Global sensing and its impact for quantum many-body probes with criticality
GGlobal sensing and its impact for quantum many-body probes with criticality
Victor Montenegro, ∗ Utkarsh Mishra, † and Abolfazl Bayat ‡ Institute of Fundamental and Frontier Sciences,University of Electronic Science and Technology of China, Chengdu 610051, PR China (Dated: February 9, 2021)Quantum sensing is one of the key areas which exemplifies the superiority of quantum technologies.Nonetheless, most quantum sensing protocols operate efficiently only when the unknown parametersvary within a very narrow region, i.e., local sensing. Here, we provide a systematic formulation forquantifying the precision of a probe for multi-parameter global sensing when there is no priorinformation about the parameters. In many-body probes, in which extra tunable parameters exist,our protocol can tune the performance for harnessing the quantum criticality over arbitrarily largesensing intervals. For the single-parameter sensing, our protocol optimizes a control field such thatan Ising probe is tuned to always operate around its criticality. This significantly enhances theperformance of the probe even when the interval of interest is so large that the precision is boundedby the standard limit. For the multi-parameter case, our protocol optimizes the control fields suchthat the probe operates at the most efficient point along its critical line. Interestingly, for an Isingprobe, it is predominantly determined by the longitudinal field. Finally, we show that even a simplemagnetization measurement significantly benefits from our optimization and moderately deliversthe theoretical precision.
Introduction.—
The emerging field of quantum sensingexploits quantum features for developing a new class ofsensors with unprecedented precision [1–6]. Originally,the superiority of quantum sensors was shown by ex-ploiting quantum superposition of GHZ-type states innon-interacting particles [7–10]. Such sensors use the re-sources (e.g., the number of particles L ) more efficientlyto enhance their precision, quantified by the variance ofthe estimation, from the usual classical standard limit(bounded by 1 /L ) to the Heisenberg limit (bounded by1 /L ) [11, 12]. However, if the particles interact, theprecision goes down [13–16]. Moreover, the GHZ statesare difficult to create and prone to decoherence [17–22].Hence, developing these types of sensors for many par-ticles, where the quantum enhancement becomes signif-icant, is extremely challenging in practice. To overcomesuch difficulties, a plethora of novel methods and systemshave been exploited for sensing purposes, including quan-tum control techniques [23–26], machine learning algo-rithms [27–29], hybrid variational methods [30], feedbackschemes [31–34], quantum chaos [35], periodically drivensystems [36, 37] and sequential measurements [38–41].Strongly correlated many-body systems are among theefficient quantum probes [42–48]. In particular, theground state of many-body systems with quantum phasetransitions is known to provide quantum-enhanced sens-ing at the vicinity of their critical point [49–58]. Theseschemes, unlike the GHZ-based quantum sensing pro-tocols, truly exploit the interaction between the parti-cles and because of operation at equilibrium they bene-fit from easier preparation and more robustness againstdecoherence. However, the quantum-enhanced sensingonly occurs at the vicinity of the critical point [49], mak-ing these sensors most suitable for local sensing, wherethe unknown parameter varies within a very narrow in- S e n s i n g p r e c i s i o n Unknown parameter
Probe withoutcontrol fi eld Probe withcontrol fi eld AntiferromagneticParamagnetic
Disordered (0,0) 21 (0,0) [ ] [ ] (a) (b)(c) [ ] [ ] AntiferromagneticParamagneticDisordered
