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Linear-Optical Generation of Eigenstates of the Two-Site X Y Model

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Much of the anticipation accompanying the development of a quantum computer relates to its application to simulating dynamics of another quantum system of interest. Here we study the building blocks for simulating quantum spin systems with linear optics. We experimentally generate the eigenstates of the XY Hamiltonian under an external magnetic field. The implemented quantum circuit consists of two CNOT gates, which are realized experimentally by harnessing entanglement from a photon source and by applying a CPhase gate. We tune the ratio of coupling constants and magnetic field by changing local parameters. This implementation of the XY model using linear quantum optics might open the door to the future studies of quenching dynamics using linear optics.
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Linear-optical generation of eigenstates of the two-site XY model
Stefanie Barz1,, Borivoje Daki´
c1,2, Yannick Ole Lipp1, Frank Verstraete1, James D. Whitfield1, Philip Walther1
1University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria
2Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
Present address: Clarendon Laboratory, Department of Physics,
University of Oxford, Parks Road, Oxford OX1 3PU, UK
Much of the anticipation accompanying the development of a quantum computer relates to its application to
simulating dynamics of another quantum system of interest. Here we study the building blocks for simulating
quantum spin systems with linear optics. We experimentally generate the eigenstates of the XY Hamiltonian
under an external magnetic field. The implemented quantum circuit consists of two CNOT gates, which are
realized experimentally by harnessing entanglement from a photon source and by applying a CPhase gate. We
tune the ratio of coupling constants and magnetic field by changing local parameters. This implementation of
the XY model using linear quantum optics might open the door to the future studies of quenching dynamics
using linear optics.
INTRODUCTION
In 1982, Richard Feynman proposed the idea for the effi-
cient simulation of quantum systems [1]. Complex systems,
whose properties cannot easily be computed with classical
computers, can be simulated by other well-controllable quan-
tum systems. In this way, an easily accessible system can
be used for reproducing the dynamics and the quantum state
of another system of study. The insight of having one con-
trollable quantum system simulate another is what forms the
foundation of quantum simulation. There are two different
approaches for simulating quantum systems that have been
implemented experimentally: analog and digital simulation.
Analog quantum simulators are designed to mimic a quantum
system by reproducing its evolution in a faithful manner [2, 3].
Alternatively, the effect of the unitary evolution of a quantum
system may be regarded as that of a quantum circuit acting
on some initial state. This inspires the approach of a digital
quantum simulator where the state of the system is encoded
into qubits and processed via quantum logic gates [4–6]. The
main challenge – apart from providing a sufficiently power-
ful quantum computer – lies in finding a way to decompose
the Hamiltonian into a suitable form. Experimentally, basic
quantum simulations of both types have been demonstrated
as proof-of-concept experiments on several quantum architec-
tures including trapped ions [7–10], optical lattices [11, 12],
nuclear magnetic resonance [13–17] and photons [18–20].
Here we exploit a scalable approach for digital quantum
simulation for strongly interacting Hamiltonians, which has
been suggested in [21]. The general idea of [21] is to construct
the explicit finite quantum circuits that transform the Hamil-
tonian into one corresponding to non-interacting particles. In
this work, we apply this method to the XY Hamiltonian for
two spins in a magnetic field. We develop a quantum circuit
that transforms product input states to the eigenstates of this
Hamiltonian. Our approach allows us to recover the whole
spectrum of certain quantum many-body problems - a distinct
advantage of our implementation. We experimentally imple-
ment this circuit in a linear optical setup and generate ground
a
b
FIG. 1. The quantum circuit for the generation of the eigenstates of
the XY Hamiltonian. aThe decomposition of the unitary Uthat
transforms the eigenstates of a non-interacting Hamiltonian to the
eigenstates of the XY Hamiltonian. Full control of the system pa-
rameter wthat specifies a particular Hamiltonian is granted by local
operations. bThe local unitaries (Y H)Xtogether with the first
CNOT transform the computational basis states into the four Bell
states and we obtain the circuit shown in (b). The asterisk (*) is to
remind us that this simplification is valid only for certain inputs.
and excited states for the two-qubit Heisenberg XY model in
a transverse external field. Our circuit consists of two CNOT
gates, where one of the gates is absorbed in the state genera-
tion and this other is implemented physically.
