(cid:104)ψ| and Pˆ(φ) = |φ(cid:105)(cid:104)φ|. (4)
ψ
Themostgeneralformofthedensitymatrixis
ρˆ= w Pˆ(1) +···+w Pˆ(m) (5)
1 m
where Pˆ(j) = |ψ (cid:105)(cid:104)ψ | are projection operators onto one-dimensional subspaces of H spanned by unit
j j
vectors |ψ (cid:105), w > 0, and w +···+w = 1. It is important to note that the subspaces spanned by |ψ (cid:105)
j j 1 m j
do not have to be mutually orthogonal.14 Note that the general density matrix ρˆis self-adjoint, ρˆ = ρˆ∗,
and that trρˆ = 1. One can utilize a decomposition of the form (5) to prepare the system to be in state ρˆ
by choosing an basis {|ψ (cid:105),··· ,|ψ (cid:105)} and then selecting a state |ψ (cid:105) (which is equivalent to selecting
0 n−1 j
Pˆ(j) = |ψ (cid:105)(cid:104)ψ |) according to a probability equal to w . Once the state ρˆhas been prepared, however,
j j j
it is an important fact that the decomposition (5) generally is not unique, which can be interpreted to
mean that the system “forgets” (or “does not know”) how it was prepared to be in state ρˆsince there are
many possible ways to do this.15 Nonetheless, given the mathematical expression (5) (which describes
a possible preparation procedure), one can interpret w to be the “prior probability” that the system was
j
preparedtobeinstate|ψ (cid:105)if infactEq.(5)diddescribetheactualpreparationprocedure(whichitmight
j
not,giventhatotherdecompositionsarepossible).
12Some authors refer to p (φ) as the transition probability because it is the probability that state |ψ(cid:105) transitions to state
ψ
|ψ(cid:105),|ψ(cid:105) → |φ(cid:105),asaconsequenceoftakingameasurement. ThethirdaxiomisknownasBorn’sRule,afterMaxBornwho
proposeditin1926.
13Foranyboundedoperators(e.g.,formatrices)AandB,
trAB =trBA.
Thisisaveryusefulresult. Hereweusethefactthatgiventwocolumnvectorsaandb,aHb=traHb=trbaH.
14Notethatwhentheunitvectors|ψ (cid:105)arenotorthogonal,thenEq.(5)isnotaspectralrepresentationofρˆ. However,when
j
theyareorthogonal,then(5)isaspectralrepresentation.
15Anotherinterpretationisthatρˆcontainsthesimultaneousquantumsuperpositionofallofthe(potentiallyinfinite)ways
that it could have been prepared into existence, which is another example of “quantum parallelism.” Discussion of the
nonuniquenessofthedensitymatrixcanbefoundinChapter2of[2]andinAppendixAbelow.
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 6
Ifthereexistsastate|φ(cid:105) ∈ Hsuchthatthegeneraldensitymatrixcanbewrittenasρˆ= Pˆ(φ) = |φ(cid:105)(cid:104)φ|,
then we say that ρˆis a pure state. if ρˆis pure, then the system is in a definite quantum mechanical state
|φ(cid:105). Ifρˆisnotpure,andmustbewritteninthegeneralform(5),thenitisamixedstate.16
Given that the quantum mechanical system has been prepared in a state ρˆ, then the expected value of
ˆ
anobservableM isgivenby,
(cid:104)M(cid:105) (cid:44) E{M} = trρˆMˆ. (6)
Representingthedensitymatrixρˆbyadecomposition(5),thisisstraightforwardtoshow:
n−1 n−1
(cid:88) (cid:88)
ˆ
trρˆM = w |ψ (cid:105)(cid:104)ψ | m |φ (cid:105)(cid:104)φ |
j j j i i i
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
j=0 i=0
Pˆ(ψj) Pˆ(φi)
n−1 n−1
(cid:88) (cid:88)
= w m |(cid:104)ψ |φ (cid:105)|2
j i i j
j=0 i=0
n−1 n−1
(cid:88) (cid:88)
= w m p (φ )
j i ψ i
j=0 i=0
n−1
(cid:88)
= w E {M}
j ψ
j=0
= E{M} = (cid:104)M(cid:105).
