of 7
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
Adiabatic quantum imaginary time evolution
Kasra Hejazi
,
1
Mario Motta,
2
and Garnet Kin-Lic Chan
1
1
Division of Chemistry and Chemical Engineering,
California Institute of Technology
, Pasadena, California 91125, USA
2
IBM Quantum
, Almaden Research Center, San Jose, California 95120, USA
(Received 7 August 2023; revised 9 April 2024; accepted 13 June 2024; published 18 July 2024)
We introduce an adiabatic state preparation protocol which implements quantum imaginary time evolution
under the Hamiltonian of the system. Unlike the original quantum imaginary time evolution algorithm, adiabatic
quantum imaginary time evolution does not require quantum state tomography during its runtime and, unlike
standard adiabatic state preparation, the final Hamiltonian is not the system Hamiltonian. Instead, the algorithm
obtains the adiabatic Hamiltonian by integrating a classical differential equation that ensures that one follows
the imaginary time evolution state trajectory. We introduce some heuristics that allow this protocol to be
implemented on quantum architectures with limited resources. We explore the performance of this algorithm
via classical simulations in a one-dimensional spin model and highlight essential features that determine its
cost, performance, and implementability for longer times, and compare to the original quantum imaginary time
evolution for ground-state preparation. More generally, our algorithm expands the range of states accessible
to adiabatic state preparation methods beyond those that are expressed as ground states of simple explicit
Hamiltonians.
DOI:
10.1103/PhysRevResearch.6.033084
I. INTRODUCTION
A central step in the quantum simulation of physical sys-
tems is to prepare a relevant initial state. Taking ground-state
simulation as an example, we typically wish to prepare a
state with sufficient overlap with the desired ground-state
|

0

of a Hamiltonian
H
. In the context of near-term quan-
tum algorithms [
1
], which minimize both qubits
/
ancillae and
gate resources, many protocols for such ground-state prepara-
tion have been proposed. Examples include variational ansatz
preparation [
2
–
7
], adiabatic state preparation (ASP) [
8
–
10
],
and variational [
11
] and quantum imaginary time evolution
(QITE) [
12
–
18
], the latter being the subject of this work.
Because ground-state preparation is in general formally
hard, all these methods rely on some assumptions. For ex-
ample, ASP starts from an initial Hamiltonian
H
(0), whose
ground state is simple to prepare, and defines an adiabatic path
H
(
s
)
,
0

s

1, with
H
(1)
≡
H
, the desired Hamiltonian
[
19
]. To prepare the ground state to sufficient accuracy,
H
(
s
)
must change slowly; an estimate of the adiabatic runtime is
[
8
,
20
]
T
∼
max
s
,
j
|

j
(
s
)
|
dH
/
ds
|

0
(
s
)
|

2
, where
|

0
(
s
)

,
|

j
(
s
)

,
with
j

1, denote the instantaneous ground and excited states
of
H
(
s
) and

(
s
) is the energy gap between the ground state
and the first excited state. For ASP to be efficient,
H
(
s
)must
be chosen such that min
s

(
s
) is not too small, e.g., at worst
1
/
poly(
L
) in system size
L
, for a polynomial cost algorithm.
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International
license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
The QITE algorithm [
12
], on the other hand, applies
e
−
H
τ
(
τ>
0) to boost the overlap of a candidate state with the
ground state of
H
; for this work, we consider Hamiltonians
that are sums of local terms
H
=
∑
α
h
α
, where
h
α
is geo-
metrically local (i.e., each term
h
α
acts on a constant number
of adjacent qubits regardless of system size). Reference [
12
]
introduced a near-term quantum algorithm to obtain the states
|

(
τ
)
=
e
−
H
τ
|

(0)

/

e
−
H
τ
|

(0)

,
(1)
without employing any ancillae or postselection. The method
is efficient if
|

