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[email protected] hs uhr otiue qal oti work. this to equally contributed authors These slarge. is nraitcseai o xeiet,qatmprobes quantum experiments, for scenario realistic In unu erlg mly unu entanglement quantum employs metrology Quantum unu erlg ihoeaxlaypril nacorrelat a in particle auxiliary one with metrology Quantum upromteohrpooaswt esresources. less ver with to dynami proposals quantum other order the the In simulate outperform to meas algorithm environment. of quantum partially-correlated precision proper developed a a the as improve in preci to prepared such estimation when states the selectively limit auxiliary analyse employing Heisenberg can We the this which approaches In situation. , it this auxiliary environment. under correlated one noise a only in with ones en scheme with unentangled estimation the parameter of of precision The environment. n nraitcmtooy nage rbsaemr sensitiv more are probes entangled metrology, realistic In nage unu rbs n a at- can one probes, quantum entangled .INTRODUCTION I. 2 colo hsc n lcrnc,HnnNra University Normal Hunan Electronics, and Physics of School a-igHe, Wan-Ting n eateto hsc,ApidOtc ejn raMjrLa Major Area Beijing Optics Applied Physics, of Department − aN Zhang, Na-Na 1 / 2 biul,teueo en- of use the Obviously, . ejn omlUiest,Biig107,China 100875, Beijing University, Normal Beijing 1 , ∗ 1 n t unu simulation quantum its and unY Guang, Huan-Yu i-igZhao, Jie-Xing n − Dtd a 4 2021) 24, May (Dated: 1 being , 1 † 1 , ∗ uGoDeng, Fu-Guo clqbt require ical ae ah oia ttsaeitoue oestablish to introduced are since However, states [26–28]. subspace logical decoherence-free bath, lated Markovian is non-Markovian. dynamics quantum or open unentangled the outperform matter utilizi no longer ones regime, no parti- this will of In probes number entangled [25]. the superdecoherence with i.e., per rate linearly cles, decoherence increase the environment, will molecule, a correlated particle in a spins nuclear with e.g. interact particles when nertsi h aeevrnetdet h inhomoge- the to due environment same the differ- in decoher- diamond, rates the and ence energies in of Zeeman centers different homogeneity manifest NV spins the for ent requires example, ver- For critically experimentally it qubits. to since difficult it be ify might it environment appealing, non-Markovian is a in entanglement utilizing the approach states. still auxiliary al- can proper we that preparing resources, when show environment. less HL we correlated use we a proposals, though in previous noise with well-designed of selectively Comparing a impact can such metrology the Using quantum offset in bath. auxiliary particle correlated one auxiliary we a only here in with resources, scheme qubit of measurement usage a the propose or- reduce In effectively experiment. to per der used resources valuable of number ueetpeiinta hti akva quantum Markovian a as in scales mea- which that dynamics, higher non- than a a precision for in surement allows probes entangled dynamics quan- Using quantum non-Markovian Markovian 23]. open to [4, the dynamics subject diamond, tum are in envi- systems photosyn- centers the natural NV quantum in and e.g. freedom complexes large, of thetic sufficiently degrees not of is relatively number ronment are the couplings or systems system-bath large, the quantum When non-Markovian open and Markovian [17–22]. of into classified dynamics usually are quantum the all, iYnLi, Zi-Yun nodrt vroedchrnei nuncorre- an in decoherence overcome to order In nteohrhn,atog h hoeia scheme theoretical the although hand, other the On age rbsi vnlwrta that than lower even is probes tangled rmn nohrevrnetmodels environment other in urement onie seilyfracorrelated a for especially noise, to e ino u ceeadfidotthat out find and scheme our of sion f u cee eapyarecently- a apply we scheme, our ify ffe h mato environmental of impact the offset uiir tt.W ute discuss further We state. auxiliary so u rpsladso htit that show and proposal our of cs 1 ae,w rps measurement a propose we paper, , ∗ 1 uQogDeng, Ru-Qiong n igAi, Qing and ua 106 China 410006, Hunan , n 2 = boratory, N n hsclqbt,i obe the doubles it qubits, physical − 3 1 / , ‡ 4 2 2] nteohrhand, other the On [24]. , 1 dbath ed N log- ng 2 neous gyromagnetic ratios. Recently, based on the bath- When the entangled probes are in uncorrelated environ- engineering technique [29, 30] and the gradient ascent t ment, we can obtain Γn(t)= n 0 dτγ(τ)= nΓ(t), where pulse engineering (GRAPE) algorithm [31, 32], we theo- γ(t) and Γ(t) are respectively the´ dephasing rate and de- retically proposed and experimentally demonstrated that coherence factor for a single qubit. Thus, the uncertainty the open with an arbitrary Hamilto- is explicitly written as [24, 37] nian and spectral density can be exactly and efficiently simulated [33, 34]. Thus, based on this algorithm, we 1 cos2 (φt) e−2nΓ(t) δφ2 = − . (4) show how to verify that our scheme can make the esti- nTt sin2 (φt) e−2nΓ(t) mation precision achieve the HL in a quantum simulation experiment [35, 36]. In order to attain the best precision, it is necessary This paper is organized as follows: Our measurement to optimize this expression of uncertainty against the scheme and its quantum dynamics for quantum metrol- duration of each single measurement t. The best in- ogy in a correlated bath are introduced in the next sec- terrogation time satisfies φte = kπ/2 with odd k and tion. Specifically, we offer an example of magnetic-field dΓ(t) 2nt t=te =1 [24], and thus yields sensing utilizing our measurement scheme. Then, in dt |

