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ACKNOWL EDGMENTS

Supported by a startup grant from Cornell University (R.A.D.),

European Research Council Starting Grant VDW-CMAT, and DFG/

SFB-951 HIOS project A10. A.A. thanks F. Toigo and P. L. Silvestrelli

for useful discussions. This research used resources of the

Argonne Leadership Computing Facility at Argonne National

Laboratory, which is supported by the Office of Science of the U.S.

Department of Energy under contract DE-AC02-06CH11357.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/351/6278/1171/suppl/DC1

Theoretical Methods

Supplementary Text

Tables S1 to S3

10 December 2015; accepted 31 January 2016

10.1126/science.aae0509

REPORTS

◥

QUANTUM OPTICS

Generation of multiphoton entangled

quantum states by means of

integrated frequency combs

Christian Reimer,

1

*Michael Kues,

1

*†Piotr Roztocki,

1

Benjamin Wetzel,

1,2

Fabio Grazioso,

1

Brent E. Little,

3

Sai T. Chu,

4

Tudor Johnston,

1

Yaron Bromberg,

5

‡

Lucia Caspani,

6

§ David J. Moss,

7

|| Roberto Morandotti

1,8

†

Complex optical photon states with entanglement shared among several modes are

critical to improving our fundamental understanding of quantum mechanics and have

applications for quantum information processing, imaging, and microscopy. We

demonstrate that optical integrated Kerr frequency combs can be used to generate

several bi- and multiphoton entangled qubits, with direct applications for quantum

communication and computation. Our method is compatible with contemporary fiber

and quantum memory infrastructures and with chip-scale semiconductor technology,

enabling compact, low-cost, and scalable implementations. The exploitation of

integrated Kerr frequency combs, with their ability to generate multiple, customizable,

and complex quantum states, can provide a scalable, practical, and compact platform

for quantum technologies.

Multi-entangled states of light hold an-

swerstofundamentalquestionsinquan-

tum physics and are the cornerstone of a

range of applications, including quantum

communications (1), computation (2–4),

and sensing and imaging with a resolution be-

yond the classical limit (5). Thus, the controllable

realization of multiple quantum states in a com-

pact platform would enable a practical and pow-

erful implementation of quantum technologies.

Although applications of frequency combs have

been mostly classical thus far, their distinctive

architecture, based on multiple interacting modes

and the phase characteristics of the underlying

nonlinear processes, has the potential to offer

new and powerful ways to achieve the gener-

ation of multiple, customizable, and complex

states of nonclassical light. The quantum pro-

perties of frequency combs have recently begun

to be investigated, revealing their potential for

the generation of large quantum states (6–8).

However, the continuous-variable nonclassical

states (squeezed vacuum) that have been dem-

onstrated with this approach have not yet achieved

the quality (amount of squeezing) required for

optical quantum computation (9). For the gen-

eration of single photons and continuous- and

discrete-variable quantum states (qubits), a wide

variety of second- and third-order nonlinear

sources, optical fibers, and gases, as well as single

quantum emitters, have been exploited (10,11).

Recent progress has focused on transferring both

classical frequency combs (12)andquantum

sources (13) to integrated optical platforms. Such

integrated approaches provide the advantages

of compact, scalable, mass-producible, and low-

cost devices (14). Demonstrated integrated de-

vices include sources of heralded single photons

(15–17) and entangled photon pairs (18), in prin-

ciple allowing implementations of quantum al-

gorithms (19,20). Here we show the parallel

generation of bi- and multiphoton entangled

states in a compact, integrated quantum fre-

quency comb source.

Our quantum frequency comb is generated in a

CMOS (complementary metal-oxide semiconductor)–

compatible, high-refractive-index glass in a four-

port microring resonator architecture [details

on device fabrication and characteristics are

presented in (21)]. The weak and anomalous

dispersion of our device enables broadband phase

matching for spontaneous four-wave mixing

(SFWM), thereby generating a broad frequency

1176 11 MARCH 2016 •VOL 351 ISSUE 6278 sciencemag.org SCIENCE

1

Institut National de la Recherche Scientifique–Énergie

Matériaux Télécommunications, 1650 Boulevard Lionel-

Boulet, Varennes, Québec J3X 1S2, Canada.