(
τ
)

has finite correlation volume
C
for all
earlier imaginary times, in which case
|

(
τ
)

can be prepared
by implementing a series of local unitaries acting on
O
(
C
)
qubits on the candidate state. By using this technique which
reproduces the imaginary time trajectory, one can also use
QITE as a subroutine in other ground-state algorithms, as well
as to prepare nonground states and thermal (Gibbs) states, for
example, by reintroducing ancillae [
17
], or by sampling [
12
].
However, to find the unitaries in QITE one needs to per-
form tomography [
12
] of the reduced density matrices of
|

(
τ
)

over regions of volume
C
. Although the measurement
and processing cost is polynomial in system size, it can still be
prohibitive for large
C
. Despite various improvements in the
QITE idea in terms of the algorithm and implementation, this
remains a practical drawback [
13
]. [We briefly note also some
other near-term imaginary time evolution algorithms, such as
the variational ansatz-based quantum imaginary time evolu-
tion, introduced in Ref. [
11
], which reproduces the imaginary
time evolution trajectory in the limit of an infinitely flexible
variational ansatz, as well as the probabilistic imaginary time
evolution algorithm (PITE) [
21
], whose probability of success
decreases exponentially with evolution time.]
2643-1564/2024/6(3)/033084(7)
033084-1
Published by the American Physical Society
HEJAZI, MOTTA, AND CHAN
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
Here, we introduce an alternative near-term, ancilla-free,
quantum method that generates the imaginary time evolution
of a quantum state without any tomography. It thus eliminates
one of the resource bottlenecks of the original QITE. The idea
is to consider the imaginary time trajectory

(
τ
) as generated
by an adiabatic process under a particular Hamiltonian
̃
H
(
τ
).
This adiabatic Hamiltonian is approximated by the solution
of an auxiliary dynamical equation that can be solved entirely
classically, i.e., without any feedback from the quantum sim-
ulation. Although propagation under
̃
H
reproduces imaginary
time evolution when performed adiabatically [i.e., one stays
in the ground state of
̃
H
(
τ
)], this is different to the usual ASP,
because
̃
H
does not approach
H
at the end of the path, even
though it shares the same final ground state. We thus refer to
this algorithm as adiabatic quantum imaginary time evolution
or A-QITE.
Like QITE, the application of A-QITE for sufficiently
long imaginary time formally prepares the ground state. But
also like QITE, the imaginary time trajectory generated by
A-QITE has other applications. The ability to reproduce
nonunitary evolution enables applications to Lindblad simu-
lation [
17
], while for finite imaginary time, it can naturally
be applied to prepare thermal states (either in a purified
formulation [
17
] or via sampling [
12
,
13
]) or be used in a
subspace expansion method for excited states [
12
]. Generally
speaking, the A-QITE algorithm turns the preparation of any
state that can be reached by the differential evolution of an
initially known ground state into an adiabatic state prepara-
tion process. In this work, however, we primarily focus on
the generation of the imaginary time trajectory itself and its
long-time limit relevant to physical ground-state preparation.
We examine the feasibility and performance of A-QITE
for the illustrative case of the Ising-like Heisenberg
XXZ
model in a transverse field. There we study the behavior
of the instantaneous gap and norm of
̃
H
as a function of
imaginary time (as these determine the cost of integrating
the classical equation to determine
̃
H
as well to implement
the adiabatic quantum simulation under
̃
H
).
̃
H
(
τ
) becomes
increasingly nonlocal with time and we introduce a geometric
locality heuristic to truncate terms in
̃
H
, which we compare
to the original inexact QITE procedure. We finish with some
observations on the cost and practical implementation of the
algorithm.
II. FORMALISM
A. General theory
Consider a lattice system described by a Hamiltonian
H
.
We desire an adiabatic Hamiltonian
̃
H
(
τ
) whose ground state
at every
τ
is given by Eq. (
1
). Consider an infinitesimally
imaginary time evolved state from
τ
to
τ
+
d
τ
:
|

(
τ
+
d
τ
)

=
|

(
τ
)

−
d
τ
QH
|

(
τ
)

+
O
(
d
τ
2
)
,
(2)
where
Q
=
1
−|

(
τ
)


(
τ
)
|
projects out the ground sub-
space. Now, suppose
|

(
τ
)

is the ground state of
̃
H
(
τ
);
one should determine
̃
H
(
τ
+
d
τ
)
=
̃
H
(
τ
)
+
δ
̃
H
such that
|

