Open quantum generalisation of Hopfield neural networks

4 MB·

by  P. Rotondo

Open quantum generalisation of Hopfield neural networks
P. Rotondo,1,2 M. Marcuzzi,1,2 J. P. Garrahan,1,2 I. Lesanovsky,1,2 and M. Mu¨ller3
1School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
2Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,
University of Nottingham, Nottingham NG7 2RD, UK
3Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, UK
We propose a new framework to understand how quantum effects may impact on the dynamics
of neural networks. We implement the dynamics of neural networks in terms of Markovian open
quantum systems, which allows us to treat thermal and quantum coherent effects on the same
footing. Inparticular,weproposeanopenquantumgeneralisationofthecelebratedHopfieldneural
network, the simplest toy model of associative memory. We determine its phase diagram and show
7
thatquantumfluctuationsgiverisetoaqualitativelynewnon-equilibriumphase. Thisnovelphase
1
ischaracterisedbylimitcyclescorrespondingtohigh-dimensionalstationarymanifoldsthatmaybe
0
regarded as a generalisation of storage patterns to the quantum domain.
2

n
a Introduction— Neural networks (NNs) [1] - artifi- ten realised via classical Monte Carlo dynamics), while
J
cialsystemsinspiredbytheneuralstructureofthebrain quantumeffectsareduetoatransversefieldHamiltonian
6
- have become essential tools for solving tasks where that turns this classical equilibrium system into a quan-

] more traditional rule-based algorithms fail. Examples tum non-equilibrium one. In this approach, the result
n are pattern and speech recognition [2], artificial intelli- of a NN computation is imprinted on the long time den-
n
gence [3, 4], and the analysis of big data [5]. All these sitymatrix,whosepropertiesasafunctionofthecontrol
-
s NNs evolve according to the laws of classical physics. parameters(temperature,quantumdriving,initialstate)
i
d It is widely believed that computational processes can determine the phase diagram of the system. This setup
. benefit by exploiting the properties of quantum mechan- reduces to the classical Hopfield NN when the quantum
t
a ics. SeminalworksbyShor[6]andGrover[7]haveproven Hamiltonian is removed. By means of mean-field meth-
m
the existence of quantum algorithms that systematically ods (which are exact for fully connected models such as
- outperform their classical counterparts. More recently, this one) we calculate the phase diagram of the model,
d
n we have seen the advent of a first generation of quan- and show that a new non-equilibrium phase, character-
o tum information processors: for instance, Shor’s algo- ized by the presence of limit cycles (LCs), arises due
c rithm for the factorization of integer numbers has been to the competition between coherent and dissipative dy-
[
implementedonatrapped-ionquantumcomputer[8]and namics. This may be regarded as a quantum generalisa-

1 D-Wave machines, based on superconducting qubits, are tionoftheretrievalphaseoftheclassicalHopfieldmodel.
v envisioned to solve a particular class of NP-hard prob- Classical Hopfield NNs— Originally, the Hopfield
7 lems via quantum annealing [9–11]. NNwasintroducedasatoymodelofassociativememory.
2
7 Animportantandtimelyquestioniswhetheritispos- In the human brain memory patterns are stored and can
1 sibletotakeadvantageofquantumeffectsinNNcomput- be retrieved by association, i.e., when a pattern similar
0 ing[12]. Todate,however,noneoftheexistingproposals enough to one of those stored is presented to the NN the
1. [13–15] allows to define a satisfactory framework to con- systemisabletoretrievethecorrectoneviaclassicalan-
0 siderquantumeffectsinNNdynamics. Theproblemisa nealing. The two fundamental ingredients to reproduce
7 conceptualone: thedynamicsofclosedquantumsystems this are: (i) a dynamics on a system of N binary spins
1 is governed by deterministic temporal evolution equa- (σi=±1,i=1,...,N),thatrepresentneuronactivity(+1
v: tions, whereas NNs are always described by dissipative firing and −1 silent); (ii) an appropriate prescription for
Xi dynamical equations, thus preventing any straightfor- the couplings Jij that connect the i-th neuron with the
ward generalization of NNs computing in quantum sys- j-th one, which must be able to store a set of p differ-
ar tems [16]. Here we overcome this obstacle, by proposing ent memory patterns ξ(µ) (i.e. fixed spin configurations)
i
a framework for quantum NNs based on open quantum withi=1,...,N, µ=1,...,p. Inthislanguage, memory
systems (OQSs). We consider, in particular, the concep- retrieval denotes a phase in which the dynamics drives
tually simplest case of Markovian dynamics, where the the system towards configurations which are closely re-
evolution of the density matrix is described by a Lind- sembling one of the ξ(µ) for some µ. It turns out that
i
blad equation [17]. thefollowingdiscretetimedynamicsfulfillstheserequire-
In order to make our ideas concrete we introduce a ments:
generalisation based on OQSs of one of the most stud-
ied NN systems, the Hopfield model [18] (see Fig. 1). σ (t+1)=sign⎛∑J σ (t)⎞, J = 1 ∑p ξ(µ)ξ(µ).
