A Quantum N-Queens Solver - Quantum Journal
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A Quantum N-Queens Solver
Valentin Torggler1 , Philipp Aumann1 , Helmut Ritsch1 , and Wolfgang Lechner1,2
1
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
2
Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
May 29, 2019
The N -queens problem is to find the po-
sition of N queens on an N by N chess
board such that no queens attack each
other. The excluded diagonals N -queens
arXiv:1803.00735v3 [quant-ph] 28 May 2019
problem is a variation where queens can-
not be placed on some predefined fields
along diagonals. This variation is proven
NP-complete and the parameter regime
to generate hard instances that are in-
tractable with current classical algorithms
is known. We propose a special purpose
Figure 1: Sketch of the setup. - (a) The N -queens
quantum simulator that implements the
problem is to place N non-attacking queens on an N
excluded diagonals N -queens completion by N board. A variation thereof is the N-queens com-
problem using atoms in an optical lat- pletion problem where some of the queens are already
tice and cavity-mediated long-range inter- placed (yellow). In addition, some excluded diagonals
actions. Our implementation has no over- are introduced (dashed blue lines) on which no queen
head from the embedding allowing to di- can be placed. (b) Each queen is represented by an
rectly probe for a possible quantum advan- atom which is trapped in an anisotropic optical potential
(blue) allowing for tunneling in x-direction only. Collec-
tage in near term devices for optimization
tive scattering of pump laser light (green arrows) into an
problems. optical resonator induces atom-atom interactions, pre-
venting atoms from aligning along the y-axis and along
the diagonals. After initial preparation in superposition
1 Introduction states delocalized in x-direction (red tubes), increasing
the interactions transfers the system into its solid phase,
Quantum technology with its current rapid ad-
which is the solution of the queens problem (black balls).
vances in number, quality and controllability of
quantum bits (qubits) is approaching a new era
with computational quantum advantage for nu-
tum computer [13–15]. In particular, adiabatic
merical tasks in reach [1–11]. While building
quantum computing [16–18] has been proposed
a universal gate-based quantum computer with
to solve computationally hard problems by find-
error-correction is a long-term goal, the require-
ing the ground state of Ising spin glasses [19].
ments on control and fidelity to perform algo-
Despite considerable theoretical [18] and experi-
rithms with such a universal device that out-
mental [20] efforts, quantum speedup in adiabatic
perform their classical counterparts are still elu-
quantum computing has not been demonstrated
sive. Building special purpose quantum comput-
in an experiment yet [21]. Thus, demonstrat-
ers with near-term technology and proving com-
ing quantum advantage by solving optimization
putational advantage compared to classical algo-
problems using quantum simulation tools is a cru-
rithms is thus a goal of the physics community
cial step towards the development of general pro-
world wide [12]. Quantum simulation with the
grammable quantum optimizers [22, 23].
aim to solve Hamiltonian systems may serve as
a building block of such a special purpose quan- Here we present a scheme that aims at solv-
ing the N -queens problem, and variations of it,
Wolfgang Lechner: wolfgang.lechner@uibk.ac.at
using atoms with cavity-mediated long-range in-
Accepted in Quantum 2019-05-28, click title to verify 1
teractions [24–28]. We note that the N -queens bination with constrained tunneling [32]. Hence
problem is not just of mathematical interest but there is no qubit overhead and the mode resources
also has some applications in computer science scale linearly with N . The required number of
[29]. In this work, variations of the problem are qubits is reduced from several hundreds to below
used as a testbed [30] to study a possible quan- 50, which is available in current experiments. By
tum advantage in solving classical combinatorial implementing our scheme with less than 50 atoms
problems in near term quantum experiments. the problem is already hard to tackle with current
Our proposed setup consists of N ultracold classical algorithms [30].
atoms in an optical lattice that represent the Light-mediated coupled tunneling gives rise to
queens on the chess board [31]. The non- non-local quantum fluctuations across the whole
attacking conditions are enforced by a combina- lattice in the intermediate stage of the transition
tion of restricted hopping [32] and interactions [46, 47]. Their non-uniform signs stemming from
between the atoms stemming from collective scat- the relative phases of the cavity fields from site
tering of pump laser light into a multi-mode cav- to site indicate that the system’s Hamiltonian
ity [33–40] (see Fig. 1). For the excluded diag- is non-stoquastic and can thus not be efficiently
onals variation of the N -queens problem, addi- simulated with path integral Monte Carlo meth-
tional repulsive optical potentials are introduced. ods on a classical computer [48]. The question
The solution of the problem (or the ground state of a quantum speed-up is thus open unless a lo-
of the many-body quantum system) is attained cal transformation to a stoquastic Hamiltonian is
via a superfluid-to-solid transition. From the found [49, 50].
measurement of photons that leave the cavity [41]
it can be determined if a state is a solution to the In our model implementation the non-local
N -queens problem. The position of the atoms qubit interactions are mediated via the field
can in addition be read out with single site re- modes of an optical resonator, which will attain
solved measurement. The final solution is a clas- non-classical atom-field superposition states dur-
sical configuration and thus easy to verify. We ing the parameter sweep. This appears to be
show that a full quantum description of the dy- an essential asset of the system as we find that
namics is required to find this solution. the ground-state is reached only with a very low
probability, when the full quantum dynamics of
Following Ref. [1], we identify a combination the fields is replaced by a classical mean-field ap-
of several unique features of the proposed model proximation.
that makes it a viable candidate to test quan-
tum advantage in near term devices. (a) The As a final feature let us point out here that the
completion and excluded diagonals problem is verification of a solution is computationally triv-
proven to be NP-complete and hard instances for ial as the final state is classical and no quantum
the excluded diagonals variant are known from tomography is needed. In principle the conver-
computer science literature [30], (b) the problem gence to a solution can be simply deduced from
maps naturally to the available toolbox of atoms the cavity outputs at the end of the sweep. With
in cavities and thus can be implemented with- this, the proposed setup may serve as a plat-
out intermediate embedding and no qubit over- form to demonstrate combinatorial quantum ad-
head, (c) the verification is computationally sim- vantage in near-term experiments.
ple and (d) the number of qubits required to solve This work is organized as follows: In Sec. 2
problems which are hard for classical computers we introduce a quantum model based on coupled
(N > 21 for the solvers used in Ref. [30]) is avail- quantum harmonic oscillators simulating the N
able in the lab. queens problem. A proposed physical implemen-
Methods such as minor embedding [23, 42], tation using ultracold atoms in optical lattices
LHZ [22, 43, 44] or nested embedding [45] always and light-mediated atom-atom interaction is de-
cause a qubit overhead. Here the intermediate scribed in Sec. 3. In Sec. 4 we present a numerical
step of embedding the optimization problem in comparison between model and implementation
an Ising model is removed by implementing the including photon loss. Finally we discuss in Sec.
infinite-range interactions with cavity-mediated 5 how light leaking out of the cavity can be used
forces tailored to the problem’s geometry in com- for read-out and we conclude in Sec. 6.