(
τ
+
d
τ
)

is its ground state: perturbation theory deter-
mines the ground state of
̃
H
(
τ
+
d
τ
)as
|

(
τ
)
+
[
̃
E
0
−
̃
H
(
τ
)]
−
1
Q
δ
̃
H
|

(
τ
)

, where
̃
E
0
is the smallest eigenvalue of
̃
H
. We will be working with evolution schemes for
̃
H
that start
with and maintain
̃
E
0
=
0 (more on this below). Thus using
(
2
) we should have
δ
̃
H
|

(
τ
)

=
d
τ
̃
HH
|

(
τ
)

+
O
(
d
τ
2
)
,
(3)
wherewehaveused
̃
HQ
=
Q
̃
H
=
̃
H
.
Equation (
3
) is the main equation the adiabatic Hamil-
tonian
̃
H
should satisfy. However, it does not uniquely
determine
δ
̃
H
and so there are many generating equations for
̃
H
. A simple choice is
d
̃
H
d
τ
=
̃
HH
+
H
̃
H
,
(4)
where we have used
̃
H
|

(
τ
)
=
0 [see above Eq. (
5
)forjus-
tification] and added the term
H
̃
H
to make the right-hand side
Hermitian. This has the formal solution
̃
H
(
τ
)
=
e
H
τ
̃
H
(0)
e
H
τ
[where
̃
H
(0) is chosen so that

(0) is its ground state with
zero eigenvalue]. The above scheme can, in principle, be
implemented as a hybrid quantum-classical algorithm, where
̃
H
is first determined by the classical integration of Eq. (
4
)
and then used to implement adiabatic state evolution quan-
tumly. [In other words, we carry out Hamiltonian dynamics
with
̃
H
(
τ
) along the adiabatic path
τ
=
0
...β
, which, if per-
formed slowly (see below), will guarantee that one remains in
the ground state of
̃
H
(
τ
); the dynamics can be translated into a
circuit using standard Hamiltonian simulation techniques [
2
]].
As is clear, this procedure does not involve any feedback from
the quantum simulation and thus does not involve tomogra-
phy, unlike the original QITE.
In addition, the above procedure can be used to gen-
erate an adiabatic Hamiltonian
̃
H
(
τ
) to prepare states
beyond
e
−
β
H
|

(0)

, generalizing to the class of states
T
e
−
∫
β
0
d
τ

L
(
τ

)
|

(0)

, where
L
(
τ

) is an arbitrary operator that
need not even be Hermitian. This can be seen by considering
each time step of propagating
L
(
τ
) separately and decompos-
ing
L
into a sum of Hermitian and anti-Hermitian parts
L
H
+
L
AH
. The time evolution reads
d
̃
H
/
d
τ
={
̃
H
(
τ
)
,
L
H
(
τ
)
}+
[
̃
H
(
τ
)
,
L
AH
(
τ
)]. Thus a wide class of states, beyond ground
states or physical imaginary time-evolved states, become ac-
cessible through an adiabatic state evolution (assuming a
nonvanishing gap).
However, this naive scheme has some potential problems.
One set is analogous to that encountered in the original
quantum imaginary time evolution scheme, namely, the time
evolution of
̃
H
renders it nonlocal (both geometrically and in
terms of the lengths of the Pauli strings in
̃
H
) and increas-
ingly complicated. For example, even if
H
=
∑
α
h
α
and
̃
H
=
∑
β
̃
h
β
are geometrically local, the time derivative introduces
geometrically nonlocal terms like
h
α
̃
h
β
and the number of
such terms grows exponentially with time (the exponent in the
rate of increase of Pauli strings with significant coefficients
depends on the details of
h
α
) up until the maximum number
of 4
L
, where
L
is the size of the lattice. This renders both the
classical determination of
̃
H
(
τ
) and the quantum implemen-
tation of state evolution under
̃
H
(
τ
) inefficient for large
τ
.
There is also a second set of problems arising from the
norm of
̃
H
and its spectrum. We use the symbol
|
̃
φ
i
(
τ
)

to denote the instantaneous eigenstate of
̃
H
(
τ
) with eigen-
value
̃
E
i
(
τ
)[if
̃
H
implements the imaginary time evolution
perfectly, then
|
̃
φ
0
(
τ
)
=|

