Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve

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by  T. Jorg

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Energy gaps in quantum first-order mean-field-like transitions:
The problems that quantum annealing cannot solve
T. Jo¨rg1, F. Krzakala2,3, J. Kurchan4, A. C. Maggs2 and J. Pujos2
1 CNRS et ENS UMR 8549, 24 Rue Lhomond, 75231 Paris Cedex 05, France, LPTENS
0 2 CNRS; ESPCI ParisTech, 10 rue Vauquelin, UMR 7083 Gulliver, Paris, France 75005, PCT
1
3 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, NM 87545 USA
0
2 4 CNRS; ESPCI ParisTech, 10 rue Vauquelin, UMR 7636, Paris, France 75005, PMMH

n
a
J
PACS 05.30.-d–Quantum statistical mechanics
8 PACS 05.70.Fh–Phase transitions: general studies
2
Abstract. - We study first-order quantum phase transitions in models where the mean-field
]
h traitmentisexact,andtheexponentiallyfastclosureoftheenergygapwiththesystemsizeatthe
p transition. We consider exactly solvable ferromagnetic models, and show that they reduceto the
- Groverprobleminaparticularlimit. Wecomputethecoefficientintheexponentialclosureofthe
t
n gap using an instantonic approach, and discuss the(dire) consequences for quantum annealing.
a
u
q
[

2
Many important practical problems involve the mini- XORSAT [9]. As we shall see, this implies the failure of
v
5 mization of a function of discrete variables. Solving such quantumannealingforthehardestoptimizationproblems.
6 combinatorial problems by temperature annealing is a The reason why quantum annealing is not an efficient
8 classical strategy in computer science [1]: the idea is to
strategy for finding the ground state across a first-order
4
use thermal fluctuations to avoid trapping the system in
transition can be understood from a simple argument.
.
2 local minima, and thereby efficiently visit the whole con-
Quantum annealing could in principle be more efficient
1 figuration space. It has been proposed to extend this ap-
than thermal annealing for certain classes of problems:
9
proachtoquantumfluctuations[2];itisthusofinterestto
0 From the WKB approximation it is well known that a
askwhether annealingby tuning downthe amplitude ofa
: quantum particle tunnels rapidly through very high (in
v quantummechanicalkinetic operatorsuchasatransverse
energy) but thin (in distance) energy barriers. Thermal
i
X magnetic field Γ can outperform the classical approach. annealingismuchbetteratlow,butdeepbarriercrossing.
In particular, can problems that normally take exponen-
r However, in a first-order transition the two states whose
a tial time be solved in only polynomial time?
freeenergiescrossaregenerallyfar fromeachotherinthe
phase space; quantum tunneling must be inefficient.
Some considerable efforthas been devoted to this ques-
To make this argument more precise, and to compute
tioninthecontextofdifficultcombinatorialproblems(see
how slow an annealing should be such that the tunelling
for instance [3]) which have a counterpart in statistical
do happens, one can consider the Landeau-Zener theory
physics where they corresponds to mean-field spin-glass
of level crossings [2]. During an avoided crossing, the
models [4,5]. However, most of the studies were purely
time needed in order to actually reach the ground state
numerical and thus restricted to very small sizes due to
is bounded by the inverse of the energy gap ∆ between
the difficulty of simulating quantum mechanics without a
these states. If the total annealing is longer τ ∆−2,
quantumcomputer. InarecentLetter[6](seealso[7]),we
≫
then the adiabtic theorem imply that at each time step,
arguedthatwiththeusualimplementationofthequantum
the systems remains in the ground state. Otherwise, the
annealing it is likely that the most difficult systems un-
system misses the crossing and is not in the ground state
dergoa quantumtransitionof the first order as the trans-
at the end of the computation. A good estimates of the
versefieldistuned;thisisagenericfeatureformanyquan-
running time of the algoritmh is thus obtain by the mini-
tumspinglasses[8]. Morerecently,afirstordertransition
mal energy gap ∆ during the annealing process [2].
