Programmable adiabatic demagnetization:
State preparation by simulated cooling
Erez Berg

Anne Matthies

Achim RoschMark Rudner

University of Cologne

University of Cologne

University of Washington

arXiv:2210.17256

Simulating Physics on Computers
N-particle QM
𝑖ℏ𝜕! 𝜓 𝒓" , … , 𝒓# , 𝑡
ℏ$ $
ℏ$ $
= −
∇" − ⋯ −∇# + 𝑉 𝒓" , … , 𝒓#2𝑚2𝑚

𝜓 𝒓" , … , 𝒓# , 𝑡“Curse of dimensionality”!!"#$! =𝐻$#,&'(,),*&'(,),*𝜎#&,,ℎ#& 𝜎#& + $ 𝐽#+ 𝜎#& 𝜎+ + ⋯#,+,&,,: Pauli matrices

Discrete models: digital computer friendlyOutline• How to find ground stateson quantum simulators?

• Simulated adiabatic
demagnetization protocol• Cooling trivial andtopological excitations

Ground States on Quantum Computers
Q: How to prepare the ground state of a given
Hamiltonian on a quantum simulator?
Noisy Intermediate-Scale Quantum (NISQ) era Preskill, 2018𝑡𝑡
𝐻 𝑡 = 1 − 𝐻. + 𝐻/𝜏𝜏• Adiabatic Preparation
E

𝐻. : “Simple” Hamiltoniane.g. 𝑯𝟏 = −𝒉 ∑𝒋 𝝈𝒛𝒋

𝐻/ : Hamiltonian of interest

÷

÷

.

• Variational Quantum Eigensolvers: prepare a wavefunction|𝜓 𝜆! , … , 𝜆"- ⟩, minimize ⟨𝜓|𝐻|𝜓⟩ over 𝜆! , … , 𝜆"Variational Quantum Eigensolver review: Tilly et al. (2021)

Adiabatic Demagnetization

cryodynamics.de
Decrease B(t) adiabatically ⟹ decrease Treset to the polarized state, and the process is repeated until convergence to a low-energy steady state iseved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol’s
ormance in the presence of noise and show how the information from the measurement of the bath spins
be used to monitor the cooling process. The performance of the algorithm depends on the nature of theations of the system; systems with non-local (topological) excitations are more difficult to cool than those
local excitations. We explore the possible mitigation of this problem by trapping topological excitations.

Simulated Adiabatic Coolingreparation on quantum simulators and comportant for the characterization of ground
quantum chemistry and material science [1–y, it can also be utilized for a variety of quanroblems [5–7]. On a quantum computer, thebe digitally implemented using a sequence
Many approaches of ground state preparationed, including variational quantum simulation
tic state preparation [13–16]. Each of theses own challenges. For example, the perfornal quantum simulation highly depends onvariational ansatz [12, 17]. At the sameate preparation is sensitive to the trajectory
pace connecting the initial and final Hamilo fail when a phase transition separates theates. Furthermore, errors that occur during
eparation are not intrinsically corrected ando detect.

Hamiltonian 𝐻0UVJ

bvbi2K
g

J
g

J
g

J
g

J
g

J
g

g

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g

g

#i?

FIG. 1.
Schematic illustration of the proposed setup. (a) Eachsystem spin si (light blue) is coupled to a bath spin i (dark blue)by switchingon theRosch,couplingEB,g(t).
System spins may be generically
ithms that mimic coolingby couplingMark
to a Rudner,Anne Matthies,
Achim
arXiv:2210.17256
coupled, as illustrated here by J. At the beginning of the protocol,
ropy bath [18–24] haveseebeenalsoproposed
S. Pollaasetal-al. (2021)
the bath is polarized and subjected to a large simulated magnetic fieldreset to the polarized state, and the process is repeated until convergence to a low-energy steady state iseved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol’s
ormance in the presence of noise and show how the information from the measurement of the bath spins
be used to monitor the cooling process. The performance of the algorithm depends on the nature of theations of the system; systems with non-local (topological) excitations are more difficult to cool than those
local excitations. We explore the possible mitigation of this problem by trapping topological excitations.