2

Department of

Physics and Astronomy, University of Sussex, Falmer,

Brighton BN1 9RH, UK.

3

State Key Laboratory of Transient

Optics and Photonics, Xi'an Institute of Optics and Precision

Mechanics, Chinese Academy of Science, Xi'an, China.

4

Department of Physics and Materials Science, City

University of Hong Kong, Tat Chee Avenue, Hong Kong,

China.

5

Department of Applied Physics, Yale University, New

Haven, CT 06520, USA.

6

School of Engineering and Physical

Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

7

School of Electrical and Computer Engineering, RMIT

University, Melbourne, Victoria 3001, Australia.

8

Institute of

Fundamental and Frontier Sciences, University of Electronic

Science and Technology of China, Chengdu 610054, China.

*These authors contributed equally to this work. †Corresponding

author. E-mail: michael.kues@emt.inrs.ca (M.K.); morandotti@

emt.inrs.ca (R.M.) ‡Present address: Racah Institute of Physics,

The Hebrew University of Jerusalem, Jerusalem 91904, Israel.

§Present address: Institute of Photonics, Department of Physics,

University of Strathclyde, Glasgow G4 0NW, UK. ||Present address:

Center for Micro-Photonics, Swinburne University of Technology,

Hawthorn, Victoria 3122,Australia.

RESEARCH

on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from on March 11, 2016Downloaded from

comb of photons emitted at thecavityresonances.

The ring resonator’s characteristics, with a high

quality (Q) factor of around 240,000 (803 MHz

linewidth and 200 GHz free spectral range; fig. S1),

lead to high field enhancement and allow for low-

power operation, while simultaneously enabling the

direct generation of bright and narrow-bandwidth

photons, without the need for spectral filtering

of the photons (22,23). The device is pumped

with a passive mode-locked fiber laser (repeti-

tion rate, 16.8 MHz), which is spectrally filtered

to excite a single ring resonance at 1556.2 nm

(192.65 THz) with a pulse duration of 570 ps

coupled into the resonator. The pulses are di-

rectly filtered by the resonator, resulting in a

perfect match between the spectral bandwidth

of the pump pulses and the excited resonator

mode. This in turn leads to the generation of

pure single-mode photons in each resonance (13),

confirmed by single-photon autocorrelation mea-

surements [fig. S2 and (21)].

Using a high-resolution tunable C-band

wavelength filter and a grating-based spectrum

analyser, the single-photon spectrum was char-

acterized at the output of the resonator (21).

Figure 1 shows the measured single-photon count

rate and the calculated photon-pair production

rate per pulse as a function of wavelength. A very

broad frequency comb of photons is emitted,

covering the full S, C, and L bands defined by the

International Telecommunication Union (wave-

lengths ranging from 1470 to 1620 nm). The SFWM

process generates a spectrum that is symmetric

in frequency, whereas the spectral asymmetry in

the measured photon counts can be explained by

Raman scattering, which could be further reduced

by cooling the chip (24). As a result of the broad

phase-matching condition, achieved through the

close-to-zero waveguide dispersion, the emitted

comb exhibits a flat and broadband spectrum

with uniform pair production rates, ranging from

0.02 to 0.04 pairs per pulse, over the full mea-

sured comb.

To demonstrate an entangled quantum fre-

quency comb, we chose time-bin entanglement,

which, among several intrinsic advantages, is

particularly suitable for information processing

and transmission (25) because of its robustness

(e.g., with respect to polarization fluctuations); it

thus can be preserved even over long propaga-

tion distances in standard fiber networks (26).

Starting from the single-mode photon pairs,

we generated time-bin–entangled qubits by pass-

ing the pulsed pump laser through a stabilized

unbalanced fiber interferometer with a 11.4-ns

delay (longer than the pulse duration of the laser),

thereby producing double pulses of equal power

with a defined relative phase difference. The tem-

poral separation of the two pulses can be arbitrarily

chosen, as long as it is larger than the temporal

duration of the single photons; this approach thus

offers considerable flexibility. The double pulses

were then coupled into the integrated microring

resonator (Fig. 2), where an average pump power

of 0.6 mW was chosen so that the probability of

creating a photon pair from both pulses simulta-

neously was low enough to be negligible.