Accepted in Quantum 2019-05-28, click title to verify 2
2 Quantum simulation of the N - Constraints (ii), (iii) and (iv) are enforced
queens problem by infinite range interactions between the atoms
with
N
Following the idea of adiabatic quantum compu- HQ = UQ
X
Aijkl n̂ij n̂kl , (3)
tation [16–18], we construct a classical problem ijkl=1
Hamiltonian Hpr such that its ground state corre-
sponds to the solutions of the N -queens problem. where n̂i,j = b†i,j bi,j and UQ > 0. The interaction
In order to find this ground state, the system is matrix is
evolved with the time-dependent Hamiltonian
3
if (i, j) = (k, l)
t Aijkl = 1 if i = k ∨ i + j = k + l ∨ i − j = k − l
H(t) = Hkin + Hpr (1)
τ
0 otherwise,
from t = 0 to t = τ . Initially at t = 0, (4)
the system is prepared in the ground state of where in the first case all three constraints are
H(0) = Hkin . During the time evolution, the broken.
second term is slowly switched on. If this pa- In order to implement variations of the N -
rameter sweep is slow enough, the system stays queens problem, we need to exclude diagonals (for
in the instantaneous ground state and finally as- the excluded diagonals problem) and pin certain
sumes the ground state of H(τ ) at t = τ . If the queens (for the completion problem). These addi-
lowest energy gap of Hpr is much larger than the tional conditions are implemented by local energy
one of Hkin , this state is close to the ground state offsets of the desired lattice sites
of Hpr and thus the solution of the optimization
N N
problem. X X
Hpot = UD Dij n̂ij − UT Tij n̂ij . (5)
In the following we construct the problem i,j=1 i,j=1
Hamiltonian Hpr and the driver Hamiltonian
Hkin . The system is modeled as a 2D Bose- For UD > 0 the first term renders occupations
Hubbard model with annihilation (creation) op- of sites on chosen diagonals energetically unfa-
erators bij (b†ij ) on the sites (i, j). A position vorable. Each diagonal (in + and − direction)
of a queen is represented by the position of an has an index summarized in the sets D+ and D− ,
atom in an optical lattice with the total number respectively, and the coefficients are
of atoms being fixed to N . The non-attacking
2
if i + j − 1 ∈ D+ ∧ i − j + N ∈ D−
condition between queens, which amounts to in-
teractions between two sites (i, j) and (k, l), is im- Dij = 1 if i + j − 1 ∈ D+ ∨ i − j + N ∈ D−
plemented with four constraints: There can not 0 otherwise.
be two queens on the same line along (i) the x- (6)
direction j = l, (ii) the y-direction i = k, the For UT > 0, the second term favors occupations
diagonals (iii) i + j = k + l and (iv) i − j = k − l. of certain sites. The sites where queens should be
Condition (i) is implemented by using an initial pinned to are pooled in the set T and therefore
state with one atom in each horizontal line at yj the coefficients are given by
and restricting the atomic movement to the x- (
direction [see Fig. 1(b)]. Thereby we use the a 1 if (i, j) ∈ T
Tij = (7)
priori knowledge that a solution has one queen in 0 otherwise.
a row, which reduces the accessible configuration
2
space size from NN to N N configurations. In this The problem Hamiltonian of the N -queens prob-
vein the restricted tunneling Hamiltonian [32] is lem with excluded diagonals is then
given by Hpr = HQ + Hpot . (8)
N
X
Hkin = −J B̂ij , (2) Note that due to the initial condition atoms never
i,j=1
meet and sites are occupied by zero or one atom
where J is the tunneling amplitude and B̂ij = only. Hence the system can be effectively de-
b†i,j bi+1,j + b†i+1,j bi,j with B̂N j = 0 are the tunnel- scribed by spin operators [31, 40], also without
ing operators. large contact interactions.
Accepted in Quantum 2019-05-28, click title to verify 3
Figure 3: Energy spectrum. - Eigenvalue spectrum of
H(t) for (a) the model Hamiltonian [Eq. (1)] and (b)
Figure 2: Time evolution of site occupations. - Each
the Hamiltonian created by the field-atom interactions
subplot shows a snapshot of the site occupations hn̂ij i
[Eq. (10)] for the instance described in Fig. 2. The or-
for the parameter sweep in Eq. (1) and a sweep time
ange and green lines show ground state and first excited
Jτ /~ = 49. This is the instance shown in Fig. 1a, where
state, respectively. The blue lines depict higher energy
the excluded diagonals are indexed by D+ = {2, 3, 6, 9}
eigenstates. The low energy sector, and especially the
and D− = {1, 2, 8, 9} and one queen is pinned at site
minimal gap, are qualitatively similar.
(3, 5), i.e. T = {(3, 5)} (see text). The final values of
the sweep are UQ = J, UD = 5J, UT = 2J while J is
kept constant. Since the sweep time is large enough, the
state of the system adiabatically converges to the unique
3 Implementation
solution of the problem, which can be easily verified.
Here we propose a specific and at least concep-
tionally simple and straightforward physical im-
plementation of Eq. (1), where the N queens are
Let us illustrate the parameter sweep in Eq.
directly represented by N ultracold atoms in an
(1) for a specific example instance with N = 5
N xN two-dimensional optical lattice. Assuming
queens (see Fig. 1). The excluded diagonals cho-
tight binding conditions the atoms are confined to
sen here restrict the ground state manifold to
the lowest band of the lattice. They can coher-
two solutions, and by biasing site (3, 5) one of
ently tunnel between sites [51] and interact via
these solutions is singled out. The time evolu-
collective light scattering within an optical res-
tion of the site occupations hn̂ij i from numerically
onator. To implement the required queens inter-
solving the time-dependent Schrödinger equation
actions via light scattering we make use of a set
is shown in Fig. 2. Initially, the atoms are
of optical field modes in a multi-mode standing-
spread out in x-direction since the ground state
wave resonator [see Fig. 1(b)]. It was shown be-
of H(0) = Hkin is a superposition of excitations
fore that this configuration in principle allows to
along each tube. After evolving for a sufficiently
implement arbitrary site to site interactions [40]
large time Jτ /~ = 49, the system is in the ground
if a sufficient number of modes is used.
state of Hpr and thus assumed the solution of the
To reduce the necessary Hilbert space without
optimization problem.
loss of generality, it is sufficient to enable tun-
The energy spectrum of the given instance is neling only along one dimension (x), e.g. the
shown in Fig. 3(a). The minimal gap between rows of the lattice and increase the lattice depth
ground state (orange) and first excited state in the column (y) direction. This confines the
(green) determines the minimum sweep time τ atoms (queens) to move only along tubes in the
to remain in the ground state according to the x-direction forming a parallel array of N 1D opti-
Landau-Zener formula. At the end of the sweep, cal lattices as routinely used to study 1D physics
the ground state closely resembles the solution to with cold atoms [52].
the excluded diagonals problem shown in Fig. 1. Luckily, for the special case of the infinite range
The Hilbert space for the atomic state corre- queens interactions, one can find a strikingly sim-
sponding to the configuration space mentioned ple and intuitive example configuration requiring
above grows exponentially as N N and thus, as only few field modes for each of the three re-
usual for quantum systems, the computational maining queens interaction directions. For this
costs get large for rather small systems. Simula- we consider the optical lattice to be placed in the
tions with significantly larger systems are hence center symmetry plane of the optical resonator,
not easily tractable. where one has a common anti-node of all sym-
Accepted in Quantum 2019-05-28, click title to verify 4
metric eigenmodes. The trapped atoms are then We show this by numerical simulations in Sec. 4.
illuminated by running plane wave laser beams While the details of the interaction are altered it
from three different directions within the lattice still sufficiently well approximates the N -queens
plane with frequencies matched to different lon- interaction.
gitudinal cavity modes.