(
τ
)

]. Taking expectation values
033084-2
ADIABATIC QUANTUM IMAGINARY TIME EVOLUTION
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
of Eq. (
4
) with
|
̃
φ
i

, the eigenvalues of
̃
H
evolve as
d
̃
E
i
d
τ
=
2
̃
E
i

̃
φ
i
|
H
|
̃
φ
i

.If
̃
H
is initialized with zero ground-state energy,
the evolution will keep it vanishing. However, the other eigen-
values evolve as
̃
E
i
(
τ
)
=
exp
[
2
∫
τ
0
d
τ


̃
φ
i
(
τ

)
|
H
|
̃
φ
i
(
τ

)

]
̃
E
i
(
τ
=
0)
.
(5)
To see the potential problems with this, consider the example
where
H
has a finite spectrum with all eigenvalues above
(or below) 0. In that case, we clearly see that the eigen-
values
̃
E
i
,
i
>
0 are always growing (shrinking) with time,
potentially exponentially fast. Thus there is the possibility for
numerical issues at long times in determining
̃
H
and imple-
menting evolution under it, as we now discuss.
The integration of the classical differential equation for
̃
H
and the corresponding time evolution under
̃
H
will carry
some finite numerical error which depends on
||
̃
H
||
.This
means that rather than obtaining the exact
̃
H
(
τ
), we ob-
tain
̃
H
(
τ
)
+
V
; if one uses, e.g., a finite-order Runge-Kutta
method, then
||
V
|| ∝ ||
̃
H
||
. Depending on the dynamics
of the eigenvalues, this may introduce a large devia-
tion from the instantaneous ground state, for example, if
|
̃
φ
0
|
V
|
̃
φ
i

/
̃

i
|
1 (where
̃

i
is the instantaneous gap to
the
i
th state of
̃
H
). Similarly, the quantum adiabatic evolu-
tion time depends on
||
̃
H
(
τ
)
||
: since max
s
||
d
̃
H
/
ds
/
̃

(
s
)
2
|| ∝
β

H

max
τ
||
̃
H
(
τ
)
/
̃

(
τ
)
2
||
, the total adiabatic evolution
time
T
∝
β

H

max
τ
||
̃
H
(
τ
)
/
̃

(
τ
)
2
||
, which can diverge if
the numerator is exponentially growing or the denominator
is exponentially decreasing. As a result, we will be focused
on studying the behavior of

̃
H
(
τ
)

and
̃

(
τ
) numerically
below (especially their behavior with total imaginary time and
length). Finally, the implementation of Hamiltonian simula-
tion under
̃
H
(
τ
) also introduces errors that grow polynomially
with
||
̃
H
||
(see, e.g., Corollary 2 of [
22
]).
The appearance of such problems might be considered
natural given that reproducing the imaginary time trajectory
as
β
→∞
is equivalent to the (formally) intractable prob-
lem of exact ground-state preparation. However, as faithful
imaginary time evolution produces an exponentially decaying
infidelity with the final state, the long-time algorithm may not
be needed if sufficient fidelity is already reached. For finite
imaginary time, as needed in thermal state simulation or to
prepare the more general class of states discussed above, the
concerns are also less relevant. In addition, various heuristics
may be used to ameliorate these above concerns. We now turn
to the discussion of heuristics, before studying the potential
issues at finite and at long imaginary times through numerical
simulations.
B. Locality heuristic
To address the growing nonlocality of
̃
H
we first write a
modified generating equation for
̃
H
, with separate differential
equations for the individual terms
̃
h
β
. We choose
|

(0)

such
that it is annihilated by each
̃
h
β
(0) and the evolution preserves
this annihilation condition, analogous to Eq. (
4
). We consider
d
̃
h
β
d
τ
=
∑
α
{
̃
h
β
,
h
α
}

