SC I ENCE ADVANCES | R E S EARCH ART I C L E

PHYS I CS

1CAS Key Laboratory of Microscale Magnetic Resonance and Department of ModernPhysics, University of Science and Technology of China (USTC), Hefei 230026, China.2Hefei National Laboratory for Physical Sciences at the Microscale, USTC, Hefei230026, China. 3Synergetic Innovation Center of Quantum Information and Quan-tum Physics, USTC, Hefei 230026, China. 4Institut für Theoretische Physik und IQST,Albert-Einstein-Allee 11, Universität Ulm, D-89081 Ulm, Germany.*These authors contributed equally to this work.†Corresponding author. Email: zhenyu.wang@uni-ulm.de (Z.-Y.W.); djf@ustc.edu.cn (J.D.)‡Present address: Department of Radiation Oncology, Johns Hopkins School of Medi-cine, Baltimore, MD 21287, USA.

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

Copyright © 2019

The Authors, some

rights reserved;

exclusive licensee

American Association

for the Advancement

of Science. No claim to

originalU.S. Government

Works. Distributed

under a Creative

Commons Attribution

NonCommercial

License 4.0 (CC BY-NC).

D

Breaking the quantum adiabatic speed limit by jumpingalong geodesicsKebiao Xu1,2,3*, Tianyu Xie1,2,3*, Fazhan Shi1,2,3, Zhen-Yu Wang4†, Xiangkun Xu1,2,3‡,Pengfei Wang1,3, Ya Wang1,3, Martin B. Plenio4, Jiangfeng Du1,2,3†

Quantum adiabatic evolutions find a broad range of applications in quantum physics and quantum technolo-gies. The traditional form of the quantum adiabatic theorem limits the speed of adiabatic evolution by theminimum energy gaps of the system Hamiltonian. Here, we experimentally show using a nitrogen-vacancycenter in diamond that, even in the presence of vanishing energy gaps, quantum adiabatic evolution is possible.This verifies a recently derived necessary and sufficient quantum adiabatic theorem and offers paths to overcomethe conventionally assumed constraints on adiabatic methods. By fast modulation of dynamic phases, we dem-onstrate near–unit-fidelity quantum adiabatic processes in finite times. These results challenge traditional viewsand provide deeper understanding on quantum adiabatic processes, as well as promising strategies for thecontrol of quantum systems.

ow

on A

ugust 29, 2021http://advances.sciencem

ag.org/nloaded from

INTRODUCTIONCoherent control on quantum systems is a fundamental element ofquantum technologies that could revolutionize the fields of informationprocessing, simulation, and sensing. A powerful and universal methodto achieve this control is the quantum adiabatic technique, which ex-hibits intrinsic robustness against control errors ensured by the quan-tum adiabatic evolution (1). Besides important applications in quantumstate engineering (2, 3), quantum simulation (4–6), and quantumcomputation (7–11), the quantum adiabatic evolution itself also pro-vides interesting properties such as Abelian (12) or non-Abelian geo-metric phases (13), which can be used for the realization of quantumgates. However, the conventional quantum adiabatic theorem (14, 15),which dates back to the idea of extremely slow and reversible change inclassical mechanics (14, 16), imposes a speed limit on the quantumadiabatic methods, that is, for a quantum process to remain adiabatic,the changes in the system Hamiltonian at all times must be muchsmaller than the energy gap of the Hamiltonian. On the other hand,to avoid perturbations from the environment, high rates of change aredesirable. This tension can impose severe limitations on the practicaluse of adiabatic methods. Despite the long history and broad applica-bility, it was found recently that key aspects of quantum adiabatic evo-lution remain not fully understood (17, 18) and that the condition inthe conventional adiabatic theorem is not necessary for quantum ad-iabatic evolution (19, 20).

In this work, we experimentally demonstrate adiabatic evolutionswith vanishing energy gaps and energy level crossings, which areallowed under a recently proven quantum adiabatic condition (20) thatis based on dynamical phases instead of energy gaps, by using a nitrogen-vacancy (NV) center (21) in diamond. In addition, we reveal that usingdiscrete jumps along the evolution path allows quantum adiabatic pro-cesses at unlimited rates, which challenges the view that adiabatic pro-

cesses must be slow. By jumping along the path, one can even avoid pathpoints where the eigenstates of the Hamiltonian are not feasible inexperiments. Furthermore, we theoretically and experimentallydemonstrate the elimination of all the nonadiabatic effects on the systemevolution of a finite evolution time by driving the system along thegeodesic that connects initial and final states, as well as combating systemdecoherence by incorporating pulse sequences into adiabatic driving.

RESULTSExperimental study of the necessary and sufficient quantumadiabatic conditionTo describe the theory for experiments, we consider a quantum systemdriven by a Hamiltonian H(l) for adiabatic evolution. In terms of its in-stantaneous orthonormal eigenstates ∣yn(l)⟩ (n = 1,2, …) and eigen-energies En(l), the Hamiltonian is written as H(l) = ∑n En(l)∣yn(l)⟩⟨yn(l)∣. For a given continuous finite evolution path, ∣yn(l)⟩ changesgradually with the configuration parameter l. In our experiments, lcorresponds to an angle in some unit and is tuned in time such thatl = l(t) ∈ [0,1]. The system dynamics driven by the Hamiltonian isfully determined by the corresponding evolution propagator U(l). It isshown that one can decompose the propagator U(l) = Uadia(l)Udia(l)as the product of a quantum adiabatic evolution propagator Uadia(l)that describes the ideal quantum evolution in the adiabatic limit anda diabatic propagator Udia(l) that includes all the diabatic errors(20). In the adiabatic limit, Udia(l) = I becomes an identity matrixand the adiabatic evolution U = Uadia(l) fully describes the geo-metric phases (12, 13) and dynamic phases accompanying the ad-iabatic evolution (that is, the deviation from adiabaticity U − Uadia

vanishes). This decomposition guarantees that both Uadia(l) andUdia(l) are gauge invariant, i.e., invariant with respect to any cho-sen state basis.

According to the result of (20), the error part satisfies the first-order differential equation (ℏ = 1)

ddl

UdiaðlÞ ¼ iWðlÞUdiaðlÞ ð1Þ

with the boundary condition Udia(0) = I. The generatorW(l) describesall the nonadiabatic transitions. On the basis of ∣yn(0)⟩, the diagonal

1 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

D

matrix elements of W(l) vanish, i.e., ⟨yn(0)∣W(l)∣yn(0)⟩ = 0. Theoff-diagonal matrix elements

⟨ynð0Þ∣WðlÞ∣ymð0Þ⟩ ¼ eifn;mðlÞGn;mðlÞ ð2Þ

are responsible for nonadiabaticity. Here, fn,m(l) ≡ fn(l) − fm(l) isthe difference of the accumulated dynamic phases fn(l) on ∣yn(l)⟩,and the geometric part Gn,m(l) = ei[gm(l) − gn(l)]gn,m(l) consists ofthe geometric functions gn;mðlÞ ¼ i ynðlÞ∣ d

dl∣ymðlÞ� �

and the ge-ometric phases gnðlÞ ¼ ∫l0 gn;nðl′Þdl′.