Depending on the size of the problem we need 3.1 Tight-binding model for atoms interacting
to add more laser frequencies in each direction via light
to avoid a periodic recurrence of the interac-
Driving far from any atomic resonance the inter-
tion within the lattice. Note that all frequencies
nal degrees of freedom of the atoms can be elim-
can be easily derived and simultaneously stabi-
inated. In this so-called dispersive limit the re-
lized from a single frequency comb with a spac-
sulting effective Hamiltonian couples the atomic
ing matched to the cavity length. Since all light
motion to the light fields [53].
frequencies are well separated compared to the
Single-particle Hamiltonian. For a single par-
cavity line width, each is scattered into a dis-
ticle of mass mA , the motion of the atoms in the
tinct cavity mode and as the cavity modes are
x-y-plane is described by [25, 54]
not directly coupled no relative phase stability
is needed [40]. The resulting position-dependent p̂2x + p̂2y
collective scattering into the cavity then intro- H1 = + VLx cos2 (kL x̂) + VLy cos2 (kL ŷ)
2mA
duces the desired infinite-range interactions be- M tot
tween the atoms. Choosing certain frequencies ˜ c,m a† am
X
2
+ Vbias |F(x̂, ŷ)| − ~ ∆ m
and varying their relative pump strengths al- m=1
lows for tailoring these interactions to simulate M tot
ηm h∗m (x̂, ŷ)am + a†m hm (x̂, ŷ) .
X
the three non-attacking conditions in the queens +~
problem discussed in the previous section. Addi- m=1
(9)
tional light sheets and local optical tweezers can
The first line contains the kinetic term with the
be used to make certain diagonals energetically
momentum operators p̂x and p̂y . Classical elec-
unfavorable (for the excluded diagonal problem)
tric fields create optical potentials with depths
or pin certain queens. Alternatively pinning can
VLx , VLy and Vbias . The first two create the optical
be achieved by selective cavity mode injection.
lattice with wave number kL and lattice spacing
In the first part (Sec. 3.1) we will now intro- a = π/kL , while Vbias is much smaller and only
duce a Bose-Hubbard-type Hamiltonian for the responsible for a bias field on certain sites, for in-
trapped atoms interacting via collective scatter- stance for excluding diagonals. Thereby F(x, y)
ing for general illumination fields in more detail. is an electric field distribution whose maximum
For this we first consider the full system of cou- is normalized to one.
pled atoms and cavity modes and later adiabati- The last two terms describe the free evolu-
cally eliminate the cavity fields to obtain effective tion of the cavity fields and atom-state-dependent
light-mediated atom-atom interactions. scattering of the pump fields into the cavity,
In the second part (Sec. 3.2) we discuss how to the atom-light interaction. The quantized elec-
implement the non-attacking conditions of the N - tric cavity fields are described by am (a†m ), the
queens problem with such light-mediated atom- annihilation (creation) operators of a photon in
atom interactions. Specifically, we consider the the m-th mode. These fields are coupled to
limit of a deep optical lattice leading to sim- the classical pump fields with mode functions
ple analytical expressions. These formulas allow hm (x, y) via the effective scattering amplitudes
us to find a specific pump configuration of wave ηm = gm Ωm /∆a,m , with the pump laser Rabi fre-
numbers and pump strengths leading to the ideal quencies Ωm , the atom-cavity couplings gm and
queens interaction Hamiltonian in Eq. (3). As the detunings between pump lasers and atomic
a deep lattice depth slows down atomic tunnel- resonance frequency ∆a,m . The effective cavity
ing it requires long annealing times. Luckily, it detunings ∆ ˜ c,m = ∆c,m − N U0,m are given by the
turns out that the pump configuration derived for detunings between pump laser and cavity mode
a deep lattice still gives the same ground state frequencies ∆c,m and the dispersive shifts of the
for moderate lattice depth with faster tunneling. cavity resonance due to the presence of the atoms
Accepted in Quantum 2019-05-28, click title to verify 5
in the cavity N U0,m [24]. The structure of the fields enters in the on-site
Note that, for example by placing the opti- and nearest neighbor atom-mode overlaps
cal lattice (x-y-plane) in a common anti-node of Z
ij
the standing wave cavity modes and exciting only vm = dx w2 (x − xi )hm (x, yj ) (13)
TEM00 modes, the atom-cavity coupling is uni- Z
form in space in our model. Thus the only spatial uij
m = dx w(x − xi )hm (x, yj )w(x − xi+1 ) (14)
dependence in the cavity term is due to the pump
fields. where yj = ja with j = 0, ..., N − 1 are the
Generalized Bose-Hubbard Hamiltonian. The tube positions and w(x) the one-dimensional
atom-atom interactions are taken into account Wannier functions in x-direction. This is be-
by introducing bosonic field operators Ψ̂(x) with cause for VLy VLx we can approximate the y-
†
R 2
H̃ = d xΨ̂ (x)H1 Ψ(x) [25, 55]. Note that we dependence of the Wannier functions by a Dirac
do not include contact interactions since atoms delta: w2D (x) = w(x)δ(y).
never meet due to the initial condition we will The last term Hpot describes all extra fields re-
use. We assume that the optical lattice with sponsible for local energy off-sets at certain sites
depths VLx and VLy is so deep, that the atoms are that stem from the weak classical fields with the
tightly bound at the potential minima and only distribution F(x, y) (Vbias VLx ). The off-sets
the lowest vibrational state (Bloch band) is occu- ought to be calculated from the overlaps of fields
and Wannier functions, analogously to vm ij . Thus
pied. Moreover, the optical potential created by
bias and cavity fields is comparably small, such the fields have to be chosen such that the result-
that the form of the Bloch wave functions only de- ing Hamiltonian resembles Eq. (5). We do not
pends on the optical lattice [54]. In this limit we detail the derivation further here and use Hpot
can expand the bosonic field operators in a local- for numerical simulations.
ized Wannier basis Ψ̂(x) = i,j w2D (x − xij )bij
P
Atom-atom interaction Hamiltonian. The
with the lowest-band Wannier functions w2D (x) main focus of this work is to show how to cre-
coming from Bloch wave functions of the lattice ate the tailored all-to-all particle interactions via
[56]. We split the resulting Hamiltonian in three collective scattering. We derive this interaction
terms by eliminating the cavity fields introduced in the
H̃ = Hkin + Hcav + Hpot , (10) previous section [54, 57, 58]. This can be done
which will be explained in the following. because the cavity fields decay through the mir-
As in the standard Bose-Hubbard model, one rors with the rates κm and thus end up in a
obtains a tunneling term Hkin as in Eq. (2). Tun- particular steady-state for each atomic config-
neling in y-direction is frozen out by ensuring uration. Assuming that the atomic motion is
VLy VLx . The other terms Hcav and Hpot much slower than the cavity field dynamics, i.e.
J/~ |∆ ˜ c,m + iκm |, this steady-state is a good
originate from the weak cavity-pump interference
fields and the bias fields introduced above and approximation at all times. The stationary cavity
should resemble Hpr [Eq. (8)]. In order to realize field amplitudes are given by
the sweep Eq. (1), the relative strength of these ηm
ast
m ≡ N Θ̂m (15)
terms and the kinetic term has to be tuned, e.g. ˜
∆c,m + iκm
by ramping up the pump laser and bias field in-
tensity (make Hcav and Hpot larger) or the lattice and thus replaced by atomic operators (see Ap-
depth (make Hkin smaller). pendix B).