4 The Boltzmann-Gibbs Distribution
4.1 Some Canonical Statistical Mechanics
ForamechanicallyisolatedquantummechanicalsystemplacedinaheatbathatconstanttemperatureT,
constantvolumeV,andconstantparticlenumberN,thedensitymatrixisgivenbythecanonicaldensity
matrix,orBoltzmann-Gibbsdensitymatrix,17
e−βHˆ
ρˆ= , (7)
tre−βHˆ
withβ = 1 ,whereforphysicalsystemsk istakentobeBoltzmann’sconstant,k = k .18 Notethat,
kT B
n−1 n−1 n−1
e−βHˆ = e−βHˆI = e−βHˆ (cid:88)|ψ (cid:105)(cid:104)ψ | = (cid:88)e−βHˆ |ψ (cid:105)(cid:104)ψ | = (cid:88)e−βEj |ψ (cid:105)(cid:104)ψ |.
j j j j j j
j=0 j=0 j=0
Withtr|ψ (cid:105)(cid:104)ψ | = (cid:104)ψ |ψ (cid:105) = 1,wehave
j j j j
n−1
tre−βHˆ = (cid:88)e−βEj = Z , (8)
β
j=0
16Toreiterate: ifρˆisamixedstate,thenitcannotbewrittenasρˆ=P(φ)forsomestate|φ(cid:105)∈H. I.e.,thesystemisnotina
definitequantummechanicalstate|φ(cid:105)∈H.
17WewillalsorefertothisformofρˆasthecanonicalstateortheBoltzmann-Gibbsstate.
18Wewilltakek =1.
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 7
whereZ isthepartitionfunction.19 ThustheBoltzmann-Gibbsdensitymatrixtakestheform,
β
(cid:88)n−1 e−βEj (cid:88)n−1 e−βEj
ρˆ= |ψ (cid:105)(cid:104)ψ | = w |ψ (cid:105)(cid:104)ψ |, w = . (9)
j j j j j j
Z Z
β β
j=0 j=0
Note that w is the Boltzmann-Gibbs probability that a mechanically isolated, constant particle and
j
constant volume statistical system in a heat bath is in the energy eigenstate |ψ (cid:105). Indeed, consider the
j
observableM representedbytheoperator,
(cid:88)
Mˆ = Pˆ(j) = |ψ (cid:105)(cid:104)ψ | = 1·|ψ (cid:105)(cid:104)ψ |+0· |ψ (cid:105)(cid:104)ψ |.
j j j j i i
i(cid:54)=j
TheobservableM correspondstoasking(andtakingameasurementtoanswer)thequestion:20
Isthesystemintheparticularenergyeigenstate|ψ (cid:105)withenergy(eigenvalue)E ?
j j
Notethatthisquestioncorrespondstothe(random)indicatorfunction,21
M = 1(E(ψ ) = E ) ⇐⇒ ρˆ= Pˆ(j) = |ψ (cid:105)(cid:104)ψ |.
j j j j
Wehave
e−βEj
p(ψ ) = E{1(E(ψ ) = E )} = (cid:104)M(cid:105) = trρˆMˆ = trρˆPˆ(j) = w = ,
j j j j
Z
(cid:124) (cid:123)(cid:122) (cid:125) β
(cid:104)Pˆ(j)(cid:105)
which is the Boltzmann-Gibbs distribution, p = p(ψ ) = Z−1e−βEj, encountered in classical statistical
j j β
mechanics.22
Having gone from the quantum mechanical statement of the canonical distribution to the classical
statement,let’sgointhereversedirection:
e−βEj (cid:88)n−1 e−βEi
p = = |(cid:104)ψ |ψ (cid:105)|2
j i j
Z Z
β β (cid:124) (cid:123)(cid:122) (cid:125)
i=0
δij
(cid:88)n−1 e−βEi
= tr |ψ (cid:105)(cid:104)ψ | |ψ (cid:105)(cid:104)ψ |
i i j j
Z
β
i=0
(cid:32) (cid:33)
(cid:88)n−1 e−βHˆ
= tr |ψ (cid:105)(cid:104)ψ | |ψ (cid:105)(cid:104)ψ |
i i j j
Z
β
i=0
19ForN (cid:29)1,thenumberofstates,n,isaverylargenumberanditisgenerallyintractabletocomputethepartitionfunction
Z .
β
20NotethatthisquestionismoreprecisethanaskingifthesystemisinanyeigenstatethathasenergyE . Thisisbecause
j
ofthepossibilitythattheenergyE couldbedegenerate(i.e., havemorethanoneeigenstatewiththesameeigenvalueE ).
j j
ThuswearenotaskingfortheprobabilitythatthesystemhasenergyE ;weareaskingfortheprobabilitythatthesystemis
j
instateψ whichisonlyoneoftheenergyE eigenstatesifenergylevelE isdegenerate.
j j j
21Thisisanimportantfact: Apurestate,Pˆj =|ψ (cid:105)(cid:104)ψ (cid:105)representsameasurementthatcorrespondstotoaskingaYes-No
j j
question,i.e.,toanindicatorfunction.
22In classical statistical mechanics, j indexes a classical phase space state, whereas in quantum statistical mechanics it
indexesa(pure)quantummechanicalstate.
