Entanglement in a quantum annealing processor

3.1 MB·

by  T. Lanting

Entanglement in a quantum annealing processor
T. Lanting∗,1 A. J. Przybysz,1 A. Yu. Smirnov,1 F. M. Spedalieri,2,3 M. H. Amin,1,4 A. J. Berkley,1
R. Harris,1 F. Altomare,1 S. Boixo†,2 P. Bunyk,1 N. Dickson‡,1 C. Enderud,1 J. P. Hilton,1
E. Hoskinson,1 M. W. Johnson,1 E. Ladizinsky,1 N. Ladizinsky,1 R. Neufeld,1 T. Oh,1
I. Perminov,1 C. Rich,1 M. C. Thom,1 E. Tolkacheva,1 S. Uchaikin,1,5 A. B. Wilson,1 and G. Rose1
1D-Wave Systems Inc., 3033 Beta Avenue, Burnaby BC Canada V5G 4M9
2Information Sciences Institute, University of Southern California, Los Angeles CA USA 90089
3Center for Quantum Information Science and Technology, University of Southern California
4Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
5National Research Tomsk Polytechnic University, 30 Lenin Avenue, Tomsk, 634050, Russia
Entanglement lies at the core of quantum algorithms designed to solve problems that are in-
tractablebyclassicalapproaches. Onesuchalgorithm,quantumannealing(QA),providesapromis-
ingpathtoapracticalquantumprocessor. WehavebuiltaseriesofscalableQAprocessorsconsisting
of networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopy
4 to measure the energy eigenspectrum of two- and eight-qubit systems within one such processor,
1 demonstratingquantumcoherenceinthesesystems. Wepresentexperimentalevidencethat,during
0
acriticalportionofQA,thequbitsbecomeentangledandthatentanglementpersistsevenasthese
2
systems reach equilibrium with a thermal environment. Our results provide an encouraging sign

n that QA is a viable technology for large-scale quantum computing.
a
J

5 I. INTRODUCTION II. QUANTUM ANNEALING
1

QA is designed to find the low energy configurations
]
h of systems of interacting spins. A wide variety of opti-
p mization problems naturally map onto this physical sys-
-
t tem [19–22]. A QA algorithm is described by a time-
n
dependentHamiltonianforasetofN spins,i=1,...,N,
a Thelastdecadehasbeenexcitingforthefieldofquan-
u tumcomputation. Awiderangeofphysicalimplementa-
q tions of architectures that promise to harness quantum 1 (cid:88)
[ (s)= (s) ∆(s)σx, (1)
mechanicstoperformcomputationhavebeenstudied[1– HS E HP − 2 i
i
1 3]. Scaling these architectures to build practical proces-
v sorswith manymillions tobillionsof qubitswillbe chal- where the dimensionless P is
0 H
lenging [4, 5]. A simpler architecture, designed to imple- (cid:88) (cid:88)
0 = h σz+ J σzσz (2)
5 ment a single quantum algorithm such as quantum an- HP − i i ij i j
3 nealing (QA), provides a more practical approach in the i i<j
1. near-term [6, 7]. However, one of the main features that and σx,z are Pauli matrices for the ith spin. The energy
i
0 makes such an architecture scalable, namely a limited scales ∆ and are the transverse and longitudinal ener-
4 number of low bandwidth external control lines [8], pro- gies of the spiEns, respectively, and the biases h and cou-
i
1 hibits many typical characterization measurements used plingsJ encodeaparticularoptimizationproblem. The
: ij
v in studying prototype universal quantum computers [9– time-dependentvariationof∆and isparameterizedby
Xi 14]. Theseconstraintsmakeitchallengingtoexperimen- s t/tf with time t [0,tf] and toEtal run (anneal) time
tally determine whether a scalable QA architecture, one t ≡. QA is performed∈by first setting ∆ , which re-
ar that is inevitably coupled to a thermal environment, is sfults in a ground state into which the spin(cid:29)s cEan be easily
capableofgeneratingentangledstates[15–18]. Ademon- initialized [6]. Then ∆ is reduced and is increased un-
stration of entanglement is considered to be a critical til ∆. At this point, the systemE Hamiltonian is
milestone for any approach to building a quantum com- domEin(cid:29)ated by , which represents the encoded opti-
P
puting technology. Herein, we demonstrate an experi- mization probleHm. At the end of the evolution a ground
mental method to detect entanglement in subsections of state of represents the lowest energy configuration
P
aquantumannealingprocessortoaddressthisfundamen- for the pHroblem Hamiltonian and thus a solution to the
tal question. optimization problem.
III. QUANTUM ANNEALING PROCESSOR
∗Electronicaddress: [email protected]
†currentlyatGoogle,340MainSt,Venice,California90291
‡currently at Side Effects Software, 1401-123 Front Street West, We have built a processor that implements S using
H
Toronto,Ontario,Canada superconducting flux qubits as effective spins [6, 7, 23,
2
FIG. 2: An illustration of entanglement between two qubits
during QA with h = 0 and J < 0. We plot calculations of
i
the two-qubit ground state wave function modulus squared
in the basis of Φ and Φ , the flux through the bodies of
q1 q2
q andq ,respectively. Thecolorscaleencodestheprobabil-
1 2
itydensitywithredcorrespondingtohighprobabilitydensity
and blue corresponding to low probability density. We used
Hamiltonian (1) and the energies in Fig 1d for the calcula-
tion. The four quadrants represent the four possible states
of the two-qubit system in the computation basis. We also
plot the single qubit potential energy (U versus Φ ) cal-
q1
culated from measured device parameters. (a) At s = 0
(∆ (cid:29) 2|J|E ∼ 0), the qubits weakly interact and are each
in their ground state √1 (|↑(cid:105)+|↓(cid:105)), which is delocalized in
2
the computation basis. The wavefunction shows no correla-
tion between q and q and therefore their wavefunctions are
1 2
separable. (b) At intermediate s (∆ ∼ 2|J |E), the qubits
ij
are entangled. The state of one qubit is not separable from
the state of the other, as the ground state of the system is
√
approximately |+(cid:105) ≡ (|↑↑(cid:105)+|↓↓(cid:105))/ 2. A clear correlation
is seen between q and q . (c) As s → 1, ∆ (cid:28) 2|J |E and
1 2 ij
thegroundstateofthesystemapproaches|+(cid:105). However,the
energygapg betweenthegroundstate(|+(cid:105))andthefirstex-
cited state (|−(cid:105)) is closing. When the qubits are coupled to
FIG. 1: (a) Photograph of the QA processor used in this
a bath with temperature T and g <k T, the system is in a
B
study. Wereportmeasurementsperformedontheeight-qubit
mixedstateof|+(cid:105)and|−(cid:105)andentanglementisextinguished.