Sec. III, we apply a recently-developed quantum algo- 2 1 2nΓ(te) rithm to simulate our measurement scheme with an aux- δφ e= 2 e , (5) | n Tte iliary qubit and show that it can approach the HL. In Appendix, we give a brief description for the open quan- where the subscript e indicates that the entangled probes tum dynamics of n qubits in a correlated bath and an are used. uncorrelated bath, respectively. When the entangled probes are in a correlated envi- ronment, the superdecoherence of the probes will modify the probability as II. MODEL AND DYNAMICS 1 2 P = 1+cos(nφt) e−n Γ(t) . (6) A. under Superdecoherence 2 h i The uncertainty of parameter φ reads In quantum metrology, using entangled probes can 2 −2n2Γ(t) obtain an increase in precision when assuming a fully- 2 1 cos (nφt) e δφ e = − . (7) coherent evolution, while in realistic scenario, there | nTt sin2 (nφt) e−2n2Γ(t) is always decoherence caused by environmental noise. 2 The optimal precision with entangled probes under By minimizing δφ e, i.e., requiring that φte = kπ/2 with decoherence was first analyzed in Markovian environ- 2 dΓ(t|) odd k and 2n t e =1, we have ment [37], and further discussed in non-Markovian en- dt |t=t vironment [24]. The best resolution in the estimation 2 2 1 2n Γ(te) was obtained with a given number of particles n and a δφ e = e . (8) | n2Tt total duration of experiment T , i.e., nT/t being the ac- e tual number of experiment. The entangled probes are When utilizing unentangled probes, the uncertainty of prepared in an n-qubit GHZ state ( 0 ⊗N + 1 ⊗N )/√2, parameter is δφ2 = 1 exp[2Γ(t )], and the best in- | i | i |u nT tu e and then evolve freely for a duration t. In the ideal terrogation time is given by φtu = kπ/2 with odd k and case, we have ( 0 ⊗N + exp( inφt) 1 ⊗N )/√2, where φ dΓ(t) 2t u =1 [24]. Following Ref. [24], we define is the parameter| i to be estimated.− However,| i in the pres- dt |t=t ence of dephasing noise, the final state becomes 0 ⊗N + δφ u ⊗N t{| i r = , (9) exp[ inφt Γn(t)] 1 /√2, with Γn (t)= dτγn(τ), | − − | i } 0 δφ e and γn(t) being the dephasing rate of n qubits.´ After a | π/2 pulse, the probability of finding these probes in the where δφ e and δφ u are the standard deviation of the initial state reads estimated| | parameter| | φ when using entangled and un- 1 entangled probes, respectively. Here, r characterizes P = 1 + cos(nφt)e−Γn(t) . (1) 2 the improved precision of measurement for using entan- h i gled probes instead of unentangled ones. For entan- The uncertainty of the measurement can be calculated as gled probes in an uncorrelated environment, we obtain 2 r = n(te/tu) exp[2Γ(tu) 2nΓ(te)], while for entan- 2 1 δφ = , (2) gled probes in a correlated− environment, we have r2 = (nT/t) F (φ) n(t /t ) exp[2Γ(t ) 2n2Γ(t )]. Obviously, r changes e u u − e where the Fisher information [38, 39] is along with the dependence of function Γ(t) on time [24], i.e., the dynamics of decoherence. For example, 2 1 ∂P t2 sin2(φt)e−2Γn(t) Γ(t) t corresponds to the Markovian dephasing dynam- F (φ) = . (3) ∝ 2 ≡ P (1 P )  ∂φ  1 cos2(φt)e−2Γn(t) ics. When Γ(t) has a quadratic behavior, i.e., Γ(t) t , it − − ∝ 3

Table I: The relative parameter resolution r varies with dif- † ferent dephasing dynamics Comman bath : Bwkb k b k k Coupling uncorrelated correlated g g a strength: gk k gk gk k environment environment ()ga = Ng k k 1 2 3 ⋯⋯ N Markovian r = 1 r = n−1/2