THEORY
Our work focuses on the simulation of the two-qubit XY
Hamiltonian in a transverse external field:
H=Jxσxσx+Jyσyσy+1
2B(σz+σz),(1)
with Jxand Jybeing coupling constants, Bthe magnetic field
with unit magnetic moment, and σithe Pauli matrices that
represent the particles’ spin in x, y or zdirection, respectively.
Our work here focuses on preparing eigenstates. Specifi-
cally, our goal is to find a unitary Uthat transforms the Hamil-
arXiv:1410.1099v1 [quant-ph] 4 Oct 2014
2
tonian, H, into one corresponding to non-interacting quasi-
particles, ˜
H, hence diagonalizing it as:
˜
H=UHU=ω1σz11+ω211σz,(2)
where ω1=E1+E2
2and ω2=E1E2
2are the quasi-particle
energies, tan w= (JxJy)/B, and E1,E2are the eigenen-
ergies. It is not difficult to verify that the desired unitary U is
given by:
U=
cos w
20 0 sin w
2
01
2
1
20
01
2
1
20
sin w
20 0 cos w
2
,(3)
and that E1=pB2+ (JxJy)2=E4,
E2=Jx+Jy=E3.
By applying Uto the computational basis states, the eigen-
states of ˜
H, we obtain the eigenstates of the Hamiltonian H:
|ψ1i=U|00i= cos w
2|00i+ sin w
2|11i(4)
|ψ2i=U|01i=1
2(|01i+|10i)(5)
|ψ3i=U|10i=1
2(|01i−|10i)(6)
|ψ4i=U|11i=sin w
2|00i+ cos w
2|11i.(7)
We directly constructed the circuit but to generalize to arbi-
trary length XY spin chain the more general case was pre-
sented in Ref. [21]. The steps for the general case are 1)
identify spins with fermionic modes, 2) Fourier transform the
fermionic modes, and 3) perform a Bogoliubov transforma-
tion to diagonalize the free fermions. The first step requires
only relabeling while the second and third steps require ac-
tual transformations. The Fourier transform over Lsites can
be done in Llog Lsteps. An additional L2gates are needed
to account for the antisymmetry in the fermionic basis [21].
Finally, the Bogoliubov transformation requires only mixing
positive and negative momenta pairwise. The parameters of
this mixing depend on Jx, Jy, B and the momenta of the two
Fourier modes being mixed. The full Bogoliubov transfor-
mation can be done with L/2gates which can all be done in
parallel. The combined procedure implements a diagonalizing
unitary similar to (2) in polynomial cost in L. Using photonic
systems, the implementation of a heralded entangling gate re-
quires two additional ancilla photons per two-qubit gate.
EXPERIMENT AND RESULTS
In our experiment, we generate the eigenstates using linear
optics. To this end, we have experimentally realized a flex-
ible optical circuit (Fig. 2) that may implement any unitary
U(w)corresponding to the two-qubit XY Hamiltonian (1)
in a transverse field with the system parameter w. Our circuit
consists of two CNOT gates and local operations, which allow
FIG. 2. Schematic of the experimental setup implementing U
(Fig. 1b). A pair of photons—polarization-encoded qubits—is both
created and initialized to a desired state via spontaneous parametric
down-conversion. Different Hamiltonians can flexibly be simulated
be tuning the parameter w= arctan((JxJy)/B)through local
unitary operations (LU) wrapped in between two CN OT operations.
The former CNOT is thus merged with the SPDC process, while the
latter is implemented by a combination of two-photon interference
and polarization dependent beam splitters [23–25].
the manipulation of w. Fig. 1a shows that the first CNOT gate
together with the preceding unitaries in a transforms the four
product state inputs |00i,|01i,|10i,|11iinto one of the Bell
states |ψ±i= (|01i ± |01i)/2,|φ±i= (|00i ± |11i)/2.
These Bell states can be naturally obtained by exploiting the
entanglement of a spontaneous parametric down-conversion
(SPDC) source. Thus, we integrate the first CNOT into the
state preparation process (see Fig. 1b). The input register in
Fig. 1b originates from a type-II SPDC source, where a β-
barium borate (BBO) crystal is pumped with a femtosecond-
pulsed laser (394.5nm, 76MHz) to emit pairs of correlated
photons at a wavelength of 789nm [22]. In our implemen-
tation, |0iand |1icorrespond to horizontal and vertical polar-
ization, respectively. In our experiment, we generate entan-
gled photons pairs in the four different Bell states and input
them in the subsequent circuit. In combination with narrow-
bandwidth filters of 3nm this procedure yields state fidelities
for the input states of 97 ±1%.