Equation 2 shows that the differences of dynamic phases fn,m aremore fundamental than the energy gaps in suppressing the nonadia-batic effects because the energies En do not explicitly appear in theseequations. According to (20), when the dynamic phase factors at dif-ferent path points add destructively

Dn;mðlÞ ¼ ∣∫l0 eifn;mðl′Þdl′∣ < D ð3Þ

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

for n ≠ m and any l ∈ [0,1] of a finite path with bounded Gn,m(l)and d

dlGn;mðlÞ, the deviation from adiabaticity can be made arbi-trarily small by reducing D with a scaling factor determined by themagnitudes of Gn,m(l) and d

dlGn;mðlÞ, that is, the operator norm‖UdiaðlÞ � I‖ <

ffiffiffiD

p ðG2tot þ G′totÞl2 þ ð ffiffiffi

Dp þ DÞGtot , where Gtot =

∑n≠m max∣Gn,m(l′)∣ and G′tot ¼ ∑n≠m max∣ ddl′Gn;mðl′Þ∣ for 0 <

l′ ≤ l (20). In the limit D→ 0, the system evolution is adiabatic alongthe entire finite path with Udia(l) → I. For a zero gap throughout theevolution path, the evolution is not adiabatic because Dn,m(l) = l isnot negligible because of the constructive interference of the dynam-ic phase factors at different path points. For a large constant gap, thedestructive interference gives a negligible Dn,m and hence an adiabaticevolution.

To experimentally verify the adiabatic condition in Eq. 3 by an NVcenter, we construct the Hamiltonian for adiabatic evolution in thestandard way (2, 3), that is, we apply a microwave (MW) field to drivethe NV electron spin states ∣ms = 0⟩ ≡ ∣ −z⟩ and ∣ms = +1⟩ ≡ ∣z⟩ (see

on August 29, 2021

http://advances.sciencemag.org/

ownloaded from

ImRe

Ener

gies

(MH

z)

A

Init. ToTT mo.Driving

H

Stat

e pr

ojec

tion

0.0

0.5

1.0

D

Stat

e pr

ojec

tion

Evolution time (µs)

0.0

0.5

1.0

0 2 4 6 8

C

ImRe

Ener

gies

(MH

z)

G

Im

Re

Ener

gies

(MH

z)

Stat

e pr

ojec

tion

0.0

0.5

1.0

Evolution time (µs)0 2 4 6 8

Evolution time (µs)0 2 4 6 8

E F

B

0.12

0.06

0.0

0.12

0.06

0.0

0.12

0.06

0.00.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 1. Quantum adiabaticity of continuous driving. (A) Illustration of experimental control. An MW resonant with an NV transition forms in the rotating frame aHamiltonian with the instantaneous eigenstates ∣y1(l)⟩ = ∣+l⟩ and ∣y2(l)⟩ = ∣−l⟩ that are separated by an energy gap W(l) proportional to the amplitude of the MWfield. (B) Evolving path (red curve with an arrowhead) on the Bloch sphere when increasing the parameter l = l(t) in the MW phase with the evolution time t. (C) Theenergies of the eigenstates for a constant gap W(l) = W0 = 2p × 6 MHz and the corresponding real (Re) and imaginary (Im) parts of eif1,2 as a function of l. (D) The measuredprojections (dots) of the system state on the ∣x⟩, ∣y⟩, and ∣z⟩ states. The system state was initialized in ∣+l = 0⟩ = ∣x⟩ [see (B)] and subsequently driven by the Hamiltonianwith the eigenenergies shown in (C) and with a changing rate dl/dt = 0.12 MHz and a path length qg = 2p. The lines show the ideal state projections of the instantaneouseigenstate ∣+l⟩. The red line is a plot of D1,2(l), i.e., the interference of eif1,2 at different path points. (E and F) Same as (C) and (D), respectively, but for a gap W(l) =W0[2 + cos (W0lT)], larger than the gap in (C). Because e1,2(l) is not negligible, the corresponding evolution in (F) is not adiabatic. (G and H) Same as (C) and (D),respectively, but for the gap W(l) = Wp(l) that has energy level crossings. D1,2(l) is negligible and induces the quantum adiabatic evolution shown in (H). The error barsin all the figures represent two SEM.

2 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

http://advances.scD

ownloaded from

Fig. 1A and Materials and Methods for experimental details). TheHamiltonian H(l) under an on-resonant MW field reads

HxyðlÞ ¼ WðlÞ2

∣y1ðlÞ½ i⟨y1ðlÞ∣� ∣y2ðlÞ⟩ y2ðlÞ∣h � ð4Þ

where the energy gap W(l) is tunable and the instantaneous eigenstatesof the system Hamiltonian ∣y1(l)⟩ = ∣+l⟩ and ∣y2(l)⟩ = ∣−l⟩. Here

∣±li≡ 1ffiffiffi2

p ð∣z⟩ ± eiqgl∣� z⟩Þ ð5Þ

is tunable by varying the MW phases qgl. We define the initial eigenstates∣±x⟩ ≡ ∣±0⟩ and the superposition states ∣± yi≡ 1ffiffi

2p ð∣z⟩ ± i∣� z⟩Þ

for convenience.In the traditional approach where the Hamiltonian varies slowly with

a nonvanishing gap, the strength of relative dynamic phase f1,2 =f1(l) − f2(l) rapidly increases with the change of the path parameterl, giving the fast oscillating factor eif1,2 with a zero mean [see Fig. 1Cfor the case of a constant gap W(l) = W0]. Therefore, the right-handside of Eq. 1 is negligible in solving the differential equation, leading tothe solution Udia(l) ≈ I. As a consequence of the adiabatic evolu-tion U ≈ Uadia(l), the state initialized in an initial eigenstate of theHamiltonian follows the evolution of the instantaneous eigenstate(Fig. 1D).