has indeed been indentified in the phase diagram of one min
ofthemoststudiedrandomoptimizationproblems,called We will see that mean-field first-order transitions have
p-1
T. J¨org, F. Krzakala,J. Kurchan, A. C. Maggs and J. Pujos
genericallyanexponentiallysmallgap∆ Ne−αN where lim 1 log eβN 1+eN(log2−β)
∝
N is the system size. This implies τ eN, that is to say: ≈ N→∞−βN
≫ (cid:16) (cid:16) (cid:17)(cid:17)
quantum annealing is an exponentially slow algorithmfor log2
= min(f ,f ) with f = 1 and f =
a mean-field system with a first-order transition1. P P F − P − β
The goalof this Letter is to illustrate these features via
One recognizes a first-order transition at β = log2 be-
a complete analytical and detailed numerical analysis for c
tween two phases that are always locally stable (no spin-
a family of models. We consider the ferromagnetic p-spin
odal): a ferromagnetic phase that consists of the classical
model, which reduces to a mean-field ferromagnet for the
configurationwhereallspins areupfor β >β andatriv-
case p = 2 and to the Grover problem when p . We c
→ ∞ ial paramagnetic phase at larger temperature.
show how to solve the thermodynamics of these models
The extreme quantum case: Γ= . When Γ is large
using standard tools of statistical physics. We perform
∞
the classical part of can be neglected; we then find, in
extensive numerical studies of the gap for the case of p
H
the σx basis, N independent classical spins in a field Γ:
finite and odd. By introducing an ansatz for the domi-
nant instantonic pathways, we show how to compute the
f = T log2 T log(coshΓ/T). (3)
QP
coefficient in the exponential decay of the gap. − −
Theentropydensityisgivenbythelogarithmofabinomial
The simplest quantum ferromagnet. – We con-
in [ ΓN,ΓN]: this is a perfect quantum paramagnet.
sider a Hamiltonian with N Pauli spins σ of the form −
The general case. For Γ = 0 and inverse tempera-
= +ΓV where is a function of the longitudinal
z z
vHaluesHσz of the spins.H is thus diagonal in the σz rep- ture β < log2 we saw that the classical model is just a
z
H model where (almost) all configurations have zero energy.
resentation. We focus on the ferromagnetic p-spin model:
Inthiscase,wethuscanignorethetwononzerolevelsand
1
= σz ...σz Γ σx (1) we expect the quantum paramagnetic free energy fQP to
H −Np−1 i1 ip − i be valid for all Γ. A simple perturbation computation
i1X,...,ip Xi
– given in the next section– shows that this is true in
Mp(σ~z)
= ΓMT(σ~x)= N mp(σ~z) ΓmT(σ~x) the low-temperature phase as well, when β > log2. The
− Np−1 − − −
system thus has two distinct phases, the first a quantum
where we have defined the longi(cid:2)tudinal magnetizati(cid:3)on paramagnetic and the second a ferromagnetic phase. A
M(σ~z)= σz and the transverse one MT(σ~x)= σx first-order transition occurs when the free energies cross
i i i i
and their magnetization by site m = M/N and mT = so that f = min(f ,f ). The phase diagram of the
P P QP F
MT/N. That sort of models were introduced initially in model is very simple: For low Γ and T, the free-energy
a spin-glass context in [4,10]. The ground state of the density is that of the classicalmodel in the ferromagnetic
classical problem, when Γ = 0, corresponds to all spins phase, while for largerΓ it jumps to the quantum param-
aligned in the same direction. Whereas both the up and agnetic free energy; a first-order transition separates the
down states are valid ground states for even p, the up twodifferentbehaviorsatthevalueΓsuchthatf =f ;
F QP
state is the unique ground state for odd p, and we will this happens on the line defined by
concentrate on this case for simplicity. The case p = 2 is
1 eβ
the usual Curie-Weiss model, where the transition is con-
Γ= arccosh (4)
tinuous [11,12]. For p > 2 however, both quantum and β 2
thermal transitions are discontinuous. Of special interest
where the magnetization jumps from 0 to 1 (see Fig. 1).
is the limit p where for p odd m( S~ )p 1 if
Thezero-temperaturebehaviorcanbeunderstoodfrom
→ ∞ { } → ±
m= 1, and zero otherwise. It leads to:
standard Rayleigh-Schr¨odinger perturbation theory [13].