Adjusting the subsequent local operations using a set of
quarter-wave and half-wave plates allows us to tune the sys-
tem parameter w.
The circuit for Uis completed by applying another CNOT
gate. In our experiment, this destructive CNOT gate uses
a polarization-dependent beam splitter (PDBS) which has a
different transmission coefficient Tfor horizontally polar-
ized light (TH= 1) as for vertically polarized light (TV=
1/3) [26]. If two vertically-polarized photons are reflected
at this PDBS, they acquire a phase shift of π. Two succes-
3
a b
c
0.00 0.25 0.50 0.75 1.00 1.25 1.50
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
cxx1
A
0.00 0.25 0.50 0.75 1.00 1.25 1.50
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
cyy1
A
0.00 0.25 0.50 0.75 1.00 1.25 1.50
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
azbz1
A
0.00 0.25 0.50 0.75 1.00 1.25 1.50
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Fidelity1
A
Fidelity
X1X2
Y1Y2
(Z1+Z2)/2
d
w w
w w
FIG. 3. Characterization of the eigenstates of the XY Hamiltonian. We show the measurement of the three components of the two-qubit
XY Hamiltonian separately. aCorrelations when measuring the qubits in the basis σxσx.bCorrelations when measuring the qubits in the
basis σyσy.cCorrelations when measuring the qubits in the basis (σz+σz)/2. This separate measurement of each part of the
Hamiltonian makes it possible to see the measured data for each part separately and thus enables a more detailed analysis of each part of the
Hamiltonian. dFidelities of the generated states. In all subfigures, the lines show a fit of the experimental data. The error bars are smaller than
the point size and hence not shown.
sive PDBSs with the opposite splitting ratios then equalize the
output amplitudes. This setup, in combination with two half-
wave plates (HWPs) (see Figure 2) implements a destructive
CNOT gate, where the success of the operation is determined
by postselection on a coincidence detection of the final output
photons [23, 25, 26]. For this CNOT gate, we experimentally
achieve a process fidelity [27] of 86.0±0.3%.
Using this setup we are able to prepare both ground and
excited states at arbitrary values of the system parameter w
by tuning the local unitaries in each input mode of the main
polarization dependent beam-splitter. Fig. 3 shows different
correlation measurements to characterize these states for sev-
eral choices of w. In the Additional Information, we show the
reconstructed density matrices of all measured states; Fig. 3d
shows the state fidelities as obtained from the density matri-
ces.
Our demonstration shows that the main features of the XY
Hamiltonian can be reproduced. The obtained fidelities lie be-
tween 0.75 and 0.9, these are the expected values when con-
sidering the fidelities of the entangled input states of 0.97 and
a process fidelity of 0.86. Since the state fidelities of the ex-
perimental states are non-perfect, the measured data deviate
from the theoretically expected values. However, as one can
see in 3, the obtained states show the same behavior as one
would expect from the theoretical eigenstates. In order to ob-
tain data even closer to the values, one would need to increase
the fidelity of the entangled input states, which are mainly
limited by higher-order emissions in the current setup and can
be increased using lower pump powers. Another limitation is
the process fidelity of 0.86, which is mainly due to the non-
perfect interference in our second CNOT gate. This interfer-
ence could be improved by making the photons spectrally and
spatially indistinguishable. In summary, the current state fi-
delities are mainly limited due to technical challenges, which
4
can be overcome.
CONCLUSION
We have demonstrated the preparation of the eigenstates for
the XY Hamiltonian under an external magnetic field. In the
original proposal [21], it was pointed out that the same ap-
proach can also be applied to prepare thermal states and sim-
ulate the dynamical evolution of any integrable model. Other
examples of integrable models are the Kitaev honey comb lat-
tice [28], the 1-D Hubbard model and the Heisenberg models.