However, a quantum evolution with a nonvanishing gap and along evolution time is not necessarily adiabatic. In Fig. 1 (E and F),we show a counterexample that increasing the energy gap in Fig. 1C to

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

W(l) = W0[2 + cos (W0lT)] ≥ W0 will not realize adiabatic evolutionbecause, in this case, the D1,2(l) in Eq. 3 and the G1,2(l) are not neg-ligible. For example, D1,2(l) = J2(1)l ≈ 0.115l (Jn being the Besselfunction of the first kind) whenever the difference of dynamic phasesis amultiple of 2p. This counterexample is different from the previouslyproposed counterexamples (17, 18, 20, 22), where the Hamiltoniancontains resonant terms that increase ∣ d

dlGn;mðlÞ∣ and hence modifythe evolution path when increasing the total time. Our counter-example also demonstrates that the widely used adiabatic condition∣ yn∣

ddt ∣ym

� �∣=∣En � Em∣≪1 (15), which is based on the energy

gap and diverges at En − Em = 0, does not guarantee quantum adiabaticevolution. On the contrary, the condition in Eq. 3 based on dynamicphases, i.e., integrated energy differences, does not diverge for anyenergy gaps. We note that fast amplitude fluctuations on the controlfields (hence energy gaps) can exist in adiabatic methods [e.g., see (23)]because of their strong robustness against control errors. By adding errorsin the energy gap shown in Fig. 1E, the adiabaticity of the evolution issubstantially enhanced (Fig. 2), showing that the situations to have non-adiabatic evolution with a fluctuating energy gap are relatively rare.

We demonstrate that adiabatic evolution can be achieved evenwhen the energy spectrum exhibits vanishing gaps and crossingsas long as Eq. 3 is satisfied for a sufficiently small D. As an exam-ple, we consider the energy gap of the form WðlÞ ¼ WpðlÞ ≡W′0½1þa cosð2W′0TlÞ�, which has zeros and crossings for ∣a∣ > 1 [see Fig.1G for the case of a ≈ 2.34, where W′0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=ð2þ a2Þp

W0 is usedto have the same average MW power in Fig. 1 (C and G) ]. Despite thevanishing gaps and crossings, the corresponding factor eif1,2 parame-terized by the parameter l = t/T is fast oscillating (Fig. 1G) with azero mean and realizes quantum adiabatic evolution for a sufficiently

on August 29, 2021

iencemag.org/

0.2 0.4 0.6 0.8 1.00.00.0

0.06

0.12

0 2 4 6 80.0

0.5

1.0

Evolution time (µs)

Stat

e pr

ojec

tion

ImRe

0 2 4 6 8Evolution time (µs)

Stat

e pr

ojec

tion

0.0

0.5

1.0

ImRe

Stat

e pr

ojec

tion

0.0

0.5

1.0

Evolution time (µs)0 2 4 6 8

Ener

gies

(MH

z)

Ener

gies

(MH

z)

Ener

gies

(MH

z)

Re Im

A B C

0.2 0.4 0.6 0.8 1.00.00.0

0.06

0.12

0.2 0.4 0.6 0.8 1.00.00.0

0.06

0.12

Fig. 2. Recovery of quantum adiabaticity by adding energy gap fluctuations. (A) The energies of instantaneous eigenstates, the real and imaginary parts of eif1,2,D1,2(l), and the measured projections (dots) of the system state on the ∣x⟩, ∣y⟩, or ∣z⟩ states. The results are the same as those in Fig. 1 (E and F) where the energy gapW(l) = W0[2 + cos (W0lT)]. (B and C) Same as (A) but changing the energy gap to W(l) → 1.1W(l) and W(l) → 0.8W(l), respectively, by adding an amplitude bias in thecontrol field of the experiments. The fluctuation in the energy gap induces random modulation on the function eif1,2. The destructive interference on eif1,2 leads to asmaller average D1,2(l) and hence improved quantum adiabatic evolution. In (A), the cyan dashed line in the plot of Dn,m(l) shows the line J2(1)l ≈ 0.115l.

3 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

on August 29, 2021

http://advances.sciencemag.org/

Dow

nloaded from

large total time T (see Fig. 1H). In fig. S1, we show how the adiaba-ticity can also be preserved when gradually introducing energy levelcrossings.

Unit-fidelity quantum adiabatic evolution within afinite timeWithout the restriction to nonzero energy gaps, it is possible to com-pletely eliminate nonadiabatic effects and to drive an arbitrary initialstate ∣Yi⟩ to a target state ∣Yt⟩ of a general quantum system by the

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

quantum adiabatic evolution of a finite time duration. We demon-strate this by driving the system along the geodesic for maximal speed[see, e.g., (24, 25) for more discussion on the geodesic in quantummechanics]. The system eigenstate ∣y1ðlÞi ¼ cos 1

2 qgl� �

∣y1ð0Þi þsin 1

2 qgl� �

∣y2ð0Þi connects ∣y1(0)⟩ and ∣y1(1)⟩ along the geodesicby varying l = 0 to l = 1, with its orthonormal eigenstate ∣y2ðlÞi ¼�sin 1

2 qgl� �

∣y1ð0Þi þ cos 12 qgl� �

∣y2ð0Þivaried accordingly (see Ma-terials and Methods). The method works for any quantum system(e.g., a set of interacting qubits) because geodesics can always be found

= 0 = 0.2 = 1

A B C

Fide

lity

D

Im

Re Re

Im ImRe

2.0 6.0 0.18.04.00.00.00.040.080.12

2.0 6.0 0.18.04.00.00.00.040.080.12

2.0 6.0 0.18.04.00.00.00.040.080.12

Fig. 3. Transition of adiabatic driving from the standard continuous protocol to the jumping protocol. (A) Relations among the energy gap W(l), path parameterl, evolution time t, phase factor eif1,2(l), and D1,2(l) for the standard adiabatic driving with a constant gap W0 = 2p × 5 MHz. (B and C) Same as (A) with the maximum gapW0 = 2p × 5 MHz but with a ratio rjump of intervals to have W(l) = 0 along the path parameter l. Therefore, the case of rjump = 0 corresponds to the standard adiabaticprotocol without a vanishing gap. A ratio rjump > 0 in (B) opens regions that have W(l) = 0. For the maximum value rjump = 1, we get in (C) the jumping protocol thatonly drives the system at discrete path points with a Rabi frequency equalling W0. For comparison, the plot of D1,2(l) in (A) is also shown in (B) and (C) by a gray dashedline. (D) Calculated fidelity to the ideal adiabatic state at the final time as a function of rjump. The fidelity increases to 100% when the driving is getting to the jumpingprotocol. The solid line shows the case that the initial state is prepared in the initial eigenstate ∣x⟩ of the Hamiltonian, while the dashed line is the result for the initial statebeing the superposition state ∣y⟩. The driving is along the adiabatic path given in Eq. 5 with qg = p and is repeated back and forth three times for a total time T = 3 ms.