The cavity-related terms in Eq. (9) give rise to In the coherent regime |∆˜ c,m | κm , the atom-
light interaction is then described by an effective
˜ c,m a† am
X
Hcav = − ~ ∆ m interaction Hamiltonian for the atoms [28]
m
(11)
N ηm Θ̂†m am + a†m Θ̂m
X
+~ ∆˜ c,m η 2
m
N 2 Θ̂†m Θ̂m .
X
eff
m Hcav =~ (16)
˜ 2 + κ2
∆
m c,m m
with the order operator of cavity mode m
N The collective, state-dependent scattering in-
1 X ij
duces interactions between each pair of sites (i, j)
Θ̂m = vm n̂ij + uij
m B̂ij . (12)
N i,j=1 and (k, l): Density-density interactions due to the
Accepted in Quantum 2019-05-28, click title to verify 6terms containing n̂ij n̂kl and a modified tunnel- dicular to the lines along which queens should
ing amplitude due an occupation or a tunneling not align, that is along the x-direction and along
event somewhere else in the lattice due to n̂ij B̂kl , the diagonals. We denote the corresponding wave
B̂ij n̂kl and B̂ij B̂kl . While the density-density vectors with kxm = (km 0 , 0)T , k+ = (k 0 , k 0 )T and
m m m
interactions constitute the problem Hamiltonian k−
m = (km
0 , −k 0 )T , respectively, with the wave
m
[see Eq. (3)], the latter cavity-induced tunneling numbers km 0 . Therefore the pump fields are given
terms lead to non-local fluctuations which might by
help to speed up the annealing process [46]. hm (x, y) = eikm x , (18)
Since the Wannier functions are localized at
ij tend to where x = (x, y)T is the position vector and km
the lattice sites, the on-site overlaps vm
is a wave vector in any of the three directions.
be much larger than the nearest-neighbor over-
With running wave pump fields, Eq. (17) can
laps uij m . Thus density-density interactions are be written as
expected to be the dominant contribution to
eff . As intuitively expected, the atoms local-
Hcav dl
X
Hcav = UQ Ãijkl n̂ij n̂kl . (19)
ize stronger for deeper lattices, where analyti- ijkl
cal expressions for the overlaps can be obtained
within a harmonic approximation of the poten- This formally corresponds to HQ [Eq. (3)], where
tial wells leading to Gaussian Wannier functions the quantities now have a physical meaning: The
with a width ∝ (VLx )−1/4 . Apart from a correc- interaction matrix is given by
tion factor due to this width, the on-site over- X
Ãijkl = fm cos(km (xij − xkl )) (20)
laps are given by the pump fields at the lat- m
tice sites. The nearest-neighbor overlaps corre-
spond to the pump fields in between the lattice with lattice site connection vectors xij − xkl and
sites, but are exponentially suppressed (see Ap-
∆˜ c,m η 2
pendix C). Consequently, in the deep lattice limit fm UQ = ~ m
(21)
(large VLx ) when the width tends to zero we get ˜
∆c,m + κ2m
2
vmij = h (x , y ) and uij = 0 [28], and an interac-
m i j m −1
with M
P
tion Hamiltonian [from Eq. (16)] m=0 fm = 1. The dimensionless param-
eters fm capture the relative strengths of the
˜ c,m η 2
X ∆ modes, determining the shape of the interaction.
dl m
Hcav =~
˜ 2 + κ2
∆ They have to be chosen such that à approximates
m c,m m
A [Eq. (4)]. The overall strength of the interac-
h∗m (xi , yj )hm (xk , yl )n̂ij n̂kl ,
X
× tion term is captured by the energy UQ , which
ijkl
(17) can be easily tuned by the cavity detunings or
which only depends on density operators, and the pump intensities to implement the parameter
hence does not include cavity induced-tunneling. sweep in Eq. (1). For the following discussion we
define an interaction function
3.2 N -queens interaction
X
Ã(r) = fm cos(km r) (22)
m
In this section we aim to find pump fields hm (x, y)
such that the interaction Hamiltonian in the which returns the interaction matrix when eval-
deep lattice limit Eq. (17) corresponds to the de- uated at lattice site connection vectors Ãijkl =
sired queens Hamiltonian HQ [Eq. (3)] containing Ã(xij − xkl ).
the non-attacking conditions. Using these pump We note that one set of parallel kµm (µ ∈
fields we later show numerically in Sec. 4, that the {x, +, −}) creates an interaction à which is con-
atom-atom interaction for realistic lattice depths stant and infinite range (only limited by the laser
[Eq. (16)], although slightly altered, still well re- beam waist) in the direction perpendicular to the
sembles the queens interaction. propagation direction r ⊥ kµm . Along the prop-
We consider three sets of M parallel running agation direction r k kµm instead, the interaction
wave laser beams with different propagation di- is shaped according to the sum of cosines, and
rections, each of which could be created by a fre- can be modified by the choice of wave numbers
quency comb. The three directions are perpen- kmµ = |kµ | and their relative strengths f .
m m
Accepted in Quantum 2019-05-28, click title to verify 7In the following we will use the example wave
numbers
2m + 1
0
km = kL 1+ (23)
2M
with m = 0, ..., M − 1 and uniform fm = 1/M .
Taking into account kxm only, the interaction
along the x-direction r k kxm at lattice site dis-
tances rj = jπ/kL = ja has the values
(
(−1)l for j = 2M l, l ∈ Z
Ã(rj ) = (24)
0 otherwise,
Figure 4: Energy penalty created by one atom. - These
as shown in Appendix D. If we guarantee that density plots show the energy penalty Ã(x − xa ) for
−2M < j < 2M this results in an interac- an atom at position x = (x, y)T created by an atom
tion which is zero everywhere apart from j = 0, at position xa = (a, a)T (red dot) for N = M = 5
i.e. at zero distance. So for repulsive interac- [see Eq. (22)]. The dots indicate lattice site posi-
˜ c,m > 0), the wave tions and the solid (dashed) contour lines indicate where
tions (UQ > 0 and thus ∆
x Ã(x−xa ) = 0 (Ã(x−xa ) = 1) is fulfilled. (a) Pumping
vectors km create the non-attacking interaction along the x-axis with the wave vectors kxm = (km 0
, 0)T
along the y-direction (Ãijkl = 1 if i = k and 0 0
with km /kL = 1.1, 1.3, 1.5, 1.7, 1.9 [according to Eq.
otherwise) as long as N ≤ 2M . This is illus- (23)] creates interactions along y. (b) Additionally in-
trated in Fig. 4(a) for M = N = 5. Analo- cluding diagonal pump lasers k+ 0
m = (km , km )
0 T
and
gously, k±m cause the non-attacking interactions k−m = (km
0
, −km
0 T
) implements the full queens inter-
along the diagonals. In a square action along diagonals and vertical lines [Eq. (3)].
√ lattice the di-
agonals have the distance rj √ / 2, which is com-
pensated by km ± = |k± | = 0 . Since there
2km
m delta at x = 0. See Appendix D for a detailed
are 2N − 1 diagonals, one has to make sure that derivation.