1/2 Auxiliary non-Markovian r = n r = 1 { System bit bits

1 !!NN Probe state: ()|1Na | 0 N + | 0 N a |1 N is the non-Markovian dephasing dynamics, which is actu- 2 ally the quantum Zeno dynamics [27, 40–42]. Hereafter, we just follow the terminology in Ref. [24]. In Tab. I, we analyze the relative resolution of the parameter r in dif- Figure 1: Schematic for N-qubits and 1 auxiliary qubit in a ferent situations, i.e., Markovian vs non-Markovian dy- correlated bath. An (N + 1)-qubit entangled system, initially namics and correlated vs uncorrelated environment. We ⊗N ⊗N at Ψ(0) =( 1 a 0 + 0 a 1 )/√2, interacts with a cor- can learn from Tab. I that using entangled probes can related| bath.i | Thei | i coupling| i constants| i between the qubits and a not improve the precision in the presence of superdeco- the kth mode of the bath satisfy gk = Ngk. herence because r 1 for n 2. ≤ ≥ † the kth mode of the environment, bk is the creation op- B. Quantum Metrology Using an Auxiliary erator of the harmonic oscillator with frequency ωk. Particle According to Eq. (A.5), the quantum dynamics of the reduced of the total system including the As shown in the pervious section, on account of su- working and auxiliary qubits is described by the quantum perdecoherence, the best precision with entangled probes master equation will no longer be superior to that with unentangled ones. ρ˙ = i[ρ,HS + HLS] Previous works [26, 27] introduced logical states which N are decoherence-free to improve the precision at the cost 1 1 + [ C (0,t)(σ(j)ρσ(i) σ(i)σ(j),ρ ) of doubled resources used per experiment. In this sec- 2 ij z z − 2{ z z } i,jX=1 tion, we present a measurement scheme with auxiliary N states, only one auxiliary qubit being used per experi- 1 + C (0,t)(σaρσ(i) σ(i)σa,ρ ) ment. When we prepare a properly-designed auxiliary ia z z − 2{ z z } qubit, the precision of parameter estimation can ap- Xi=1 N proach the HL. 1 + C (0,t)(σ(i)ρσa σaσ(i),ρ ) As illustrated in Fig. 1, we initially prepare an (N +1)- ai z z − 2{ z z } qubit entangled state with N being the number of work- Xi=1 a a ing qubits, i.e., +Caa(0,t)(σ ρσ ρ)], (12) z z − where the time correlation functions are 1 ⊗N ⊗N Ψ(0) = 1 a 0 + 0 a 1 , (10) | i √2 | i | i | i | i A B sin ωkt  CAB(ω,t)=2 gk gk [2n(ωk,T ) + 1] ωk=ω(13) ωk | where the subscript a indicates the auxiliary qubit. Xk

The total system is governed by the Hamiltonian H = for A, B = i,j,a with n(ωk,T ) being the Bose-Einstein HS + HB + Hint, HS and HB being the Hamiltonian of distribution at temperature T , A, ρ = Aρ + ρA is the probe and bath, Hint being the interaction Hamiltonian anti-commutator, { }

N N N 1 (i) 1 a † (i) (j) (i) a H = Ω σ + ω σ + ω b b HLS = Fij (0,t)σ σ + Fia(0,t)σ σ 2 0 z 2 a z k k k z z z z Xi=1 Xk i,jX=1 Xi=1 N N (i) (i) a a † a (i) I + (gk σz + gk σz )(bk + bk), (11) + Fai(0,t)σz σz + Faa(0,t) , (14) Xk Xi=1 Xi=1 where ωa is the frequency of the auxiliary qubit, we as- is the Hamiltonian with the Lamb shift sume identical frequency Ω0 for the N working qubits, A B cos ωkt 1 (i) a FAB (ω,t)= gk gk − ωk=ω (15) σz and σz are the Pauli operators of the working and ωk | a Xk auxiliary qubits, respectively, gk (gk ) denotes the cou- pling constant between the working (auxiliary) qubit and for A, B = i,j,a, I is the identity . 4

1. Correlated Environment

Suppose entangled probes are physically close, i.e., (j) gk = gk for j = 1, 2, N. They may suffer from superdecoherence with the··· time correlation function Cij (0,t) = γ(t) for i, j = 1, 2, N. As illustrated in Fig. 1, in our measurement scheme,··· we prepare a proper auxiliary qubit, whose coupling constant with the kth a mode in the bath reads gk = Ngk. Thus, we have 2 Cia(0,t)= Cai(0,t)= Nγ(t), and Caa = N γ(t). Equa- tion (12) is rewritten as