4 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

httD

ownloaded from

(24, 25). An example of the geodesic path for a single qubit is given inEq. 5, which intuitively can be illustrated by the shortest path on theBloch sphere (see Fig. 1B). We find that, along the geodesic, the nonzeroelements g2,1(l) and g1,2(l) are constant. We adopt the sequencetheoretically proposed in (20) that changes the dynamic phases at Nequally spaced path points l = lj ( j = 1,2, … , N). By staying at eachof the points

lj ¼ ð2NÞ�1ð2j� 1Þ ð6Þ

for a time required to implement a p phase shift on the dynamicphases, we have Udia(1) = I because W(l) commutes and

D1;2ð1Þ ¼ ∣∫1

0 eif1;2ðlÞdl∣ ¼ 0

In other words, by jumping on discrete points lj, the system evolu-tion at l = 1 is exactly the perfect adiabatic evolutionUadia although theevolution time is finite, and an initial state ∣Yi⟩ will end up with theadiabatic target state ∣Yt⟩ =Uadia∣Yi⟩. To realize the jumping protocol,we apply rectangular p pulses at the points lj without time delay be-tween the pulses because, between the points lj, the Hamiltonian hasa zero energy gap and its driving can be neglected (see Fig. 3C). Thesimulation results in Fig. 3 show how the transition from the standard

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

continuous protocol to the jumping one gradually increases the fidelityof adiabatic evolution.

We experimentally compare the jumping protocol with the contin-uous one along the geodesic given in Eq. 5 by measuring the fidelitybetween the evolved state and the target state ∣Yt⟩ that follows the idealadiabatic evolution Uadia. The continuous protocol has a constant gapand a constant sweeping rate as in Fig. 1C. As shown in Fig. 4 (A and B),for the case of a geodesic half circle (qg = p), the jumping protocolreaches unit fidelity within the measurement accuracy, while the stan-dard continuous driving has much lower fidelity at short evolutiontimes. The advantage of the jumping protocol is more prominent whenwe traverse the half-circle path back and forth [see Fig. 4 (C to F) for theresults of a total path length of 6qg]. We observe in Fig. 4 that the con-stant gap protocol provides unit state transfer fidelity only when the ini-tial state is an eigenstate of the initial Hamiltonian ∣Yi⟩ =∣x⟩ and whenthe relative dynamic phase accumulated in a single half circle is f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2kpÞ2 � ðqgÞ2q

(k = 1,2,…) (see Materials and Methods). However,

the phase shifts on the system eigenstates accompanying adiabatic evo-lution cannot be observed when the initial state is prepared in one of theinitial eigenstates. Therefore, in Fig. 4, we also compare the fidelity forthe initial state ∣Yi⟩ = ∣y⟩, which is a superposition of the initial eigen-states ∣± x⟩. The results confirm that the jumping protocol achieves ex-actly the adiabatic evolutionUadia within the experimental uncertainties.

on August 29, 2021

p://advances.sciencemag.org/

ytil edi Fytil edi F

Evolution time (µs)0.0 0.5 1.0 1.5 2.0 2.5 3.0

Total time (µs)

0.0 0.5 1.0 1.5 2.0Total time (µs)

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.00 2 4 6 8 10 12

A

ytil edi Fytil edi F

B

C

D

E

F

Continuous

Jumping

Continuous

1.0 1.0

Jumping Jumping

Continuous

Fig. 4. Performance of adiabatic protocols along geodesics. (A) Measured fidelity (blue dots) between the final state and the target adiabatic state as a function ofthe total time by using a continuous driving protocol with a constant gapW0 = 2p × 5 MHz. The inset indicates the control path where the instantaneous eigenstate ∣y1(l)⟩ ofthe Hamiltonian proceeds from ∣x⟩ to the ∣ − x⟩ via a geodesic half circle. The initial state ∣y1(0)⟩ = ∣ Yi⟩ is prepared in the initial eigenstate ∣x⟩ (top) or ∣y⟩ (bottom), theequal superposition of initial eigenstates. The target state is defined as the state driven by an ideal, infinitely slow adiabatic evolution. The red lines show the numericalsimulation that has taken experimental noise sources into consideration (see Materials and Methods). The horizontal gray lines indicate the level of unit fidelity. (B) Same as (A)but for a jumping protocol where the p pulses have the same amplitude as the continuous protocol but are applied at the path points ljwithout time delay. The crosses in theinset illustrate the path points for N = 5 pulses. (C and D) Same as (A) and (B), respectively, but for a longer path containing six half circles by using three times of back-forwardmotion. (E and F) Fidelity during the evolution for a total time T = 3 ms using the protocols in (C) and (D).

5 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

http://advances.scienD

ownloaded from

Robustness of quantum adiabatic evolution via jumpingTo demonstrate the intrinsic robustness guaranteed by adiabatic evo-lutions, in Fig. 5, we consider large randomdriving amplitude errors inthe jumping protocol. We add random Gaussian distributed errorswith an SD of 50% to the control amplitude. To simulate white noise,we change the amplitude after every 10 ns in an uncorrelated manner.Despite the large amplitude errors, which can even cause energy levelcrossings, during the evolution (see Fig. 5A for a random time trace), achange of fidelity is hardly observable in Fig. 5B. Additional simula-tions in fig. S2 also demonstrate the robustness to amplitude fluctua-tions with different kinds of noise correlation, i.e., Gaussian whitenoise, Ornstein-Uhlenbeck process modeled noise, and static randomnoise. The robustness of the jumping protocol can be further enhancedby using a larger number N of points along the path (fig. S3).

While it is different from dynamical decoupling (DD) (20, 26), thejumping protocol can suppress the effect of environmental noisethrough amechanism similar toDD. Therefore, the fidelity is still highevenwhen the evolution time ismuch longer than the coherence time,T*2 ¼ 1:7ms, of theNV electron spin (fig. S4). This evidence is useful to

design adiabatic protocols that provide strong robustness against bothcontrol errors and general environmental perturbations.

Avoiding unwanted path points in adiabatic evolutionWithout going through all the path points, the jumping protocol hasadvantages to avoid path points (i.e., Hamiltonian with certain eigen-states) that cannot be realized in experiments. As a proof-of-principleexperiment, we consider the Landau-Zener (LZ) Hamiltonian (27)

HðlÞ ¼ HLZðlÞ ≡ BzðlÞ sz2 þ Dsx2

ð7Þ

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

with sa (a= x, y, z) being the Pauli matrices. Because D is nonzero inthe LZ Hamiltonian, tuning the system eigenstates to the eigen-states ∣±z⟩ of sz requires Bz → ± ∞. Therefore, for a perfect statetransfer from ∣Yi⟩ = ∣−z⟩ to ∣Yt⟩ = ∣z⟩, by using the standard con-tinuous protocol, it is required to adiabatically tune Bz from −∞ to +∞(see insets of Fig. 6A). The experimental implementation of Bz = ± ∞however requires an infinitely large control field, which is a severe lim-itation. In our experiment, a largeBz field can be simulated by going tothe rotating frame of the MW control field with a large frequency de-tuning. The experimental realization of Bz → ± ∞ can be challengingin other quantum platforms. For example, for superconducting qubitswhere D/(2p) could be as large as 0.1 GHz, the tuning range of Bz/(2p)is usually limited to a couple of gigahertz or even of the same order ofmagnitude asD/(2p) (28). For a two-level quantum system comprisingBose-Einstein condensates in optical lattices, the maximum ratio ofBz/D is determined by the band structure (29). For singlet-triplet qubitsin semiconductor quantum dots, the exchange interaction for thecontrol ofBz is positively confined (30).On the contrary, with the jump-ing approach, one can avoid the unphysical points such as Bz = ±∞ asan infinitely slow and continuous process is not required and canachieve high-fidelity state transfer as shown in Fig. 6.