Simulated Adiabatic Cooling𝐻 𝑡 = 𝐻0 + 𝐻1 (𝑡) + 𝐻2 (𝑡)reparation on quantum simulators and comportant for the characterization of ground
*
𝐻
𝑡
=
−𝐵(𝑡)
$
𝜎
+ [1–
quantum 1
chemistry and material sciencey, it can also be utilized for a variety+ of quanroblems [5–7]. On a quantum computer, thebe digitally implemented using a sequence
Many approaches of ground state preparation
) )
𝐻
𝑡
=
−𝑔(𝑡)
$
𝜎
ed, including
2 variational quantum simulation+ 𝑠+
tic state preparation [13–16]. Eachof these+s own challenges. For example, the perfornal quantum
simulation
highly depends on
Initialstate:
variational ansatz [12, 17]. At the sameate preparation is sensitive to the trajectory𝜓 0 the=initial
𝜓 and
⋯↑1
pace connecting
final↑ Hamil0⊗o fail when a phase transition separates theates. Furthermore, errors that occur during
eparation are not intrinsically corrected ando detect.

UVJ

bvbi2K
g

J
g

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g

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g

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g

J
g

g

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g

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#i?
(b)

(c)

systemsystemcooling cycle
bathinpolarized
state

bath

measurement& resetFIG. 1.
Schematic illustration of the proposed setup. (a) Eachsystem spin si (light blue) is coupled to a bath spin i (dark blue)by switchingon theRosch,couplingEB,g(t).
System spins may be generically
ithms that mimic coolingby couplingMark
to a Rudner,Anne Matthies,
Achim
arXiv:2210.17256
coupled, as illustrated here by J. At the beginning of the protocol,
ropy bath [18–24] haveseebeenalsoproposed
S. Pollaasetal-al. (2021)
the bath is polarized and subjected to a large simulated magnetic fieldreset to the polarized state, and the process is repeated until convergence to a low-energy steady state iseved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol’s
ormance in the presence of noise and show how the information from the measurement of the bath spins
be used to monitor the cooling process. The performance of the algorithm depends on the nature of theations of the system; systems with non-local (topological) excitations are more difficult to cool than those
local excitations. We explore the possible mitigation of this problem by trapping topological excitations.

Simulated Adiabatic Cooling𝐻 𝑡 = 𝐻0 + 𝐻1 (𝑡) + 𝐻2 (𝑡)reparation on quantum simulators and comportant for the characterization of ground
*
𝐻
𝑡
=
−𝐵(𝑡)
$
𝜎
+ [1–
quantum 1
chemistry and material sciencey, it can also be utilized for a variety+ of quanroblems [5–7]. On a quantum computer, thebe digitally implemented using a sequence
Many approaches of ground state preparation
) )
𝐻
𝑡
=
−𝑔(𝑡)
$
𝜎
ed, including
2 variational quantum simulation+ 𝑠+
tic state preparation [13–16]. Eachof these+s own challenges. For example, the perfornal quantum
simulation
highly depends on
Initialstate:
variational ansatz [12, 17]. At the sameate preparation is sensitive to the trajectory𝜓 0 the=initial
𝜓 and
⋯↑1
pace connecting
final↑ Hamil0⊗o fail when a phase transition separates theates. Furthermore, errors that occur during
eparation are not intrinsically corrected ando detect.

UVJ

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g

J
g

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g

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g

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g

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g

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(b)

(c)

systemsystemcooling cycle
bath

bath

Time evolve:inpolarized
statemeasurement& resetFIG. 1.
Schematic illustration of the proposed setup. (a) Eachsystem spin si (light blue) is coupled to a bath spin i (dark blue)by switchingon theRosch,couplingEB,g(t).
System spins may be generically
ithms that mimic coolingby couplingMark
to a Rudner,Anne Matthies,
Achim
arXiv:2210.17256
coupled, as illustrated here by J. At the beginning of the protocol,
ropy bath [18–24] haveseebeenalsoproposed
S. Pollaasetal-al. (2021)
the bath is polarized and subjected to a large simulated magnetic fieldSimulated Adiabatic Cooling
Why does this cycle cool?
Entropy conserved
𝑺𝑺 (𝒕) + 𝑺𝑩 𝒕 = 𝒄𝒐𝒏𝒔𝒕