ρ˙ = i[ρ,HS + HLS] 1 N 1 + [ γ(t)(σ(j)ρσ(i) σi σ(j),ρ ) 2 z z − 2{ z z } i,jX=1 N 1 +N γ(t)(σaρσ(i) σ(i)σa,ρ ) z z − 2{ z z } Xi=1 N 1 +N γ(t)(σ(i)ρσa σaσ(i),ρ ) z z − 2{ z z } Xi=1 +N 2γ(t)(σaρσa ρ)]. (16) z z − ⊗N For simplicity, we denote 1 0a 1 and 2 ⊗N | i ≡ | i| i | i ≡ 1a 0 . The diagonal terms of the density matrix are constant| i| i in time, while the off-diagonal term is given by

−i(NΩ0−ωa)t ρ12(t)= e ρ12(0). (17) As in the conventional quantum metrology scheme, af- ter a free evolution for a time interval t and a π/2 pulse, the probability of finding the probes in the initial state reads 1 P = [1 + cos(NΩ ω )t]. (18) 2 0 − a We investigate Eq. (18) for different decoherence dynam- ics in Fig. 2. According to Eq. (8), the entangled probes have a higher precision when experiencing a longer evo- Figure 2: The uncertainty of the measurement δΩ0 for (a) lution time te. Since the best interrogation time for the a Markovian dephasing dynamics with Γ(t) = αt, (c) a non- correlated bath is much shorter than that for the uncor- Markovian dynamics with Γ(t) = βt2. Equation (18) for (b) related bath, the precision for the former is much worse. a Markovian dephasing dynamics, (d) a non-Markovian dy- However, in our scheme, the HL can be recovered because namics. The dashed green line indicates the uncorrelated de- the effects of the noises on the working qubits have been coherence. The solid red line indicates the superdecoherence effectively canceled due to the auxiliary qubit. We fur- in correlated environment. And the dotted blue line indicates ther compare the case in non-Markovian dynamics with our scheme. The green triangle, red circle and blue square show the best interrogation time for uncorrelated decoher- the one in Markovian dynamics. The best interrogation ence, superdecoherence and our scheme, respectively. We use −1 times are longer for the former, which is consistent with the following parameters, n = 4, Ω0 = 2π 5 Hz, α = 1 s , the prediction in Ref. [24]. β = 1 s−2, T = 1 s. × According to Eq. (8), the best resolution in the esti- mation satisfies (NΩ0 ωa)te = kπ/2 with odd k. The uncertainty in our scheme− reads For unentangled probes, each probe needs a properly- designed auxiliary qubit to cancel the effects of the noises 1 1 1 on it. The uncertainty of the measurement is δω2 = = , (19) 0 |e N 2Tt 2 ∝ 2 1 2 1 e (n 1) Tte (n 1) δω2 = = , (20) − − 0 |u N ′Tt nTt ∝ n where n = N +1. N is smaller than n due to the auxiliary u u qubit. where the best interrogation time t is given by (Ω u 0 − 5 7 follows the evolution

6 2 ′ ′ i(NΩ0−ωa)t −(N+K ) γ(t )dt ρ12(t)= ρ12(0)e e ´ . (22) 5 Since N +K2 >N, using auxiliary qubits can not reduce 4 but increase the dephasing of probes. As a consequence, our measurement scheme can not improve the precision r of measurement in the case of uncorrelated environment. 3 HL Markovian noise 2 Non-Markovian noise Our scheme 3. Partially-Correlated Environment

1 In this subsection, we consider a partially-correlated environment. Assume that all qubits, including N work- 0 5 10 15 20 25 30 35 40 45 50 ing qubits and one auxiliary qubit, are spatially ar- n ranged in a linear array, as shown in Fig. 1. Following Ref. [43], the real-valued homogeneous correlation func-

Figure 3: The relative ratio r = δφ u/δφ e of the frequency tions Cij (0,t), Cia(0,t) and Cai(0,t) read respectively resolutions of the entangled and unentangled| | probes when −x|i−j| suffering from supercoherence. The black solid line repre- Cij (0,t) = e γ(t), (23) sents the noise-free HL. When there are no auxiliary qubits, −x(N+1−i) the gray dashed line shows the case of Markovian noise, and Cia(0,t) = Cai(0,t)= Ke γ(t), (24) the red dash-dotted line shows the case of non-Markovian dy- namics. And the blue dotted line shows our scheme with only where x = d/ξ with d being the spatial distance between one auxiliary qubit. two adjacent qubits, ξ is the environmental correlation length [43]. For an uncorrelated bath, we have ξ =0 and Cij = δij γ(t), while ξ = and Cij = γ(t) for any i and ′ ∞ ωa)tu = kπ/2 with odd k. In this case, n = 2N and j in a fully-correlated bath. Generally, a finite but non- thus only half of the qubits play the role as the working vanishing ξ corresponds to a partially-correlated bath. qubits. The off-diagonal term of the density matrix is given by