unit cell indicated. The bodies of the qubits are extended
loops of Nb wiring (highlighted with red rectangles). Inter-
qubit couplers are located at the intersections of the qubit
bodies. (b) Electron micrograph showing the cross-section ure1cshowsthecircuitschematicofapairoffluxqubits
of a typical portion of the processor circuitry (described in with the magnetic flux controls Φx and Φx . The an-
qi ccjj
moredetailinAppendixA).(c)Schematicdiagramofapair nealing parameter s is controlled with the global bias
of coupled superconducting flux qubits with external control Φx (t) (see Appendix A for the mapping between s and
biases Φxqi and Φxccjj and with flux through the body of the Φcxcjj and a description of how Φx is provided for each
ithqubitdenotedasΦ . Aninductivecouplingbetweenthe ccjj qi
qi qubit). The strength and sign of the inductive coupling
qubits is tuned with the bias Φx . (d) Energy scales ∆(s)
co,ij between pairs of qubits is controlled with magnetic flux
andE(s)inHamiltonian(1)calculatedfromanrf-SQUID(Su-
perconductingQuantumInterferenceDevice)modelbasedon Φxco,ij that is provided by an individual on-chip digital-
themedianofindependentlymeasureddeviceparametersfor to-analog converter for each coupler [8]. The parame-
these eight qubits. See Appendix A for more details. (e),(f) ters h and J are thus in situ tunable, thereby allowing
i ij
The two and eight-qubit systems studied were programmed the encoding of a vast number of problems. The time-
tohavethetopologiesshown. Qubitsarerepresentedasgold
dependent energy scales ∆(s) and (s) are calculated
spheres and inter-qubit couplers, set to J =−2.5, are repre- E
from measured qubit parameters and plotted in Fig. 1d.
sented as silver lines.
Wecalibratedandcorrectedtheindividualfluxqubitpa-
rameters in our processor to ensure that every qubit had
a close to identical ∆ and (the energy gap ∆ is bal-
E
24]. Figure1ashowsaphotographoftheprocessor. Fig- anced to better than 8% between qubits and to better
E
3
than 5%). See Appendix A for measurements of these the evolution, g becomes less than k T, where T char-
B
energy scales. The interqubit couplers were calibrated acterizes the temperature of the thermal environment to
asdescribedinRef.[25]. Theprocessorstudiedherewas which the qubits are coupled. At this point, we expect
mountedonthemixingchamberofadilutionrefrigerator the system to evolve into a mixed state of + and
| (cid:105) |−(cid:105)
held at temperature T =12.5 mK. and the entanglement will vanish with g for sufficiently
long thermalization times. At the end of QA, s = 1,
∆ 0,andHamiltonian(1)predictstwodegenerateand
∼
IV. FERROMAGNETICALLY COUPLED localized ground states, namely the FM ordered states
INSTANCES ... and ... .
|↑ ↑(cid:105) |↓ ↓(cid:105)
Theexperimentsreportedhereinfocusedononeofthe
eight-qubit unit cells of the larger QA processor as indi- V. MEASUREMENTS
cated in Fig. 1a. The unit cell was isolated by setting
all couplings outside of that subsection to J = 0 for
ij In order to experimentally verify the change in spec-
all experiments. We then posed specific instances
P tral gap in the two- and eight-qubit systems during QA,
H
withstrongferromagnetic(FM)couplingJ = 2.5and
ij weusedqubittunnelingspectroscopy(QTS)asdescribed
−
h = 0 to that unit cell as illustrated in Figs. 1e and f.
i in more detail in Ref. [33] and Appendix B. QTS allows
These configurations produced coupled two- and eight-
us to measure the eigenspectrum and level occupation
qubit systems, respectively. Hamiltonian (1) describes
of a system during QA by coupling an additional probe
the behaviour of these systems during QA.
qubit to the system. We performed QTS on the two-
Typical observations of entanglement in the quantum and eight-qubit systems shown in Figs. 1e and f. Fig-
computing literature involve applying interactions be- ures 3a and b show the measured energy eigenspectrum
tween qubits, removing these interactions, and then per- for the two- and eight-qubit systems, respectively, as a
forming measurements. Such an approach is well suited function of s. The measurements are initial tunneling
to gate-model architectures (e.g. Ref. [11]). During QA, rates of the probe qubit, normalized by the maximum
however, the interaction between qubits is determined observed tunneling rate. Peaks in the measured tunnel-
by the particular instance of P, in this case a strongly ing rate map the energy eigenstates of the system under
H
ferromagnetic instance, and cannot be removed. In this study [33]. As the system evolves (increasing s), ∆(s) in
way,systemsofqubitsundergoingQAhavemuchmorein Hamiltonian (1) decreases and the gap between ground
common with condensed matter systems, such as quan- and first excited states closes. The spectroscopy data in
tum magnets, for which interactions cannot be turned Fig. 3a reveal two higher energy eigenstates. We observe
off. Indeed, a growing body of recent theoretical and a similar group of higher energy excited states for the
experimental work suggests that entanglement plays a eight-qubit system in Fig. 3b. Note that g closes earlier
centralroleinmanyofthemacroscopicpropertiesofcon- in the QA algorithm for the eight-qubit system as com-
densed matter systems [26–32]. Here we introduce other pared to the two-qubit system. In all of the panels of
approaches to quantifying entanglement that are suited Fig. 3, solid curves indicate the theoretical energy levels
to QA processors. We establish experimentally that the predicted by Hamiltonian (1) using the measured ∆(s)
two-andeight-qubitsystems,comprisingmacroscopicsu- and (s). The agreement between the experimentally
perconducting flux qubits coupled to a thermal bath at obtaiEnedspectrumandthetheoreticalspectrumisgood.