Adiabatic
demagnetization:
As 𝐵(𝑡) decreases, theentropy of the bathspins increases
Microsopic picture:
Imagine system with3 states, single bath spin
No system-bathcoupling (𝒈 = 𝟎)

Entropy of the systemdecreases⇒""/ 2 d)

i.if

,

11 d)

12 ) •,

13,1^3/ 2. T )-•B

11,1^7

i.if

Simulated Adiabatic Cooling 13,1^3

11 d)12 ) •,Why does this cycle cool?
Adiabatic
demagnetization:
As 𝐵(𝑡) decreases, theentropy of the bathspins increases
Microsopic picture:
Imagine system with3 states, single bath spin𝒈≠𝟎
Steady state (many cycles):𝝆 → |𝝍𝟎 ⟩⟨𝝍𝟎 |/ 2. T )Entropy conserved
𝑺𝑺 (𝒕) + 𝑺𝑩 𝒕 = 𝒄𝒐𝒏𝒔𝒕

-

•

⇒

B

11,1^7

Entropy of the systemdecreases

÷É:÷

/ 2 d),

B

11,1^7

used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field
acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured
and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state isachieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol’s
performance in the presence of noise and show how the information from the measurement of the bath spins
can be used to monitor the cooling process. The performance of the algorithm depends on the nature of theexcitations of the system; systems with non-local (topological) excitations are more difficult to cool than thosewith local excitations. We explore the possible mitigation of this problem by trapping topological excitations.

UV
ion on quantum simulators and comJ
J
J
J
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g
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Why
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n also be utilized for a variety of quanms [5–7]. OnGrounda quantum
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itally implemented
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state properties in quantum chemistry and material science [1–generally, it can also be utilized for a variety of quan(b)(c)
pproaches 4].
ofMoregroundstate preparation
tum information problems [5–7]. On a quantum computer, theHamiltonian can be digitally implemented using a sequence
uding variational
quantum simulationof unitary gates. Many approaches of ground state preparation
have been proposed, including variational quantum simulation
te preparation[13–16].
Each
of
these
Perturbation
theoryin 𝒈(𝒕)[8–12] and adiabatic
state preparation[13–16]. Each of thesesyst
approaches has its own challenges. For example, the perforchallenges.
For example, the perfor(Assuming:mance of variational quantum simulation highly depends on
the quality of the variational ansatz [12, 17]. At the same
system,depends onantumtheormalizing
simulation
highlytime, adiabatic state preparation is sensitive to the trajectorybain Hamiltonian space connecting the initial and final Hamiltime
evolution)
tional slow
ansatz
[12,
17].
At
the
same
tonians. It tends to fail when a phase transition separates theand final states. Furthermore, errors that occur during
paration isinitial
sensitive
to the trajectory
adiabatic state preparation are not intrinsically corrected andin
FIG. 1.
Schematic illustration of the proposed setup. (a) Eachmay be difficult to detect.
system spin s (light blue) is coupled to a bath spin(dark blue)
onnecting the initial and final Hamilpolarby switching on the coupling g(t). System spins may be generically
Recently, algorithms that mimic cooling by coupling to aas illustrated here by J. At the beginning of the protocol, sta
simulated
low-entropy bath separates[18–24] have been proposed
when a phasetransition
the as al- coupled,
the
bath
is
polarized and subjected to a large simulated magnetic fieldternative routes that may overcome some ofthese challenges.
''
$
̇
B(t),
while
in a random state. Panel (b) shoes
𝐻% that≈such−4𝜋𝑔
𝑡 during∫ 𝑑𝜔𝑛&they𝜔 𝜔𝜒% 𝜔the 𝛿system𝜔may−begin2𝐵(𝑡)The keyadvantagesofcooling algorithms
are that
Furthermore,errorsoccur
the time dependence of the parameters, and panel (c) illustrates thecan be run cyclically without any special requirements oncooling cycle. A magnetic field B(t) is applied to the bath spins, andstates and do not requireprior knowledgeof the taron are not initialintrinsicallycorrected
andlinearly decreased from B to B until time t = 3T /4, after which
get ground state. Simulated cooling schemes also do not re- FIG.
Schematic
illustrationof th
it is held1.
constant until
time T , the total durationof the sweep. The
quire the target state to be adiabaticallyconnected
to
a
prod'
'
t.
system-bath coupling g(t) is slowly switch on until t = T /4, kept
𝜒 %%provides
𝑡 =inherent* 𝑖𝜃(𝑡)⟨[𝑠
𝑡 , 𝑠 0 ]⟩uct state. The cyclic operation
robustnessUVJ