2N − 1 ≤ 2M . Upon combining all wave vectors The last line allows for a simple interpretation:
from three directions we finally obtain the full The interaction consists of peaks repeating with
queens interaction, as shown in Fig. 4(b), which a spatial period R = 2π/∆k. Each of these peaks
is realized with Mtot = 3M = 3N frequencies in has the shape of the real part of the Fourier trans-
our example. form of the envelope function f (k) with a width
Note that there are several combinations of corresponding to the inverse of the mode band-
wave numbers and mode strengths which, at least width σ ∼ 2π/∆kBW .
approximately, create the desired line-shaped in- For the (approximate) non-attacking condition
teractions perpendicular to the light propagation (Ã(ja) ≈ 1 for j = 0 and |Ã(ja)| 1 other-
direction. For this it is insightful to reformulate wise) there are two conditions. Firstly, at most
the interaction as a Fourier transform. To deal one peak should be within the region of the
with continuous functions, we define an envelope atoms. Thus the period has to be larger than
f (k) with f (km ) = fm , which is sampled at the the (diagonal) size of the optical lattice R ≥ N a
√
wave numbers n∆k with n ∈ Z containing all (R ≥ 2N a). Secondly, the width of one peak
kmµ . Considering one illumination direction for
has to be smaller than the lattice spacing σ . a.
simplicity, the interaction [Eq. (22)] along r k kµm Combining these conditions to R & N σ, we see
with r = |r| can be written as that the minimum number of modes per direction
"
√
( ∞
X
) # M scales linearly with N
Ã(r) = Re 2πF f (k) δ(k − n∆k) (r)
n=−∞ M ≈ ∆kBW /∆k & N. (26)
∞ √
2π
Therefore, with only ∼ N modes this quite gener-
X
= Re 2πF{f } r − l ,
l=−∞
∆k ically allows for creating an interaction along lines
R∞ √ (25) perpendicular to the light propagation. Note
where F{f }(r) = −∞ dkf (k)eikr / 2π is the however, that the second condition also implies
Fourier transform of f (k) and δ(x) is a Dirac that the spatial frequency spread has to be at
Accepted in Quantum 2019-05-28, click title to verify 8least on the order of the lattice wave number
∆kBW & kL .
4 Numerical justification of assump-
tions
We compare the ideal model Hamiltonian de-
scribed in Sec. 2 [Eq. (1)] to the physically mo-
tivated tight-binding Hamiltonian for finite lat-
tice depths introduced in Sec. 3 [Eq. (10)]. As
for the ideal model in Fig. 2, we consider the
time evolution during a slow linear sweep of UQ ,
UT and UD by numerically integrating the time-
dependent Schrödinger equation. Physically, this
sweep can be realized by ramping up the pump
and the bias field intensities. Moreover, we show
that evolving the system using a classical approx-
imation for the cavity mode fields does not result
Figure 5: Comparison of the time evolution with quan-
in a solution to the N -queens problem, and ad- tum and classical fields. - Lattice site occupations for
dress the effect of dephasing by photon loss with a time evolution during a nearly adiabatic sweep using
open system simulations. the cavity Hamiltonian including cavity-assisted tunnel-
In the following we use a realistic lattice depth ing for the same parameters as in Fig. 2. Subplot (a)
of VLx = 10ER with the recoil energy ER = shows the dynamics using the full quantum interaction
~2 kL2 /(2mA ). For example, for rubidium 87 Rb Hamiltonian [Eq. (16)]. It closely resembles the results
from the model Hamiltonian in Fig. 2. Subplot (b) shows
and λL = 785.3 nm it is ER /~ = 23.4 kHz [26].
the time evolution with the classical approximation of
The chosen lattice depth leads to a tunneling am- the cavity fields [Eq. (27)]. The state does not converge
plitude J ≈ 0.02ER , which can be obtained from to the solution, also not for much larger sweep times.
the band structure of the lattice. We consider our The modes for both cases where chosen as in Fig. 4(b).
cavity model in Eq. (10) for N = 5. The pump
modes are as in Sec. 3.2 and Fig. 4. While in
the limit of a deep lattice this would result in the mode functions by the finite width Wannier func-
ideal model interactions, here they depend on the tions (see Appendix C). Moreover, we consider
overlaps between Wannier functions and pump the time evolution during the nearly adiabatic
modes [Eq. (13)] and are thus altered. In the sweep for Jτ /~ = 49 for the same parameters
following the overlaps are calculated with Wan- by integrating the time-dependent Schrödinger
nier functions which where numerically obtained equation. Snapshots of the site occupations hn̂ij i
from the band structure of the lattice. It turns for several times are shown in Fig. 5(a), where
out, that the deviation from the ideal overlaps we observe a similar behavior as for the model
does not qualitatively change the interaction for in Fig. 2. This suggests that the system is robust
the realistic parameters used. against the errors introduced by the moderate lat-
tice depth. Note that the physical Hamiltonian
used here is non-stoquastic in the occupation ba-
4.1 Coherent dynamics sis due to cavity-induced tunneling.
The energy spectrum of the Hamiltonian for fi-
nite lattice depths in Eq. (16) is shown in Fig. 4.2 Classical cavity fields
3(b) and is qualitatively of the same form as for
the ideal model in Fig. 3(a). In comparison the In the following we show that quantum correla-
eigenvalue gaps tend to be smaller at the end tions are crucial for the efficiency of the sweep.
of the sweep. This is because the on-site atom- In particular, if we substitute the field operators
mode overlaps decrease for shallower lattices and ast
m by its expectation values representing classi-
less localized atoms due to a smoothing of the cal cavity fields the solution is not found. We
Accepted in Quantum 2019-05-28, click title to verify 9consider the semi-classical Hamiltonian
˜ c,m η 2
X N 2∆
class m
Hcav =~
˜ 2 + κ2
∆
m c,m m
× Θ̂†m hΘ̂m i + hΘ̂†m iΘ̂m − |hΘ̂m i|2 ,
(27)
where the expectation values have to be calcu-
lated self-consistently with the current atom state
vector. This substitution amounts to considering
Figure 6: Comparison to open system dynamics. - (a)
only first order fluctuations around the mean of The figure shows the fidelity Fsol between the solution
Θ̂m in Eq. (16). and the instantaneous eigenstate (black dotted), and be-
Consequently, the dynamics are described by tween solution and dynamical state from the time evo-
a differential equation which is non-linear in the lution using the Schrödinger equation (blue), classical
state vector |ψi. We numerically solve this equa- fields (green) and the open system (red). The open sys-
tem fidelity depends on ∆ ˜ c /κ, where we present results
tion by self-consistently updating the expectation
value in each time step. It turns out that even for for the values 5, 50, 100, 200, 500, 1000 (from light red
to dark red). (b) The fidelity between solution and the
very long sweep times, using classical fields does ˜ c /κ for the
final state after the sweep as a function of ∆
not lead to a solution of the queens problem. The curves of (a), where the colors are the same as in (a).
time evolution for Jτ /~ = 49 is depicted in Fig. The parameters used here are as in Fig. 5.