The dissipative part of the dynamics corresponds to the i ⎝j≠i ij j ⎠ ij N α=1 i j
thermal stochastic dynamics of the classical model (of- (1)
2
(a) where we define a set of jump operators as follows:
�
j J
ij �i |ei Li±=Γi±σi±, Γi±= (2coes∓hβ(/β2∆∆EEii))12 . (3)
⌦ �
i
± Here β = 1/T is the inverse temperature, ∆E =
i
g
| i ∑j≠iJijσjz thechangeinenergyunderflippingofthei-th
spin, and σ± =(σx±iσy)/2, with σx,y,z the Pauli matri-
⌦ = 0 i i i
e �E ces. Quantum effects are included by a uniform trans-
�
(b) ⇢ss = ⌦ = 0 verse field in the x-direction, corresponding to a Hamil-
Z 6
tonian,
E
⇢ss = tlim eLt⇢in H =Ω∑N σx. (4)
= !e1��E i=1 i
� 6 Z
⇠(1) ⇠(2) ⇠(3) In the absence of this term, Eq. (2) describes a
classical stochastic dynamics: any initial density ma-
FIG. 1. (a) In the Hopfield model neurons (dots) are bi- trix that is diagonal in the σz basis remains diago-
nary spins describing the activity of the neurons (+1 firing, nal under the evolution and Eq. (2) reduces to P˙ =
-1 silent). The OQSs framework allows us to study the com- ∑Ni=1∑τ=±Γ2iτ[σiτ − 21(1+τσiz)]P, where P is the prob-
petitionbetweenthermalandquantumeffects. Inparticular,
ability vector formed by the diagonal of ρ. The rates
the i-th neuron changes its activity state at a rate Γi± as in Γ2 obeydetailedbalancewithrespecttotheBoltzmann
theclassicalmodelorundergoesaquantumstatechange,due iτ
distribution for energy E at temperature T, so that this
to the coherent driving introduced in Eq. (4). (b) If Ω = 0,
thestationarystateisatthermalequilibrium. Thequalitative is the master equation for the classical Hopfield NN, as
behavioroftheenergyfunctionoftheclassicalNNissketched shown in the supplementary material (SM) [22].
in a one dimensional projection of the configurational space. Mean-field solution— Since the NN is defined in
Memory patterns are stored as the energy minima of the en- terms of fully-connected interactions, the mean-field ap-
ergy function. Whenever the NN is initialized close enough
proximation should be exact [22]. We consider the equa-
(close in the sense of the Hamming distance between spin tions of motion for the time-dependent operators σz,
configurations) to a specific memory pattern, the dynamics ± i
σ . These 3N equations can be reduced to a closed
in Eq. (1) allows to retrieve the corresponding stored pat- i
tern. In the presence of quantum effects (Ω≠0), the nature set of equations for the 2p collective operators sαµ =
of the stationary state can be non-trivial, i.e. it may be non- (1/N)∑Ni=1ξi(µ)σiα (α = z,y). The mean-field approxi-
thermal,duetothecompetitionbetweenquantumcoherence
mation considers the evolution of the averaged collective
and irreversible classical dynamics. variables mz,y = ⟨sz,y⟩ neglecting correlations between
µ µ
them [22]. The evolution equations then read:
Eq. (1) describes a zero temperature Monte Carlo dy- 1 N
m˙ z =2Ωmy+ ∑ξξξ tanh(βξξξ ⋅mz)−mz, (5)
namics that can be easily generalized to include thermal N i=1 i i
effects [19]. Moreover it can be proven that this dynam-
1
ics minimizes the energy function E = −21∑i≠jJijσiσj, m˙ y =−2Ωmz− 2my, (6)
namelyanIsingmodelwithpattern-dependentcouplings.
Techniques used in the statistical physics of disordered where we introduced a vectorial notation for both the
systems enable to investigate Hopfield NNs (and more collective operators and the memory patterns, m˙ α =
general types of NNs) quantitatively [19–21]. In sta- (mα,...,mα). mz represents the overlap between the
1 p µ
tistical physics language, the retrieval phase is the low µ-th memory pattern and the neuronal configuration
temperaturephasecorrespondingtoanenergylandscape of the system (in the z direction), and thus can be
where memory patterns are stable states of the NN, i.e. used as an order parameter: if in the stationary state
the thermal equilibrium stationary states; see Fig. 1. mz ≈ (1,0,...,0), this means that the NN correctly re-
1
Open quantum Hopfield NNs—To introduce trieved the first pattern stored (and similarly for the
other p−1 patterns). Setting m˙ z =m˙ y =0 in Eqs. (5,6)
quantum effects, we employ a description of the NN dy-
we get the equations for the stationary solutions:
namicsintermsofopenquantummasterequations. The
starting point of our analysis is a master equation in
(1+8Ω2)mz =ξξξtanh(βξξξ⋅mz), (7)
Lindblad form for the density matrix ρ:
where we have assumed self-averaging, typical of
ρ˙=−i[H,ρ]+∑N ∑(L ρL† − 1{L† L ,ρ}), (2) disordered systems [23] in the large N limit, i.e.,
i=1τ=± iτ iτ 2 iτ iτ 1/N∑Ni=1f(ξξξi) → f(ξξξ), where (⋅) is the average over
3
1 0.6 thisnewfeatureistoconsiderahigh-T expansionofEqs.
