lating the transverse magnetization of the Ising
model [17]: the main difference respect to the
work proposed in that paper is we have access to
the whole energy spectrum, which allows us to
simulate time and temperature evolution as well.
In this work, we implement a four-qubit experi-
ment that could be interesting both as a proposal
for testing and comparing devices quality and for
its implications in condensed matter physics. We
perform the exact simulation of a spin chain pro-
posed in Ref.[18] with an Ising-type interaction.
The Ising model is one of the most famous ex-
actly solvable models, i.e. those models that are
integrable. Actually, the steps to find a quan-
tum circuit that diagonalizes the Ising Hamilto-
nian follow the same strategy than the analytical
solution of the model. Therefore, the method can
be extended to other integrable models like the
Kitaev-honeycomb model, which a circuit has al-
ready been proposed [19]. As we are performing
an exact simulation, we have access to the whole
spectrum and not only to ground state: time evo-
lution and thermal states can be simulated ex-
actly as well. This provides a new approach in
quantum simulation if an exact circuit is found
for those non trivial models, such as Heisenberg
model, which have an ansatz to be solved. In
particular, for one-dimensional spin chains, the
Bethe ansatz [20] is the most successful method
and several proposals exist to simulate and ex-
tend it to two-dimensions using tensor network
techniques [21]. As the one-dimensional Ising
model has analytic solutions for arbitrary num-
ber of spins and the circuit proposed in this paper
can be efficiently generalized to larger number of
qubits, the methods outlined in this work can be
used to benchmark a quantum computer by see-
ing how this compares against known solutions.
The paper is structured as follows. In section
2 we describe the method proposed in Ref.[18] to
construct an efficient circuit that diagonalizes the
Ising Hamiltonian: the number of gates scales as
n
2
and the circuit depth as n log n. In section 3
we explain briefly the basic concepts of time evo-
lution in quantum mechanics and give a specific
example to be simulated using the circuit derived
in the previous section. In section 4, we propose
two methods to simulate the expected value of an
operator for finite temperature. Section 5 sum-
marizes the properties of the three devices used
for this work, two from IBM and one from Rigetti,
and in section 6 we present the results of ground
state transverse magnetization and the time evo-
lution of | ↑↑↑↑i state and compare the three de-
vices according to them. Finally, the conclusions
are exposed in section 7.
2 Quantum circuit for the Ising Hamil-
tonian
Let’s consider the existence of a quantum cir-
cuit that disentangles a given Hamiltonian and
transforms its entangled eigenstates into product
states. This circuit will be represented by an uni-
tary transformation U
dis
e
H = U
†
dis
HU
dis
, (1)
where H is the model Hamiltonian and
e
H is a
noninteracting Hamiltonian that can be written
as
e
H =
P
i
i
σ
z
i
. This diagonal Hamiltonian con-
tains the energy spectrum
i
of the original one
and its eigenstates correspond to the computa-
tional basis states. Then, we will have access to
the whole spectrum of the model by just prepar-
ing a product state and applying U
dis
.
There exist infinite U
dis
gates for a given
Hamiltonian. In general, to find these disentan-
gling unitaries will be a hard task, probably as
hard as finding a method to diagonalize analyti-
cally the Hamiltonian. However, for some mod-
els we can follow a kind of recipe to construct a
disentangling gate. For the case of Ising Hamilto-
nian, the steps to obtain the U
dis
quantum gate
are based on the analytical solution of the model
[22, 23]: i) Implement the Jordan-Wigner trans-
formation to map the spins into fermionic modes.
ii) Perform the Fourier transform to get fermions
to momentum space. iii) Perform a Bogoliubov
transformation to decouple the modes with op-
posite momentum. Thus, the construction of the
disentangling gate can be done by pieces:
U
dis
= U
JW
U
F T
U
Bog
. (2)
In the following subsections, we derive the quan-
tum gates needed to implement the above trans-
formation.
Accepted in Quantum 2018-12-18, click title to verify 2