Based on the quantum dynamics shown in Fig. 2, we ′ ′ i(NΩ0−ωa)t −A(N,x) γ(t )dt compare the uncertainty of measurement with and with- ρ12(t)= ρ12(0)e e ´ . (25) out auxiliary qubits in the presence of supercoherence. In Fig. 3, we consider two cases, i.e., the noise is fully where the factor Markovian with Γ(t) t or non-Markovian dynamics A(N, x) = (K a)2 + b a2, (26) with Γ(t) t2. When the∝ noise is of non-Markovian dy- n − − namics, r ∝remains unity for different n’s. It implies that a = exp[ x(N +1 i)], (27) the precision of measurement can not be improved by us- − − Xi=1 ing the entangled probes. When the noise is Markovian, r n approaches zero as n increases, which suggests that using b = exp( x i j ). (28) the entangled probes may even worsen the precision in − | − | i,jX=1 the presence of superdecoherence. However, if one auxil- iary qubit is employed to cancel the effects of noise, the Here, A(N,a)γ(t) represents the total dephasing rate of precision in our scheme can approach the HL. the working and auxiliary qubits. In order to improve the precision of measurement, we can always choose a proper K to minimize the factor A(N, x). Therefore, 2. Uncorrelated Environment employing auxiliary qubits can also improve the precision of a measurement for the entangled probes in a partially- When all probes are placed in an uncorrelated envi- correlated environment. ronment, e.g. spatially separated, the correlation func- tion becomes Cij (0) = γ(t)δij . And the quantum master Eq. (12) is simplified as C. Example for Magnetic-Field Sensing

1 (i) (i) ρ˙ = i[ρ,HS + HLS]+ [ γ(t)(σ ρσ ρ) As known to all, the direction and magnitude of the 2 z z − Xi geomagnetic field are closely related to the position on a a the earth. Thus, various schemes have been put for- +Caa(0,t)(σz ρσz ρ)]. (21) − ward to measure the magnetic field in order to utilize a Let K = gk /gk. Consequently, we have Caa(0,t) = geomagnetic for navigation, e.g., radical pair mechanism K2γ(t). The off-diagonal element of the density matrix [4] for avian navigation and the decoherence behaviors 6 of the NV center in diamond [44–48]. In this section, auxiliary qubits. The total system is governed by the as an example, we show how to utilize our measurement Hamiltonian [33, 34] scheme with one auxiliary qubit in magnetic-field sens- N N ing. The initial state of the entangled probes with N 1 1 H = Ω σ(i)+ ω σa+β (t) σ(i)+β (t)σa, (33) working qubits and one auxiliary qubit is prepared in 2 0 z 2 a z 1 z 2 z ⊗N ⊗N Xi=1 Xi=1 Ψ(0) = ( 1 a 0 + 0 a 1 )/√2, where 0 and 1 |refer toi the| i parallel| i and| i antiparallel| i states| i with re-| i with spect to the magnetic field, respectively. Then, let the J probe be exposed to a magnetic field B. After a time β1(t) = b1 ωjF (j)cos(ωj t + ψ1j ), (34) interval t, it evolves into Xj=1 J 1 ⊗N ⊗N ρ(t)= [ 1 a 1 a( 0 0 ) + 0 a 0 a( 1 1 ) β2(t) = b2 ωjF (j)cos(ωj t + ψ2j ), (35) 2 | i h | | ih | | i h | | ih | Xj=1 i(Nγ0B−γaB)t −Γ(t) ⊗N +e e 1 a 0 a( 0 1 ) | i h | | ih | where b1 and b2 are the noise amplitudes perceived +e−i(Nγ0B−γaB)te−Γ(t) 0 1 ( 1 0 )⊗N ],(29) | iah |a | ih | by the working and auxiliary qubits, respectively. ω0 (ωJ = Jω0) is the base (cutoff) frequency with ωj = jω0. where γ0 (γa) is the gyromagnetic ratio of the working F (j) is the function which determines the type of noise, (auxiliary) qubit. By choosing an appropriate auxiliary and ψ (i = 1, 2) is a random number. All the param- a ij qubit which satisfies gk = Ngk, we have Γ(t)=0. The eters above can be manipulated manually. Here, we ap- reduced density matrix of the working qubits at time t is ply identical time-dependent magnetic fields to all of the given by working qubits to simulate the superdecoherence. We further assume b2 = nb1 and ψ1j = ψ2j to cancel the 1 ⊗N ⊗N ρ(t) = [ 1 a 1 a( 0 0 ) + 0 a 0 a( 1 1 ) effects of the noise by the auxiliary qubit. 2 | i h | | ih | | i h | | ih | We divide the Hamiltonian (33) into two parts, i.e., the i(Nγ0B−γaB)t ⊗N +e 1 a 0 a( 0 1 ) N (i) a | i h | | ih | control Hamiltonian Hc = (Ω0 i=1 σz + ωaσz )/2, and −i(Nγ0B−γaB)t ⊗N N (i) +e 0 a 1 a( 1 0 ) ]. (30) P a | i h | | ih | the noise Hamiltonian H0(t)= β1(t) i=1 σz + β2(t)σz . In the Schrödinger picture, the propagatorP of this dy- After a pi/2 pulse, the probability of finding the probes namics is given by in their initial state reads P = [1+cos(Nγ Bt γ Bt)]/2. 0 a ˜ Here, the Fisher information is given by − U(t) = Uc(t)U(t) (36)