12.5 mK, become entangled during the QA algorithm.
The data presented in Figs. 3a and 3b indicate that
To illustrate the evolution of the ground state of these the spectral gap between ground and first excited state
instances during QA, a sequence of wave functions for decreases monotonically with s when all h = 0. Under
i
the ground state of the two-qubit system is shown in these bias conditions, these systems possess Z symme-
2
Fig. 2. A similar sequence could be envisioned for the try between the states ... and ... . The degen-
eight-qubit system. We consider these systems subject eracy between these st|a↑tes i↑s(cid:105)lifted|↓by fi↓n(cid:105)ite ∆(s). To
to zero biases, hi = 0. For small s, ∆ (cid:29) 2|Jij|E, and explicitly demonstrate that the spectral gap at hi =0 is
the ground state of the system can be expressed as a due to the avoided crossing of ... and ... , we
product of the ground states of the individual qubits: have performed QTS at fixed s|↑as a↑(cid:105)functio|↓n of ↓a(cid:105)“di-
i⊗nNit=er1m√1e2d(|i↑a(cid:105)tie+s,|↓∆(cid:105)i)<w2heJre N, =and2,t8he(segeroFuingd. 2aan)d. fiFrosrt aqgunboitsst,ict”hubsiassweheip(cid:54)=ing0tthheastywstaesmusntihforormuglhy daepgpelnieedratcoyaaltl
ij
excited states of the∼pro|ces|sEor are approximately the de- h = 0. As a result, either the state ... or ...
i
|↑ ↑(cid:105) |↓ ↓(cid:105)
localized superpositions ( ... ... )/√2 becomes energetically favored, depending upon the sign
|±(cid:105) ≡ |↑ ↑(cid:105) ± |↓ ↓(cid:105)
(Fig. 2b). The state + is the maximally entangled Bell of h . Hamiltonian (1) predicts an avoided crossing, as
i
| (cid:105)
(or GHZ, for eight qubits) state [17]. As s 1, the en- a function of h , between the ground and first excited
i
→
ergy gap g between the ground and first excited states states at degeneracy, where h = 0, with a minimum en-
i
approaches g (E E ) ∆(s)N/(2J (s))N−1 and ergy gap g. The presence of such an avoided crossing is
2 1 ij
≡ − ∝ | |E
vanishes as ∆(s) 0 (Fig. 2c). At some point late in a signature of ground-state entanglement [14, 34]. For
→
4
10 1 10 1
8 0.8 8 0.8
6 0.6 6 0.6
4 4
0.4 0.4
2 2
0.2 0.2
0 0
0
0.3 0.35 0.4 0.2 0.25 0.3 0.35
6 6 6 6
4 4 4 4
2 2 2 2
0 0
0 0
−2 −2
−2 −2
−4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4
FIG. 3: Spectroscopic data for two- and eight-qubit systems plotted in false colour (colour indicates normalized qubit tunnel
spectroscopy rates). A non-zero measurement (false colour) indicates the presence of an eigenstate of the probed system at a
givenenergy(ordinate)ands(abscissa). Panel(a)showsthemeasuredeigenspectrumforthetwo-qubitsystemasafunctionof
s. Panel(b)showsasimilarsetofmeasurementsfortheeight-qubitsystem. ThegroundstateenergyE hasbeensubtracted
1
from the data to aid in visualization. The solid curves indicate the theoretical expectations for the energy eigenvalues using
independentlycalibratedqubitparametersandHamiltonian(1). Weemphasizethatthesolidcurvesarenotafit,butrathera
prediction based on Hamiltonian (1) and measurements of ∆ and E. The slight differences between the high-energy spectrum
prediction and measurements are due to the additional states in the rf-SQUID flux qubits. A full rf-SQUID model that is in
agreement with the measured high energy spectrum is explored in the Supplementary Information. Panel (c) and (d) show
measured eigenspectra of the two-qubit system vs. h = h ≡ h for two values of annealing parameter s, s = 0.339 and
1 2 i
s=0.351 from left to right, respectively. Notice the avoided crossing at h =0. Panel (e) and (f) show analogous measured
i
eigenspectra for the eight-qubit system with (with h = ... = h ≡ h ). Because the eight-qubit gap closes earlier in QA for
1 8 i
this system, we show measurements for smaller s, s=0.271 and s=0.284 from left to right, respectively.
large gaps, g >k T, there is persistent entanglement at estimates for the energy gaps are derived from the un-
B
equilibrium(seeRefs.[18,26,28,29,31]andtheSupple- certainty in extracting the centroids from the rate data.