We end this paper with a discussion on the extension to dy-
namical studies. The importance of generating eigenstates is
underlined in the context of quenching where the Hamilto-
nian of a system is instantaneously changed and the dynam-
ics of a quantum system is examined. Recently, this prob-
lem has attracted significant interest [29, 30] and it is a dif-
ficult task to simulate the quantum dynamics classically. In
the quantum setting, utilizing algorithms such as the one im-
plemented in the current work, initial states can be prepared
and one could then perform evolution under a different Hamil-
tonian and observe the quenching properties for polynomial
costs with a quantum computer. Since the XY model exhibits
critical phases and quantum phase transitions, both adiabatic
quenches through a phase transition and quenched dynam-
ics can be studied using the present work as a starting point.
While we did not explore dynamics in the present work, fu-
ture work might begin with the preparation of eigenstates and
proceed to break integrability e.g. by including an additional
magnetic field in the Xdirection and observing the dynam-
ics of various observables. This will require the subsequent
application of further entangling gates. However, currently
the maximum number of subsequent photonic gates that has
been demonstrated experimentally is two [31], which can be
increased to three [32] when using entangling input states as
demonstrated here.
ACKNOWLEDGMENTS
This work was supported from the European Commis-
sion, Q-ESSENCE (No. 248095), QUILMI (No. 295293),
EQUAM (No. 323714), PICQUE (No. 608062), GRASP (No.
613024), and the ERA-Net CHISTERA project QUASAR,
the John Templeton Foundation, the Ford Foundation, the Vi-
enna Center for Quantum Science and Technology (VCQ), the
Austrian Nano-initiative NAP Platon, the Austrian Science
Fund (FWF) through the SFB FoQuS (F4006-N16), START
(Y585-N20) and the doctoral programme CoQuS, the Vienna
Science and Technology Fund (WWTF, grant ICT12-041),
and the United States Air Force Office of Scientific Research
(FA8655-11-1-3004).
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Additional information: Analysis of the output states
Here, we show the density matrices of the theoretical as well as the measured output states.
HH
HV
VH
VV
HH
HV
VH
VV
0.5
0.0
0.5
1.0
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HV
VH
VV
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VV
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ab
cd
ef
gh
i j
1.0
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-0.5
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VV
VH
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-0.5
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HH
HV
HV VH VV
VV
VH
FIG. 4. Measured density matrices corresponding to the output state: |ψ1i= cos (w/2) |00i+ sin (w/2) |11ifor different values of w(real
parts). a, w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1), g,
w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
7
HH
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FIG. 5. Theoretical density matrices corresponding to the output state: |ψ1i= cos (w/2) |00i+ sin (w/2) |11ifor different values of w(real
parts). a, w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1), g,
w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
8
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FIG. 6. Measured density matrices corresponding to the output state: |ψ2i= (|01i+|10i)/2for different values of w(real parts). a,
w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1), g,
w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
9
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FIG. 7. Measured density matrices corresponding to the output state: |ψ3i= (|01i−|10i)/2for different values of w(real parts). a,
w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1), g,
w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
10
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FIG. 8. Theoretical density matrix corresponding to the output state: |ψ2i= (|01i+|10i)/2for all values of w(real parts).
HH
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FIG. 9. Theoretical density matrix corresponding to the output state: |ψ3i= (|01i − |10i)/2for all values of w(real parts).
11
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FIG. 10. Measured density matrices corresponding to the output state: |ψ4i=sin (w/2) |00i+ cos (w/2) |11ifor different values of w
(real parts). a, w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1),
g, w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
12
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FIG. 11. Theoretical density matrices corresponding to the output state: |ψ4i=sin (w/2) |00i+ cos (w/2) |11ifor different values of w
(real parts). a, w= arctan (0), b, w= arctan (1/16), c, w= arctan (1/8), d, w= arctan (1/4), e, w= arctan (1/2), f, w= arctan (1),
g, w= arctan (2), h, w= arctan (4), i, w= arctan (4), j, w= arctan (16).
... Photonic circuits have been used for realising varied phenomena such as bosonic and fermionic quantum walks 25,26 , quantum-classical differences in complexity [27][28][29][30] , quantum chemistry 31 and localisation in transport 32 . The ground states of some few-body spin systems [33][34][35] , as well as some analogues of quantum state transfer (QST) through spin chains using coherent optics 36,37 , have been realised in photonics. ...
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