2 2 2 where (Nγ0 γa) t sin (Nγ0Bt γaBt) i (i) a − (Ω t P σ +ωatσ ) F (B)= − 2 − . (31) 2 0 i z z 1 cos (Nγ0Bt γaBt) Uc(t) = e , t i t a − − −i dτβ (τ) P σ( ) −i dτβ (τ)σ U˜(t) = e 0 1 i z e 0 2 z . (37) Since the best interrogation time for the entangled probes ´ ´ is determined by (Nγ0B γaB)te = kπ/2 with odd k, The initial state of the total system is Ψ(0) = − ⊗N ⊗N | i we calculate the uncertainty in the estimated value of B [ 1 a 0 + 0 a 1 ]/√2. Let it evolve for a time in- as terval| i | it under| thei | Hamiltoniani (33). Thus, we have

2 1 ψ(t) = U(t) ψ(0) δB e = 2 . (32) | i | i | (Nγ0 γa) Tt 1 −iφ(t) ⊗N iφ(t) ⊗N − = (e 1 a 0 + e 0 a 1 ),(38) √2 | i | i | i | i When N is large enough, the precision of measurement with will approach the HL. φ(t) = φA(t)+ φB (t), 1 φ (t) = (NΩ ω )t, III. QUANTUM SIMULATION OF QUANTUM A 2 0 − a METROLOGY 1 t t φB (t) = [N dτβ1(τ) dτβ2(τ)]. (39) 2 ˆ0 − ˆ0 In the NMR platform, we can use bath-engineering technique and GRAPE algorithm to simulate the quan- In order to mimic the effect of decoherence, we prepare tum dynamics under superdecoherence. By this quantum a large number of ensemble which evolve under different simulation, we can further prove that our measurement Hamiltonians characterized by a set of the random num- bers ψ , ψ . Finally, the probability of finding the scheme with one auxiliary qubit can approach the HL, { 1j 2j } even in the case of superdecoherence. Utilizing the bath- probes in the initial state is over the ensemble as engineering technique, we apply a time-dependent mag- 1 P (t)= [1 + cos 2φ cos2φ sin 2φ sin 2φ ]. (40) netic field to the total system including the working and 0 2 Ah Bi− Ah Bi 7