5(b). The discrepancy shows the necessity of en-
tangled light-matter states in our procedure.
with the rates
4.3 Dephasing due to cavity field loss ˜ c,m (|αµ |2 − |αν |2 )
X
Ωµν = ∆ m m (30)
m
Finally, motivated by experimental considera- X
µ ν 2
Γµν = κm |αm − αm | . (31)
tions, we consider the open system including pho- m
ton decay through the cavity mirrors. This sys-
tem can be described by a master equation for While the energy gaps ~Ωµν describe the coherent
the atoms [54] dynamics we considered up to now, the dephasing
rates Γµν stem from photon loss. Hence a super-
i eff position of two scattering eigenstates |µi and |νi
ρ̇ = − [Hkin + Hcav , ρ]
~ looses its coherence depending on the difference
X N 2 η 2 κm
of the scattered fields, or how distinguishable the
m † †
+ 2 2
2 Θ̂m ρΘ̂m − { Θ̂ Θ̂
m m , ρ} ,
m ∆c,m + κm states are by field measurement.
(28) We now return to the example we had before,
where curly brackets denote the anti-commutator using uniform mode strengths fm = 1/M , de-
(see Appendix B). The model is suitable for ana- cay rates κ and detunings ∆ ˜ c . Figure 6 com-
lyzing the dephasing close to the coherent regime pares the coherent Schrödinger time evolution
before any steady state is reached. from Sec. 4.1, the mean-field approximation with
More insight can be gained by rewriting it classical cavity fields from Sec. 4.2 and the open
in the basis of scattering eigenstates |νi with system dynamics with dephasing. For the lat-
Θ̂m |νi = θm ν |νi, which scatter a field αν = ter the master equation is approximated by us-
m
N ηm ν . These states converge to the occu-
˜ c,m +iκm θ m ing Monte-Carlo wave function simulations. As
∆
pation states in the deep lattice limit. The time a measure of similarity between states we use
evolution for the matrix elements ρµν = hµ|ρ|νi the fidelity. For q two mixed states it is defined
√ √
reads as F (ρ, σ) = Tr( ρσ ρ) and reduces to the
overlap |hψ|φi| for pure states.
ρ̇µν =(−Γµν − iΩµν )ρµν The open system dynamics is depicted for dif-
J X
ferent detunings ∆ ˜ c /κ while keeping UQ fixed.
−i hµ|B̂|kiρkν − hk|B̂|νiρµk
~ k This can be achieved by adjusting the pump
(29) strength η correspondingly. In this case the de-
Accepted in Quantum 2019-05-28, click title to verify 10phasing rates by an atomic state |ψi
† st
X
P (|ψi) = 2κm h(ast
m ) am i
~Γµν κ 2 X
µ ν 2 m
= N |θm − θm | (32)
UQ ˜
∆c ˜ c,m η 2 N 2
X 2κm ∆
m
= m
hΘ̂†m Θ̂m i
˜ c,m ∆
∆ ˜ 2 + κ2
m c,m m
go to zero for ∆ ˜ c /κ 1. Thus, as ex- ζX ζ dl
≈UQ Ãijkl hn̂ij n̂kl i = hHcav i.
pected, the open system converges to the coher- ~ ijkl ~
ent Schrödinger dynamics in this limit (see Fig. (33)
6(b)). In the last line we assumed that ζ = 2κm /∆ ˜ c,m
Note that the coherence between states creat- does not depend on m and a deep lattice.
ing similar fields is preserved much longer than Since P is proportional to the energy expec-
for other states, which is expected to be impor- tation value, the ground state, i.e. the solution
tant at the late stage of the sweep. For states with of the queens problem, causes a minimal photon
fixed similarity (e.g. one atom moved), |θm µ −θ ν |2
m
flux at the detector P0 = 3N UQ ζ. It stems from
−2
is on the order of N , and thus the dephasing the on-site terms (i, j) = (k, l), where the fac-
rates do not scale with N for such states. tor 3 comes from the three pump directions. In
contrast, each pair of queens violating the non-
attacking condition in à leads to an increase of
the photon flux by ∆P = 2UQ ζ. The two atoms
5 Read-out create an energy penalty for one another, explain-
ing the factor 2. The relative difference of the
photon flux due to a state with L attacking pairs
After the parameter sweep we need to determine and a solution is given by
if the obtained state is a solution or not. This
can in principle be done by reading out the fi- L∆P 2L
= . (34)
nal atomic state with single site resolution using P0 3N
a quantum gas microscope [59, 60]. However, as
we consider an open system with the cavity out- As this scales with 1/N it is difficult to distin-
put fields readily available, we will show that by guish solutions from other states via measure-
proper measurements on the output light we can ment of the intensity for large N . Note that
˜ c,m , photons from different
for non-uniform κm /∆
directly answer this question without further ad-
ditions. Note that after the sweep at the stage modes have to be distinguished.
of the read-out, quantum coherences do not have
to be preserved since the solution is a classical 5.2 Field measurement
state. This gives the freedom to increase the lat- Measurement of the output field quadratures, for
tice depth to some high value in the deep lattice example by homodyne detection, gives insight
regime to suppress further tunneling, and to in- about the absolute position of the atoms pro-
crease the pump power or decrease the detunings jected onto the pump laser propagation direction.
in order to get a stronger signal at the detector. For the three directions used in our setup, this
yields the occupations of each column hN̂ix i and
each diagonal hN̂i+ i and hN̂i− i. Since a solution
of the queens problem has maximally one atom
5.1 Intensity measurement
on each diagonal and exactly one atom on each
column, it must fulfill
For uniform cavity detunings, a state correspond-
ing to the solution of the N -queens problem scat- hN̂ix i = 1 ∧ hN̂i+ i ≤ 1 ∧ hN̂i− i ≤ 1. (35)
ters less photons than all other states. Thus the
measurement of the total intensity in principle The output field quadratures can thus be used to
allows one to distinguish a solution from other determine if a classical final state is a solution or
states. To illustrate this we consider the total not, which is the answer to the blocked diagonals
rate of photons impinging on a detector scattered decision problem we aim to solve. The signatures
Accepted in Quantum 2019-05-28, click title to verify 11total occupations of the columns N̂ix =
P
j n̂ij .
For at least N modes (M ≥ N ) this system
of equations can be inverted yielding the column
occupations hN̂ix i. By measuring the cavity out-
put field quadratures scattered from the diagonal
pump light we obtain the occupations of each di-
agonal hN̂i+ i and hN̂i− i. Inverting the system of
equations for diagonals demands at least as many
pump modes as diagonals, that is 2N − 1. Inver-
sion can also be done efficiently and intuitively
by using a discrete Fourier transform and its in-
verse. To reveal the Fourier relation, one has to
express the vmij ’s in Eq. (36) within the harmonic
(or in the deep lattice) approximation (see Ap-
pendix C). The so obtained approximate inver-
sion formula also works well for realistic lattice
depths.