(5,6) up to the first non-linear order. In this case the
0.5
equationsbecomeanalogoustoaLotka-Volterra(LV)dy-
0.5 Limit
0 Cycles namical system, widely studied in the literature on eco-
(LC) logical systems [24]. Under appropriate conditions, a LV
-0.5 LC + FM
0.4 system is known to have limit cycles (LCs) solutions.
-1 ⌦ Theobservationabovesuggeststhatouropenquantum
-1 -0.5 0 0.5 1
0.3 Paramagnetic HopfieldNNcanfeatureatleastthreepossiblephasesin
1 (PM) the (T,Ω) plane: (i) a paramagnetic phase where the
dynamics converges to the trivial solution mz =0; (ii) a
0.5 0.2
retrieval phase where the attractor of the system is one
my 0 Retrieval of the non-trivial solutions of Eq. (7); (iii) a novel time-
0.1 (FM) dependentstationaryphase, emergingfromthecompeti-
-0.5
tion between the dissipative and coherent dynamics.
-1 Phase diagram—Tocharacterizethephasediagram
-1 -0.5 m0z 0.5 1 0 0.5 0.6 T0.7 0.8 0.9 1 of this OQS in the (T,Ω) plane, we follow two different
routes: (i) we solve numerically the mean-field equations
FIG.2. PhasediagramoftheOQSgeneralizationoftheHop- at low p by averaging over disorder using the factor-
field model in the (T,Ω) plane. The NN displays a param- ized probability distribution P(ξξξ) = ∏pα=1p(ξ(α)) with
agnetic phase in the high temperature regime, with a stable p(ξ) = 1/2δ(ξ−1)+1/2δ(ξ+1); (ii) we perform a Lya-
attractorinmz =0. Theboundaryoftheparamagneticphase punov linear stability analysis [25], namely we study the
is established by looking at the stability of this attractor. In
dynamical stability of the stationary points under small
the low T and Ω regime the system is in the ferromagnetic
perturbations (see [22] for details). Through either ap-
(retrieval) phase, whereas for low T but sufficiently large Ω,
proachweidentifytheboundariesofthedifferentphases.
the NN displays a LC (top left), which is the only stable
attractor once the ferromagnetic solutions become unstable. The phase diagram of t√he NN is summarized in Fig.
However the LC appears well below this line (and above the 2. For T > 2/3 and Ω > (1−T)/8T the system is in
dotted line), producing a small region of phase coexistence the paramagnetic phase. Retrieval is possible in the low
(seefluxdiagramatbottomleft). Abovethedashedlineand temperature and low Ω regime, whereas the dynamics
for T <2/3, the ferromagnetic solutions disappear.
features LCs as a stationary manifold in the low tem-
perature and high coherence regime. At the boundary
between the retrieval and the LC phase, we recognize a
the disorder distribution. Equations (5,6) allow to study small region where the two phases coexist: initializing
theinterplaybetweentheretrievalandtheparamagnetic the system close enough to (mz,my)=(0,0), drives the
phasesoftheNN.Itisworthremarking, however, thata NN towards the retrieval solutions, whereas initial con-
more general approach, involving the replica method, is ditions chosen outside a critical hypervolume centered
required to deal with the spin glass phase arising when around (0,0) converge to the LC (see insets in Fig 2).
an extensive number of memory patterns is loaded into For a single pattern, p = 1, we are able to rigorously
the NN [20]. In the following we focus on the retrieval
prove the presence of a LC phase, since Eqs. (5) and (6)
phaseofourOQS,whichamountstoconsideringthecase can be recast in Li´enard form [26], (see also [22]) for
of a finite number of patterns p. T <2/3 and T >(8Ω2+1)−1. The Li´enard theorem [27]
As a first step, we notice that Eq. (7) has the same guarantees the existence, uniqueness and stability of a
formofthemean-fieldequationobtainedfortheclassical LC (which wraps around the origin).