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#i?
(b)

(c)

systemsystemcooling cycle

bath

bath

in
polarized
statemeasurement& reset

i

i

0

f

2

system spin s (light blue) is coupled
to weak noise by automatically correcting some errors and reof each sweep, the bath spins are measured and reset.
&
unwanted excitations
on subsequent cycles.
that mimicmoving
coolingby couplingtoaFurther- by switching on the coupling g(t). Sys$

more, in simulated cooling on a quantum computer, monitor-

1& constant& at g(t) = g0 until
i time t2 , and then switched off. At the end

Ground state preparation on quantum simulators and computers is very important for the characterization of groundstate properties in quantum chemistry and material science [1–4]. More generally, it can also be utilized for a variety of quantum information problems [5–7]. On a quantum computer, theHamiltonian can be digitally implemented using a sequenceof unitary gates. Many approaches of ground state preparation
have been proposed, including variational
quantum simulation𝑺[8–12] and adiabatic state preparation [13–16]. Each of theseapproaches has its own challenges. For example, the performance of variational quantum simulation highly depends on
the quality of the variational ansatz [12, 17]. At the sametime, adiabatic state preparation is sensitive to the trajectoryin Hamiltonian space connecting the initial and final Hamiltonians. It tends to fail when a phase transition separates theinitial and final states. Furthermore, errors that occur during
adiabatic state preparation are not intrinsically corrected andmay be difficult to detect.

UVJ

bvbi2KJJJJSimulated Adiabatic Cooling
g

g

g

g

(b)

cooling cycle

0Here, we define a simple, scalable protocol for low-energy
state preparation of an arbitrary gapped Hamiltonian on aquantum computer or programmable quantum simulator. We
draw an analogy between our protocol and adiabatic demag-inpolarized
statemeasurement& reset| FIG.
↑↑↑↓↑↑⋯⟩
1.
Schematic illustration of the proposed setup. (a) Each(b)system spin si (light blue) is coupled to a bath spin i (dark blue)by switching on the coupling g(t). System spins may be genericallycoupled, as illustrated here by J. At the beginning of the protocol,
the bath is polarized and subjected to a large simulated magnetic field
B(t), while the system may begin in a random state. Panel (b) shoes
the time dependence of the parameters, and panel (c) illustrates thecooling cycle. A magnetic field B(t) is applied to the bath spins, andlinearly decreased from B0 to Bf until time t2 = 3T /4, after which
it is held constant until time T , the total duration of the sweep. The
system-bath coupling g(t) is slowly switch on until t1 = T /4, kept
constant at g(t) = g0 until time t2 , and then switched off. At the endof each sweep, the bath spins are measured and reset.

fidelity

fidelity

E0

0
6
3

E

cted via resoenergy splitystem. Thus,
h-energy andm the system.
dt c , can be
integral overextracted en-