Figure 7: Signature of an atomic state in the cavity out-
put. - In left and right column the signature of two Strictly speaking the condition in Eq. (35) is
different atomic (pure) states are compared, whose oc- sufficient only for classical configurations, like oc-
cupations are shown on the top (a). On the left there is a cupation number basis states |φν i. Some super-
solution to the N -queens problem, while on the right one positions |ψi = ν cν |φν i which are no solu-
P
atom was moved. (b) The polar plot shows the cavity tions might also fulfill the above criterion, be-
field expectation value hast
m i in the complex plane. Mea- cause summands in the field expectation val-
suring the fields by homodyne detection yields a certain
ues hast mi =
2 st
ν |cν | hφν |am |φν i can cancel each
P
quadrature of the field depending on the phase angle
φ. An example is illustrated by the black line. (c) The other. For instance, for UT = 0 the solution
measurable quadratures are shown for the example an- from our example in Fig. 2 |ψsol i = |1, 4, 2, 5, 3i
gle. The sequences can be Fourier transformed to obtain scatters the same fields hast m i as the √ superpo-
the occupations of columns and of the diagonals. The 1 2
sition |ψnosol i = (|ψnosol i + |ψnosol i)/ 2 with
parameters used here are UQ = 5J and ∆ ˜ c /κ = 10. We 1 2
|ψnosol i = |1, 3, 2, 5, 4i and |ψnosol i = |1, 4, 5, 2, 3i,
used M = 2N − 1 = 9 modes per direction with wave both of which are no solution. In this notation
numbers as in Eq. (23). Different colors encode the dif-
the state |i1 , i2 , ..., iN i has one atom on each site
ferent directions of the pump modes: kxm (blue), k+ m
(green) and k− (ij , j). However, these macroscopic superposi-
m (red). The lighter the color the larger
the wave number |kx,± tions are highly unstable. Even theoretically the
m |.
measurement back-action [41, 61] projects super-
positions of states scattering different fields (such
of two example states in the cavity fields are de- as |ψnosol i) to one of its constituents. The inclu-
picted in Fig. 7. sion of measurement back-action due to contin-
Let us illustrate the measurement by consid- uous measurement might thus lead to intriguing
ering only light scattered from the x-direction phenomena beyond those presented here and is
with incident wave vectors kxm . Since these plane subject to future work.
waves are constant in y-direction, the atom-field
overlaps do not depend on j. Neglecting cavity-
induced tunneling, the field quadratures for a We emphasize again that the measurements de-
phase difference φ are scribed above answer the question if we found a
! solution or not, which is the answer to the com-
−iφ
X ηm e−iφ binatorial decision problem. The exact configu-
Re(hast
m ie )= Re v i1 hN̂ix i,
˜ c,m + iκm m
∆ ration of the final state can be measured with
i
(36) single site resolution as demonstrated in several
revealing that cavity fields are determined by the experiments [59, 60].
Accepted in Quantum 2019-05-28, click title to verify 126 Conclusions NP-complete and numerically hard [30].
Acknowledgments. We thank I. Gent, C. Jef-
We present a special purpose quantum simulator ferson and P. Nightingale for fruitful discus-
with the aim to solve variations of the N -queens sions. Simulations were performed using the open
problem based on atoms in a cavity. This com- source QuantumOptics.jl framework in Julia [64]
binatorial problem may serve as a benchmark to and we thank D. Plankensteiner for related dis-
study a possible quantum advantage in interme- cussions. V. T. and H. R. are supported by Aus-
diate size near term quantum experiments. From trian Science Fund Project No. I1697-N27. W.
the algorithmic point of view, the problem is in- L. acknowledges funding by the Austrian Sci-
teresting for quantum advantage as it is proven ence Fund (FWF) through a START grant under
NP-hard and instances can be found that are not Project No. Y1067-N27 and the SFB BeyondC
solvable with current state-of-the-art algorithms. Project No. F7108-N38, the Hauser-Raspe foun-
From the implementation point of view, the pro- dation, and the European Union’s Horizon 2020
posed quantum simulator implements the queens research and innovation program under grant
problem without overhead and thus a few tens agreement No. 817482 PasQuanS.
of atoms are sufficient to enter the classically in-
tractable regime. The proposed setup of atoms in
a cavity fits the queens problem naturally as the
required infinite range interactions arise there in-
herently. We find that by treating the light field
classically the simulation does not find the solu-
tions suggesting that quantum effects like atom-
field entanglement cannot be neglected. More-
over, we investigate the influence of photon loss
on the coherence time.
The queens problem is formulated as a decision
problem, asking whether there is a valid config-
uration of queens or not given the excluded di-
agonals and fixed queens. Remarkably, to answer
the decision problem, a read-out of the atom posi-
tions is not required as the necessary information
is encoded in the light that leaves the cavity. To
determine the position of the queens requires sin-
gle site resolved read-out, which is also available
in several current experimental setups [59].
In this work we concentrated on the coherent
regime. The driven-dissipative nature of the sys-
tem provides additional features which can be ex-
ploited for obtaining the ground state. For cer-
tain regimes, cavity cooling [24, 62] can help to
further reduce sweep times and implement error
correction. Moreover, the back action of the field
measurement onto the atomic state can be used
for preparing states [41].
Note that an implementation of the N -queens
problem for a gate-based quantum computer was
proposed in Ref. [63] aiming to find a solution
of the unconstrained N -queens problem. Our
work in contrast employs an adiabatic protocol
and intends to answer the question if a solution
exists given constraints of blocked diagonals or
already placed queens, which was shown to be
Accepted in Quantum 2019-05-28, click title to verify 13References Martinis, and Hartmut Neven. Character-
izing quantum supremacy in near-term de-
[1] Aram W Harrow and Ashley Montanaro. vices. Nature Physics, 14(6):595, 2018. DOI:
Quantum computational supremacy. Nature, 10.1038/s41567-018-0124-x.
549(7671):203–209, 2017. DOI: 10.1038/na- [12] John Preskill. Quantum computing and
ture23458. the entanglement frontier. arXiv:1203.5813
[2] J Ignacio Cirac and Peter Zoller. Goals and [quant-ph], 2012.
opportunities in quantum simulation. Nat. [13] Hannes Bernien, Sylvain Schwartz, Alexan-
Phys., 8(4):264–266, apr 2012. ISSN 1745- der Keesling, Harry Levine, Ahmed Omran,
2473. DOI: 10.1038/nphys2275. Hannes Pichler, Soonwon Choi, Alexander S
[3] Rainer Blatt and Christian F Roos. Quan- Zibrov, Manuel Endres, Markus Greiner,
tum simulations with trapped ions. Na- et al. Probing many-body dynamics on a 51-
ture Physics, 8(4):277–284, 2012. DOI: atom quantum simulator. Nature, 551(7682):
10.1038/nphys2252. 579, 2017. DOI: 10.1038/nature24622.
[4] Immanuel Bloch, Jean Dalibard, and Sylvain
[14] Kihwan Kim, M-S Chang, Simcha Korenblit,
Nascimbene. Quantum simulations with ul-
Rajibul Islam, Emily E Edwards, James K
tracold quantum gases. Nat. Phys., 8(4):
Freericks, G-D Lin, L-M Duan, and Christo-
267–276, apr 2012. ISSN 1745-2473. DOI:
pher Monroe. Quantum simulation of frus-
10.1038/nphys2259.
trated ising spins with trapped ions. Nature,
[5] Alán Aspuru-Guzik and Philip Walther.
465(7298):590–593, 2010. DOI: 10.1038/na-
Photonic quantum simulators. Na-
ture09071.
ture Physics, 8(4):285–291, 2012. DOI:
[15] Peter Schauß, Marc Cheneau, Manuel En-
10.1038/nphys2253.
dres, Takeshi Fukuhara, Sebastian Hild,
[6] Sylvain De Léséleuc, Sebastian Weber, Vin-
Ahmed Omran, Thomas Pohl, Christian
cent Lienhard, Daniel Barredo, Hans Pe-
Gross, Stefan Kuhr, and Immanuel Bloch.
ter Büchler, Thierry Lahaye, and Antoine
Observation of spatially ordered structures
Browaeys. Accurate mapping of multi-
in a two-dimensional rydberg gas. Nature,
level rydberg atoms on interacting spin-
491(7422):87–91, 2012. DOI: 10.1038/na-
1/2 particles for the quantum simulation
ture11596.
of ising models. Physical review letters,
120(11):113602, 2018. DOI: 10.1103/Phys- [16] Tadashi Kadowaki and Hidetoshi Nishimori.