This pump configuration transforms the orig-

inally single-mode photon pairs into entangled

states, where the photons are in a superposition

of two temporal modes. In particular, the en-

tangled state jytimebin i¼ 1

ﬃﬃ

2

pðjSs;SiiþjLs;LiiÞ

is generated, where the signal (s) and idler (i)

photons are in a quantum superposition of the

short (S)andlong(L)timebins(21). Most im-

portantly, these entangled qubits are generated

over all the microring resonances, thus leading to

a quantum frequency comb of time-bin–entangled

photon pairs.

To characterize the degree of entanglement,

the generated signal and idler photons were each

passed individually through a different fiber

interferometer with an imbalance identical to

SCIENCE sciencemag.org 11 MARCH 2016 •VOL 351 ISSUE 6278 1177

1

2

3

4

5

6

1480 1500 1520 1540 1560 1580 1600

0.01

0.02

0.03

S-Band C-Band

Pump

Entangl. Visibility

s1-i1: 83.0% (98.5%)

s2-i2: 84.1% (93.2%)

s3-i3: 86.5% (96.0%)

s4-i4: 82.4% (96.0%)

s5-i5: Stabilization

s6-i6: 83.1% (99.5%)

Single photon count rate (kHz)

SFWM photon pairs per pulse

Wavelength (nm)

s6 s5 s4 s3 s2s1 i1 i2 i3 i4 i5 i6

Pump

L-Band

SFWM

Asymmetry

Fig. 1. Measured single-photon spectrum of the integrated quantum frequency comb. Single-photon

spectrum (red circles) emitted by the microring resonator, measured using a grating-based spectrum

analyzer and a high-resolution digital tunable filter in the C band (bottom inset). The S, C, and L bands are

indicated. The red curve shows the symmetric contribution generated through SFWM, whereas the blue

curve shows the spectral asymmetry, which can be explained by Raman scattering.The channels used in the

entanglement measurements are shown in the bottom inset; the measured raw entanglement visibilities

(with background-corrected values in parentheses) for the individualchannel pairs are shown in the top inset.

Fig. 2. Quantum frequency comb. A pulsed laser is passed through an unbalanced fiber Michelson

interferometer, generating double pulses with a phase difference ϕ. The pulses are fed into the microring

resonator, exciting one microring resonance and generating time-bin–entangled photon pairs on a fre-

quency comb through SFWM. For the purposes of analysis [entanglement verification (Fig. 3, A to D) or

quantum state tomography (Fig. 4, A and B)], each photon of the spectrally filtered photon pair is

individually passed through an interferometer with the temporal imbalance equal to the time-bin sep-

aration and then detected with a single-photon detector. For the four-photon measurements (Fig. 3E and

Fig. 4, C to D), four frequency modes that are symmetric to the excitation field are collected, passed

through the interferometers, and spectrally filtered before detection.

RESEARCH |REPORTS

that used for the pump laser (Fig. 2). This setup

allowed the measurement of the quantum inter-

ference between the signal and idler photons. For

the resonances closest to the excitation frequency,

we measured a photon coincidence rate of 340 Hz,

which gives an estimated pair production rate of

302 kHz per channel (0.018 pairs per double pulse),

accounting for system and detection losses of

14.75 dB (21). We selected five different frequency

channel pairs within the C band (marked in Fig. 1)

and recorded quantum interference with raw

visibilities above 82.4%, which, being greater than

1

ﬃﬃ

2

p≈71% (Fig. 1, top inset, and Fig. 3), confirm en-

tanglement through the violation of the Clauser-

Horne-Shimony-Holt (Bell-like) inequality (27). Af-

ter subtracting the measured background (Fig. 3),

the visibility was found to be above 93.2% on all

channel pairs (Fig. 1, top inset).