0.5

bath

bath

1

0.5

E0

1

Recently, algorithms that mimic cooling by coupling to asimulated low-entropy bath [18–24] have been proposed as alternative routes that may overcome some of these challenges.
The key advantages of such cooling algorithms are that they
can be run cyclically without any special requirements on
initial states and do not require prior knowledge of the target ground state. Simulated cooling schemes also do not require the target state to be adiabatically connected to a product state. The cyclic operation provides inherent robustness
to weak noise by automatically correcting some errors and removing unwanted excitations on subsequent cycles. Furthermore, in simulated cooling on a quantum computer, monitoring the bath spins during the cooling protocol can provide information on the process of approaching the system’s groundstate without observing the system directly and may be used
to identify the presence of errors.

gsystemsystem0
6
3

netization [25, 26], known from solid-state systems. For example, with nuclear adiabatic demagnetization, one can reachtemperatures in the µK range [27, 28] by coupling the system
to nuclear moments polarized in a large magnetic field andadiabatically ramping down the field.
Fig. 1 illustrates the cooling scheme. We consider a genericsystem comprised of N qubits (light blue), coupled to an additional set of Nbath “bath” qubits (dark blue). We take the

25 50 75
# of cycles

EBoth
at the effec-

(a)

g

(c)

Measured bath state:| ↑↑↑↑↑↑ ⋯ ⟩

g

g

J

#i?

𝑯 : Quantum Ising modelsi (0)]i.

g

J

100

0

25 50
# of cy

NoiseRandom errors (rate 𝜼𝒆 )SystemHeat (rate C)
Bath!0 = Tr 𝜌H𝐻!0 > 𝐸6
Steady state 𝜌H with 𝐸 = 𝐻<(. 8=Energy density of steady state: =
8:
; 8:
89

= −𝐶 𝑒 − 𝑒6 + 𝜂9 = 0𝑒 − 𝑒* ∝ 𝜂+

“Thermalizing” Hamiltonain:B< ( /A?=𝜌 → 𝜌9> ∝ 𝑒
for 𝐶, 𝜂9 → 0 with fixedC)NoiseDepolarizing noise rate 𝜼𝒆 = 𝟐 ⋅ 𝟏𝟎?𝟐

1

Measured bath state:| ↑↑↑↓↑↑ ⋯ ⟩| ↑↑↑↑↑↑ ⋯ ⟩
(b)

0.5

E0

fidelity

3

0
6

E

3
25 50 75
# of cycles

100

0
25 50 75
# of cycles

100

Improve energy by post selection:
Measuresystembath measurement
is | ↑↑↑↑↑↑ ⋯ ⟩ngle run of the
protocolforonly
thewhennon-integrable(J =outcome1,

z = 0.2, Ns = 8, N⌧ = 101, T = 6, Bi = 5, Bf = 0.7,

Noise0.5
less than 1 excitation

0
0

(a)

(b)

1

2

# of errors, ⌘eless than 1 excitation

0
0

0.02

# of errors, ⌘eenergy density

1(a)energy density

energy density

Ferromagnet
Paramagnet
0.15

0
4

6

syste

FIG. 3. (E E0 )/E0 as a function of the average number of errors
FIG. 4. Energy d𝜂:averagenumberofdepolarizing
errors(phase/bit
flip)+⌘e : (a) The ferromagnetic case (J = 1, hx = 0.5, hz = 0) is shownfor (a) the paramagn
per
spin
per
cycle
in red squares, the paramagnetic case (J = 0.5, hx = 1, hz = 0) inan average number
blue dots. Within the grey shaded area, the energy is below the lowest
per sweep and spin
excitation gap . (b) For small noise levels, post-selection (empty
erage energy is belo
red squares: FM, empty blue circles: PM) improves the outcome ofthe energy density gin the ferromagnetic
the protocol, i.e. it lowers the energy of the system. See text a thedescription of the post-selection protocol. (Parameters: Ns = 8,
linearly with system
N = 101, N = 100, N= 1000, T = 6, B = 5, B = 0.7,
size increases with

NoiseFM w/post selection
PM w/post selection

(a)

0.5
less than 1 excitation

0
0

(a)

(b)