As a remark, we find that our jumping protocol with N = 1 (i.e., aRabi pulse) specializes to the optimized composite pulse protocol (29)but has the advantage of requiring no additional strong p/2 pulses at

on August 29, 2021

cemag.org/

1.0

0.0

0.80.60.40.2

0.0 0.1 0.2 0.3 0.4 0.5

A

B

0.0 0.1 0.2 0.3 0.4 0.5

0

5

10

Total time (µs)

Evolution time (µs)

Fide

lity

Fide

lity

Evolution time (µs)

1.0

0.8

0.6

0.4

0.2

0.0 0.5 1.0 1.5 0.30.2 5.20.0

Fig. 5. Robustness of the jumping protocol. (A) Exemplary time trace of thedriving Rabi frequency. The amplitudes of the Rabi frequency are randomly gen-erated by the Gaussian distribution with a mean of 2p × 5 MHz and an SD of 2p ×2.5 MHz. The amplitudes are uncorrelated at every slices of duration of 10 ns.(B) Fidelity to the final adiabatic state as a function of the total time using theamplitudes as in (A) and the initial state prepared in the eigenstate ∣x⟩. Theinset of (B) shows the fidelity during the evolution time t for N = 5. The fidelityis measured by comparing the experimental state with the ideal state under aninfinitely slow adiabatic evolution.

A

B

Evolution time (µs)

Popu

latio

nF

idel

ity

Δ

Eig

enen

ergi

es1.0

0.8

0.6

0.4

0.2

0.00.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

1.0

0.5

0.0

Fig. 6. Avoiding unphysical points in the LZ model. (A) Measured fidelity(dots) to the adiabatic state during the evolution time along the path of the LZmodel by using the jumping protocol of N = 5 pulses. Left inset is the eigenener-gies of the LZ model with an avoided crossing D = 2p × 5 MHz. The path (rightinset) was set as from the initial eigenstate ∣−z⟩ to ∣z⟩ (blue arrowheads) andback from ∣z⟩ to ∣−z⟩ (purple arrowheads). The red circles indicate the unphysicalpath points (∣±z⟩) that require an infinitely large Bz and eigenenergies in theHamiltonian. The jumping protocol avoids the use of the unphysical points andadiabatically transfers the states ∣±z⟩ with high fidelity (dots) during the evolutionby jumping only on the path points indicated by red crosses. (B) Population at ∣−z⟩during the evolution time along the path. The red lines are the numerical simulations,and the target state is defined as the ideal adiabatic state driven under the controlwith an infinite number of p pulses.

6 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

the beginning and the end of the evolution. Moreover, by applying thejumping protocol with N = 1 to the adiabatic passage proposed in (31),we obtain the protocol that has been used to experimentally generateFock states of a trapped atom (32). When, instead of a single targetpoint, high-fidelity adiabatic evolution along the path is also desired,we can use the jumping protocol with a larger N.

on August 29, 2021

http://advances.sciencemag.org/

Dow

nloaded from

DISCUSSIONIn summary, our experiments demonstrated that energy level crossingsand vanishing gaps allow and can even accelerate quantum adiabaticevolutions, challenging the traditional views that adiabatic control mustbe slow and that unit-fidelity adiabatic processes require an infiniteamount of evolution time. By experimentally verifying a recently derivedquantumadiabatic condition,wehave shown that the quantumdynamicphases are more fundamental than energy gaps in quantum adiabaticprocesses. Owing to rapid changes of these phases, nonadiabatic transi-tions can be efficiently suppressed and fast-varying Hamiltonians canstill realize quantum adiabatic evolutions. Our results break the limit im-posed by the conventional adiabaticmethods that originate from the tra-ditional concept of extremely slow change in classicalmechanics (14, 16),allowing fast quantum adiabatic protocols with unit fidelity within finiteevolution times. In addition, the freedom of using vanishing gaps pro-vides the ability to avoid unphysical points in an adiabatic path andallows incorporating pulse techniques (26) into a quantumadiabatic evo-lution to suppress environmental noise for long-term robust adiabaticcontrol. While it is possible to mimic the infinitely slow quantum adia-batic evolution by using additional counterdiabatic control, i.e., shortcutsto adiabaticity (29, 33–37), the implementation of the counterdiabaticcontrol can be exceedingly intricate because it may need interactionsabsent in the system Hamiltonian (36, 37). Furthermore, the counter-diabatic control unavoidably changes the eigenstates of the initialHamiltonian and introduces additional control errors (36, 37). How-ever, because our protocol uses the intrinsic adiabatic path that followsthe eigenstates of the Hamiltonian, no additional control is required.As a consequence, our methods avoid the use of difficult or unavail-able control resources and share the intrinsic robustness of adiabaticmethods. With the removal of the prerequisites in the conventionaladiabatic conditions, namely nonzero gaps and slow control, our resultsprovide new directions and promising strategies for fast, robust controlon quantum systems.

MATERIALS AND METHODSAdiabatic evolution along the geodesics of a generalquantum systemFor two arbitrary states (e.g., entangled states and product states) ofa general quantum system, ∣Yi⟩ and ∣Yt⟩, one can write ⟨Yi∣Yt⟩ ¼cos 1

2 qg� �

eifi;t , with fi,t and qg being real. Here, qg is the path lengthconnecting ∣Yi⟩ and ∣Yt⟩ by the geodesic, and we set fi, t = 0 by aproper gauge transformation (24). The geodesic (24, 25) that con-nects ∣Yi⟩ and ∣Yt⟩ by varying l = 0 to l = 1 can be written as∣y1(l)⟩ = ci(l)∣Yi⟩ + ct(l)∣Yt⟩, where the coefficients ciðlÞ ¼cos 1

2 qgl� �� sin 1

2 qgl� �

cot 12 qg� �

and ctðlÞ ¼ sin 12 qgl� �

=sin 12 qg� �

for

sin 12 qg� �

≠ 0. To describe ∣y1(l)⟩ in terms of the system eigenstates,we chose an orthonormal state ∣y2(0)⟩ º (I −∣Yi⟩⟨Yi∣) ∣Yt⟩ ifsin 1

2 qg� �

≠ 0. When ∣Yt⟩ is equivalent to ∣Yi⟩ up to a phase factor

[i.e., sin 12 qg� � ¼ 0], ∣y2(0)⟩ can be an arbitrary orthonormal state. Then,

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

the geodesic and its orthonormal state can be written as ∣y1ðlÞi ¼cos 1

2 qgl� �

∣y1ð0Þi þ sin 12 qgl� �

∣y2ð0Þi and ∣y2ðlÞi ¼ �sin 12 qgl� �

∣y1ð0Þi þ cos 12 qgl� �

∣y2ð0Þi, respectively. Along the geodesic, wehad g2;1ðlÞ ¼ �g1;2ðlÞ ¼ i 12 qg being a constant and gn,m = 0 forother combinations of n and m. Along the geodesic, if one changesthe dynamic phases with a p phase shift only at each of the N equallyspaced path points lj = (2N)−1(2j − 1), with j = 1,2, … , N, then theoperatorsW(l) at different l commute and we have ∫10 e

if1;2dl ¼ 0. Asa consequence,Udiað1Þ ¼ exp½i∫10 WðlÞdl� ¼ I and the quantum evo-lution U = Uadia does not have any nonadiabatic effects.