In addition to confirming time-bin entangle-

ment, the quantum interference also reveals the

phase characteristic of the nonlinear generation

process. This phase dependency has been well

described in theory (28) and has been exploited,

for example, in optical squeezing (29). Here we

show that this phase dependency is also man-

ifested at the single-photon level, with a clear

difference between second- and third-order non-

linear interactions. When spontaneous parametric

down-conversion (SPDC) in second-order non-

linear media is used to generate the entangled

photon pairs, quantum interferenceisexpected

to be proportional to 1 –Vcos(a+b+ϕ)(26),

where Vis the fringe visibility, ϕis the pump

interferometer phase, and aand bare the phases

for the signal and idler interferometers, respec-

tively. In contrast, for photons generated through

SFWM (as in this work), quantum interference is

expected to be of the form 1 –Vcos(a+b–2ϕ)for

degenerate SFWM or 1 –Vcos(a+b–ϕ

1

–ϕ

2

)for

nondegenerate SFWM, where ϕ

1

and ϕ

2

are the

phases of the two pump fields (21). As shown in

Fig. 3, we confirmed this important difference in

phase dependency between SPDC and SFWM

through five separate quantum interference mea-

surements, where the phases of the interferom-

eters are either tuned separately or simultaneously

in a symmetric and an antisymmetric way. The

expected behavior for SPDC and SFWM is plotted

in Fig. 3, A to E, with dashed and solid lines,

respectively. The measurements confirm the dif-

ference between the two processes, which can be

explained through the additional photon involved

in SFWM. In the generation of entangled photon

pairs, the phase of the excitation photon (or photons)

can be used to adjust the quantum state. If a single

photon is involved (as in second-order processes),

only the phase of this photon can be used as a

control parameter. In third-order nonlinear pro-

cesses, however, two photons generate the quan-

tum state, enabling an additional control parameter

(the relative phase between the two photons).

Although in this study only a single excitation field

was used to demonstrate the effect, exploiting

two distinct nondegenerate pump fields could

lead to an additional degree of freedom for the

generation of all-optical reconfigurable quan-

tum states.

The distinctive multimode characteristic of the

frequency comb architecture presented here can

be extended to create multiphoton entangled

quantum states. By selecting two different signal-

idler pairs, we can generate two-photon qubit

states, given by jy1i¼ 1

ﬃﬃ

2

pðjSs1;Si1 iþei2ϕjLs1;Li1 iÞ

and jy2i¼ 1

ﬃﬃ

2

pðjSs2 ;Si2 iþei2ϕjLs2 ;Li2 iÞ.Bypost-

selecting four-photon events with one photon

on each frequency channel, these two states are

multiplied, resulting in a four-photon time-bin–

entangled state, given by jy4photoni¼jy1ijy2i¼

1

2ðjSs1;Si1 ;Ss2 ;Si2 iþei2ϕjSs1;Si1 ;Ls2;Li2 iþei2ϕj

Ls1 ;Li1 ;Ss2 ;Si2 ;iþei4ϕjLs1;Li1 ;Ls2 ;Li2 iÞ.Forthe

generation of a four-photon state, the coherence

length of both photon pairs has to be the same

and must be matched to the excitation field’s

coherence time. This requirement is intrinsically

fulfilled through the resonant characteristics

(equal resonance bandwidths) of the ring cavity,

in combination with the excitation scheme de-

scribed above. By setting the pump power to 1.5 mW,

we measured a quadruple detection rate of 0.17 Hz,

which corresponds to a calculated generation rate

of 135 kHz, taking into account the system and

detection losses of 14.75 dB. We then performed

four-photon quantum interference measurements

(Fig. 3F). Four-photon interference generally is

not present for two completely independent

1178 11 MARCH 2016 •VOL 351 ISSUE 6278 sciencemag.org SCIENCE

2

Phase (π)

0 0.8 10.60.40.2 1.2 1.81.61.4

0

1

Normalized counts

0.5

0

1

Normalized counts

0.5

0

1

Normalized counts

0.5

0

1

Normalized counts

0.5

0

1

Normalized counts

0.5

Measured signal

Measured background Theory SPDC

Entangl.

Visibility

Theory SFWM

φ=0

α=β∈[0,2π]

φ∈[0,2π]

α=β=0

α∈[0,2π]

φ=β=0

α=β=φ∈[0,2π]

α=β=–φ∈[0,2π]

2

Phase (π)

0 0.8 10.60.40.2 1.2 1.81.61.4

0

1

Normalized counts

0.5

2-Photon quantum interference

4-Photon quantum interference

Measured Theory 4-Photon Theory 2-Photon

Fig. 3. Entanglement

and phase control by means

of SFWM. (Ato E)To

demonstrate the difference

between the phase character-

istics of SPDC and SFWM, five

different quantum interference

measurements were per-

formed. Three interferometer

phases were adjusted: ϕ,a,

and b, being the phases of the

pump, signal, and idler interfer-

ometers, respectively

(conditions are specified in

each panel). The error bars

represent the standard devia-

tion of seven measurements.