,
(6)
where
{
a
,
b
}

denotes the anticommutator of
a
and
b
if they do
not commute and zero if they do. Equation (
6
) also satisfies
the adiabatic trajectory of Eq. (
3
) because (
4
) and (
6
) only
differ by terms that annihilate
|

(
τ
)

; so far no heuristic has
been introduced. The expression
{
̃
h
β
,
h
α
}

means that
̃
H
no
longer contains geometrically nonlocal terms: each
̃
h
β
grows
its support from the contribution of terms
h
α
that overlap with
the boundary of its support at every step.
We can then introduce a heuristic to control the width of
support. In particular, we can truncate summation over
α
in
Eq. (
6
) so that only a subset of terms
h
α
are retained for each
̃
h
β
. More precisely, a neighborhood block is assigned to every
̃
h
β
term which is a region with a given spatial extent
w
that
surrounds the location of
̃
h
β
at
τ
=
0; every
h
α
term that lies
in the neighborhood block of
̃
h
β
is retained in Eq. (
6
). In this
approximation,
̃
H
remains strictly
w
-geometrically local with
time. We note that this above heuristic is different from the
locality approximation in the original QITE algorithm, as it is
a direct restriction on the operator, rather than the correlation
length of the imaginary-time-evolved state. The relationship
between the two is studied numerically below.
C. Gauge degree of freedom
We can introduce other modifications to the generating
equation of
̃
H
which do not modify the imaginary time evolu-
tion trajectory but which can, in principle, affect
||
̃
H
||
and its
spectrum. Consider, for example,
d
̃
h
β
d
τ
=
∑
α
{
̃
h
β
,
h
α
}

+
f
(
̃
h
β
)
,
(7)
where
f
(
̃
h
β
(
τ
)
)
annihilates

(
τ
). This ensures that the ground
state of
̃
H
is a zero eigenstate for all
τ
(although it does
not ensure that it is always the ground state). We can then
optimize
f
(
̃
h
β
) to control the gap and norm. Specifically, in
tests below, we consider
f
(
̃
h
β
)
=
u
1
̃
h
β
−
u
2
̃
h
2
β
.The
u
1
term is
equivalent to adding a constant shift to
H
in Eq. (
4
). While we
do not expect this gauge choice to remove the fundamental
difficulties of preparing states for infinite imaginary times,
we can expect to improve the preparation of states for finite
imaginary time. In practice, we do not know how to choose
f
(
u
) ahead of time, but it may be chosen in a heuristic manner
or
f
(
u
) may be chosen as part of a variational ansatz for

(
τ
)
at finite
τ
.
III. NUMERICAL SIMULATIONS
We now study the imaginary time trajectory generated by
the
̃
H
dynamics, including the locality and gauge modifi-
cations described above. We classically propagate
̃
H
by a
second-order Runge-Kutta method [
23
] with a finite time step
[error
O
(
3
) per time step; note we use
rather than
d
τ
here
to emphasize a finite step] and then diagonalize
̃
H
(
τ
) to study
the trajectory of the ground state
0
(
τ
). We then monitor
various quantities, such as
||
̃
H
(
τ
)
||
,thegap
̃

(
τ
), the infi-
delity with the exact imaginary time propagated state
I
τ
(
τ
)
=
1
−|
̃
φ
0
(
τ
)
|

(
τ
)
|
2
, and the infidelity with the ground state
of
H
,
I
∞
(
τ
)
=
1
−|
̃
φ
0
(
τ
)
|

(
∞
)
|
2
, i.e., the outcome of the
infinite imaginary time evolution.
033084-3
HEJAZI, MOTTA, AND CHAN
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
FIG. 1. Adiabatic evolution under the Hamiltonian
̃
H
for a
Heisenberg
XXZ
system of length
L
=
8.
̃
H
is generated by integrat-
ing Eq. (
4
) with second-order Runge-Kutta and time-step
,whichis
varied here.
I
∞
,
I
τ
: infidelities with the ground state of
H
and with
the exact imaginary time evolved state;
̃