HopfieldNN[19]: inparticularasuitablerescalingofmz The LC phase can be interpreted as a new quantum
withaneffectivetemperatureTeff =T(1+8Ω2)establishes retrievalphase: forlowenoughtemperatures, infact, in-
their equivalence. This means that the structure of the dependently from the initial conditions the NN is always
stationary points of the dynamics is equivalent to the driven towards a LC with a large oscillation amplitude.
classical one up to a rescaling of the temperature. In Forinstance, thecasereportedinFig.3a(corresponding
particular, the retrieval solutions mz ≈ (1,...,0), the toT =0.15)showsoscillationsreachingamaximumover-
paramagneticsolutionmz =0,aswellasthewholesetof lap of ∼0.8 with the single stored pattern. Limit cycles
metastable states (spurious memories) [19] are still fixed of this kind (involving mostly a single overlap) also ex-
points of the quantum dynamics. Closer inspection of istforgenericp>1. Numericalevidencesuggeststhatat
Eqs. (5,6) reveals, however, that quantum driving does longtimesoscillationsinvolveatmosttwooverlaps,while
more than just rescale temperature. all the remaining p−2 asymptotically vanish. More im-
Beyond these stationary solutions, the mean-field portantly, when starting from an initial condition with
equations (5,6) also display time-dependent periodic so- largeoverlapwithoneofthepatterns(say, theµ-thone)
lutions at long times. A simple way to gain insight on andsignificantlysmalleroverlapwiththeremainingones
4
(a)1 1 1 where L is the generator of the open quantum dynamics
0.5 0.5 0.5 [i.e., Lρ is shorthand for the r.h.s. of Eq. (2)], and ρin is
my0 0 0 the initial state. Assuming L to be diagonalizable, the
density matrix can be further expanded as [34]:
-0.5 -0.5 -0.5
-1 -1 -1 22N
-1 -0.5 m0z 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 ρ(t)= ∑ cletλlRl, (8)
l=1
⌦
0 0.5 1 where λ are the eigenvalues, R are the right eigenma-
l l
(b) decay modes Im(�) (c) 25 trices of L, i.e. LRl = λlRl, and cl the components of
the initial state on these eigenmodes. Because of preser-
20 mˆz(k)
| | vation of probability and positivity, the eigenvalues of L
15
havenon-positiverealpartsandareeitherrealorcomein
Re(�) 0 10 complexconjugatepairs. Thestationarymanifoldiscon-
5 structed from all the R for which Re(λ )=0; we define
stationary l l
manifold 00 20 40 60 80 100 nitsdimensionandreordertheeigenvaluessuchthatthe
k zero ones appear first. If λ = 0 ∀ l = 1,...,n then each
l
initial state maps to a state in the stationary manifold
FIG. 3. (a) From left to right flux diagrams of the mean-
asymptotically and no time dependence survives at long
field dynamics at T = 0.15, Ω = 0.1,0.58,0.8. The different
attractorsarehighlightedineachpanel. InthebulkoftheLC times. To allow for limit cycles, L must display at least
phase the maximum overlap with one of the stored patterns apairofconjugate,purely-imaginaryeigenvalues±iω,as
is∼0.8,independentlyfromtheinitialconditions,suggesting sketched in Fig. 3b. The long time evolution due to the
that also this phase can be interpreted as a retrieval phase. presenceoftheseeigenvaluesisunitaryandonecouldfor-
(b) Sketch of the spectrum of a plausible Liovillian super-
mallydefineacorrespondingreducedHamiltonianacting
operator describing the LC shown in (a). Time-dependent
coherently on the stationary subspace [33–35].
periodic stationary states require at least a conjugate pair of
A Fourier analysis of the LCs (see Fig 3c) justifies a
purelyimaginaryeigenvalues. (c)ThestructureoftheFourier
components ∣mˆz(k)∣ of the top right LC, justifies the single single frequency approximation for the stationary state,
frequency approximation made in the main text. leadingtoρss(t)∼R0+R1+R2eiωt+R2†e−iωt, thatallows
to obtain a quantitative agreement with the numerics by
requiring that the Ri’s are such that: Tr(sαR0/1) = 0,
seemstoalwaysleadtooscillationsinthe(mzµ,myµ)plane Tr(szR2) = m¯z and Tr(syR2) = m¯yei(cid:15). This choice
and vanishing ones in the others. This hints at the pos- reproduces a LC in the (mz,my) plane parameterized
sibility of associating to any stable fixed point attractor as mz(t) = Tr(szρ (t)) ∼ m¯zcos(ωt) and my(t) =
ξ(µ) of the NN a corresponding LC attractor. It is more- Tr(syρ (t))∼m¯ycosss(ωt+(cid:15))(with(cid:15)aproperrealphase).
i ss
over noteworthy that the fundamental properties of the Summary and Outlook—Weproposedaframework
Hopfield dynamics are rather robust against the intro- based on open quantum systems to investigate quantum
ductionofacoherenttermsuchasthetransversefieldwe effects in the dynamics of neural networks. We applied
use in this work. It is worth remarking that this robust- thisapproachtoanopensystemquantumgeneralization
ness is closely related to symmetry arguments: indeed ofthecelebratedHopfieldmodel. Asintheclassicalcase
the generalized Z symmetry broken in the FM phase of it is possible to use a mean-field treatment to determine
2
the classical model is still broken in the LC phase. Here, the phase diagram. We identified a retrieval phase with
additionally, the stationary state spontaneously breaks fixedpointsassociatedtoclassicalpatternsandquantum
the time translation symmetry, which should correspond effects can be accounted for by an effective temperature.