(F) Four-photon entanglement

measurement with all photon

phases tuned simultaneously,

showing clear four-photon

quantum interference with a

visibility of 89%. The solid line

indicates the expected function;

the dashed line shows the

cosine interference in the two-

photon case.

RESEARCH |REPORTS

two-photon qubit states. The interference is ex-

pected to be proportional to 3 + cos(4a–4ϕ)+

4cos(2a–2ϕ), where ais the phase of all four

entangled photons and ϕis the pump interfer-

ometer phase (21). Our data follow the expected

relation, having a visibility of 89% without compen-

sation for background noise or losses. Further-

more, we repeated the four-photon measurement

by selecting different combinations of four modes,

always finding four-photon entanglement [fig. S3

and (21)].

Lastly, to fully characterize the entangled states,

we performed quantum state tomography (30).

This method measures the state density matrix,

fromwhichitispossibletoextractimportantchar-

acteristics such as the fidelity, which describes

how close the measured state is to the ideal en-

tangled state (21). We first measured the two-

photon qubits generated on comb lines that were

symmetric with respect to the pump wavelength

(Fig.4,AandB)andfoundafidelityof96%,con-

firming that our generated quantum states are

of high quality and very close to the ideal en-

tangled state. For the four-photon entangled

state (Fig. 4, C and D), we obtained a fidelity of

64% without compensation for background

noise or interf erometer imperfections, which is

comparable to the fidelity measured for non-

integrated four-photon states used for practical

applications (3).

Key characteristics of our quantum frequency

comb include the intrinsic and simultaneous

operation over many modes, the generation of

high-purity photon pairs, the high-quality bi- and

multiphoton entanglement shared among these

modes, and the inherent compatibility with fiber

technology. Because of these features, it has ver-

satile and immediate applications in areas such as

quantum communications and quantum compu-

tation. For example, our source can be implemented

into both single-photon and entanglement-based

quantum communication protocols. The broad-

band nature of the quantum comb is particularly

attractive for multichannel applications, where

the amount of transmitted data can be increased

through the use of multiple, equally well-performing

channels. We repeated the two-photon tomography

measurement after adding 40 km of fiber and

measured a fidelity of 87% [fig. S4 and (21)], dem-

onstrating that the entanglement is preserved

after long fiber propagation.

Furthermore, two-photon time-bin–entangled

qubits have been used successfully for linear

universal quantum computation (4,25), and

the parallel generation and processing of mul-

tiple qubits can directly enhance such proto-

cols where the information capacity scales with

the number of comb lines used, as theoretically

predicted (25). Even though the demonstrated

multiphoton entangled states are separable,

because they are generated as a product of bi-

photon Bell states, it is conceivable that—through

the use of multiple excitation fields (6) or con-

trolled phase gates (31)—nonseparable multiphoton

cluster states could be constructed and used for

measurement-based quantum computation (3).

Further device integration of the frequency

comb will lead to more compact and stable sys-

tems with higher performances, resulting in

better detection rates. All the components that

were used in our setup, such as the laser, filters,

interferometers, and detectors (connected via

optical fibers), could be integrated on a single

chip (13)toreducesizeandlosses(currentlyat

14.75 dB). An easily realized decrease in losses by

5 dB would increase the four-photon detection

rate by a factor of 100, and an achievable loss

reduction of 10 dB would increase it to the useful

kilohertz range.

Our results indicate that integrated quantum

frequency comb sources based on third-order non-

linearities can open up new venues for the gen-

eration and control of complex quantum states,

thus providing a scalable and practical platform

for optical quantum information processing.