Hamiltonian of the NV center under MW controlUnder a magnetic field bz along the NV symmetry axis, the Hamil-tonian of the NV center electron spin without MW control readsHNV ¼ DS2z � gebzSz , where Sz is the electron spin operator, D ≈ 2p ×2.87 GHz is the ground-state zero field splitting, bz is the magnetic field,and ge = − 2p × 2.8MHzG−1 is the electron spin gyromagnetic ratio (21).Following the standard methods to achieve a controllable Hamilto-nian for quantum adiabatic evolution (2, 3), we applied an MW fieldffiffiffi2

pWðlÞ cos wMWt þ ϑðlÞð �½ to the NVms = 0 andms = 1 levels to form

a qubit with the qubit states ∣z⟩≡∣ms = 1⟩ and ∣ − z⟩≡∣ms = 0⟩. TheMW frequency wMWmay also be tuned by the parameter l to realizea controllable frequency detuning d(l) with respect to the transitionfrequency ofms = 0 andms = 1 levels. In the standard rotating frameof the MW control field, we had the general qubit Hamiltonian un-der the MW control (21)

HðlÞ ¼ dðlÞ sz2þWðlÞ cosϑðlÞ sx

2þ sinϑðlÞ sy

2

h ið8Þ

where theMWphase ϑ(l), MWdetuning d(l), andMWRabi frequen-cy W(l) are all tunable and can be time dependent in experiment. Theusual Pauli operators satisfy sz∣±z⟩ = ±∣±z⟩ and [cos (qgl)sx + sin(qgl)sy] ∣±l⟩ = ±∣±l⟩, where the states ∣±l⟩ are given in Eq. 5.

By setting the MW detuning to d(l) = 0, we achieved the Hamil-tonian in Eq. 4, which in terms of the Pauli operators reads

HxyðlÞ ¼ WðlÞ cosðqglÞ sx2 þ sinðqglÞsy2

h i

By varying the parameter l, the system eigenstates follow the geo-desics along the equator of the Bloch sphere where the north and southpoles are defined by the states ∣ ± z⟩. Here, the energy gap W(l) isdirectly controlled by the amplitude of the MW field. On the otherhand, by using a constant Rabi frequencyW(l) = D and a tunable fre-quency detuning d(l) = Bz(l), we obtained the LZHamiltonianHLZ(l)given in Eq. 7.

Adiabatic evolution by continuous driving with aconstant gapConsider a conventional adiabatic driving where a constant ampli-tude driving field rotates around the z axis, with the HamiltonianHðlÞ ¼ 1

2We�i12szqglsqei12szqgl, which is parameterized by l = t/T along

a circle of latitude with sq = szcosq + sxsinq in a total time T. The dif-ference of the accumulated dynamic phases at l = 1 on the two eigen-states is f =WT. One can show that the system evolution at l = 1 reads

U ¼ e�i12qgszexp �i12ðfsq � qgszÞ

� �ð9Þ

7 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

on August 29, 2021

http://advances.sciencemag.org/

Dow

nloaded from

The ideal adiabatic evolution is obtained by using Eq. 9 in theadiabatic limit T → ∞ (i.e., f → ∞)

Uadia ¼ limT→∞

U ¼ e�i12qgsz ei12qgcosqsqe�i12fsq

Without the part of dynamic phases, Uadia describes geometric evo-lution, and for a cyclic evolution (i.e., qg = 2p), the geometric evolution isdescribed by the Berry’s phases ±p(cosq − 1). By comparingUadia andUor by using the results of (20), the nonadiabatic correction is given by

Udia ¼ exp i12ðf� qgcosqÞsq

� �U ′ ð10Þ

with

U ′ ¼ exp �i12ðfsq � qgszÞ

� �

In the adiabatic limit T → ∞ (i.e., f → ∞), Udia = I is the identifyoperator. We note that, when the phase factor of the state is irrelevant,one can perform perfect state transfer by this driving if the initial state isprepared in an initial eigenstate of the drivingHamiltonianH(l) (i.e., aneigenstates of sq). From Eq. 10,Udia is diagonal on the basis of sq whenU′ º I. As a consequence, when U′ º I and ∣Yi⟩ is prepared as aneigenstate of sq [and henceH(l = 0)], the evolved stateU∣Yi⟩matchesthe target state Uadia∣Yi⟩ up to a phase factor. For the case of the evo-

lution along the geodesic (e.g., q = p/2) andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2 þ q2g

q¼ 2kp (k =

1,2,…), we had U′º I and therefore a perfect population transferfor the initial eigenstates of sx.

Numerical simulationsIn the simulations, we modeled dephasing noise and random fluc-tuations by adding them to the Hamiltonian (Eq. 8) via d(l) →d(l) + d0 and W(l) → W(l)(1 + d1). Here, d0 is the dephasing noisefrom static and time-dependent magnetic field fluctuations with aT*2 ¼ 1:7 ms. d1 is the random static changes in the driving ampli-

tude. d0 follows the Gaussian distribution with the mean value m =0 and the SD s = 2p × 130 kHz. The probability density of d1 hasthe Lorentz form f (d1, g) = 1/{pg[1 + (d1/g)

2]} with g = 0.0067. Allthe parameters in the distribution function were extracted fromfitting the free induction decay and the decay of Rabi oscillation.