1

2

# of errors, ⌘eless than 1 excitation

0
0

0.02

# of errors, ⌘eenergy density

1⋅⋅

energy density

energy density

Ferromagnet
Paramagnet

0.15

0
4

6

syste

FIG. 3. (E E0 )/E0 as a function of the average number of errors
FIG. 4. Energy d𝜂:averagenumberofdepolarizing
errors(phase/bit
flip)+⌘e : (a) The ferromagnetic case (J = 1, hx = 0.5, hz = 0) is shownfor (a) the paramagn
FM
is
more
per
difficult
spin
per
cycle
to
cool
in red squares, the paramagnetic case (J = 0.5, hx = 1, hz = 0) inan average number
blue dots. Within
the grey shaded area,
the energy is below
the lowest
per sweep and spin
- Post-selection
decreases
energyexcitation gap . (b) For small noise levels, post-selection (empty
erage energy is belo
red squares: FM, empty blue circles: PM) improves the outcome ofthe energy density gin the ferromagnetic
the protocol, i.e. it lowers the energy of the system. See text a thedescription of the post-selection protocol. (Parameters: Ns = 8,
linearly with system
N = 101, N = 100, N= 1000, T = 6, B = 5, B = 0.7,
size increases with

Topological vs. Trivial excitations
FM: Domain wall excitations

Cannot be removed by local coupling to bath!

Energy of steady state:

89
8:

=

<(
. 8=
; 8:

= −𝐶 𝑒 − 𝑒6 / + 𝜂9 = 0𝑒 − 𝑒* ∝ 𝜂+

Finite 𝒅 = 𝟏 system (# of excitations = O(1)):
𝑒 − 𝑒* ∝ 𝜂+ 𝐿

Topological vs. Trivial excitations
4

PM

(b)

0.15

energy densitys than 1 excitationenergy density

(a)

0

0.02

4

of errors, ⌘e

6

8

0.15

𝜂+ = 0.02
𝜂+ = 0.02
𝜂+ = 0.002

0

10

4

system size

PM

FM

6

8

10

system size

QCP

FM

5

𝑝(𝐸, )
occupation

ge number of errors
FIG. 4. Energy(a)density(E E0 )/E
as a function
system size
J/hx = 0.5 (b)J/h0x = 1(c) J/hx = 2⌦ z z ↵for (a) the paramagnetic
5, hz = 0) is showncase and (b) the ferromagnetic case and for1
si si+1
hx = 1, hz = 0) inan average number of errors ⌘e = 0, 2 · 10 3 , 10 2 and 2 · 10 𝜂2! = 2
5
y is below the lowest 1 per sweep10andspin (light to dark). In the shaded grey area the av𝜂! = 0.2
ost-selection (empty
erage energy is below the system’s gap. In the paramagnetic phase,𝜂! = 0.02
10
10
oves the outcome ofthe energy density grows with ⌘e but not with system size, whereas
𝜂! = 0phase the energy density grows approximately
tem. See text a the 0.5 in the ferromagnetic
10 15
ameters: Ns = 8,
linearly with system size. The slope of the energy density vs. system-7 increasing0
7 ⌘ .-7The 0observed
7 behavior-7
0 is consis7
Bi = 5, Bf = 0.7,
size increases witheenergyenergy model, Eqs.
energytent with the predictionof the rate equation(6) and (8).
0Parameters are the same as in Fig. 3.

Topological vs. Trivial excitations“Trapping” topological excitations

𝑱’ < 𝑱

Reduce the number of mobile excitations(b)(b)(a)20 𝑱’(c)= 𝟎. 𝟑𝟑 𝑱
0 (c)

time
timetimesi si+1si si+1

⌦ z 1z ↵ 1
si si+1

(b)

1

00
20

20

⌦z z

(c)

0.5

(d)

(d)

0.5

0.5

(d)

0

bond
0 1 2 3 4 5 6 7
1 2 3 bond4 5 6 7

FIG. 5.