: instantaneous gap of
̃
H
;
||
̃
H
||
: norm of
̃
H
. The same color is used in all of the plots for each
value of
, even though we have shown each
only once.
We study the antiferromagnetic Heisenberg
XXZ
model in
one dimension with open boundary conditions:
H
λ
z
=
∑
j
S
x
j
S
x
j
+
1
+
S
y
j
S
y
j
+
1
+
λ
z
S
z
j
S
z
j
+
1
,
(8)
with
λ
z
=
2, which results in an Ising-like anisotropy. To
generate a simple initial state, we initialize the adiabatic
Hamiltonian
̃
H
as a staggered transverse field:
̃
H
(
τ
=
0)
=
∑
j
[(
−
1)
j
S
x
j
+
1
2
] (where each term in brackets separately
annihilates the ground state); the resulting initial state is thus
a product state. We then determine
̃
H
under the dynamics
generated by Eqs. (
4
), (
6
), and (
7
), and study various prop-
erties of
̃
H
and its instantaneous ground state
̃
φ
0
(
τ
), which is
intended to produce the imaginary-time trajectory.
̃
H
has the
symmetry
∏
i
S
x
i
, under which
̃
φ
0
(
τ
) has a definite eigenvalue.
Note that
H
λ
z
=
2
has symmetry broken ground states with long
range order [
24
,
25
], but we take
|

(
∞
)

to belong to the same
symmetry sector as
̃
φ
0
(
τ
).
In Fig.
1
we first consider
̃
H
generated by Eq. (
4
) and the
infidelities
I
(
τ
),
I
∞
(
τ
),
||
̃
H
(
τ
)
||
, and
̃

(
τ
)asafunctionof
time-step
used in the classical integration of Eq. (
4
). As
seen, for all step sizes, at early times the infidelity with the
ground state of
H
,
I
∞
(
τ
), decreases exponentially quickly, to
10
−
2
or less. [Although this is a small system, we note that
achieving finite infidelities of
O
(10
−
1
) is already important
for ground-state preparation applications as it enables variants
of quantum phase estimation adapted to the near-term setting
[
26
]]. The achieved infidelities decrease as
is decreased
towards an infinitesimal step size in the classical integrator,
which would give
I
τ
(
τ
)
=
0 at all times and
I
∞
(
τ
)
→
0at
long times. For a finite step size
,wesee
I
∞
(
τ
) reaches a
minimum value, while
I
τ
(
τ
) first increases with time, reaches
a maximum, and then plateaus. The time for the minimum
I
∞
is close to (slightly after) the time for the maximum
I
τ
;
FIG. 2. Adiabatic evolution under
̃
H
generated by Eq. (
6
), for
the Heisenberg
XXZ
system as a function of chain length
L
.
̃
H
is
generated using second-order Runge-Kutta and a time step of
=
0
.
05. Quantities are the same as in Fig.
1
.
we refer to this (approximate) time as
τ
c
.
||
̃
H
||
increases
exponentially, while
̃

decreases exponentially until about
τ
c
, before slowly increasing again. The above is consistent
with our analysis that for a finite time step it is only possible
to determine the adiabatic Hamiltonian accurately up to a
maximum time
τ
c
.
We next study
̃
H
generated by the exact Eq. (
6
), i.e., with-
out enforcing locality, as a function of system size
L
and a
fixed classical time step
=
0
.
05 in Fig.
2
. The results are
similar to those using
̃
H
; both formally generate the exact
imaginary-time trajectory for infinitesimal step size. Using a
finite time step, examining
I
∞
,
I
τ
, we see the same behavior
of reaching a minimum
I
∞
and a maximum
I
τ
at some ap-
proximate time
τ
c
. Surprisingly, this does not seem to change
the maximum reliable propagation time,
τ
c
≈
1
.
8, between
L
=
6 and 12.
The above two numerical studies illustrate the formal ex-
actness of the A-QITE procedure, as well as the numerical
challenges in its implementation arising from the behavior of
||
̃
H
(
τ
)
||
and
̃