±
Consider the set of eigenvalues E and eigenvectors k of
k
= N11 σz =N +Γ σx (2) the unperturbed model, when Γ = 0. The series fo|rithe
H − Xi ! Xi lowest perturbed eigenvalue Emin(Γ) reads
wherethefunction11(x)is1ifxistrueandzerootherwise. Γ2V V
min k k min
We now specialize to this p= limit. Emin(Γ)=Emin+ΓVii+ +... .
∞ Emin Ek
k6=min −
The p= limit. – X
(5)
∞
The classical case: Γ=0. Thep= modelistrivial Since V = 0 if and only if the two configuration i and j
ij
∞ 6
in the limit Γ 0 where there are only two levels with differ by a single spin flip, odd orders do not contribute
→
nonzeroenergiesE =N andE = N. Thepartitionsum in Eq. (5). Noting that V 2 reduces to a sum
− k6=n| nk|
is thus Z =2N 2+2coshβN so that over the N levels connected to E by a single spin flip,
i
− P
and using the fact that all E = 0 (except E = N),
1 k min
f = lim log 2coshβN +2N 2 succesivestermsareeasylycomputedandonefinds,−to all
N→∞−βN −
(finite) orders (see [6] for a similar computation):
(cid:0) (cid:1)
1Infinitedimensionsoneexpectsthatnucleationwillhelp. How-
ever,optimizationproblemsarenotfinitedimensionalgenerically. Emin(Γ)= N Γ2+o(1). (6)
− −
p-2
Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve
The expansion can also be performed using now ΓV as passing through unity. All interior roots to the function
a starting point and with as perturbation. Consider are thus bracketed by a comb of poles separated by 2Γ.
0
H
the eigenvalue NΓ. In the base N corresponding to In the small Γ phase this rigorously brackets almost all
− | i
the eigenvalues of ΓV 2, we obtain the eigenvalues near λ = 0. The exception is the lowest
eigenvaluewhichcansplitofffromthe comb,asignofthe
k N 2
E(Γ)= NΓ+ N N + |h |H0 | i| +... . phase transition in the large N limit.
0
− h |H | i NΓ Ek Intheparamagneticphase,thelowesteigenvalueisvery
k6=n − −
X
(7) close to λ= ΓN. In this case ΓN λ is very small so
− − −
Denoting a(l) the elements of the vector N in the z ba- thatwecanwriteλ= ΓN+η. Inadditiontheoverwelm-
sis, the first-order term in this expansion|reaids Na2(1). ing majority of eigenva−lues ǫi are close to zero 3; Eq. (14)
−
Since the a(l) are of order 2−N/2 the first-order shift is then implies, at the transition when Γ=1
tiny. The next term involves a sum over the 2N 1 levels
− N 1 2N 1 N 1 2N 1
1 = + − = − ,
a(1)k(1)2 2−N k(1)2 2N N λ λ 2N −η − η N
| | = | | . (8) (cid:20)− − − (cid:21) (cid:20) − (cid:21)
ΓN Ek ΓN Ek so that finally η2 =N2/2N at the critical point and
k6=min− − k6=min− −
X X
The lastsumis entropicallydominatedby the states with ∆min =2N2−N/2. (15)
E = 0 and therefore gives a negligible contribution (as
k The gap closes exponentially fast at the transition. We
one can check term by term). Subsequent terms are
have an extremely simple model with a first-order mean-
treated similarly. This yields the ground-state energy:
field transition and most of the physics discussed in this
E = N Γ2+o(1) for Γ<Γ (9) Letterisalreadypresentinthis model: difficultproblems,
GS c
− − suchasthisonewhereonlyonein2N configurationshasa
E = ΓN +o(1) for Γ>Γ (10)
GS − c lowenergy,manifestthemselvesbyafirst-ordertransition
with Γc =1+O(1/N). (11) in the quantum annealing path, and consequently by an
exponentially small gap.