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 8
(cid:32) (cid:33)
e−βHˆ (cid:88)n−1
= tr |ψ (cid:105)(cid:104)ψ | |ψ (cid:105)(cid:104)ψ |
i i j j
Z
β (cid:124) (cid:123)(cid:122) (cid:125)
i=0
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) P(j)=Mˆ
ρˆ
Iˆ
= trρˆMˆ = (cid:104)M(cid:105) = (cid:104)Pˆ(j)(cid:105) = w .
j
4.2 Free Energy for a Finite-Dimensional Quantum Mechanical System
TherelevantThermodynamicPotentialforthecanonicaldistributionisthe(Helmoltz)freeenergy,which
isclassicallygivenby
F (p) = U(p)−TS(p),
T
where p = (p ,··· ,p )T, (cid:80) p = 1, denotes the classical canonical distribution.23 In classical
0 n−1 i i
statisticalmechanics,thequantity
n−1
(cid:88)
S = S(p) = − p lnp = −E{lnp}
j j
j=0
isthe(Gibbs)thermodynamicentropy(settingk = 1)and
n−1
(cid:88)
U = U(p) = E{E} = (cid:104)E(cid:105) = p E
j j
j=0
istheaverageenergy(internalenergy)ofthesystem. Itisevidentthatthefreeenergyisafunctionofthe
temperatureT (or,equivalently,ofβ = 1/T)andoftheBoltzmann-Gibbsdistributionp,F = F (p).
T
It is a fundamental fact of equilibrium statistical mechanics that the free energy takes its minimum
value at thermal equilibrium, a fact which is true both for classical and quantum statistical mechanics.24
Note that minimizing the free energy corresponds to a balance between reducing the internal energy U
andincreasingtheentropyS. Classicalsimulatedannealing(CA),drawsMCMCsamplesfromasystem
having the Boltzmann-Gibbs distribution as its stationary distribution while taking T → 0 very slowly.
Thisultimatelyresultsinsamplesbeingdrawnfrom,ornear,theT = 0energygroundstateofthesystem.
Quantummechanically,thefreeenergyforthecanonicalstategiveninEquationsρˆ(7)and(9)is
F (ρˆ) = U(ρˆ)−TS(ρˆ)
T
where
ˆ
U(ρˆ) = (cid:104)E(cid:105) = trρˆH and S(ρˆ) = −trρˆlnρˆ.
In this form, S = S(ρˆ) is known as the quantum mechanical, or von Neumann, entropy. Thus, quantum
mechanically,
ˆ
F (ρˆ) = trρˆH +T trρˆlnρˆ. (10)
T
23But again note the equivalence of the classical and quantum mechanical Boltzmann-Gibbs distribution, p = w =
j j
Z−1e−βEj,andseethenextparagraph.
β
24Thishasinterestingconsequences. Forexample,itallowsforanalternativederivationoftheclassicalBoltzmann-Gibbs
(cid:80)
distributionasthedistributionthatminimizesthefreeenergywithrespecttopsubjecttotheconstraintthat p =1.
j j
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 9
One can readily see that this yields the classical free energy as we can show that U(ρˆ) = U(p) and
S(ρˆ) = S(p)asfollows:
e−βHˆ (cid:88)n−1
ˆ
U(ρˆ) = trρˆH = tr E |ψ (cid:105)(cid:104)ψ |
j j j
Z
β
j=0
n−1 n−1
= tr 1 (cid:88)E e−βHˆ|ψ (cid:105)(cid:104)ψ | = tr 1 (cid:88)E e−βEj|ψ (cid:105)(cid:104)ψ |
j j j j j j
Z Z
β β
j=0 j=0
(cid:88)n−1 e−βEj (cid:88)n−1
= E tr|ψ (cid:105)(cid:104)ψ | = p E = U(p).
j j j j j
Z
β (cid:124) (cid:123)(cid:122) (cid:125)
j=0 j=0
(cid:124) (cid:123)(cid:122) (cid:125) =(cid:104)ψj|ψj(cid:105)=1
=wj=pj
Note that the canonical state ρˆ can be written in terms of the orthonormal energy eigenvectors |ψ (cid:105) as
j
showninEq.(9). Forthisreasonw = p and|ψ (cid:105)areeigenvalue/eigenvectorpairsfortheoperatorρˆ,
j j j
ρˆ|ψ (cid:105) = p |ψ (cid:105),
j j j
andEq.(9)givesthespectralrepresentationofρˆ. Therefore(seeEq.(2))
(cid:16) (cid:17) (cid:16) (cid:17)
ρˆlnρˆ |ψ (cid:105) = p lnp |ψ (cid:105)
j j j j
and(seeEq.(3))
n−1
(cid:88)(cid:16) (cid:17)
ρˆlnρˆ= p lnp |ψ (cid:105)(cid:104)ψ |.