(
τ
). We now study the impact of heuristics at
controlling these quantities. We first apply the locality heuris-
tic for the generation of
̃
H
, implemented using Eq. (
6
) with
finite width
w
=
5inFig.
3
. We study a range of system sizes
and integration time steps. Although the finite width means
that we no longer follow the imaginary-time trajectory exactly
(even for infinitesimal time step), we still see exponential
decay of the infidelity with the ground state to useful values of
∼
10
−
2
or less. Overall, the infidelity dynamics shows similar
behavior to previous examples and at longer times there is a
τ
c
corresponding to a change in the behavior of the infidelities
and gaps. However, as the time step goes to 0, the minimum
I
∞
does not keep decreasing, because of the finite width error.
From the perspective of the classical integration and quantum
implementation, the dynamics generated under the locality
heuristic is much more favorable than under the exact A-QITE
̃
H
dynamics. In particular,

̃
H

grows orders of magnitude
033084-4
ADIABATIC QUANTUM IMAGINARY TIME EVOLUTION
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
FIG. 3. Adiabatic evolution under
̃
H
generated by Eq. (
6
), using a locality truncation with block size
w
=
5, for Heisenberg
XXZ
chains
of various lengths. Panels (a) and (b) show infidelities for different values of
and
L
=
8. Note that the infidelities are defined in the same way
as in Fig.
1
. Panels (c) and (d) show the behavior of the infidelities for different lengths, second-order Runge-Kutta,
=
0
.
05. Panels (e) and
(f) depict the gap and norm for different lengths; the same colors as the second column are used. Panels (g) and (h) denote the norm and the
gap of the adiabatic Hamiltonian as a function of length. The data is consistent with power-law behaviors for both.
more slowly with
τ
and (for fixed
τ
) close to linearly with
L
,
while
̃

appears to decrease like poly(1
/
L
).
In Fig.
4
,weshow
̃
H
generated by Eq. (
6
), using the
locality heuristic with widths 3
,
5
,
7, and time step
=
0
.
05,
as well as a protocol where the width is increased at increasing
times. For comparison, we also show data from the original
quantum imaginary time evolution scheme with widths 2
,
4
,
6
and for time step
=
0
.
08. As expected, the best achievable
fidelity with the exact state
I
∞
(
τ
) increases with
w
; however,
the best
I
(
∞
) in fact decreases moving from
w
=
5to
w
=
7.
Compared to the original QITE scheme, the achievable in-
fidelity appears to be better (i.e., lower) using the adiabatic
FIG. 4. Infidelity with the desired ground state for both the adi-
abatic QITE and the original QITE algorithm. Even values of
w
correspond to the original QITE algorithm and the odd
w
values and
the varying case corresponds to the adiabatic QITE. The time steps
are taken as 0.05 for A-QITE and 0.08 for QITE. The varying
w
case
starts with
w
=
3, then changes to
w
=
5, and ultimately to
w
=
7.
The positions of the transitions are shown by vertical dotted lines.
Hamiltonian
̃
H
with a similar locality definition. Further work
is required to establish a more precise relationship between
these approximations.
We additionally consider the modified
̃
H
generated in a
different gauge using Eq. (
7
). As an illustrative example, we
consider the case
u
1
=
1
/
2,
u
2
=
2, with the locality con-
straint
w
=
5. In Fig.
5
, we see that, although the behavior
of the infidelities
I
∞
and
I
τ
are qualitatively similar to what
we have seen previously (corresponding to
u
1
=
u
2
=
0) with
a similar
τ
c
, the detailed form is different; for example, the
infidelity has a flatter plateau region. In addition,
||
H
||
grows
even more slowly with
τ
(with a flattened region between
∼
1
<τ <
3) while still being close to linear in
L
, while