Exponential closure of the gap. Near the transition
The reader could at this point argue that we have not
thetreatmentmustberefined: Thereisan(avoided)level
shown that all choices of the quantum interaction lead to
crossing at Γ =1 in the large N limit between the para-
c this result; perhaps a more intelligent choice would turn
magnetic and the ferromagnetic ground state. We now
the transition to second order, and make the gap polyno-
compute the behaviorofthe quantumgaparoundΓ =1.
c mial? We know that for this precise model, this is just
We write the Hamiltonian in the σx basis:
impossible. In fact, this model is nothing else than the
Grover problem [14], that is: searching for a minimum
=Γε δ +E a a (12)
ij i i,j c i j
H value in an unsorted database. The best algorithm is
where ~a is the state corresponding to all spins aligned known, and it is an exponential one [14]. It is obtained
in the z direction expressed in the x basis. E = N by adjusting the evolution rate of the Hamiltonian in the
c
−
and ε s are the (binomially distributed) energies due to quantumannealingprocesssoastokeeptheevolutionadi-
i
the quantum interaction. With an appropriate conven- abaticoneachinfinitesimaltimeinterval. Indoingso,the
tion for the eigenvectors we can take for a the vector total running time can be τ ∆−1 [15], which is still ex-
∝
2−N/2(1,1,1,....1). In this basis, on multiplying with an ponential. There is thus no way to avoid the exponential
eigenvector~v of eigenvalue λ, we find gap in this situation.
ai Behavior for general p. – We now consider finite
0 = (Γε λ)v +E a (~a.~v)=v +E (~a.~v)
i− i c i i cΓε λ value of p and begin by calculating the phase diagram in
i
−
the static approximation. We then consider closureof the
Multiplying again by~a, we find
gap using numerical diagonalization and an instantonic
a2(~a.~v) calculation which we then compare.
(~a.~v)+E i =0 (13)
c Γε λ Phase diagram. We shall first use the Suzuki-Trotter
i
Xi − formula in order to map onto a classical model with an
so that additional “time” dimension:
N 1
2N Γεi λ =1. (14) Z= e−βHz+βΓ iσix =limTr{~σ}e−NβsHzeNβsΓ iσix Ns
Xi − Ns→∞
The qualitative behavior of the eigenvalues can now be X{~σ}(cid:16) P (cid:17) h P i
understoodgraphically: Betweeneachpole in the denom- = lim ~σ e− Nα=s1 NβsHz(α)e Nα=s1NβsΓ iσix(α) ~σ .
inator of Eq. (14) the function passes from −∞ to +∞ Ns→∞X{~σ}h | P P P | i
2Note that inthe σz basis the ground-state vector |Ni has ele- 3Systematiccorrectionstothisapproximationdonotchangethe
ments±2N/2. result.