If we further assume a Gaussian noise, then we have φ2m−1(t) =0 for any positive integer m [33], and thus h B i 1 1 cos2 [(Nω ω ) t] yields δΩ2 = = − 0 − a . (42) 0 2 2 (T/t) F (φ) N Tt sin [(Nω0 ωa) t] 1 −2χ(t) − P0(t)= [1 + cos 2φA(t)e ], (41) 2 By similar derivation to Eq. (19), we have δΩ2 0 |e∝ (n 1)−2. We find the scaling law for our scheme can where approach− the HL when N is large enough, even in the +∞ 2 4 dω 2 ωt case of superdecoherence. χ(t)= φB (t) = 2 S(ω) sin , h i 2π −∞∫ ω 2 We apply the quantum simulation algorithm to sim- ulate our scheme as shown in Fig. 4. We use the same 1 2 S(ω)= [N S11(ω) NS12(ω) NS21(ω)+ S22(ω)]. types of lines as Fig. 2 to represent the probability. Here, 4 − − the number of random realizations in ensemble we used 3 +∞ in this simulation is 2 10 . The time steps between Here, Sij (ω) = −∞ dt βi(0)βj (t) exp(iωt) (i, j = 1, 2) × is the Fourier transform∫ h of the two-timei correlation func- the dots are 5 ms. Without loss of generality, we as- tion β (t )β (t ) , which depends on the time interval sume the model of white noise, i.e., F (j)=1/j [30]. The h i 1 j 2 i base and cutoff frequencies for the Markovian noise are t2 t1. And S(ω) is the total power spectral density of the− noise, which describes the energy distribution of the ω0 = 0.2 Hz and ωc = 140 Hz, respectively, while for the non-Markovian noise, we use the following parame- stochastic signal in the frequency domain [33]. −3 ters ω0 = 10 Hz and ωc = 0.18 Hz. We can learn We utilize the relation that b2 = Nb1 and ψ1j = ψ2j to obtain the two-time correlation functions as from Fig. 4 that the results of stochastic Hamiltonian simulation are in quite-good agreement with the results J of the Bloch-Redfield equation. The blue square dots b2ω2 β (t + τ)β (t) = 1 0 j2F (j)2 cos(ω τ), represent our scheme, which implies that employing one h 1 1 i 2 j Xj=1 auxiliary qubit can cancel the effects of noise. As a re- β (t + τ)β (t) = β (t + τ)β (t) sult, the precision in our scheme can approach the HL. In h 1 2 i h 2 1 i J order to explore the underlying physical mechanism ex- Nb2ω2 = 1 0 j2F (j)2 cos(ω τ), plicitly, we also plot the corresponding decoherence factor 4 j Γ(t) in Fig. 5. For the Markovian noise, the coherence Xj=1 decays with a constant rate, and thus using entangled J N 2b2ω2 probes will not improve the precision of the estimation. β (t + τ)β (t) = 1 0 j2F (j)2 cos(ω τ). h 2 2 i 2 j For the non-Markovian noise, since the coherence decays Xj=1 quadratically with time, the use of entangled probe can offer a better estimation, but the precision is still lower Thus, the power spectral densities are respectively given by than the HL. However, for a correlated bath, because all qubits suffer from the collective noise, by properly arranging the auxiliary qubit the noises on the working πb2ω2 J S (ω) = 1 0 j2F (j)2[δ(ω ω )+ δ(ω + ω )], qubits can be effectively canceled and thus the noise-free 11 2 − j j Xj=1 measurement can be performed.

S12(ω) = S21(ω) J Nπb2ω2 IV. CONCLUSION = 1 0 j2F (j)2[δ(ω ω )+ δ(ω + ω )], 2 − j j Xj=1 In this paper, we propose a quantum metrology scheme 2 2 2 J N πb1ω0 2 2 with auxiliary states. The auxiliary states are properly S22(ω)= j F (j) [δ(ω ωj )+δ(ω + ωj )]. 2 − designed and can selectively offset the impact of envi- Xj=1 ronmental noise. On account of superdecoherence, we Obviously, the total power spectral density reads analyze the optimal precision with and without auxiliary qubits. We find out that when the auxiliary qubits are 2 S (ω)= N S11 (ω) NS12 (ω) NS21 (ω)+ S22 (ω)=0, not employed, the precision can not be improved by us- − − ing the entangled probes no matter whether the noise and the decoherence function also vanishes, i.e., is Markovian or non-Markovian. However, utilizing an auxiliary qubit can make the precision approach the HL +∞ 4 dω 2 ωt in our scheme. We further discuss the cases of uncorre- χ(t)= 2 S (ω) sin =0. 2π −∞∫ ω 2 lated and partially-correlated environment, and find out that employing auxiliary qubits can improve the preci- We calculate the uncertainty of the measurement by sion of measurement in the partially-correlated environ- the Fisher information as ment but it fails in the uncorrelated environment. As 8

1 (a) 1 Non-Markovian noise 0.8 Markovian noise 0.8 Our scheme 0.6

P 0.6 0.4 (t) 0.2 0.4

0 0 0.2 0.4 0.6 0.8 1 0.2 t (s) 0 (b) 1 0 0.2 0.4 0.6 0.8 1 0.8 t (s)

0.6 Figure 5: The simulated decoherence factor Γ(t) for different

P schemes: the gray solid line for Markovian environment, the 0.4 red dash-dotted line for non-Markovian environment, and the blue dotted line for our scheme. 0.2