mentary Information). We discuss the actual source of the underlying Gaussian
We experimentally verified the existence of avoided widths (the observed level broadening) below. For both
crossings at multiple values of s in both the two- and the two- and eight-qubit system, we confirmed that the
eight-qubitsystemsbyusingQTSacrossarangeofbiases expectationvaluesofσz foralldeviceschangesignasthe
h 4,4 . In Fig. 3c we show the measured spectrum systemmovesthroughtheavoidedcrossing(seeFigs. 1-3
i
of t∈he{−two-q}ubit system at s = 0.339 up to an energy of of the Supplementary Information and [34])
6 GHz for a range of bias h . The ground states at the Figures 3e and f show similar measurements of the
i
far left and far right of the spectrum are the localized spectrum of eight coupled qubits at s = 0.271 and
states and , respectively. At h = 0, we observe s = 0.284 for a range of biases h . Again, we observe
i i
|↓↓(cid:105) |↑↑(cid:105)
an avoided crossing between these two states. We mea- an avoided crossing at h = 0. The measured energy
i
sure an energy gap g at zero bias, h = 0, between the gaps at s = 0.271 and 0.284 are g/h = 2.2 0.08 GHz
i
±
groundstateandthefirstexcitedstate,g/h=1.75 0.08 and g/h = 1.66 0.06 GHz, respectively. Although the
± ±
GHz by fitting a Gaussian profile to the tunneling rate eight qubit gaps in Figs. 3e and f are close to the two
dataatthesetwolowestenergylevelsandsubtractingthe qubit gaps in Figs. 3c and d, they are measured at quite
centroids. Here h (without any subscript) is the Planck different values of the annealing parameter s. As ex-
constant. Figure 3d shows the two-qubit spectrum later pected, the eight-qubit gap is closing earlier in the QA
in the QA algorithm, at s = 0.351. The energy gap has algorithm as compared to the two-qubit gap. The solid
decreased to g/h=1.21 0.06 GHz. Note that the error curves in Figs. c-f indicate the theoretical energy levels
±
5
predicted by Hamiltonian (1) and measurements of ∆(s) with the avoided crossings shown in Fig. 3, and high oc-
and (s). Again, the agreement between the experimen- cupation fraction of the ground state, confirmed early in
E
tallyobtainedspectraandthetheoreticalspectraisgood. QA by the measurements of P 1 shown in Fig. 4.
1
(cid:39)
For the early and intermediate parts of QA, the en- We then performed measurements of all available linear
ebrogtyh gthape tgwois- alanrdgeerigthhta-qnubteitmspyesrtaetmurse.,Wge(cid:29)expkeBcTt,thfoart cerxopsesc-tsautsicoenptviabliulietioefsσχzijfo≡rdth(cid:104)eσizit(cid:105)h/dqh˜ujb,itwahnedreh˜(cid:104)σ=iz(cid:105) ishthies
i j E j
ifweholdthesystemsattheses,thentheonlyeigenstate a bias applied to the jth qubit. The measurements are
with significant occupation will be the ground state. We performed at the degeneracy point (in the middle of the
confirmed this by using QTS in the limit of long tun- avoidedcrossings)wheretheclassicalcontributiontothe
neling times to probe the occupation fractions. Details cross-susceptiblity is zero.
are provided in Appendix C. Figures 4a and b show the From these measurements, we calculated as de-
χ
measured occupation fractions of the ground and first fined in Ref. [34] (see Appendix D for more dWetails). A
excited states as a function of s for both the two- and non-zero value of this witness detects ground-state en-
eight-qubit systems. The solid curves show the equilib- tanglement, and global entanglement in the case of the
rium Boltzmann predictions for T =12.5 mK and are in eight-qubitsystem(meaningeverypossiblebipartitionof
good agreement with the data. the eight-qubit system is entangled). Figures 4c and d
The width of the measured spectral lines is domi- show for the two- and eight-qubit systems. Note
χ
W
nated by the noise of the probe device used to perform that for two qubits at degeneracy, coincides with
χ
W
QTS [33]. The probe device was operated in a regime in ground-state concurrence. These results indicate that
which it is strongly coupled to its environment, whereas the two- and eight-qubit systems are entangled midway
the system qubits we studied are in the weak coupling through QA. Note also that a susceptibility-based wit-
regime. The measured spectral widths therefore do not ness has a close analogy to susceptibility-based measure-
represent the intrinsic width of the two- and eight-qubit ments of nano-magnetic systems that also report strong
energy eigenstates. During the intermediate part of QA, non-classical correlations [29, 31].
the ground and first excited states are clearly resolved.
TheoccupationfractionmeasurementsshowninFig.4
The ground state is protected by the multi-qubit en-
indicatethatmidwaythroughQA,thefirstexcitedstate
ergy gap g k T, and these systems are coherent.
B ofthesesystemsisoccupiedastheenergygapgbeginsto
(cid:29)
At the end of the annealing trajectory, the gap between
approach k T. The systems are no longer in the ground
B
the ground state and first excited state shrinks below
state, but, rather, in a mixed state. To detect the pres-
the probe qubit line width of 0.4 GHz. An analysis of
ence of mixed-state entanglement, we need knowledge
thespectroscopydata,whichestimatestheintrinsiclevel
about the density matrix of these systems. Occupation
broadening of the multi-qubit eigenstates, is presented
fraction measurements provide measurements of the di-
in the Supplementary Information. The analysis shows
agonalelementsofthedensitymatrixintheenergybasis.
thattheintrinsicenergylevelsremaindistinctuntillater
We assume that the density matrix has no off-diagonal
in QA. The interactions between the two- and eight-
elementsintheenergybasis(theydecayontimescalesof
qubit systems and their respective environments repre-
severalns). Werelaxthisassumptionbelow. Populations
sent small perturbations to Hamiltonian (1), even in the
P and P plotted in Figs. 4a and 4b indicate that the
1 2
regime in which entanglement is beginning to fall due to
system occupies these states with almost 100% probabil-
thermal mixing.
ity. This means that the density matrix can be written
(cid:80)2
in the form ρ = P ψ ψ where ψ represents
i=1 i| i(cid:105)(cid:104) i| | i(cid:105)
the ith eigenstate of Hamiltonian (1).
VI. ENTANGLEMENT MEASURES AND We use the density matrix to calculate standard en-
WITNESSES
tanglement measures, Wootters’ concurrence, [18], for
C
the two-qubit system, and negativity, [16, 35], for the
N
The tunneling spectroscopy data show that midway two- and eight-qubit system. For the maximally entan-
through QA, both the two- and eight-qubit systems had gledtwo-qubitBellstatewenotethat =1and =0.5.