to theappearance of aGoldstone mode, as foundin [28]. We moreover found a novel phase characterized by limit
Quantum nature of the LC phase— Examples cycles which are a consequence of the quantum driving.
of LC phases in classical Hopfield NNs with asymmetric OurapproachisanaturalextensionoftheNNparadigm
couplings exist [29, 30]. However, the LC phase that we intothedomainofopenquantumsystems. Itshowsthat
findhereisintimatelyrelatedtothecompetitionbetween the resulting phase structure of such systems can indeed
coherentanddissipativedynamics. Thepersistenceofos- be richer than that of their classical counterparts. Fu-
cillations in the long time limit implies the survival – to tureinvestigationsareneededinordertoclarifywhether
a degree – of quantum coherence. To substantiate this practical applications of NNs can benefit from quantum
claim,weargueinthefollowingwhatthestructureofthe effects, similar to their rule-based counterparts, the clas-
stationary manifold of the Lindblad equation should be, sical computers.
followingtheclassificationofRefs.[31–33]forthecaseof
systems with finite-dimensional Hilbert spaces. The research leading to these results has received
By formally integrating Eq. (2) we get ρ(t) = etLρ , funding from the European Research Council un-
in
1
der the European Union’s Seventh Framework Pro- 335266 (ESCQUMA), the H2020-FETPROACT-2014
gramme (FP/2007-2013) / ERC Grant Agreement No. Grant No.640378 (RYSQ), and EPSRC Grant No.
EP/M014266/1.
Supplemental Material to “Open quantum generalisation of Hopfield neural networks”
CONTENTS
Basic definitions 1
Derivation of the mean field equations for the open quantum Hopfield NN 2
Classical limit of the quantum master equation 2
Addition of a quantum term 3
Dynamical mean field equations and stationary state equations 6
Mapping out the phase diagram: stability analysis of the stationary state solutions 7
Stability analysis of the paramagnetic solution 7
Appearance and stability of the ferromagnetic solutions 9
Role of the effective temperature 10
Limit cycle phase: exact results and numerical solution 11
Representation of the p=1 case in terms of a Li´enard equation 11
Representation of the p=1 case in terms of a single N/2-spin 12
The 2-memories case 12
Decoupling of the equations in the 2-memories case 13
The 3-memories case 14
Bounds to the appearance of limit cycles 16
References 18
BASIC DEFINITIONS
TheHopfieldmodel[18]isastochasticprocesswithratesengineeredtosatisfydetailedbalancewithenergyfunction
E =− 1 ∑p ∑N ξ(µ)ξ(µ)σ σ =− 1 ∑p (∑N ξ(µ)σ )2 , (S1)
2N µ=1i,j=1 i j i j 2N µ=1 i=1 i i
where N is the number of spin variables (or lattice sites) in the system and p the number of patterns (memories)
one is trying to store in the network. Each of these memories is encoded in terms of an N-dimensional vector ξξξ(µ),
whose components are independent, identically-distributed random variables, and are usually taken to be classical
Ising spins which assume the values ±1 with equal probability. For later convenience, we define the coupling matrix
and the overlap of the spin configurationσσσ with the µ-th memory pattern as
J = 1 ∑p ξ(µ)ξ(µ), m = 1 ∑N ξ(µ)σ , (S2)
ij N µ=1 i j µ N i=1 i i
respectively. This makes it possible to re-express the energy as
N p
E =− ∑m2. (S3)
2 µ=1 µ
2
In the absence of metastable minima, the perfect retrieval of one of the stored patterns can be achieved via a proper
zero-temperature Monte Carlo dynamics, e.g., with discrete updates
σ (t+1)=sign(∆E (t)), ∆E (t)=∑J σ (t), (S4)
i i i ij j
j≠i
onthesystemconfiguration,where∆E (t)istheenergydifferenceduetoasinglespin-flipofσ atstept. Asmentioned
i i
above, this dynamics can be easily generalized to include thermal fluctuations at a fixed temperature T =1/β. In this
case, the probability of performing a single Monte Carlo update σ (t)→σ (t+1) is given by
i i
eβ(σi(t+1)∆Ei(t))
P(σ (t+1))= . (S5)
i 2cosh(βσ (t+1)∆E (t))
i i
DERIVATION OF THE MEAN FIELD EQUATIONS FOR THE OPEN QUANTUM HOPFIELD NN
A first step towards the inclusion of quantum effects in the Hopfield neural network is to reformulate the classical
dynamicsasapurelydissipativequantummasterequation. ForMarkovianevolution,thedensitymatrixofthesystem
ρ evolves under the Lindblad equation
ρ˙=Lρ, L=∑L (⋅)L† − 1{L†L ,(⋅)}, (S6)
k k 2 k k
k
with L the superoperator, L the Linblad jump operators labeled by an index k and {A,B} = AB+BA denoting
k
anticommutation. A possible strategy to reproduce the classical dynamics of the Hopfield model is to construct the
jump operators in such a way that they generate the same local processes of the aforementioned heat-bath dynamics.