0 [17, 43, 50, 51] 0 0.2 0.4 0.6 0.8 1

t (s) n n 1 H = ω σ(i) + ω b† b + σ(i) g(i)(b† + b ). 2 0 z k k k z k k k Figure 4: Quantum simulation of population dynamics in Xi=1 Xk Xi=1 Xk Fig. 2. The dashed green line, the solid red line and the dotted (A.1) ~ blue line are the same as Fig. 2, while the green triangles, the Assuming that = 1, the system Hamiltonian is HS = n (i) red circle and the blue squares are obtained by the quantum ω0 i=1 σz /2, and the bath Hamiltonian is HB = simulation approach [33, 34] for the uncorrelated decoherence, † kPωkbkbk, and the interaction Hamiltonian is HI = the superdecoherence and our scheme, respectively. n (i) (i) † Pi=1 σz k gk (bk + bk), where ω0 is the energy sep- † Paration betweenP the ground and excited states, bk (bk) is the creation (annihilation) bath operator, gk is the cou- an example, we show how to utilize our scheme with one pling constant between the qubit and kth mode of bath, auxiliary qubit in magnetic-field sensing. Finally, we use which is assumed to be real for simplicity. the bath-engineering technique and GRAPE to simulate The interaction Hamiltonian is the product of the the quantum dynamics and demonstrate that assisted by system operators and the bath operators, i.e., HI = an auxiliary qubit our scheme can approach the HL in n (i) (i) † siBi with si = σz and Bi = g (b + bk). the case of superdecoherence. i=1 k k k PWe decompose the system operators si Pinto several parts ′ We thank valuable discussions with M.-J. Tao. in the eigenspace ǫ i as si = ω si(ω) ǫ i ǫ with {| i }′ | i h | This work is supported by the National Nat- si(ω) = ǫ′−ǫ=ω ǫ si ǫ i. The summationP in si(ω) is h | | i ′ ural Science Foundation of China under Grant extendedP over all energy eigenvalues ǫ and ǫ of HS with (i) Nos. 11674033, 11474026, 11505007, and Beijing Natu- a fixed energy difference of ω [38, 43]. Because si = σz , ral Science Foundation under Grant No. 1202017. we have s (ω ) = s ( ω ) = 0 and s (0) = 1 for i 0 i − 0 i ∓ 0 i and 1 i, respectively. Introducing these eigenopera- tor| i decompositions,| i the Bloch-Redfield equations can be rewritten as

n Appendix: Dynamics of n-qubit Decoherence (j) (i) 1 (i) (j) ρ˙ = i[ρ,HS]+ D (0,t)(σ ρσ σ σ ,ρ ), ij z z − 2{ z z } i,jX=1 (A.2) The quantum dynamics of open system is described t by the well-known spin-boson model [49]. Since the time where Dij (ω,t) = dτ exp(iωτ) Bi(τ)Bj (0) are the 0 h i scale of dephasing is generally much smaller than that of spectral functions, which´ define both temporal and spa- e e longitudinal relaxation, we consider the pure-dephasing tial correlations of the pure-dephasing noise environment (i) † dynamics for simplicity. The spin-boson Hamiltonian of [52]. Bi(τ) = gk [bk exp(iωkτ)+ bk exp( iωkτ)] are the n qubits coupled to a common bath can be written as operators of environment in the interaction− picture, and e 9

Bi(τ)Bj (0) are the correlation functions, i.e., For n =1, since the total Hamiltonian is H = ω0σz/2+ h i † † ωkb bk + σz gk(b + bk), Eq. (A.5) is simplified as e e (i) (j) k k k k B (τ)B (0) = g g [2n(ω ,T )cos ω τ + e−iωk τ ]. h i j i k k k k P P Xk e e 1 (A.3) ρ˙ = i[ρ, ω σ ]+ C(0,t)(σ ρσ ρ), (A.6) 2 0 z z z − Thus, the spectral functions are explicitly given as where the single-qubit dephasing rate is given by γ(t)= sin ω t (i) (j) k C(0,t). Dij (ω,t) = gk gk [2n(ωk,T ) ωk Xk However, the n-qubit dephasing rate varies with the 1 e−iωk t environmental model. For simplicity, we assume that + − ] ωk=ω, (A.4) (i) (j) iωk | gk = gk = gk. When n qubits are in an uncorre- lated environment, e.g. the distance between the qubits † where n(ωk,T ) = b bk denotes the average occupa- are far beyond the correlation length of the environment h k i tion number of mode k at temperature T . The real [43], we can neglect all spatial correlations of the bath, part of D (0,t) causes dephasing while its imaginary n ij i.e., Cij (0,t)= γ(t)δij . Hence γn(t)= i,j=1 Ci,j (0,t)= part corresponds to Lamb shift [43]. Let Dij (0,t) = nγ(t). In contrast, when n qubits areP in a correlated 1 n (i) (j) LS 2 Cij (0,t)+ iFij (0,t), and H = i,j=1 Fij (0,t)σz σz bath, i.e., the correlation length of the environment is is the Lamb-shift Hamiltonian. Thus,P Eq. (A.2) is rewrit- much bigger than the qubits’ spatial separation, the n- ten as qubit dephasing rate is n2 times that of single qubit, i.e., 2 γn(t) = n γ(t), resulting from Cij (0,t) = γ(t) for all ρ˙ = i[ρ,HS + HLS] (A.5) i and j. This phenomenon is called superdecoherence, 1 n 1 mainly due to collective entanglement between qubits + C (0,t)(σ(j)ρσ(i) σ(i)σ(j),ρ ). and environment [53]. 2 ij z z − 2{ z z } i,jX=1

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