1 2 3 4 5 6 70

1 2bond3 4 5 6 7
1 2 3 4bond
5 6 7

10
10
10 5

15
10 10
10

(a) J/h
x
5
10

10 10
10 15

-7

-7
0

1 2 3 4 5 6 7

510 1(a) J/hx(a
1

10 15

0

0

1

occupation

20

(a)
(a)

⌦ ↵z z ↵occupation
occupationtime
time time20
20

𝜂! = 0z
Bond bondcorrelation function hszi sbond
i+1 i for the ferromagnetic

0

-7
energ
0

energ
FIG. 6. Occupation⌘e = 0,0.02,6.0.2,Occ
2p
FIG.
paramagnetic
= 0.
FIG.
6.⌘e Occupation= (J0, 0.02
h0.02,hz2 =x = 1,⌘(Je == 1,
0,paramagnetic
0.2,
p
h
=
0.5,
h
=
0).
x
paramagnetic(J h=x 0.
(J = z1,=

Topological vs. Trivial excitations“Trapping” topological excitations

𝑱’ < 𝑱

Reduce the number of mobile excitations(b)(b)(a)20 𝑱’(c)= 𝟎. 𝟑𝟑 𝑱
0 (c)

time
timetimesi si+1si si+1

⌦ z 1z ↵ 1
si si+1

(b)

1

00
20

20

⌦z z

(c)

0.5

(d)

(d)

0.5

0.5

(d)

0

bond
0 1 2 3 4 5 6 7
1 2 3 bond4 5 6 7

FIG. 5.

1 2 3 4 5 6 70

1 2bond3 4 5 6 7
1 2 3 4bond
5 6 7

10
10
10 5

15
10 10
10

(a) J/h
x
5
10

10 10
10 15

-7

-7
0

1 2 3 4 5 6 7

510 1(a) J/hx(a
1

10 15

0

0

1

occupation

20

(a)
(a)

⌦ ↵z z ↵occupation
occupationtime
time time20
20

𝜂! = 0.02

𝜂! = 0z
Bond bondcorrelation function hszi sbond
i+1 i for the ferromagnetic

0

-7
energ
0

energ
FIG. 6. Occupation⌘e = 0,0.02,6.0.2,Occ
2p
FIG.
paramagnetic
= 0.
FIG.
6.⌘e Occupation= (J0, 0.02
h0.02,hz2 =x = 1,⌘(Je == 1,
0,paramagnetic
0.2,
p
h
=
0.5,
h
=
0).
x
paramagnetic(J h=x 0.
(J = z1,=

acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured
and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state isachieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol’s
performance in the presence of noise and show how the information from the measurement of the bath spins
can be used to monitor the cooling process. The performance of the algorithm depends on the nature of theexcitations of the system; systems with non-local (topological) excitations are more difficult to cool than thosewith local excitations. We explore the possible mitigation of this problem by trapping topological excitations.

Summary

Topological excitations are harder
to cool:
𝟏/𝑴
𝒆 − 𝒆𝟎 ∝ 𝜼𝒆
𝑴: # of excitations that can beremoved locally

J
g

J
g

J
g

J
g

J
g

J
g

g

J
g

g

#i?
(b)

(c)

systemsystemcooling cycle

bath

bath
in
polarized
state

PM

measurement& reset4FMenergy density

FIG. 1.
Schematic illustration of the proposed setup. (a) Eachsystem spin si (light blue) is coupled to (b)a bath spin i (dark blue)
(a)
(a)
(b)
by switching on the coupling g(t). System spins may be generically
Recently, algorithms
that mimic cooling by coupling to 0.15
a
1
0.15
less
than 1have
excitation
coupled, as illustrated here by J. At the beginning of the protocol,
simulated low-entropy bath
[18–24]been proposed as althe bath is polarized and subjected to a large simulated magnetic fieldternative routes that may overcome some of these challenges.
B(t), while the system may begin in a random state. Panel (b) shoes
The key advantages of such cooling algorithms are that they
the time dependence of the parameters, and panel (c) illustrates thecan be run cyclically without any special requirements on
0.5
cooling cycle. A magnetic field B(t) is applied to the bath spins, andless than 1 excitation
initial states and do not require prior knowledge of the tarlinearly decreased from B0 to Bf until time t2 = 3T /4, after which
get ground state. Simulated cooling schemes also do not reit is held constant until time T , the total duration of the sweep. The
quire the target state to be adiabatically connected to a prodsystem-bath coupling g(t) is slowly switch on until t1 = T /4, kept
0
0 operation provides inherent robustness 0 constant at g(t) = g until time t , and 0then switched off. At the end
uct state. The cyclic
20
1 to weak2noise by automatically
0
0.02
4
6
8 0 1046810correcting someerrors and reof each sweep, the bath spins are measured and reset.