Unknown longitudinal parameter U n k n o w n t r a n s v e r s e p a r a m e t e r Bx/J B z / J FIG. 1. (a) Schematic of the full phase diagram for an Isingmodel in the presence of a skew magnetic field (see Ref. [69]for details). (b) By optimizing the control field, the probeis shifted in its phase diagram such that the sensing intervalis located around the criticality, where the quantum Fisherinformation is maximum. (c) For multi-parameter sensing,the optimization of the probe also displaces the sensing area.However, the location depends on the size of the sensing area. terval. Hence, tuning the system to operate optimallyat its quantum phase transition point can be very elu-sive and practically demanding, e.g., adaptive sensingstrategies with complex measurement types have to beemployed [2, 59–66]. A key open question is whether onecan employ such sensors for global sensing, where the un-known parameter varies over a wide range. While in theparticular case of temperature, there have been some ef-forts for the formulation of global thermometry [67, 68],the problem is still open for general quantum sensing.In this letter, we formulate a systematic approach formulti-parameter global sensing, where the unknown pa-rameters can vary over arbitrarily large intervals. Ourprotocol applies to any sensing protocol and provides asystematic approach for optimizing the probe. In partic-ular, for many-body sensors, we show that one can gen-uinely exploit the criticality as a resource for enhancing a r X i v : . [ qu a n t - ph ] F e b the global multi-directional magnetometry precision. Parameter estimation.—
Every sensing protocol con-tains three essential steps: (i) choosing an appropriateprobe; (ii) gathering data through repeatedly perform-ing specific types of measurements on the probe; and(iii) feed the gathered data into an estimator to inferthe value of the unknown parameters. The precision ofthe estimation of an unknown parameter h =( h , h , . . . )obeys the Cram´er-Rao inequality [70, 71]Cov[ h ] ≥ M − inv [ F C ( h )] ≥ M − inv [ F Q ( h )] , (1)where Cov[ h ] is the covariance matrix whose elements are[Cov[ h ]] µ,ν = (cid:104) h µ h ν (cid:105)−(cid:104) h µ (cid:105)(cid:104) h ν (cid:105) , M is the total number ofmeasurements, and F C ( h ) and F Q ( h ) are the classicaland quantum Fisher information (QFI) matrices, respec-tively [72]. For a given quantum probe, with density ma-trix ρ ( h ), and a specific POVM measurement { Π k } , thebound is given by the classical Fisher information matrix[ F C ( h )] µ,ν = (cid:80) k p k ( h ) [ ∂ µ log p k ( h )] [ ∂ ν log p k ( h )], where p k ( h )=Tr[ ρ ( h )Π k ] is the probability of measurementoutcome k , and ∂ ν := ∂/∂h ν . By optimizing overall possible POVMs, one can tightens the bound tobe given by the QFI F Q ( h ) [71, 72], which canbe simplified for pure states ρ ( h )= | Φ( h ) (cid:105)(cid:104) Φ( h ) | as[ F Q ( h )] µ,ν =4Re [ (cid:104) ∂ µ Φ | ∂ ν Φ (cid:105) + (cid:104) Φ | ∂ µ Φ (cid:105)(cid:104) Φ | ∂ ν Φ (cid:105) ] [73]. Inthe case of single parameter, the Eq. (1) reduces to δh ≥ M − F Q ( h ) − , (2)in which the QFI provides the ultimate bound for thevariance of the estimation. Note that, for any given probeone can indeed saturate the inequality of Eq. (2) by usingan appropriate measurement setup computed throughsymmetric logarithmic derivatives [72, 74, 75] and an op-timal estimator (which for large data set is known tobe Bayesian [76–80]). However, the optimal measure-ment basis, in general, depend on the unknown parame-ter h which varies over an interval [ h min , h max ]. The opti-mal sensing is thus applicable only when ∆ h = h max − h min is small, which is called local sensing. In this sit-uation, the optimal measurement can be chosen for h cen =( h max + h min ) /
2. For large ∆ h , the criteria for thebest sensing protocol is indeed unknown. The situationfor multi-parameter estimation is more complex as evenfor local sensing due to the non-commutativity of the op-timal POVMs for different parameters, the Cram´er-Raobound may not be achievable [75]. Single parameter global sensing.—
We first introducethe concept of global sensing for a single-parameter esti-mation, and later in the paper, we generalize it for multi-parameter cases. As mentioned before, in the case of lo-cal sensing (i.e., small ∆ h ), the Cram´er-Rao bound inEq. (2) can always be saturated. Nonetheless, the sens-ing procedure might still be highly sub-optimal due to thebad choice of the probe. Hence, an optimal local sensingalgorithm requires optimization of F Q ( h cen ) with respect to the parameters of the probe. For large ∆ h , which iscalled global sensing, the situation is more complex as:(i) in general, the optimal measurement basis varies over∆ h and no measurement setup can saturate the Cram´er-Rao bound over the entire interval; and (ii) it is not clearwhich quantity