C N
avoided crossings with the expected gap g kBT and Figure 4c shows as a function of s. Midway through
(cid:29) C
had ground state occupation P1 1. While observation QAwemeasureapeakconcurrence =0.53 0.05,indi-
(cid:39) C ±
of an avoided crossing is evidence for the presence of an cating significant entanglement in the two-qubit system.
entangledgroundstate(seeRef.[34]andtheSupplemen- Thisvalueof correspondstoanentanglementofforma-
C
tary Information for details), we can make this observa- tion E =0.388 (see Refs. [16, 18] for definitions). This
f
tion more quantitative with entanglement measures and is comparable to the level of entanglement, E = 0.378,
f
witnesses. obtained in Ref. [11], and indeed to the value E =1 for
f
We begin with a susceptibility-based witness, , the Bell state. Because concurrence is not applicable
χ
W C
which detects ground state entanglement. This witness to more than two qubits, we used negativity to de-
N
does not require explicit knowledge of Hamiltonian (1), tect entanglement in the eight-qubit system. For N >2,
but requires a non-degenerate ground state, confirmed is defined on a particular bipartition of the sys-
A,B
N
6
A 2QSystem B 8QSystem
1 1
n n
o o
ati ati
ul0.5 ul0.5
p p
o o
P P1 P P1
P2 P2
0 0
C D
0.8
N N
C 0.6 Wχ
0.6 Wχ
0.4
0.4
0.2 0.2
0 0
0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31
s s
FIG.4: (a),(b)Measurementsoftheoccupationfraction,orpopulation,ofthegroundstate(P )andfirstexcitedstate(P )
1 2
ofthetwo-qubitandeight-qubitsystem,respectively,versuss. Earlyintheannealingtrajectory,g(cid:29)k T,andthesystemisin
B
thegroundstatewithP1 <∼1. ThesolidcurvesshowtheequilibriumBoltzmannpredictionsforT =12.5mK.(c)Concurrence
C,negativityN andwitnessW versussforthetwo-qubitsystem. EarlyinQA,thequbitsareweaklyinteracting,thusresulting
χ
in limited entanglement. Entanglement peaks near s=0.37. For larger s, the gap between the ground and first excited state
shrinks and thermal occupation of the first excited state rises, thus extinguishing entanglement. Solid curves indicate the
expectedtheoreticalvaluesofeachwitnessormeasureusingHamiltonian(1)andBoltzmannstatistics. (d)NegativityN and
witness W versus s for the eight-qubit system. For all s shown, the nonzero negativity N and nonzero witness W report
χ χ
entanglement. Fors>0.39ands>0.312forthetwo-qubitandeight-qubitsystems,respectively,theshadedgreydenotesthe
regimeinwhichthegroundandfirstexcitedstatescannotberesolvedviaourspectroscopicmethod. Solidcurvesindicatethe
expected theoretical values of each witness or measure using Hamiltonian (1) and Boltzmann statistics.
tem into subsystems A and B. We define to be the constraintsonthedensitymatrixofthesystem,ρ(s). We
N
geometric mean of this quantity across all possible bi- then obtain an upper bound on Tr[ ρ(s)] by search-
AB
W
partitions. A nonzero indicates the presence of global ing over all ρ(s) that satisfy these linear constraints. If
N
entanglement. Figures 4c and d show the negativity cal- this upper bound is < 0, then we have shown entangle-
culated with measured P and P (and with the mea- ment for the bipartition A-B [36]. Figure 5 shows the
1 2
suredHamiltonianparameters∆and J )asafunction upper limit of the witness Tr[ ρ(s)] for the eight-
ij AB
E W
ofsforthetwoandeight-qubitsystems. Theeight-qubit qubitsystem. Weplotdataforthebipartitionthatgives
system has nonzero for s<0.315, thus indicating the the median upper limit. The error bars are derived from
N
presence of mixed-state global entanglement. Both con- a Monte-Carlo analysis wherein we used the experimen-
currence and negativity decrease later in QA where tal uncertainties in ∆ and J to estimate the uncertainty
C N
the first excited state approaches the ground state and in Tr[ ρ]. We also plot data for the two partitions
AB
W
becomesthermallyoccupied. Theexperimentalvaluesof that give the largest and smallest upper limits. For all
these entanglement measures are in agreement with the values of the annealing parameter s, except for the last
theoretical predictions (solid lines in Fig 4). The error two points, upper limits from all possible bipartitions of
bars in Figures 4c and d represent uncertainties in the the eight-qubit system are below zero. In this annealing
measurements of occupation fractions, ∆(s) and (s). range, the eight-qubit system is globally entangled.
E
As stated above, the calculation of and relies on
C N
theassumptionthattheoff-diagonaltermsinthedensity
matrix decay on times scales of several ns. We remove VII. CONCLUSIONS
this assumption and demonstrate entanglement through
the use of another witness , defined on some bi- To summarize, we have provided experimental evi-
AB
W
partition A-B of the eight-qubit system. The witness, dence for the presence of quantum coherence and en-
described in Appendix D, is designed in such a way tanglement within subsets of qubits inside a quantum
that Tr[ σ] 0 for all separable states σ. When annealingprocessorduringitsoperation. Ourconclusion
AB
W ≥
Tr[ ρ(s)] < 0, the state ρ(s) is entangled. Measure- is based on four levels of evidence: a. the observation
AB
W
ments of populations P and P provide a set of linear of two- and eight- qubit avoided crossings with a multi-
1 2
7
Improved designs of this device will allow much larger
0.2 systems to be studied. Our measurements represent an
median partition effective approach for exploring the role of quantum me-
0.1 lowest and highest partitions chanics in QA processors and ultimately to understand-
]
ρ ing the fundamental power and capability of quantum
B
A annealing.