Experimental setupThe experiments were performed with a home-built optically detectedmagnetic resonance platform, which consists of a confocal microscopeand an MW synthesizer (fig. S5). A solid-state green laser with awavelength of 532 nm was used for initializing and reading out theNV spin state. The light beam was focused on the NV center throughan oil immersion objective (numerical aperture, 1.4). The emitted fluo-rescence from the NV center was collected by a single-photon countingmodule (avalanche photodiode). Here, we used an NV center em-bedded in a room-temperature bulk diamond grown by chemical vapordeposition with [100] faces. It has 13C isotope of natural abundance andnitrogen impurity less than 5 parts per billion. To lift the degeneracy ofthe ∣ms = ± 1⟩ states, a static magnetic field of 510 G was provided by apermanent magnet. The magnetic field was aligned by adjusting the

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

three-dimensional positioning stage onwhich themagnet wasmountedand by simultaneously monitoring the counts of the NV center. Thedirection of themagnetic field waswell alignedwhen the counts showedno difference between with and without the magnet. Manipulation ofthe NV center was performed by MW pulses applied through a home-made coplanar waveguide. The MW pulses were generated by the I/Qmodulation of the Agilent arbitrary wave generator (AWG) 81180Aand the vector signal generator (VSG) E8267D and then amplified bythe Mini Circuits ZHL-30W-252+. An atomic clock was used to syn-chronize the timing of the two. TheAWGsupplies the I andQdatawitha frequency of 400MHz, and the VSG generates the 3898-MHz carrier.The output frequency is 4298 MHz, which matches the transition fre-quency between the NV ms = 0 and ms = + 1 states.

Experimental sequencesAs themagnetic field is 510G, we first applied the green laser for 3 ms toinitialize the NV center electronic spin to the level ofms = 0 and to po-larize the adjacent 14N nuclear spin simultaneously (38). The prepara-tion of the NV electron spin in an equal superposition state of ms = 0andms = 1was realized by applying anMW px/2 (py/2) pulse, i.e., by therotation around the x (y) axis with an angle of p/2. Then, the NV elec-tron spin was driven according to a desired path. To experimentallycharacterize the evolution path, we sampled the pathwith several pointsand measured the spin state through tomography. px/2 or py/2 pulseswere applied to read out the off-diagonal terms. Last, the spin state wasread out by applying the laser pulse again and measuring the spin-dependent fluorescence. Typically, the whole sequence was repeated105 times to get a better signal-to-noise ratio. The schematic diagramof the pulse sequence is shown in fig. S6.

In driving the NV electron spin along the path given in Eq. 5, weused an on-resonant MW field and swept the MW phase qgl withthe path parameter l. In driving the NV electron spin along the pathof the LZHamiltonian (Eq. 7), theMWphase was a constant, D was setby the Rabi frequency, and Bz was the MW frequency detuning thatvaried as Bz = − Dcot(qgl). For continuous driving, the path parameterl varies with a constant rate dl/dt = frot. In the jumping protocol, ljumps from point to point: l = lj = (2N)−1(2j − 1) with j = 1,2, … ,N. In this work, the jumping protocol had a constant driving Rabi fre-quencyW0 and l = lj if ( j − 1)T/N≤ t < jT/N for a path with N pulsesapplied in a total time T. In the experiments with the back-forwardmo-tion along the geodesic, we reversed the order of the parameter l in thebackward path, that is, in the jumping protocol, we repeated the subse-quent parameters (l1, l2,… , lN − 1, lN, lN, lN − 1,… , l2, l1), while forthe standard protocol of continuous driving, we used the rate dl/dt= frotfor a forward path and the rate dl/dt = − frot for a backward path andrepeated the process.

We removed the irrelevant dynamic phases if the initial state was notprepared in an initial eigenstate to reveal the geometric evolution. At thebeginning of state readout, we compensated the dynamic phases by ap-plying an additional driving with anMW p phase shift (i.e.,W→ −W) atthe point of the target state for a time equalling to the time for adiabaticevolution. This additional driving did not change the geometric phasesand state transfer because it was applied at the final path point.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/6/eaax3800/DC1Fig. S1. Transition of adiabatic control from a constant gap to a gap with energy level crossings.Fig. S2. Robustness of the jumping protocol against different kinds of control noise.

8 of 9

SC I ENCE ADVANCES | R E S EARCH ART I C L E

Fig. S3. Enhancing the robustness of jumping protocol in the presence of control noise.Fig. S4. Coherence protection during adiabatic evolutions.Fig. S5. Sketch of the experimental setup.Fig. S6. Experimental pulse sequence.

on August 29, 2021

http://advances.sciencemag.org/

Dow

nloaded from

REFERENCES AND NOTES1. A. M. Childs, E. Farhi, J. Preskill, Robustness of adiabatic quantum computation.

Phys. Rev. A 65, 012322 (2001).2. N. V. Vitanov, A. A. Rangelov, B. W. Shore, K. Bergmann, Stimulated Raman adiabatic

passage in physics, chemistry, and beyond. Rev. Mod. Phys. 89, 015006 (2017).3. N. V. Vitanov, T. Halfmann, B. W. Shore, K. Bergmann, Laser-induced population transfer

by adiabatic passage techniques. Annu. Rev. Phys. Chem. 52, 763–809 (2001).4. A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, M. Head-Gordon, Simulated quantum

computation of molecular energies. Science 309, 1704–1707 (2005).5. K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin,

L.-M. Duan, C. Monroe, Quantum simulation of frustrated Ising spins with trapped ions.Nature 465, 590–593 (2010).

6. J. D. Biamonte, V. Bergholm, J. D. Fitzsimons, A. Aspuru-Guzik, Adiabatic quantumsimulators. AIP Adv. 1, 022126 (2011).

7. E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Quantum computation by adiabaticevolution. arXiv:0001106 [quant-ph] (28 January 2000).

8. J. A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Geometric quantum computation usingnuclear magnetic resonance. Nature 403, 869–871 (2000).

9. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, D. Preda, A quantum adiabaticevolution algorithm applied to random instances of an NP-complete problem.Science 292, 472–475 (2001).

10. R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezzacapo, U. Las Heras, R. Babbush,A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, E. Lucero,A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan,D. Sank, A. Vainsencher, J. Wenner, T. C. White, E. Solano, H. Neven, J. M. Martinis,Digitized adiabatic quantum computing with a superconducting circuit. Nature 534,222–226 (2016).

11. K. Xu, T. Xie, Z. Li, X. Xu, M. Wang, X. Ye, F. Kong, J. Geng, C. Duan, F. Shi, J. Du,Experimental adiabatic quantum factorization under ambient conditions based on asolid-state single spin system. Phys. Rev. Lett. 118, 130504 (2017).

12. M. V. Berry, Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. AMath. Phys. Sci. 392, 45–57 (1984).

13. F. Wilczek, A. Zee, Appearance of gauge structure in simple dynamical systems.Phys. Rev. Lett. 52, 2111–2114 (1984).

14. M. Born, V. Fock, Beweis des Adiabatensatzes. Z. Phys. 51, 165–180 (1928).15. A. Messiah, Quantum Mechanics, vol. 2 (North-Holland, 1962).16. K. J. Laidler, The meaning of “adiabatic”. Can. J. Chem. 72, 936–938 (1994).17. K.-P. Marzlin, B. C. Sanders, Inconsistency in the application of the adiabatic theorem.