has to be optimized to find the optimalprobe. In the following, we address this problem.To formulate the global sensing, we first quantify theaverage uncertainty of the estimation via (cid:82) ∆ h δh f ( h ) dh ,where f ( h ) is the prior information, i.e. the probabilitydistribution, of the unknown parameter h over the sens-ing interval ∆ h of interest. From Eq. (2), one can easilyshow that this average uncertainty is bounded by g ( B ) := (cid:90) ∆ h f ( h ) F Q ( h | B ) dh, (3)where B =( B , B , . . . ) are external tunable parametersinteracting with the probe. We define the minimiza-tion of g ( B ) with respect to control parameters B asa figure of merit for finding the optimal probe, namely g ( B ∗ ):=min B [ g ( B )]. Throughout this paper, we as-sume no prior information about the unknown param-eter h , and thus, f ( h ) takes a uniform distribution,namely f ( h )=1 / ∆ h . For local sensing, i.e. small ∆ h ,the QFI is almost constant over the interval, and thus, g ( B ) ≈ / F Q ( h cen | B ). Therefore, the minimization of g ( B ) is equivalent to maximize F Q ( h cen | B ). Many-body probe for magnetometry.—
To illustrate therelevance of our general formulation for global sensing,which it can be applied to any sensing protocol indepen-dent of the choice of the probe, we exploit a chain of L interacting spin − / y compo-nent of the magnetic field is zero. The Hamiltonian is H = J L (cid:88) i =1 σ ix σ i +1 x − L (cid:88) i =1 [( B x + h x ) σ ix + ( B z + h z ) σ iz ] , (4)where, σ iα ( α = x, z ) is the Pauli operator at site i , J > B =( B x , B z ) is the controlmagnetic field which can be tuned, h =( h x , h z ) is the ran-dom field to estimate, and periodic boundary conditionis imposed. The ground state | Φ (cid:105) , for which the phasediagram is shown in Fig. 1(a), can be used for detect-ing the unknown field h . See Refs. [81–85] for the linkbetween the characterization of quantum phase transi-tion via metric tensors. The phase diagram is uniquelydetermined by ( h z + B z ) /J and ( h x + B x ) /J . Thus, bytuning B one can shift the phase diagram for the un-known parameter h . For example, in the absence of lon-gitudinal field (i.e. B x = h x =0), the ground state | Φ (cid:105) canbe solved analytically using Jordan-Wigner transforma-tion and is known to have a quantum phase transition at h crit := B z + h z = ± J [86, 87], which leads to an enhanced B z / J g ( B z ) L = 1000 (a) h cen z =0.0 J , h z =0.0 Jh cen z =0.1 J , h z =0.002 Jh cen z =0.1 J , h z =0.1 Jh cen z =0.9 J , h z =0.002 Jh cen z =0.9 J , h z =0.1 J h z / J g ( B z )
1e 2 L = 1000 h cen z = 0.1 J ( d ) B z =0.0 B z =0.5 JB z = B * z J L g ( B * z ) h cen z = 0.1 J (b) h z =0.002 Jh z =0.004 Jh z =0.006 J h z =0.008 Jh z =0.1 Jh z =0.3 J ( l o g s c a l e ) ( l o g s c a l e ) h z / J b a g ( B * z , h z ) aL b + c ( c ) (log scale) FIG. 2. (a) Average uncertainty g ( B z ) as function of B z /J for different values of interval widths ∆ h z and centers h cen z .The optimal probe can always be found by tuning the controlfield such that g ( B ∗ z )=min B z [ g ( B z )]. (b) g ( B ∗ z ) (shown bymarkers) and its corresponding fitting function aL − b + c (solidlines) as a function of L for various choices of ∆ h z . (c) Fittingcoefficients a and b versus the controlled field B z /J . (d) g ( B z )is plotted as a function of ∆ h z /J for various choices of B z . QFI with scaling ∼ L [49, 51] (see the SupplementaryMaterial for details). Interestingly, away from criticality,the QFI scales as ∼ Lξ − , where ξ ∼| ( B z + h z ) − h crit | − isthe correlation length [49]. Since the quantum-enhancedsensing is lost when the probe operates away from criti-cality, our general formulation for global sensing standsas the most suitable for these type of sensors. Example 1: Transverse field Ising probe.—