p-3
T. J¨org, F. Krzakala,J. Kurchan, A. C. Maggs and J. Pujos
We then introduce N closure relations 11= {~σ}|~σih~σ|: 1.8 p=Infinity
p=3
Ns P 1.6 p=5
Z = σ(~α)e−NβsHz(α)eNβsΓ iσix(α) σ(α~+1) 1.4 Paramagnet p=p=159
h | | i p=25
{σX(~α)}αY=1 P 1.2
= Ns e−NβsHz(α) Ns σ(~α)eNβsΓ iσix(α) σ(α~+1) T 1
h | | i 0.8
{σX(~α)}αY=1 αY=1 P
0.6
~ ~ Ferromagnet
with the convention that σ(Ns+1) = σ(1). Applying 0.4
N times the integral representation of the delta function
s
dmδ(Nm M( S~ ))f(Nm)=f(M( S~ )), one finds: 0.2
− { } { }
0
R Ns Ns βN Ns 0.2 0.4 0.6 Γ 0.8 1 1.2 1.4
Z = dm(α) dλ(α)exp m(α)p
Z αY=1 αY=1 Ns αX=1 !× Fig.1: Phasediagramoftheferromagneticp-spinferromagnet
N Ns Ns β Γσx(α)+λ(α)σz(α) for different values of p. A first-order transition separates the
exp − λ(α)m(α)+NlogTr e Ns Ns . ferromagnetic and quantumparamagnetic phases.
N
" s #
α=1 α=1 (cid:2) (cid:3)
X Y
The saddle point condition imposes that λ(α) =
are N levels with an energy gap ∆E = 2 Γ2+p2m2p−2
βpmp−1(α). Writing t=βα/N and performing the limit
s where ∆E is discontinuous at the transition.
N we obtain: p
s →∞ This is, however, only a crude description of the phe-
Z= m(t)eN 0βdt(1−p)mp(t)+NlogTre 0βdtΓσx(t)+pmp−1(t)σz(t). nleovmelenisolcologsyinogf atthethleowt-rlayninsigtiosnta,ttehs.enIfweinedxepedectontlhyeoenne-
ZD R R (16) ergy to behave as E = EGS +∆Ee−β∆E, and therefore
oneneedstocomputetheO(1)correctiontotheenergyin
We now use the “static” approximation, which we also
order to take this into account. The former computation
check numerically [11,12], and remove all “time” indices
thus misses this behavior and indeed, numerical results
for m to finally obtain:
show that the first excited state is unique. Worse, we
expect the energy gap between the ground state and the
Z = dme−βNf(β,Γ,m) (17)
excited state to close exponentially fast at the transition,
Z 1 and therefore, in order to be able to investigate this be-
f(β,Γ,m) =(p 1)mp log2cosh β Γ2+p2m2p−2 .havior, we should be looking for an exponentially small
− −β
(cid:16) p (cid:17) gap: in that case we thus need to look for exponentially
All thermodynamic quantities can now be computed. For small correction to the free energy! Fortunatly, there is a
instance, the self-consistent equation for the magnetiza- waytodealwiththisproblem: wenowturntoanumerical
tion m reads (for p>2) study of the gap and to the instantonic approach.
Closure of the gap. –
tanh β Γ2+p2m2p−2
m= pmp−1. (18) Numerical methods. We use two complementary
 (cid:16)Γp2+p2m2p−2 (cid:17)
methods to study the spectrum of the p-spin model for
 p  3 p 31. ThefullmatrixrepresentationoftheHamilto-
Itiseasytocheckthattheformerexpressionleadstofirst- ≤ ≤
nianisasparseoperatorofdimension2N. Forsuchsparse
order(quantumandclassical)transitionswhenitsminima
matrices Laczos methods are particularly useful and can
cross. In particular, the free energy for p is simply
→∞ beusedtoextractextremaleigenvaluesfromthespectrum
f= 1 for m=1 and f = 1 log2cosh(βΓ) otherwise, as
− −β forN 21. WenoteinparticularthatthatforN 21the
weobtainedinthefirstsection. Thephasediagramofthe ≤ ≤
transition occurs between two states with the maximum
model is plotted in Fig. 1.
possible angular momentum l=N/2.