# of errors,⌘e unwanted excitations# ofonerrors,⌘ecycles. Furthermoving
subsequent•UV
bvbi2K

energy density

energy density•

Simulated adiabatic cooling:
Robust method to find low-energy
states of generic Hamiltonians.

energy density•

Ground state preparation on quantum simulators and computers is very important for the characterization of groundstate properties in quantum chemistry and material science [1–4]. More generally, it can also be utilized for a variety of quantum information problems [5–7]. On a quantum computer, theHamiltonian can be digitally implemented using a sequenceof unitary gates. Many approaches of ground state preparation
have been proposed, including variational quantum simulation[8–12] and adiabatic state preparation [13–16]. Each of theseapproaches has its own challenges. For example, the performance of variational quantum simulation highly depends on
the quality of the variational ansatz [12, 17]. At the sametime, adiabatic state preparation is sensitive to the trajectoryin Hamiltonian space connecting the initial and final Hamiltonians. It tends to fail when a phase transition separates theinitial and final states. Furthermore, errors that occur during
adiabatic state preparation are not intrinsically corrected andmay be difficult to detect.

system size

system size

more, in simulated cooling on a quantum computer, monitoring the bath spins during the cooling protocol can provide in[25, 26],(EknownEfrom
solid-state systems. For exFIG. 3. (E E0 )/E0formation
as a function
the average
number of
FIG. 4. netization
Energy density0 )/E0 as a function system sizeon theofprocessof approachingtheerrors
system’s groundample, with nuclear adiabatic demagnetization, one can reach⌘e : (a) The ferromagnetic(J =1, hx =the
0.5,hz =directly0) is shown(a) the paramagnetic case and (b) the ferromagnetic case and forstatecasewithout
observingsystemand may beforusedtemperatures in the µK range [27, 28] by3 coupling
the system
in red squares, the paramagneticcase(J = 0.5,
hx = 1, hz = 0) inan average
number of errors ⌘e = 0, in
2 ·a 10, 10 2 and 2 · 10 2
to identify thepresenceof errors.
to nuclear moments polarized
large magnetic field andblue dots. Within the greyHere,shadedarea, the energy is below the lowest
per sweepadiabaticallyand spin (lightto dark).
In field.
the shaded grey area the avramping
down thewe define a simple, scalable protocol for low-energy
excitation gap . (b) state
For smallnoise oflevels,post-selection
is 1below
the system’sgap.
In thephase,
preparation
an arbitrary
gapped (empty
Hamiltonianerage
on a energyFig.
illustrates
the coolingscheme.
Weparamagnetic
consider a generic
red squares: FM, empty
bluecircles:PM)
improves
theoutcomeoftheenergydensity
grows
with
⌘
but
not
with
systemsize,
whereasequantum computer or programmable quantum simulator. We
system comprised of N qubits(light blue), coupled to an adin the ferromagnetic
phase
energydensity
the protocol, i.e. it lowers
energy betweenof the system.
See andtext adiabatica the demagdraw thean analogy
our protocol
ditional set of
Nbaththe“bath”
qubits(darkgrowsblue). approximately
We take theReduce # of mobile topologicalexcitations by trapping

description of the post-selection protocol. (Parameters: Ns = 8,N⌧ = 101, Nc = 100, Ninit = 1000, T = 6, Bi = 5, Bf = 0.7,
g0 = 0.5).

Thank you.

linearly with system size. The slope of the energy density vs. systemsize increases with increasing ⌘e . The observed behavior is consistent with the prediction of the rate equation model, Eqs. (6) and (8).
Parameters are the same as in Fig. 3.