In this sec-tion, we use the Ising many-body probe, given in Eq. (4),for single-parameter sensing when only transverse fieldexists, namely B x = h x =0. To see the performance of thisprobe for global sensing, we numerically evaluate g ( B z )for various ∆ h for a system size L =1000. In Fig. 2(a),we plot the global sensing performance g ( B z ) as functionof B z /J for different values of interval widths ∆ h z andcenters h cen z . For every h cen z and ∆ h z , the average uncer-tainty g ( B z ) has always a minimum which takes place ata particular B z = B ∗ z , showing that one can always makethe probe optimal by this choice of control field. Interest-ingly, the minimum value of g ( B z ) is independent of h cen z ,and is only determined by ∆ h z . On the other hand, theoptimal control field B ∗ z is almost independent of ∆ h z and only depends on h cen z , such that h cen z + B ∗ z ≈ h crit .This means that, the control field tends to shift the probein its phase diagram such that the interval of sensing islocated almost symmetrically around the critical point.This has been shown schematically in Fig. 1(b). To deter-mine how the average uncertainty scales for the optimalprobe, in Fig. 2(b), we plot g ( B ∗ z ) as a function of L for various choices of ∆ h z . For small ∆ h z , the averageuncertainty scales as g ( B ∗ z ) ∼ /L , which is expected for local sensing. Remarkably, by increasing ∆ h z , the scalinggoes towards the standard limit, namely g ( B ∗ z ) ∼ /L . Toquantify the transition from quantum enhanced sensingto the standard limit, we fit g ( B ∗ z ) with a function of theform aL − b + c , c →
0. In Fig. 2(c), we plot the fitting coef-ficients a and b as a function of the width ∆ h z /J . As seenfrom the figure, when ∆ h z →
0, one recovers the Heisen-berg scaling, g ( B ∗ z ) ∼ F − Q ∼ /L , from the local sensingstrategy. The quantum-enhanced sensing is captured by b > b = 1. Remarkably, the region of quantum enhancedsensing is extended only until ∆ h z ≤ . J , beyond whichthe standard limit is restored. However, it is crucial tonote that the optimization of the probe is still beneficialfor sensing even though there is no quantum-enhancedadvantage in the scaling. This can be seen in Fig. 2(d)where g ( B z ) is plotted as a function of ∆ h z /J for vari-ous choices of B z . As the figure shows, by the optimalchoice of B z = B ∗ z the average uncertainty remains lowerthan non-optimal values of the control field for all valuesof ∆ h z . In other words, for large ∆ h z , while b =1, the a coefficient becomes smaller by optimizing B z . Multi-parameter global sensing.—
Thanks to the aboveanalysis, one can readily generalize the global sensing forthe multi-parameter case. To set the performance of dif-ferent multi-parameter estimators, we recast the matrixbounds in Eq. (1) into scalar bounds. To do so, we intro-duce a (positive and real) weight matrix W such that [75]Tr [ W Cov[ h ]] ≥ M − Tr [inv [ F Q ( h | B )] W ] . (5)Throughout this work, we consider W to be identity,namely W = I . This W choice makes the left-hand side ofthe above inequality to be the sum of the variances of theunknown parameters. Inspired by the single-parametercase, we define the average uncertainty for the multi-parameter case as g ( B ) := (cid:90) ∆ h f ( h )Tr[inv[ F Q ( h | B )]] d h , (6)where f ( h ) is the prior probability distributions, which isassumed to be uniform, for magnetic field h , and the in-tegration is performed over the volume of all parameters.For the single-parameter case, the above equation reducesto Eq. (3). To optimize the probe, one has to minimize g ( B ) with respect to B , i.e. g ( B ∗ ):=min B [ g ( B )]. Example 2: Skew field Ising probe.—
In this section,we consider the probe of Eq. (4), when both h x and h z are sensed together. Since the system is not solvableanymore, we are restricted to short chains and exact di-agonalization. Unlike the transverse field Ising model forwhich the criticality happens exactly at one point, here,there is a line of criticality in the plane of ( h x , h z )[69], seeFig. 1(a). Thus, the optimization of the probe is highlynon-trivial as the control fields ( B x , B z ) can shift the -0.15 -0.16 -0.25 -0.35 -0.52-0.74 -0.11 -0.12 -0.16 -0.27 0.05-0.42 FIG. 3. (a) g ( B x , B z ) as a function of control fields ( B x , B z )for the case of an almost local sensing ∆ h x =∆ h z =0 . J . (b)By increasing the widths to ∆ h x =∆ h z =0 . J , one finds a non-trivial optimal probe along its critical line. The total systemis L = 16 and centers are chosen to be h cen x = h cen z =0 . J . Thedashed red lines represent the critical line and the numbersshow the value of g ( B x , B z ) at that point in the phase dia-gram. The global minimum is depicted in orange. phase of the probe to operate anywhere along the criticalline. For the case of local sensing, in which ∆ h x and ∆ h z are very small, the minimization of g ( B ) reduces to themaximization of Tr[inv[ F Q (( h cen x , h cen z ) | B )]].In contrast to the single-parameter case, our analysisshows that, the optimal control fields not only depend onthe location of ( h cen x , h cen z ), but also change by the choiceof the widths (∆ h x , ∆ h z ). Interestingly, the probe is al-ways shifted somewhere near the critical line which is pre-dominantly controlled by the longitudinal field. To havea quantitative analysis, in Fig. 3(a), we plot g ( B x , B z )as a function of control fields for the case of an almostlocal sensing with h cen x = h cen z =0 . J and the widths of∆ h x =∆ h z =0 . J . As the figure shows, the optimal con-trol fields are given by B ∗ x =1 . J and B ∗ z = − . J . InFig. 3(b), we increase the width to ∆ h x =∆ h z =0 . J (i.e.global sensing). Interestingly, the optimal control fieldsshift to B ∗ x =1 . J and B ∗ z =0 . J , which are different fromthe previous case. This shows that, the optimization ofthe probe for multi-parameter sensing is non-trivial andthe optimal control fields depend on the sensing range.This is schematically explained in Fig. 1(c). Sensing protocol.—
While our formulation for globalsensing systematically provides a bound for the averageuncertainty, it is not obvious whether this bound canbe saturated, as no measurement basis can be optimalthroughout the whole region. Here, we choose a simplemeasurement basis, namely magnetization M = (cid:80) i σ iz ,which is independent of the unknown parameters. Ameasurement already available in ion traps [88] and su-perconducting quantum devices [89, 90]. Although, thischoice of measurement basis is not necessarily optimal,it has practical advantages due to its simplicity. Infact, measuring the ground state results in L +1 out-comes for which one can compute the classical Fisher FIG. 4. (a) Efficiency of the magnetization measurementTr[inv[ F Q ( h | B ∗ )]] / Tr[inv[ F C ( h | B ∗ )]] as a function of h x /J and h z /J . One can achieve the ultimate precision between22% −
70% for all values of h x /J and h z /J . (b) Efficiency com-parison between the optimal probe against the non-optimalone Tr[inv[ F C ( h | B = )]] / Tr[inv[ F C ( h | B ∗ )]] as a functionof h x /J and h z /J . The performance for the optimal probeexceeds considerably the non-optimal one. Other values are: L =10, h cen x =0 . J, h cen z =0 . J , and ∆ h x =∆ h z =0 . J information matrix. We consider a two parameter sens-ing with h cen x =0 . J, h cen z =0 . J , and ∆ h x =∆ h z =0 . J .One can optimize the probe of Eq. (4) using the algo-rithm above, i.e. minimizing g ( B x , B z ), which results in B ∗ x =1 . J and B ∗ z = − . J . Using this optimal probe,we perform the magnetization measurement and com-pute Tr[inv[ F C ]] over the whole interval. To quantifythe efficiency of this measurement, in Fig. 4(a), we plotTr[inv[ F Q ( h | B ∗ )]] / Tr[inv[ F C ( h | B ∗ )]] as a function of h x /J and h z /J to compare the obtainable precision withthe ultimate bound. Interestingly, despite using this sim-ple measurement, the ultimate precision bound rangesbetween 22% −
70% for all values of h x /J and h z /J . Tosee the importance of optimizing the probe, in Fig. 4(b),we plot the Tr[inv[ F C ( h | B = ))]] / Tr[inv[ F C ( h | B ∗ ]] as afunction of h x /J and h z /J to compare the performanceof the magnetization measurement for the optimal versusthe non-optimal probe, given by B x =0 , B z =0. As seenfrom the figure, higher performance can be achieved byusing the optimal probe. This shows that our procedurefor global sensing can optimize the probe such that evenwith a simple measurement, one can harness the critical-ity to improve the precision significantly. Conclusions.—
We present a formulation for multi-parameter global sensing which not only provides abound for the average uncertainty, but also allows forsystematic optimization of the probe. By applying ourprotocol to an Ising many-body probe, we show that onecan indeed tune external control fields to harness the crit-icality for enhancing the sensing precision, even when theintervals of interests are so large that the Heisenberg limitis absent. While the optimal measurement basis remainsan open problem, we show that a simple magnetizationmeasurement can hugely benefit from our optimization.