The energy is given by e= ∂ βf, and thus at low T:
∂β Considerableimprovementsinefficiencyareobtainedby
realazingthat the total angular momentum L2 commutes
e(β) eGS+2 Γ2+p2m2p−2e−2β√Γ2+p2m2p−2 with . Thus the transition occurs in a subspace of di-
≈ H
with eGS(Γ,m)p=(p 1)mp Γ2+p2m2p−2. mension2l+1=N+1. Inthis subspacethe Hamiltonian
− − has diagonalelements correspondingto different values of
In the low-temperature T, the energpy of a system with Lz. Standardmethodsfromthetheoryofangularmomen-
N excited states with an energy gap ∆E is E = E + tum show that the off-diagonal elements of the matrix in
GS
N∆Ee−β∆E, andthis computation thus shows that there this subspaceareonlythoselabeledby (m ,m 1). The
z z
±
p-4
Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve
6 1
0.01
5 p=3
0.0001
4
1e-06
N
∆min 3 /min
∆ 1e-08
p=3
2 N=10 p=5
N=20 1e-10 p=7
N=30 p=11
1 N=60 p=15
N=90 1e-12 p=21
N=120 p=31
N=150 2-N/2
0 1e-14
0 0.5 1 1.5 2 0 20 40 60 80 100 120 140
Γ N
Fig. 2: Numerical computation of the gap versus Γ for p = 3 Fig. 3: Minimum Gap versus N from exact diagonalization
computed using the method described in the text. Very close of the ferromagnetic p-spin model for some values of p on a
tothetransitionatΓ (theblackverticalline),inaregionthat linear-logscale. Oneclearly seesthatforeachpthegap closes
c
shrinksas N increases, thegap is closing exponentially fast. exponentially with N, so that ∆ ∝N2−Nα.
min
matrix is symmetric with off-diagonalelements at the transition, one has ǫ = e−Nβ(finst−fferro) = e−NG
with G = O(1) in the zero-temperature limit. Since we
H =Γ l(l+1) m (m +1) (19)
mz,mz+1 − z z are summing overall periodic paths, one should now take
The resulting tri-diagonpalmatrix an be treated with very into accountall suchinstantonic paths that jump aneven
number of times to compute the correction to Eq. (18).
high efficiency allowing one to study systems of N 100
∼ Each of these jumps can occur at any time t [0,β] and
in just a few seconds. The limiting factor in the study of
∈
thesaddle-pointcomputationthusreads,atthetransition:
even larger systems is the reduction of the gap to double
precisionmachineaccuracysothatfloatingpointround-off β2 β4 β6
errors dominate the results. Fig. 3 shows the dependence Z = 2e−βF +2e−βF ǫ2+ ǫ4+ +...
2 4! 6!
of the minimum gapfor some values of p. We see that for (cid:18) (cid:19)
βk
all p 3 the gap closes exponentially in N. = 2 e−βFεk, (20)
Fig≥.2 showsthe dependence ofthe gap∆as a function k!
k even
X
function of Γ for p = 3 and different N. ∆ indeed closes
wherethefactorβk/k!comesfromthecountingofallpos-
fast at the transition that arises exactly at the critical
sible paths with k jumps. One then recognizes the series
value predicted analytically. The region where the gap
expansion of an effective two level system:
closes is getting narrow as N increases, and one has to
be very careful in scanning Γ in order not to miss it: this
F ǫ
is an importantmessagefor future numericalsimulations. Z =Tr e−βHeff, with Heff = ǫ F . (21)
Fig. 3 shows the dependence of the minimum value of the (cid:18) (cid:19)
gap ∆ as a function of N for some values of p. For all Diagonalizingthe effective Hamiltonian at T =0 one sees
min
p 3, the gap decays exponentially as ∆ N2−Nα. immediately that the gap goes as ∆ ǫ = e−NG: the
≥ min ∝ ∝
The different values of α are given in Table 1. As we energycostoftheinstantonthusprovidestheexponentof
expected, the gap closes exponentially fast at the first- the gap at the transition.