Phys. Rev. Lett. 93, 160408 (2004).18. D. M. Tong, K. Singh, L. C. Kwek, C. H. Oh, Quantitative conditions do not guarantee the

validity of the adiabatic approximation. Phys. Rev. Lett. 95, 110407 (2005).19. J. Du, L. Hu, Y. Wang, J. Wu, M. Zhao, D. Suter, Experimental study of the validity of

quantitative conditions in the quantum adiabatic theorem. Phys. Rev. Lett. 101,060403 (2008).

20. Z.-Y. Wang, M. B. Plenio, Necessary and sufficient condition for quantum adiabaticevolution by unitary control fields. Phys. Rev. A 93, 052107 (2016).

21. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, L. C. L. Hollenberg,The nitrogen-vacancy colour centre in diamond. Phys. Rep. 528, 1–45 (2013).

22. J. Ortigoso, Quantum adiabatic theorem in light of the Marzlin-Sanders inconsistency.Phys. Rev. A 86, 032121 (2012).

23. J. Jing, L.-A. Wu, T. Yu, J. Q. You, Z.-M. Wang, L. Garcia, One-component dynamicalequation and noise-induced adiabaticity. Phys. Rev. A 89, 032110 (2014).

24. J. Anandan, Y. Aharonov, Geometry of quantum evolution. Phys. Rev. Lett. 65, 1697–1700(1990).

Xu et al., Sci. Adv. 2019;5 : eaax3800 21 June 2019

25. D. Chruscinski, A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics(Birkhäuser, 2004).

26. W. Yang, Z.-Y. Wang, R.-B. Liu, Preserving qubit coherence by dynamical decoupling.Front. Phys. 6, 2–14 (2011).

27. S. N. Shevchenko, S. Ashhab, F. Nori, Landau–Zener–Stückelberg interferometry.Phys. Rep. 492, 1–30 (2010).

28. G. Sun, X. Wen, M. Gong, D.-W. Zhang, Y. Yu, S.-L. Zhu, J. Chen, P. Wu, S. Han,Observation of coherent oscillation in single-passage Landau-Zener transitions.Sci. Rep. 5, 8463 (2015).

29. M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio,V. Giovannetti, R. Mannella, O. Morsch, High-fidelity quantum driving. Nat. Phys. 8,147–152 (2012).

30. S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, A. Yacoby, Universal quantum control oftwo-electron spin quantum bits using dynamic nuclear polarization. Nat. Phys. 5, 903–908(2009).

31. J. I. Cirac, R. Blatt, P. Zoller, Nonclassical states of motion in a three-dimensional ion trapby adiabatic passage. Phys. Rev. A 49, R3174(R) (1994).

32. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, D. J. Wineland, Generation ofnonclassical motional states of a trapped atom. Phys. Rev. Lett. 76, 1796–1799 (1996).

33. M. Demirplak, S. A. Rice, Adiabatic population transfer with control fields. J. Phys. Chem. A107, 9937–9945 (2003).

34. M. V. Berry, Transitionless quantum driving. J. Phys. A: Math. Theor. 42, 365303 (2009).35. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin,

A. Ruschhaupt, X. Chen, J. G. Muga, Shortcuts to adiabaticity. Adv. Atom. Mol. Opt. Phys.62, 117–169 (2013).

36. S. Deffner, C. Jarzynski, A. del Campo, Classical and quantum shortcuts to adiabaticity forscale-invariant driving. Phys. Rev. X 4, 021013 (2014).

37. B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. C. Jerger, A. Auer, G. Burkard,A. A. Clerk, D. D. Awschalom, Accelerated quantum control using superadiabaticdynamics in a solid-state lambda system. Nat. Phys. 13, 330–334 (2017).

38. R. J. Epstein, F. M. Mendoza, Y. K. Kato, D. D. Awschalom, Anisotropic interactions of asingle spin and dark-spin spectroscopy in diamond. Nat. Phys. 1, 94–98 (2005).

AcknowledgmentsFunding: The authors at USTC were supported by the National Key Research andDevelopment Program of China (grant nos. 2018YFA0306600, 2013CB921800,2016YFA0502400, and 2017YFA0305000), the NNSFC (grant nos. 81788101, 11227901,11722544, 91636217, and 11775209), the CAS (grant nos. GJJSTD20170001 and QYZDY-SSW-SLH004), the Anhui Initiative in Quantum Information Technologies (grant no. AHY050000),the CEBioM, the IPDFHCPST (grant no. 2017FXCX005), the Fundamental Research Funds forthe Central Universities, and the Thousand Young Talents Program. The authors of UlmUniversity were supported by the ERC Synergy grant BioQ, the EU projects HYPERDIAMONDand AsteriQs, the BMBF projects NanoSpin and DiaPol, and the IQST that is financiallysupported by the Ministry of Science, Research and Arts Baden-Württemberg. Authorcontributions: J.D. supervised the entire experiment. Z.-Y.W. and M.B.P. formulated thetheory. J.D. and F.S. designed the experiments. K.X., X.X., and P.W. prepared the setup. K.X. andT.X. performed the experiments. T.X., Z.-Y.W., and K.X. performed the simulation. Z.-Y.W., K.X.,T.X., M.B.P., Y.W., and J.D. wrote the manuscript. All authors discussed the results andcommented on the manuscript. Competing interests: All authors declare that they have nocompeting interests. Data and materials availability: All data needed to evaluate theconclusions in the paper are present in the paper and/or the Supplementary Materials.Additional data related to this paper may be requested from the authors.

Submitted 18 March 2019Accepted 15 May 2019Published 21 June 201910.1126/sciadv.aax3800

Citation: K. Xu, T. Xie, F. Shi, Z.-Y. Wang, X. Xu, P. Wang, Y. Wang, M. B. Plenio, J. Du, Breakingthe quantum adiabatic speed limit by jumping along geodesics. Sci. Adv. 5, eaax3800 (2019).

9 of 9

Breaking the quantum adiabatic speed limit by jumping along geodesics

DuKebiao Xu, Tianyu Xie, Fazhan Shi, Zhen-Yu Wang, Xiangkun Xu, Pengfei Wang, Ya Wang, Martin B. Plenio and Jiangfeng

DOI: 10.1126/sciadv.aax3800 (6), eaax3800.5Sci Adv 

ARTICLE TOOLS http://advances.sciencemag.org/content/5/6/eaax3800

MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2019/06/17/5.6.eaax3800.DC1

REFERENCES

http://advances.sciencemag.org/content/5/6/eaax3800#BIBLThis article cites 35 articles, 2 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.Science AdvancesYork Avenue NW, Washington, DC 20005. The title (ISSN 2375-2548) is published by the American Association for the Advancement of Science, 1200 NewScience Advances

License 4.0 (CC BY-NC).Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of

on August 29, 2021

http://advances.sciencemag.org/

Dow

nloaded from