More specifically, if we generalize our classical spins to quantum ones ↑→∣↑⟩=(1,0)⊺, ↓→∣↓⟩=(0,1)⊺, we can set
Li±=√γ(2coes±hβ(∆βE∆i/E2i))12σi± with ∆Ei=j∑≠iJijσjz (S7)
where γ is an overall amplitude determining the frequency of the jumps and σ± and σz are the Pauli matrices acting
i i
on the i-th spin which, in the representation chosen above, read
σz =(1 0), σ+=(0 1), σ−=(σ+)⊺, (S8)
0 −1 0 0
and satisfy the standard (anti-)commutation relations:
[σz,σ±]=±2σ±δ , [σ+,σ−]=σzδ , {σ±,σz}=0, {σ+,σ−}=1 . (S9)
i j i ij i j i ij i i i i i
The dynamics generated by Eq. (S6) features a special symmetry: working in the eigenbasis of the σzs (i.e., on states
i
of the form ∣↑↑↓...⟩), if we express the generic matrix element ρ =⟨a∣ρ∣b⟩ and define S ={ρ ∶b−a=h} one can
ab h ab
check that the evolution equations for the matrix elements close within each subspace S . Therefore, if one starts
h
in the classical subspace S ={ ∣a⟩⟨a∣ } the dynamics remains confined within it and no overlap over non-classical
0
states is generated in time. Therefore, the dynamics is exactly reduced to a stochastic process over the probabilities
p of the “configurations” ∣a⟩⟨a∣; below we show that this reconstructs a continuous-time variant of the heat-bath
a
process described above.
Classical limit of the quantum master equation
For simplicity, we are going to analyze the evolution of observables, instead of states. By our discussion above, we
can safely restrict ourselves to the identity and the σzs. The former does not evolve, while the latter obey the adjoint
i
Lindblad equation
O˙ =∑L†OL − 1{L†L ,O}, (S10)
k k 2 k k
k
3
which, once specialized to the case of interest, reads
σ˙z =∑L† σzL − 1{L† L ,σz} (S11)
i ls i ls 2 ls ls i
l,s
with s=±.
Remark: the Lindblad operators L have the following form:
iσ
Li±=√γΓi±σi±, Γi±= (2coes±hβ(/β2∆∆EEii))21, ∆Ei(σ1z,...,σNz )=j∑≠iJijσjz (S12)
implyingthatΓi± triviallyactsastheidentityonthei-thsubspaceandhence[Γi±,σi±]=0. Furthermore,sincetheΓs
are entirely defined in terms of the σizs, we have [Γi±,σjz]=0 ∀i,j. This allows us to rearrange terms in the Lindblad
equation to get, for the “flipping-up” (+) process,
L+σiz =∑N L†l+σizLl+− 12{L†l+Ll+,σiz}=γ∑Γ2l+[σl−σizσl+− 12{σl−σl+,σiz}]=
l=1 l
=γΓ2i+[σi−σizσi+− 21{σi−σi+,σiz}]+γ∑Γ2l+[σl−σizσl+− 12{σl−σl+,σiz}], (S13)
l≠i ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶
=0
where we used the fact that [σ±,σz]=0 for i≠l. The only term remaining can be thus recast as
l i
L+σiz =γ(1−σiz)Γ2i+. (S14)
Following an analogous procedure, one can calculate
L−σiz =−γ(1+σiz)Γ2i−. (S15)
Adding these terms we get the evolution equation for σz:
i
σ˙iz =γ[(1−σiz)Γ2i+−(1+σiz)Γ2i−]=−γσiz+γtanh(β∆Ei), (S16)
where for the last equality we have employed the identities
Γ2i++Γ2i−=1 and Γ2i+−Γ2i−=tanh(β∆Ei) (S17)
Employing a mean field approximation and looking for the stationary properties (i.e., setting σ˙z =0) of the system,
i
the equations above yield the naive mean field equations of the Hopfield model (exact in the retrieval phase):
⎛ ⎞
z =tanh ∑J z , (S18)
i ⎝ ij j⎠
j≠i
with z =⟨σz⟩.
i i
Addition of a quantum term
The action of an Hamiltonian inducing a coherent evolution is accounted for by completing the Lindblad equation
with a commutator:
ρ˙=−i[H,ρ]+∑L ρL† − 1{L†L ,ρ},
k k 2 k k
k
(S19)
O˙ =i[H,O]+∑L†OL − 1{L†L ,O}.
k k 2 k k
k
We choose a coherent term of the form H =Ω∑Ni=1σix which couples classical and non-classical states. The equation
of motion for σz acquires a new term
i
σ˙z =2Ωσy−γσz+γtanh(β∆E ), (S20)
i i i i
4
which in the absence of dissipation (γ =0) would simply describe Rabi oscillations of frequency 2Ω about the x axis.