j j j j
j=0
Wehave,
n−1
(cid:88)(cid:16) (cid:17)
S(ρˆ) = −trρˆlnρˆ= −tr p lnp |ψ (cid:105)(cid:104)ψ |
j j j j
j=0
n−1 n−1
(cid:88) (cid:88)
= − (p lnp )tr|ψ (cid:105)(cid:104)ψ | = − p lnp = S(p).
j j j j j j
(cid:124) (cid:123)(cid:122) (cid:125)
j=0 j=0
(cid:104)ψj|ψj(cid:105)=1
5 Optimization via Classical and Quantum Annealing
Asmentionedabove,acanonicalstatisticalmechanicalsystem25 hasafreeenergy,
ˆ
F (ρˆ) = U(ρˆ)−TS(ρˆ) = trρˆH +T trρˆlnρˆ,
T
whichisaminimum. Forlowenoughtemperature,T ≈ 0,minimumfreeenergyeffectivelycorresponds
tominimuminternalenergy,U = (cid:104)E(cid:105). Further,atT ≈ 0,energyfluctuationsarenegligibleandtheinter-
nalenergyisapproximatelyequaltothegroundstateenergyU = (cid:104)E(cid:105) ≈ E ≈constant. Exploitingthese
0
twofacts,wecan“anneal”thesystemintoanequilibriumconfigurationthatapproximatesaground-state
configuration. IntheD-Wavecomputer,annealingisnotonlycontrolledbythetemperatureparameterT
25I.e., amechanicallyisolated, constantparticle, constantvolume, systeminthermalequilibriumwithaheatbathattem-
peratureT inthestate(9).
QuantumMechanics&AnnealingLectureNotes—K.Kreutz-Delgado—VersionPRL-QMN-2016.v1.2b 10
in an initial “classical annealing” phase used to initialize the system, but also by the subsequent use of
magneticcouplingofspin-1 qubitstoanexternaldrivingfieldandtoeachotherviainteractionfieldsina
2
second “quantum annealing” phase that adiabatically evolves the Hamiltonian into a desired final form,
allthewhilemaintainingthesysteminagroundstateconfiguration.
Specifically,intheD-Wavecomputerwetake,26
ˆ ˆ ˆ ˆ
H = H(Γ ,Γ ) = Γ H +Γ H ,
P D P P D D
ˆ
where Γ and Γ take positive real number values. H is a “problem Hamiltonian operator”, which en-
D P P
ˆ
codes an optimization loss-function that we wish to minimize, and H is the initial, or disorder, Hamil-
D
tonianoperatorthatisusedtopreparethestateoftheD-Wavecomputerintoamaximallynon-committal
(“maximally disordered”) initialization configuration. The real scalar Γ ∈ [0,1] is a control parame-
P
ter that allows one to “turn on” the problem Hamiltonian H . The real scalar Γ ∈ [0,1] is a control
P D
ˆ
parameterthatallowoneto“turnoff”thedisorderHamiltonianH . WiththisformforH,wehave
D
ˆ ˆ ˆ
F (ρˆ;Γ ,Γ ) = trρˆH(Γ ,Γ )+T trρˆlnρˆ= Γ trρˆH +Γ trρˆH +T trρˆlnρˆ.
T P D P D P P D D
Toperformatwo-stage(classicalplusquantum)annealingprocess,westartwithΓ = 0andΓ = 1,
P D
andworkwiththesystem
ˆ ˆ
F (ρˆ;0,1) = trρˆH(0,1)+T trρˆlnρˆ= trρˆH +T trρˆlnρˆ.
T D
We then slowly cool (thermally anneal) the system to a very low temperature, T → T ≈ 0 (around 15
0
millikelvin). This essentially corresponds to a classical annealing (CA) step which gets us (close to) the
groundstateofH ,whichisdesignedtobeamaximallydisorderedsuperpositionofalloftheaccessible
D
statesofthesystem.
NowassumingthatT = T ≈ 0,wenextconsiderthesystem
0
ˆ ˆ
F (ρˆ;Γ ,Γ ) = Γ trρˆH +Γ trρˆH +T trρˆlnρˆ.
T0 P D P P D D 0
We very slowly (adiabatically) take Γ from 1 to 0 while simultaneously taking Γ from 0 to 1. This is
D P
the quantum annealing (QA) stage. This is done very slowly (“adiabatically”) so that the system all the
whilehasahighprobabilityofbeingmaintainedinagroundstateconfiguration. Thisadiabatictransition
ˆ ˆ
fromH toH isformallyindicatedby
D P
ˆ
lim F (ρˆ;Γ ,Γ ) = trρˆH +T trρˆlnρˆ.
T0 P D P 0
ΓD→0
ΓP→1
In this limit, the free energy being at a minimum for a system in thermal equilibrium, we should be in a
ˆ
(near) ground-state configuration of the system with (final) Hamiltonian H , a configuration which can
P
bereadfromthesystemregisters.
26Herewefollowthediscussionin[22].