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0.25

0.5

-0.5

0

0.5

<SS|

<SL|<LS|

<LL|

|SS>

|SL>

|LS>

|LL>

Re ( )

ρ

Im ( )

ρ

<LL|

<LS|

<SL|

<SS|

|LL>

|LS>

|SL>

|SS>

0

0.25

0.5

-0.5

0

0.5

<SS|

<SL|<LS|

<LL|

|SS>

|SL>

|LS>

|LL>

2-Photon qubit state - measured

<LLLL|

<LLSS|

<SSLL|

<SSSS|

|LLLL>

|LLSS>

|SSLL>

|SSSS>

0.2

0.1

0

-0.2

-0.

1

0

0.1

0.2

<SSSS|

<SSLL|

<LLSS|

<LLLL|

|SSSS>

|SSLL>

|LLSS>

|LLLL>

<LLLL|

<LLSS|

<SSLL|

<SSSS|

|LLLL>

|LLSS>

|SSLL>

|SSSS>

0.1

0.2

0<LLLL|

<LLSS|

<SSLL|

<SSSS|

|LLLL>

|LLSS>

|SSLL>

|SSSS>

-0.2

0.2

0

-0.1

0.1

Re ( ) Im ( )

ρ

4-Photon qubit state - ideal

4-Photon qubit state - measured

Re ( )

ρ

Im ( )

ρ

Re ( )

ρ

Im ( )

ρ

Fig. 4. Quantum state characterization by means of tomography. A quantum state can be fully described by its density matrix r

^

. The real (Re) and imaginary

(Im) parts of theideal density matrices of a two- and four-photon entangled qubit stateare shown in (A) and (C), respectively, represented in the time-bin basis

(|SSi,|SLi,|LSi,|LLi)and(|SSSSi,|SSSLi,…|LLLLi).The measured density matrix of the two-photon state (B) agrees very well with the ideal state, confirmed by

a fidelity of 96%. The measured density matrix of the four-photon entangled qubit state (D) reaches a fidelity of 64%, which is comparable in quality to other

nonintegrated four-photon states (3).

RESEARCH |REPORTS

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ACKNOW LEDGMEN TS

This work was supported by the Natural Sciences and Engineering

Research Council of Canada (NSERC) through the Steacie

Memorial Fellowship and Discovery Grants programs and by

the Australian Research Council Discovery Projects program.

C.R. and P.R. acknowledge the support of a NSERC Vanier Canada

Graduate Scholarship and a NSERC Alexander Graham Bell

Canada Graduate Scholarship (Master’s), respectively. M.K.

acknowledges support from FRQNT (Fonds de Recherche du

Québec–Nature et Technologies) through the MELS fellowship

program (Merit Scholarship Program for Foreign Students;

Ministère de l’Éducation, de l’Enseignement Supérieur et de la

Recherche du Québec). M.K. also acknowledges funding from

the European Union’s Horizon 2020 research and innovation

program under Marie Sklodowska-Curie grant agreement no.

656607. We acknowledge support from the People Programme

(Marie Curie Actions) of the European Union’s 7th Framework

Programme for Research and Technological Development

[B.W. under Research Executive Agency (REA) grant agreement

no. 625466 and L.C. under REA grant agreement no. 627478].

F.G. acknowledges support from Mitacs through the Accelerate

Program. S.T.C. acknowledges support from the CityU SRG-Fd

(Strategic Research Grant for fundable Competitive Earmarked

Research Grant) program #7004189. We thank R. Helsten

for the mechanical design of the interferometer housing;

P. Saggu for assisting in the measurement of the single-photon

spectrum; L. La Volpe for assisting in the evaluation of the

tomography measurement; J. Azaña, T. A. Denidni, S. O. Tatu,

and L. Razzari for providing some of the required experimental

equipment; and A. Tavares, A. Yurtsever, and M. A. Gauthier

for helpful discussions. Special thanks go to QuantumOpus

and N. Bertone of OptoElectronics Components for their help

and for providing us with state-of-the-art photon detection

equipment.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/351/6278/1176/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

References (32–37)

12 November 2015; accepted 5 February 2016

10.1126/science.aad8532

SPIN MODELS

Simple universal models capture all

classical spin physics

Gemma De las Cuevas

1

*and Toby S. Cubitt

2

Spin models are used in many studies of complex systems because they exhibit

rich macroscopic behavior despite their microscopic simplicity. Here, we prove that

all the physics of every classical spin model is reproduced in the low-energy sector

of certain “universal models,”with at most polynomial overhead. This holds for classical

models with discrete or continuous degrees of freedom. We prove necessary and

sufficient conditions for a spin model to be universal and show that one of the simplest

and most widely studied spin models, the two-dimensional Ising model with fields,

is universal. Our results may facilitate physical simulations of Hamiltonians with

complex interactions.