This clearly shows that the decoupling of the classical subspace does not hold any more: in fact, the equations for σi
z
do not close and we need to write additional ones for σx/y or, equivalently, σ±. These are generically more involved,
i i
since these operators do not commute with the Γs. To simplify the respective Lindblad equations, we will make use
in the following of the anticommutation relations, which in particular imply that
f(σz,⋯,σz,⋯,σz )σ±=σ±f(σz,⋯,−σz,⋯,σz ), (S21)
1 j N j j 1 j N
+
for any function f with a power series representation. Let us consider the equation for σ :
i
σ˙i+=−iΩσiz+γ∑σl−Γl+σi+Γl+σl+− 12{σl−Γl+Γl+σl+,σi+}+(+↔−). (S22)
l
Our aim is now to move all the Γ’s to the right. This is done applying property (S21). Introducing the notation
Γ(i)=Γ (σz,⋯,−σz,⋯,σz ). (S23)
ls ls 1 i N
equation (S22) can be written in a compact form as
σ˙i+=−iΩσiz− γ2σi+(Γ2i++Γ2i−)+ γ2σi+∑(1−σlz)(Γ(l+i)Γl+− 12Γ(l+i)Γ(l+i)− 12Γl+Γl+)+
l≠i
+ γ2σi+∑(1+σlz)(Γ(l−i)Γl−− 12Γ(l−i)Γ(l−i)− 12Γl−Γl−). (S24)
l≠i
−
Theequationforσ issimplyobtainedviahermitianconjugationoftheoneabove. Theseequationscanbesimplified
i
recognizing that Γ and Γ(i) depend on two configurations which differ by a single spin (the i-th one). It could be
thereby reasonably expected that Γ ≈ Γ(i) up to finite-size corrections which scale as 1/N. More precisely, singling
out the dependence on the i-th spin Γl±(σ1z,σ2z,...σNz ) ≡ Fli±(βJliσiz) (with no sum over repeated indices) we can
generically write Fli±(βJliσiz)=Eli±+Oli±σiz with
Eli±= Fli±(βJli)+Fli±(−βJli) (S25)
2
Oli±= Fli±(βJli)−Fli±(−βJli) (S26)
2
the even and odd parts of Fli± calculated in βJli. This can be understood by expanding Fli± as a power series in σiz
and then grouping odd powers in O and even ones in E exploiting the well-known property
⎧
⎪⎪1 (n even)
(σz)n=⎨ (S27)
i ⎪⎪⎩σz (n odd),
i
provided that the series expansion has a radius of convergence ≥∣βJ ∣, or otherwise taking the analytic continuation
li
up to ±βJli. Clearly, Γ(l±i)=Fli±(−βJliσiz) and we can write
Γ(l±i)=Γl±(gli±+fli±σiz) (S28)
with
gli±= EEl22i±+−OOl22i± and fli±=−E2E2li−±OOli2± . (S29)
li± li± li± li±
Defining ∆Eli=∑j≠i,lJljσjz we can write Fli± as
Fli±(+βJli)= √2ceo±shβ2(β∆(E∆liE+Jli)+J ) and Fli±(−βJli)= √2ceo±shβ2(β∆(E∆liE−Jli)−J ) (S30)
li li li li
5
and consequently
¿ ¿
⎧ ⎫
gli±= 21⎪⎪⎨⎪⎪⎩e∓βJli``(cid:192)ccoosshhββ((∆∆EEllii+−JJllii)) +e±βJli``(cid:192)ccoosshhββ((∆∆EEllii−+JJllii))⎪⎪⎬⎪⎪⎭, (S31)
¿ ¿
⎧ ⎫
fli±= 21⎪⎪⎨⎪⎪⎩e∓βJli``(cid:192)ccoosshhββ((∆∆EEllii+−JJllii)) −e±βJli``(cid:192)ccoosshhββ((∆∆EEllii−+JJllii))⎪⎪⎬⎪⎪⎭. (S32)
Now, according to the definition (S2),
J = 1 ∑ξ(µ)ξ(µ); (S33)
li N l i
µ
since ξ(µ)ξ(µ) for l≠i are independent random variables which take the values ±1 with equal probability, as long as
l i √
p is large we can apply the central limit theorem to argue that J ∼ p/N. If p is kept fixed, or taken to scale slower
li
than N2 these contributions will hence vanish in the thermodynamic limit and only yield small corrections at finite
– albeit large – size. We can thus perform a power series expansion
gli±=gl(i0±)+βJligl(i1±)+(βJli)2gl(i2±)+... (S34)
fli±=fl(i0±)+βJlifl(i1±)+(βJli)2fl(i2±)+..., (S35)
where summation on repeated indices is not intended. The first few terms read
g(0)=1 f(0)=0
li± li±
g(1)=0 f(1)=∓1+tanhβ∆E (S36)
li± li± li
g(2)= 1(1∓tanhβ∆E )2 f(2)=0.