order transition point. We want now to show how the Computing the Instanton. We can consider various
coefficient in the exponent can be computed analytically. ansa¨tzetocomputetheoptimalinstanton,allofthemgiv-
The Instantonic approach. It is well known that the ing lower bounds on the coefficient. The simplest one is
tunnelingbetweenquantumstatescanbecomputedusing justasharpwallwhenm(t)jumpsabruptlyfromthevalue
an instantonic approach [16]. Let us briefly explain how mQ to mF. The gap thus reads in this approximation:4
this canbe understoodviacorrectionsto the saddle-point
∆= F Q N =eNloghF|Qi (24)
computation. At the transition, two solutions (the ferro- h | i
magnetic one m=meq and the paramagnetic one m=0) 4ThiscanbeseeninthediscreteSuzuki-Trotterformalismwhere
havethesamefreeenergiesfm =fQP. Letusassumenow Ns
thatweareabletofindanothertime-dependentpathm(t) Z= hσ(~α)|eNβs(Γσx(α)+hσz)|σ(α~+1)i. (22)
—whichweshallcallinstantonic—thatspendssometime {σX(~α)}αY=1
τ in the ferromagneticstate and then jumps to the para-
1 Eachtermbutonecanbewritteninitsrespectivediagonalbase(1)
magneticstatewhereitspendsatimeτ2,andthatexactly or(2)andbecomputedwiththestaticapproach. However,thereis
p-5
T. J¨org, F. Krzakala,J. Kurchan, A. C. Maggs and J. Pujos
p Γ m αGap αGap αGap thispaperwehaveshownthateveninsystemswithtrivial
c c sharp tanh simu
energy landscapes quantum annealing can fail (and there
3 1.2991 0.8660 0.2075 0.1251 0.126(3) is thus no need for more complex phenomena to explain
5 1.1347 0.9682 0.3390 0.2686 0.270(3) this failure, as for instance in [17]). Already the p = 3
7 1.0874 0.9860 0.3888 0.3335 0.335(3) ferromagnet exhibits a first-order phase transition with
9 1.0647 0.9921 0.4150 0.3699 0.370(3) an exponentially closing gap: A scenario which is very
11 1.0514 0.9959 0.4318 0.3929 0.395(3) pessimistic for the success of the quantum annealing al-
13 1.0426 0.9965 0.4422 0.4105 0.410(3) gorithm. We have also shown that the p = limit of
∞
15 1.0364 0.9974 0.4502 0.4224 0.421(3) theferromagneticmodelisrelatedtotheGroverproblem.
17 1.0318 0.9980 0.4564 0.4315 0.431(3) This is a clear indication that these first-order transition
19 1.0282 0.9985 0.4620 0.4387 0.439(3) carry the signature of the most difficult problems.
21 1.0253 0.9987 0.4648 0.4445 0.445(3) Models presented in this Letter allow a complete ana-
23 1.0230 0.9990 0.4679 0.4493 0.450(3) lytical and numerical treatment. Their disordered coun-
25 1.0211 0.9991 0.4705 0.4534 0.454(3) terpartcan be studied using the generalizedinstanton in-
31 1.0168 0.9994 0.4763 0.4623 0.462(3) troduced in [6,7]. It would be interesting to extend this
... 1+ 1 1 1 1 log2 ... 1 1.15... approach to dilute mean-field system and random opti-
2p − 2p2 2 − p 2 − p
1 1 1 1 1 mization problems, using the quantum cavity of [9,12].
∞ 2 2 2
Table 1: First-order transition in the p-spin ferromagnet at
∗∗∗
zerotemperature: ThecriticalvaluesforthefieldΓ andmag-
c
netization mc are given. The gap at the transition decays ex- WethankP.Boniface,S.Franz,A.Rosso,G.Semerjian,
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results from exact diagonalization αGap , the estimates with
simu
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aremainingtermoftheform
hσ~1|eNβs(Γσx(α)+hσz)|σ~2i=hσ~1|σ~2ihσ~1|eNβs(Γσx(α)+hσz)|σ~1i (23)
p-6