The description of systems with many inter-

acting degrees of freedom is a ubiquitous

problem across the natural and social sci-

ences. Be it electrons in a material, neurons

interacting through synapses, or specula-

tive agents in a market, the challenge is to sim-

plify the system so that it becomes tractable

while capturing some of the relevant features of

therealsystem.Spinmodelsareonewayofad-

dressing this challenge. Originally introduced

in condensed matter physics in order to study

magnetic materials (1–4), they have permeated

many other disciplines, including quantum grav-

ity (5), error-correcting codes (6), percolation the-

ory (3), graph theory (7), neural networks (8),

protein folding (9), and trading models in stock

markets (10).

Spin models are microscopically simple, yet

their versatile interactions lead to a very wide

variety of macroscopic behavior. Formally, a spin

model is specified by a set of degrees of freedom,

the “spins,”and a cost function, or “Hamilto-

nian,”Hthat specifies the interaction pattern as

well as the type and strength of interactions

among the spins. (In physics, the Hamiltonian

specifies the energy of each possible spin con-

figuration; in other contexts, this energy value

mayquantifyamoreabstract“cost”associated

with a configuration.)

Thisdefinitionencompassesawiderangeof

models, including attractive and/or repulsive in-

teractions, regular and irregular interaction pat-

terns, models in different spatial dimensions,

models with different symmetries (for example,

“conventional”spin models with global symme-

tries versus models with local symmetries, such

as lattice gauge theories), many-body interactions

[such as vertex models and edge models (11)], and

more.Wewillusetheword“model”to refer to a

(generally infinite) family of spin Hamiltonians.

Different Hamiltonians within the same model

are typically related in some natural way. For ex-

ample, the “two-dimensional (2D) Ising model

with fields”is the family of Hamiltonians of the

form

HGðsÞ¼Xhi;jiJijsisjþXirisið1Þ

where s=s

1

,s

2

,...,s

n

is a configuration of Ising

(two-level) spins si∈f−1;1gon a 2D square lat-

tice, hi;jidenotes neigbouring spins, and J

ij

and r

i

are real numbers specifying the coupling

strengths and local fields, respectively.

Here, we show that there exist certain spin

models, which we call universal, whose low-energy

sector can reproduce the complete physics of

any other classical spin model. What does it

mean to “reproduce the complete physics”? In-

formally, we say that a spin model with Ha-

miltonian Hsimulates a target Hamiltonian

H′if ( i) t he energy levels of Hbelow a threshold

Dreproduce the energy levels of H′;(ii)thereisa

fixed subset Pof the spins of H—which we call

the “physical spins”—whose configuration for

each energy level below Dreproduces the spin

configuration of the corresponding energy level

of H′; and (iii) the partition function of Hre-

produces that of H′. From the partition function,

one can derive all equilibrium thermodynamical

properties of the system. A universal model is

then a model that can simulate any other spin

model.

We denote spin degrees of freedom—

discrete or continuous—by a string of spin

states s=s

1

,s

2

, ..., s

n

.Forq-level Ising spins

(discrete degrees of freedom with a finite

number qof distinct states), we can label the

states arbitrarily by integers: si∈f1;…;qg.For

continuous spins, a spin state is represented by a

1180 11 MARCH 2016 •VOL 351 ISSUE 6278 sciencemag.org SCIENCE

1

Max Planck Institute for Quantum Optics, Hans-Kopfermann-

Strasse 1, 85748 Garching, Germany.

2

Department of Computer

Science, University College London, Gower Street, London WC1E

6EA, UK.

*Corresponding author. E-mail: gemma.delascuevas@mpq.mpg.de

RESEARCH |REPORTS

DOI: 10.1126/science.aad8532

, 1176 (2016);351 Science et al.Christian Reimer

integrated frequency combs

Generation of multiphoton entangled quantum states by means of

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