li± 2 li li±
Stopping at first order, we find
Γ(l±i)=Γl±[1+βJli(∓1+tanh(β∆Eli))+O(pN−2)]= (S37)
=Γl±[1+βJli(∓1+tanh(β∆El))+O(pN−2)],
√
where in the last passage we used the fact that ∆E =∆E +O( p/N). We now perform this substitution in the last
li l
addend of Eq. (S24), which yields
γ2σi+∑(1+σiz)Γ2l−{1+βJli(1+tanhβ∆El)− 12[1+βJli(1+tanhβ∆El)]2− 12 +O(pN−2)}=
l≠i
(S38)
= γ2σi+∑(1+σiz)Γ2l−{−12[βJli(1+tanhβ∆El)]2+O(pN−2)}= γ2σi+∑(1+σiz)Γ2l−{O(pN−2)}=O(pN−1)
l≠i l≠i
andthesameholdsfortheotherterminvolvingΓl+. Therefore,wecansafelyneglectthesetermsinthethermodynamic
limit, which produces a considerably simplified form of the dynamical equations
σ˙±=∓iΩσz− γσ±. (S39)
i i 2 i
The equations can be further simplified by expressing them in the (x,y,z) basis:
σ˙z =2Ωσy−γσz+γtanh(β∆E ), (S40a)
i i i i
γ
σ˙y =−2Ωσz− σy, (S40b)
i i 2 i
γ
σ˙x=− σx. (S40c)
i 2 i
Interestingly enough, the equation for σx decouples from the others and one can conveniently restrict to the y and z
i
componentsonly. Atthislevel,itisalreadypossibletoappreciateastrikingdifferencewiththeclassicalcase. Indeed,
6
inordertodescribethestateofthesystem, itisnotenoughtospecifythevalueofN spins, butN vectorsneedtobe
specified. Although it will be proved in the next section that this does not dramatically affect the properties of the
stationary state, this may give rise to new intriguing phases of the neural netwok, where it is possible to appreciate
competing effects between coherence and dissipation, i.e. a hallmark of the quantum behavior. In the following
dynamical mean field equations are derived for this open quantum system. As the geometry of the model is fully
connected,inthefollowingweshallemployamean-fielddecouplingondifferentsites,whichshouldyieldexactresults
in the thermodynamic limit.
Dynamical mean field equations and stationary state equations
Before proceeding, it is useful to generalize the definition of the overlaps (S2) to this case:
sα= 1 ∑N ξ(µ)σα µ=1,⋯,p; α=x,y,z. (S41)
µ N i=1 i i
These observables can be thought as the overlap between the µ-th memory and the α component of the spins. The
equations for these collective variables read
γ N
s˙z =2Ωsy+ ∑ξξξ tanh(βξξξ ⋅sz)−γsz, (S42)
i i
N i=1
γ
s˙y =−2Ωsz− sy, (S43)
2
wherethevectorialnotationnowgroupsthepatternindices,notthepositionalones: inotherwords,thesearep-uples
of operators sα=(sα,...sα)⊺ and numbersξξξ =(ξ(1),...ξ(p))⊺.
1 p i i i
Takingtheexpectationvalue⟨sα⟩=mα,itbecomesnaturalatthispointtointroducethemeanfieldapproximation,
by discarding correlations between the collective variables ⟨mαmβ⟩→⟨mα⟩⟨mβ⟩. Assuming that the memories ξ(µ)
µ ν µ ν
arequenchedidenticallydistributedrandomvariables,theseequationscanbefurthersimplifiedusingtheself-averaging
hypothesis: denoting by ⟪⋅⟫ the average over different realizations of the disorder
⟪⋅⟫=∫ dξξξ(⋅)P(ξξξ) (S44)
with P(ξξξ) the probability distribution function of a patternξξξ, we assume that mα→⟪mα⟫ for p→∞, so that
m˙ z =2Ωmy+γ⟪ξξξtanh(βξξξ⋅mz)⟫−γmz, (S45)
γ
m˙ y =−2Ωmz− my. (S46)
2
The equations for the stationary state are then straightforwardly obtained setting m˙ α=0:
2Ωmy+γ⟪ξξξtanh(βξξξ⋅mz)⟫−γmz =0, (S47)
γ
2Ωmz+ my =0, (S48)
2
and thus:
8Ω2+γ2
mz =⟪ξξξtanh(βξξξ⋅mz)⟫. (S49)
γ2
This equation clearly shows that the role of the coherent driving Ω is qualitatively similar to that played by the
temperature in the classical case, i.e. at large enough Ω the non-trivial solutions of the equation disappear, as in the
classical case. This also shows that introducing a coherent driving does not change the mean field stationary phase
diagram of the Hopfield model, as already shown in [36]. Nevertheless, due to the coupled and non-linear nature of
theaboveequations,newoscillating phases,i.e.limitcycles,couldappear. Inthenextsectionthepossibleemergence
of such solutions is investigated.
To explicitly map out the properties of the model, a specific form for the disorder distribution must be fixed. Our
choice, which has already been considered in the literature [19], is:
P(ξξξ)= ∏p p(ξ(µ)), p(ξ)= 1δ(ξ−1)+ 1δ(ξ+1). (S50)
µ=1 2 2