W 0
[
r
T−0.1 ACKNOWLEDGEMENTS
n
o
d We thank C. Williams, P. Love, and J. Whittaker
n−0.2
u for useful discussions. We acknowledge F. Cioata and
o
b P. Spear for the design and maintenance of electron-
er−0.3 ics control systems, J. Yao for fabrication support, and
p
p D. Bruce, P. deBuen, M. Gullen, M. Hager, G. Lamont,
u
−0.4 L. Paulson, C. Petroff, and A. Tcaciuc for technical sup-
port. F.M.S. was supported by DARPA, under contract
FA8750-13-2-0035.
−0.5

0.24 0.26 0.28 0.3
s
Appendix A. QA Processor Description
FIG. 5: Upper limit of the quantity Tr[WABρ] versus s for Chip Description
several bipartitions A−B of the eight-qubit system. When
this quantity is < 0, the system is entangled with respect
Theexperimentsdiscussedinhereinwereperformedon
to this bipartition. The solid dots show the upper limit on
Tr[W ρ] for the median bipartition. The open dots above a sample fabricated with a process consisting of a stan-
AB
and below these are derived from the two bipartitions that dard Nb/AlOx/Nb trilayer, a TiPt resistor layer, pla-
give the highest and lowest upper limits on Tr[W ρ], re- narized SiO dielectric layers and six Nb wiring layers.
AB 2
spectively. Forthepointsats>0.3,themeasurementsofP1 Thecircuitdesignrulesincludedaminimumlinewidthof
and P do not constrain ρ enough to certify entanglement.
2 0.25 µm and 0.6 µm diameter Josephson junctions. The
processor chip is a network of densely connected eight-
qubitunitcellswhicharemoresparselyconnectedtoeach
qubit energy gap g k T; b. the witness χ, calcu-
(cid:29) B W other (see Fig. 1 for photographs of the processor). We
lated with measured cross-susceptibilities and coupling
report measurements made on qubits from one of these
energies,whichreportsgroundstateentanglementofthe
unitcells. Thechipwasmountedonthemixingchamber
two-andeight-qubitsystem. Noticethatthesetwolevels
of a dilution refrigerator inside an Al superconducting
of evidence do not require explicit knowledge of Hamil-
shield and temperature controlled at 12.5 mK.
tonian (1); c. the measurements of energy eigenspectra
and equibrium occupation fractions during QA, which
allow us to use Hamiltonian (1) to reconstruct the den- Qubit Parameters
sity matrix, with some weak assumptions, and calculate
concurrence and negativity. These standard measures
The processor facilitates quantum annealing (QA)
of entanglement report non-classical correlations in the
of compound-compound Josephson junction rf SQUID
two- and eight-qubit systems; d. the entanglement wit-
(radio-frequency superconducting quantum interference
ness ,whichiscalculatedwiththemeasuredHamil-
AB device) flux qubits [37]. The qubits are controlled via
W
tonian and with constraints provided by the measured the external flux biases Φx and Φx which allow us to
qi ccjj
populations of the ground and the first excited states.
treat them as effective spins (see Fig. 1). Pairs of qubits
This witness reports global entanglement of the eight-
interact through tunable inductive couplings [25]. The
qubit system midway through the QA algorithm.
system can be described with the time-dependent QA
The observed entanglement is persistent at thermal Hamiltonian,
equilibrium,anencouragingresultasanypracticalhard-
 
ware designed to run a quantum algorithm will be in- N N
(cid:88) (cid:88) 1 (cid:88)
evitably coupled to a thermal environment. The ex- HS(s)=E(s)− hiσiz+ Jijσizσjz−2∆(s) σix,
perimental techniques that we have discussed provide i i<j i
measurements of energy levels, and their populations, (3)
for arbitrary configurations of Hamiltonian parameters where σx,z are Pauli matrices for the ith qubit, i =
i
∆,h ,J during the QA algorithm. The main limitation 1,...,N. The energy scales ∆ and are the transverse
i ij
E
ofthetechniqueisthespectralwidthoftheprobedevice. and longitudinal energies of the spins, respectively, and
8
qubit parameter median measured value
10
critical current, Ic 2.89 µA 10
qubit inductance, L 344 pH Coherent Incoherent
q ← →
qubit capacitance, C 110 fF
q 9
10
TABLE I: Qubit Parameters.
z)108
the unitless biases h and couplings J encode a par- H
i ij (
ticular optimization problem. We define h˜ h and h
J˜ij Jij. We have mapped the annealinig≡paErami eter ∆/107 q1
≡ E
s for this particular chip to a range of Φx with the q2
ccjj q3
relation
6 q4
10 q5
s (Φx (t) Φx )/(Φx Φx )=t/t , q6
≡ ccjj − ccjj,initial ccjj,final− ccjj,initial f q7
(4) q8
5
where t is the total anneal time. We implement QA for 10
f 0.2 0.3 0.4 0.5 0.6
this processor by ramping the external control Φx (t) s
ccjj
from Φx = 0.596 Φ (s = 0) at t = 0 to
ccjj,initial 0
Φx =0.666 Φ (s=1) at t=t . The energy scale
ccjj,final 0 f
E ≡Meff|Iqp(s)|2 issetbythes-dependentpersistentcur- 1.5
rentofthequbit Ip(s) andthemaximummutualinduc-
| q |
tance between qubits M =1.37 pH [8]. The transverse
eff
terminHamiltonian(3),∆(s),istheenergygapbetween
thegroundandfirstexcitedstateofanisolatedrfSQUID
1
atzerobias. ∆alsochangeswithannealingparameters.