RevLett.120.113602. Quantum annealing in the transverse ising
[7] Andrew A Houck, Hakan E Türeci, and Jens model. Phys. Rev. E, 58:5355–5363, Nov
Koch. On-chip quantum simulation with su- 1998. DOI: 10.1103/PhysRevE.58.5355.
perconducting circuits. Nature Physics, 8(4): [17] Edward Farhi, Jeffrey Goldstone, Sam Gut-
292–299, 2012. DOI: 10.1038/nphys2251. mann, and Michael Sipser. Quantum compu-
[8] I. M. Georgescu, S. Ashhab, and Franco tation by adiabatic evolution. arXiv:quant-
Nori. Quantum simulation. Rev. Mod. ph/0001106, 2000.
Phys., 86:153–185, Mar 2014. DOI: [18] Tameem Albash and Daniel A Lidar. Adi-
10.1103/RevModPhys.86.153. abatic quantum computation. Reviews of
[9] Mark Saffman. Quantum computing with Modern Physics, 90(1):015002, 2018. DOI:
atomic qubits and rydberg interactions: 10.1103/RevModPhys.90.015002.
progress and challenges. Journal of Physics [19] Andrew Lucas. Ising formulations of many
B: Atomic, Molecular and Optical Physics, np problems. Frontiers in Physics, 2:5, 2014.
49(20):202001, 2016. DOI: 10.1088/0953- DOI: 10.3389/fphy.2014.00005.
4075/49/20/202001. [20] Sergio Boixo, Troels F Rønnow, Sergei V
[10] John Preskill. Quantum computing in the Isakov, Zhihui Wang, David Wecker,
nisq era and beyond. Quantum, 2:79, 2018. Daniel A Lidar, John M Martinis, and
DOI: 10.22331/q-2018-08-06-79. Matthias Troyer. Evidence for quantum
[11] Sergio Boixo, Sergei V Isakov, Vadim N annealing with more than one hundred
Smelyanskiy, Ryan Babbush, Nan Ding, qubits. Nature Physics, 10(3):218, 2014.
Zhang Jiang, Michael J Bremner, John M DOI: 10.1038/nphys2900.
Accepted in Quantum 2019-05-28, click title to verify 14[21] Philipp Hauke, Helmut G Katzgraber, Wolf- ter Nightingale. Complexity of n-queens
gang Lechner, Hidetoshi Nishimori, and completion. Journal of Artificial Intelli-
William D Oliver. Perspectives of quantum gence Research, 59:815–848, 2017. DOI:
annealing: Methods and implementations. 10.1613/jair.5512.
arXiv:1903.06559 [quant-ph], 2019. [31] Itay Hen. Realizable quantum adia-
[22] Wolfgang Lechner, Philipp Hauke, and Pe- batic search. EPL (Europhysics Letters),
ter Zoller. A quantum annealing architecture 118(3):30003, 2017. DOI: 10.1209/0295-
with all-to-all connectivity from local inter- 5075/118/30003.
actions. Science Advances, 1(9), 2015. DOI: [32] Itay Hen and Federico M Spedalieri. Quan-
10.1126/sciadv.1500838. tum annealing for constrained optimization.
[23] Vicky Choi. Minor-embedding in adia- Physical Review Applied, 5(3):034007, 2016.
batic quantum computation: I. the param- DOI: 10.1103/PhysRevApplied.5.034007.
eter setting problem. Quantum Informa- [33] Peter Domokos and Helmut Ritsch. Collec-
tion Processing, 7(5):193–209, 2008. DOI: tive cooling and self-organization of atoms
10.1007/s11128-008-0082-9. in a cavity. Physical review letters, 89
[24] Helmut Ritsch, Peter Domokos, Ferdinand (25):253003, 2002. DOI: 10.1103/Phys-
Brennecke, and Tilman Esslinger. Cold RevLett.89.253003.
atoms in cavity-generated dynamical opti- [34] Adam T Black, Hilton W Chan, and Vladan
cal potentials. Reviews of Modern Physics, Vuletić. Observation of collective friction
85(2):553, 2013. DOI: 10.1103/RevMod- forces due to spatial self-organization of
Phys.85.553. atoms: from rayleigh to bragg scattering.
[25] Christoph Maschler and Helmut Ritsch. Physical review letters, 91(20):203001, 2003.
Cold atom dynamics in a quantum opti- DOI: 10.1103/PhysRevLett.91.203001.
cal lattice potential. Physical review letters, [35] Varun D Vaidya, Yudan Guo, Ronen M
95(26):260401, 2005. DOI: 10.1103/Phys- Kroeze, Kyle E Ballantine, Alicia J Kol-
RevLett.95.260401. lár, Jonathan Keeling, and Benjamin L Lev.
[26] Renate Landig, Lorenz Hruby, Nishant Do- Tunable-range, photon-mediated atomic in-
gra, Manuele Landini, Rafael Mottl, To- teractions in multimode cavity qed. Phys-
bias Donner, and Tilman Esslinger. Quan- ical Review X, 8(1):011002, 2018. DOI:
tum phases from competing short-and long- 10.1103/PhysRevX.8.011002.
range interactions in an optical lattice. Na- [36] Subhadeep Gupta, Kevin L Moore, Kater W
ture, 532(7600):476, 2016. DOI: 10.1038/na- Murch, and Dan M Stamper-Kurn. Cav-
ture17409. ity nonlinear optics at low photon num-
[27] Igor B Mekhov and Helmut Ritsch. Quan- bers from collective atomic motion. Physi-
tum optics with ultracold quantum gases: cal review letters, 99(21):213601, 2007. DOI:
towards the full quantum regime of the 10.1103/PhysRevLett.99.213601.
light–matter interaction. Journal of Physics [37] Sarang Gopalakrishnan, Benjamin L Lev,
B: Atomic, Molecular and Optical Physics, and Paul M Goldbart. Frustration and
45(10):102001, 2012. DOI: 10.1088/0953- glassiness in spin models with cavity-
4075/45/10/102001. mediated interactions. Physical review
[28] Santiago F Caballero-Benitez, Gabriel Maz- letters, 107(27):277201, 2011. DOI:
zucchi, and Igor B Mekhov. Quantum 10.1103/PhysRevLett.107.277201.
simulators based on the global collective [38] Sarang Gopalakrishnan, Benjamin L Lev,
light-matter interaction. Physical Review and Paul M Goldbart. Exploring mod-
A, 93(6):063632, 2016. DOI: 10.1103/Phys- els of associative memory via cavity
RevA.93.063632. quantum electrodynamics. Philosophical
[29] Jordan Bell and Brett Stevens. A survey Magazine, 92(1-3):353–361, 2012. DOI:
of known results and research areas for n- 10.1080/14786435.2011.637980.
queens. Discrete Mathematics, 309(1):1–31, [39] Sebastian Krämer and Helmut Ritsch. Self-
2009. DOI: 10.1016/j.disc.2007.12.043. ordering dynamics of ultracold atoms in mul-
[30] Ian P Gent, Christopher Jefferson, and Pe- ticolored cavity fields. Physical Review A,
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