Acknowledgments.—
This work is supported by theNational Key R&D program of China (Grant No.2018YFA0306703) and National Science Foundation ofChina (Grants No. 12050410253, No. 92065115, andNo. 12050410251). U.M. acknowledges funding fromthe Chinese Postdoctoral Science Fund for Grant No.2018M643437. V.M. thanks the Chinese PostdoctoralScience Fund for Grant No. 2018M643435. ∗ [email protected] † [email protected] ‡ [email protected][1] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod.Phys. , 035002 (2017).[2] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Mar-zolino, M. W. Mitchell, and S. Pirandola, Rev. Mod.Phys. , 035006 (2018).[3] G. T´oth and I. Apellaniz, Journal of Physics A: Mathe-matical and Theoretical , 424006 (2014).[4] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho-tonics , 222 (2011).[5] F. Albarelli, M. A. C. Rossi, M. G. A. Paris, and M. G.Genoni, New Journal of Physics , 123011 (2017).[6] V. Giovannetti, S. Lloyd, and L. Maccone, Nature ,417 (2001).[7] A. N. Boto, P. 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Victor Montenegro , Utkarsh Mishra , and Abolfazl Bayat Institute of Fundamental and Frontier Sciences,University of Electronic Science and Technology of China, Chengdu 610051, PR China
In this section, we analytically derive the solution in obtaining the ground state | Φ (cid:105) of the Hamiltonian of Eq. (4)in the absence of any longitudinal field present in the system. In addition, we will also show how to evaluate the QFIfor this case.To derive the solution we consider the Hamiltonian H = J L (cid:88) i =1 σ ix σ i +1 x − L (cid:88) i =1 h z σ iz , (S1)where, σ iα ( α = x, z ) is the Pauli operator at site i , J > h z is the transverse field,and periodic boundary condition is imposed. The ground state of the Hamiltonian in Eq. (S1) can be obtained byfirst writing H in c-fermionic and then to Bogolyubov basis. Following the treatment of diagonalization of H as inRef. [87, 91, 92], we define raising and lowering operator at a site i by σ i ± = ( σ xi ± σ yi ) /
2. The Hamiltonian in termsof spin lowering and raising operator then becomes H = J L (cid:88) i =1 ( σ i + σ i +1+ + σ i − σ i +1 − + σ i + σ i +1 − + σ i − σ i +1+ ) − L (cid:88) i =1 h z σ iz . (S2)The next step is to apply Jordan-Wigner transformation which transforms the Hamiltonian to c-fermionic basis.The Jordan-Wigner transformation is defined as σ j − = exp[ − i (cid:80) j − (cid:96) =1 σ (cid:96) + σ (cid:96) − ] c j , σ j + = c † j exp[ i (cid:80) j − (cid:96) =1 σ (cid:96) + σ (cid:96) − ], and σ iz =2 c † i c i − I . The c-fermions follow the fermionic commutation relation as { c i , c † j } = δ ij and { c i , c j } = 0. Now byexploiting the translational invariance of the original Hamiltonian, the above Hamiltonian can be written in Fourierspace as follow H = (cid:88) k> (cid:0) c † k c − k (cid:1) (cid:18) h z + J cos( k ) i sin( k ) − i sin( k ) − ( h z + J cos( k ) (cid:19) (cid:18) c k c †− k . (cid:19) (S3)In the above equation, we used c (cid:96) = √ N (cid:80) k c k e ik(cid:96) and the basis of the matrix is {| (cid:105) , | k, − k (cid:105)} with k = ± π , ± π , . . . , ± ( N − πN . Now we introduce a matrix S = (cid:18) cos( θ k / e iφ sin( θ k / e − iφ sin( θ k /
2) cos( θ k / (cid:19) such that (cid:18) γ k γ † − k . (cid:19) = S (cid:18) c k c † − k . (cid:19) . By substituting S in Eq. (S3), we arrive at the diagonal form of the Hamiltonian expressed as H = (cid:88) k> (cid:0) γ † k γ − k (cid:1) (cid:18) (cid:15) k − (cid:15) k (cid:19) (cid:18) γ k γ † − k . (cid:19) . (S4)The operators { γ † k , γ k } are known as the Bogoliubov operators. The ground state energy of the system is given by E = − (cid:80) k> (cid:15) k = − (cid:80) k> (cid:113) ( h z + J cos( k )) + J sin ( k ). The ground state is annihilated by all the operators γ k and can be written as | Φ (cid:105) = Π k (cid:16) cos( θ k /
2) + sin( θ k / c † k c †− k (cid:17) | vac (cid:105) , (S5)where, | vac (cid:105) is annihilated by all c k operators. Furthermore, (cid:16) sin( θ k / , cos( θ k / (cid:17) = (cid:16) J sin( k ) (cid:15) k , h z + J cos( k ) (cid:15) k (cid:17) .For the above ground state with the tuning parameter h z , we can define ground state fidelity as F = (cid:104) Φ( h z ) | Φ( h z + ∆ h z ) (cid:105) , (S6)where, ∆ h z is the small shift in h z . By performing Taylor expansion for ∆ h z (cid:28)