)
Φx(t)isprovidedbyaglobalexternalmagneticfluxbias A
qi µ
alongwithlocalinsitutunabledigital-to-analogconvert- (
ers (DAC) that tune the coupling strength of this global pq|
I
bias into individual qubits and thus allow us to specify | q1
0.5 q2
individual biases hi. The coupling energy between the q3
ith and jth qubit is set with a local in situ tunable DAC q4
q5
that controls Φxco,ij. q6
The main quantities associated with a flux qubit, ∆ q7
q8
and|Iqp|, primarilydependonmacroscopicrfSQUIDpa- 00 0.2 0.4 0.6 0.8 1
rameters: junction critical current Ic, qubit inductance s
L , and qubit capacitance C . We calibrated all of these
q q
parameters on this chip as described in [6, 8]. We cali-
bratedallinter-qubitcouplingelementsacrosstheiravail- FIG. 6: (a) ∆(s) vs s. We show measurements for all eight
able tuning range from 1.37 pH to 3.7 pH as described qubits studied in this work. We used a single qubit Landau-
− Zener experiment to measure ∆/h<100 MHz [38]. We used
inRef.[25]. Wecorrectedforvariationsinqubitparame-
qubit tunneling spectroscopy (QTS) to measure ∆/h > 1
terswithon-chipcontrolasdescribedin[8]. Thisallowed
GHz[33]. Theredlineshowsthetheoreticalpredictionforan
us to match Ip and ∆ across all qubits throughout the
| q| rf SQUID model employing the median qubit parameters of
annealing trajectory. Table I shows the median qubit the eight devices. The vertical black line separates coherent
parameters for the devices studied here. (left)andincoherent(right)evolutionasestimatedbyanaly-
eigFhitguqureb6itss.ho∆wswmaseamseuarseumreendtwsoitfh∆sianngdle|Iqqpu|bvits.Lsanfodraaul-l ssihsowofmsineagsleurqeumbeinttsspfeocrtraalll leiingehtshqaupbeist.s s(tbu)di|eIqdp|(ins)tvhsissw.oWrke.
Weusedatwo-qubitcoupledfluxmeasurementwiththeinter-
Zenermeasurementsfroms=0.515tos=0.658[38]and
qubitcouplingelementsetto1.37pH[8]. Theredlineshows
withqubittunnelingspectroscopy(QTS)froms=0.121
the theoretical prediction for an rf SQUID model employing
tos=0.407[33]. Theresolutionlimitofqubittunneling
the median qubit parameters of the eight devices.
spectroscopy and the bandwidth of our external control
lines during the Landau-Zener measurements prevented
us from characterizing ∆ between s = 0.4 and s = 0.5,
respectively. Ip was measured by coupling a second
| q|
probequbittothequbitq withacouplingofM =1.37
i eff
pH and measuring the the flux M Ip(s) as a function
eff| qi |
of s. Ip is matched between qubits to within 3% and
| q|
∆(s) is matched between qubits to within 8% across the annealing region explored in this study.
9
Appendix B. Qubit Tunneling Spectroscopy (QTS)
QTSallowsonetomeasuretheeigenspectrumofanN-
qubit system governed by Hamiltonian . Details on
S
H
the measurement technique are presented elsewhere [33].
For convenience in comparing with this reference, we de-
fine a qubit energy bias (cid:15) 2h˜. Measurements are per-
i i
≡
formed by coupling an additional probe qubit q , with
P
qubit tunneling amplitude ∆ ∆, J˜, to one of the N
P
(cid:28) | |
qubits of the system under study, for example q . When
1
we use a coupling strength J˜ between q and q and
P P 1
apply a compensating bias (cid:15) = 2J˜ to q , the resulting
1 P 1
system + probe Hamiltonian becomes
H =H [J˜ σz (1/2)(cid:15) ](1 σz). (5)
S+P S − P 1 − P − P
Foroneofthelocalizedstatesoftheprobequbit, , FIG. 7: Typical waveforms during QTS. We prepare the
|↑(cid:105)P
for which an eigenvalue of σz is equal to +1 (i.e. the initialstatebyannealingprobeandsystemqubitsfroms=0
P
to s=1 in the presence of a large polarization bias (cid:15) . We
probe qubit in the right well), the contribution of the pol
thenbiasthesystemqubitq (towhichtheprobeisattached)
probe qubit is exactly canceled, leading to H =H , 1
S+P S to a bias (cid:15) and the probe qubit to a bias (cid:15) . With these
with composite eigenstates n, = n and eigen- 1 P
| ↑(cid:105) | (cid:105)⊗|↑(cid:105)P biases asserted, we then adjust the system qubits’ annealing
values EnR = En, which are identical to those of the parametertoanintermediatepoints∗ andtheprobequbitto
original system without the presence of the probe qubit. a point s and dwell for a time τ. Finally, we complete the
P
Here, n is an eigenstate of the Hamiltonian H (n = anneal s→1 and read out the state of the qubits.
S
| (cid:105)
1,2,...,2N).
For the other localized state of the probe qubit, ,
when this qubit is in the left well, the ground sta|t↓e(cid:105)Pof qubit. Figure 7 summarizes these waveforms during a
HELS++P(cid:15)is,|wψh0Le,r↓e(cid:105) ψ=L|ψi0sL(cid:105)th⊗e|g↓r(cid:105)oPu,nwdistthateeigoefnHvalue2E(cid:101)J˜0Lσ=z typWicealpQerTfoSrmmetahsisurmemeaesnutr.ement for a range of τ which
an0d ELPis its eig|en0v(cid:105)alue. We choose J˜ Sk−T sPuch1 allows us to measure an initial rate of tunneling Γ from
that th0e state |ψ0L,↓(cid:105) is well separate|dPf|r(cid:29)om tBhe next r|ψan0Lg,e↓(cid:105)oftoth|ψe,p↑r(cid:105)o.bWe qeurbeiptebaitasth(cid:15)is.mPeeaaskusreimneΓntcoorfreΓspfoornda
excited state for ferromagnetically coupled systems, and P
toresonancesbetweentheinitiallypreparedstateandthe
thus system + probe can be initialized in this state to
state n, , thus allowing us to map the eigenspectrum
high fidelity.
| ↑(cid:105)
Introducing a small transverse term, 1∆ σx, to of the N-qubit system.