(at
τ
c
) again appears to have a poly(1
/
L
) behavior. This indicates
that a suitable choice of
f
(
u
) can meaningfully modify the
dynamics and the achievable infidelities at finite times.
We now briefly discuss the implications for implementa-
tions in the near-term setting. For a given width
w
and Trotter
time step, the gate cost of implementing an A-QITE time step
is the same as that of QITE, namely it is the cost to imple-
ment a time step under a
w
-qubit unitary and importantly
the A-QITE time step requires no measurements. However,
one difference with the original QITE is the need to maintain
adiabaticity during the A-QITE simulation, which governs the
time of Hamiltonian simulation for the given
τ
. The (limited)
numerical data above on the scaling of
̃
H
and
̃

when using
the heuristic modifications in the Ising problem suggest that
(for given
τ
) the total adiabatic evolution time is polyno-
mial in system size [as
||
H
||
,
||
̃
H
||
scale like poly(
L
) and
̃

∼
poly(1
/
L
)]. The Hamiltonian simulation is still likely
the most challenging aspect of the algorithm for near-term
hardware. However, for modest widths, e.g.,
w
=
3, imple-
menting the A-QITE protocol is similar to implementing ASP
with geometric three-local Hamiltonians, which has already
been demonstrated on near-term hardware using error mitiga-
tion and circuit recompilation [
27
]. Alternatively, it may be
033084-5
HEJAZI, MOTTA, AND CHAN
PHYSICAL REVIEW RESEARCH
6
, 033084 (2024)
FIG. 5. Adiabatic evolution under
̃
H
generated by Eq. (
7
) with
u
1
=
2
,
u
2
=
0
.
5, block size
w
=
5, second-order Runge-Kutta,
=
0
.
05,
for the Heisenberg
XXZ
system for various lengths. Quantities same as in Fig.
1
. Panels (e) and (f) denote the norm and the gap of the adiabatic
Hamiltonian as a function of length. The data is consistent with power-law behaviors for both.
beneficial to relax the adiabaticity requirement by employing
A-QITE within a version of the quantum approximation op-
timization algorithm (QAOA) [
4
,
28
], where
̃
H
provides the
form of the operators in the ansatz.
IV. CONCLUSIONS
We have described an adiabatic state preparation protocol
that implements the imaginary time evolution trajectory with-
out any need for quantum tomography or ancilla resources.
This hybrid algorithm involves a classical time integration to
generate the adiabatic Hamiltonian, but does not require any
measurements on the quantum system. When implemented
faithfully, the algorithm leads to an exponential decrease of
the infidelity with the ground state of a desired Hamiltonian
with adiabatic time. However, the cost of evolving exactly to
long imaginary times grows rapidly with imaginary time both
in the classical and quantum parts of the protocol. The growth
in cost as a function of imaginary time arises from several
sources, including the nonlocality of the derived adiabatic
Hamiltonian. We introduce a heuristic to control this nonlocal-
ity by truncating terms in the adiabatic Hamiltonian. Another
source of growing cost at long imaginary time is related to the
norm of the adiabatic Hamiltonian and its gap. We show that
modifying the generating equation of the adiabatic Hamilto-
nian can be used to control these quantities at finite imaginary
times. Both heuristics enable one to propagate for short times
and to observe a large improvement in the approximate ground
state.
In the near-term quantum hardware setting, one advantage
of the current approach is that each time step can be imple-
mented without measurements. On the other hand, preserving
the strict adiabatic formulation may lead to long Hamiltonian
simulation times for some problems.
More generally, the A-QITE procedure we have introduced
extends the types of states that be generated using an adiabatic
simulation protocol; the target state need not be expressed
as the ground state of a known Hamiltonian, but rather can
be expressed implicitly through the differential evolution of
a known ground state. In addition, with respect to standard
ground-state adiabatic state preparation, A-QITE expands the
set of possible adiabatic paths. This potentially allows for
introduction of new quantum adiabatic routines consisting
of composite adiabatic paths (with A-QITE as one of them)
with applications such as the introduction of novel adiabatic
catalysts [
8
,
29
] as directions to be explored in the future.
ACKNOWLEDGMENTS
We thank Y. Tong, A. Dalzell, and A. Chen for helpful
discussions. K.H. and G.K.C. were supported by the U.S. De-
partment of Energy, Office of Science, Basic Energy Sciences,
under Grant No. DE-SC-0019374. G.K.C. acknowledges sup-
port from the Simons Foundation.
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