−2 P P For the plots in the main paper, measurements of Γ
Hamiltonian (5) results in incoherent tunneling from
the initial state ψL, to any of the available n, are normalized to [0,1] by dividing the maximum value
| 0 ↓(cid:105) | ↑(cid:105) across a vertical slice to give a visually interpretable re-
states [39]. A bias on the probe qubit, (cid:15) , changes the
P
sult. Figure 8b shows a typical raw result in units of
energy difference between the probe and mani-
folds. We can thus bring ψL, into|↓re(cid:105)Psonanc|e↑(cid:105)wPith any µs−1.
| 0 ↓(cid:105) We posed ferromagnetically coupled instances of the
of n, states (when E(cid:101)L = ER) allowing resonant tun-
| ↑(cid:105) 0 n form
neling between the two states. The rate of tunneling out
tohfethloecaintiitoinasllyofpnre,par.ed state |ψ0L,↓(cid:105) is thus peaked at HP =−(cid:88)hiσiz+(cid:88)Jijσizσjz (6)
| ↑(cid:105) i i<j
The measurement of the eigenspectrum of an N-qubit
systemthusproceedsasfollows. Wecoupleanadditional with J < 0 for two and eight qubit subsections of the
ij
probe qubit to one of the N-qubits (say, to q1) with cou- QA processor. Figure 8a shows typical measurements of
plingconstantJ˜ . WepreparetheN+1-qubitsystemin Γ for a two qubit subsection at several biases h and at
P i
the state ψL, by annealing from s=0 to s=1 in the s = 0.339 (J˜ < 0). We assembled multiple measure-
| 0 ↓(cid:105) P
presenceoflargebias(cid:15)pol <0onallthesystemandprobe ments to produce the spectrum shown in Figure 8b.
qubits. We then adjust s for the N-qubit system to an
intermediate point s∗ [0,1] such that ∆ k T/h and
B
s for the probe qubit t∈o s =0.612 such t(cid:29)hat ∆ /h 1 Appendix C. Equilibrium Distribution of System
P P
∼
MHz (here h is the Planck constant). We assert a com-
pensating bias (cid:15) = 2J˜ to this qubit. We dwell at this Inadditiontotheenergyeigenspectrum,QTSalsopro-
1 P
point for a time τ, complete the anneal s 1 for the vides a means of measuring the equilibrium distribution
→
system+probe, and then read out the state of the probe of an N-qubit system with a probe qubit. Suppose we
10
centering
0.05 0.05 0.05
0.04 0.04 0.04
0.03 0.03 0.03
0.02 0.02 0.02
0.01 0.01 0.01
0 0 0
0 5 10 15 0 5 10 15 0 5 10 15
6
0.05
4 0.04
)
z
H 0.03
2
G
(
E 0.02
0
0.01
−2
0
−4 −2 0 2 4
h (GHz)
FIG. 8: Spectroscopy data for two FM coupled qubits at J˜ < 0. (a) Measurements of tunneling rate Γ for three values of
P
h = h ≡ h . These data were taken at s = 0.339. Peaks in Γ reveal the energy eigenstates of the two-qubit system. (b)
1 2 i
Multiple scans of Γ for different values of h assembled into a two-dimensional color plot. For better interpretability, we have
i
subtracted off a baseline energy with respect to (a) such that the ground and first excited levels are symmetric about zero.
Notice the avoided crossing at h =0. The peak tunneling rate Γ∼|∆ (cid:104)ψL|n(cid:105)|2 [33]. The solid black and white curves plot
i P 0
thetheoreticalexpectationsfortheenergyeigenvaluesusingindependentmeasurementsshowninFigure6andHamiltonian(1).
are in the limit J˜ k T such that there is only one spectrum, we can choose an (cid:15) such that ψL
| P| (cid:29) B P | 0(cid:105) ⊗ |↓(cid:105)P
accessiblestateinthe manifold: ψL . Asde- and n are degenerate. Since the occupation of
|↓(cid:105)P | 0(cid:105)⊗|↓(cid:105)P | (cid:105)⊗|↑(cid:105)P
scribedabove,theotheravailablestatesinthesystemare thestatedependsonitsenergy,weexpectthat,afterlong
thecompositeeigenstates n inthe manifold evolutiontimes,thesetwodegeneratestatesareoccupied
| (cid:105)⊗|↑(cid:105)P |↑(cid:105)P
where n isaneigenstateoftheN-qubitsystemwithout with equal probability, PL((cid:15) =E ) = PR. Aligning the
| (cid:105) P n n
the probe qubit attached. Energy levels ER of the state ψL withallpossible2N states n we
n |↑(cid:105)P | 0(cid:105)⊗|↓(cid:105)P | (cid:105)⊗|↑(cid:105)P
manifold coincide with the energy levels E of the sys- obtain a set of relative probabilities PR. These relative
n n
tem, ER =E , even in the presence of coupling between probabilities characterize the population distribution in
n n
the probe qubit and the system. We make the assump- the system since they are uniquely determined by the
tionthatthepopulationofaneigenstatedependsonlyon energy spectrum E . However, as follows from Eq. (7),
n
its energy. Degenerate states have the same population. the set PR is not properly normalized. The probability
n
Let PL represent the probability of finding the distribution of the system itself is given by:
probe+system in the state ψL and PR represent
| 0(cid:105)⊗|↓(cid:105)P n PR
the probability of finding the probe+system in the state P (E )= n , (8)
n . At any point in the probe+system evolution n n (cid:80)2N PR
| (cid:105)⊗|↑(cid:105)P i=1 i
we expect:
(cid:80)2N
where P = 1. At every eigenenergy, (cid:15) = E ,
n=1 n P n
the denominator of Eq. (8) can be found from Eq. (7),
2N
PL+(cid:88)PR =1 (7) so that the population distribution of the system Pn has
i the form
i=1
PR PL((cid:15) =E )
As described in the previous section, we can alter the Pn = 1 nPL = 1 PLP((cid:15) =nE ). (9)
energy of |ψ0L(cid:105)⊗|↓(cid:105)P with the probe bias (cid:15)P. Based on − − P n
the spectroscopic measurements of the N-qubit eigen- Thus,theprobabilityP tofindthesystemofN qubitsin
n