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Sci Rep
Sci Rep
Scientific Reports
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Nature Publishing Group UK London

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10.1038/s41598-024-71365-8
Article
Harnessing graph state resources for robust quantum magnetometry under noise
Nguyen Phu Trong 1
Le Trung Kien 26
Nguyen Hung Q. 3
Ho Le Bin binho@fris.tohoku.ac.jp

45
1 grid.267849.6 0000 0001 2105 6888 Department of Advanced Material Science and Nanotechnology, University of Science and Technology of Hanoi, Vietnam Academy of Science and Technology, Hanoi, 11307, Vietnam
2 grid.133342.4 0000 0004 1936 9676 Department of Physics, University of California, Santa Barbara, Santa Barbara, USA
3 grid.267852.c 0000 0004 0637 2083 Nano and Energy Center, University of Science, Vietnam National University, Hanoi, 120401 Vietnam
4 https://ror.org/01dq60k83 grid.69566.3a 0000 0001 2248 6943 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, 980-8578 Japan
5 https://ror.org/01dq60k83 grid.69566.3a 0000 0001 2248 6943 Department of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai, 980-8579 Japan
6 https://ror.org/00f54p054 grid.168010.e 0000 0004 1936 8956 Present Address: Department of Applied Physics, Stanford University, Stanford, USA
4 9 2024
4 9 2024
2024
14 205288 12 2023
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Precise measurement of magnetic fields is essential for various applications, such as fundamental physics, space exploration, and biophysics. Although recent progress in quantum engineering has assisted in creating advanced quantum magnetometers, there are still ongoing challenges in improving their efficiency and noise resistance. This study focuses on using symmetric graph state resources for quantum magnetometry to enhance measurement precision by analyzing the estimation theory under time-homogeneous and time-inhomogeneous noise models. The results show a significant improvement in estimating both single and multiple Larmor frequencies. In single Larmor frequency estimation, the quantum Fisher information spans a spectrum from the standard quantum limit to the Heisenberg limit within a periodic range of the Larmor frequency, and in the case of multiple Larmor frequencies, it can exceed the standard quantum limit for both noisy cases. This study highlights the potential of graph state-based methods for improving magnetic field measurements under noisy environments.

Subject terms

Quantum metrology
Qubits
MEXT | Japan Society for the Promotion of Science (JSPS)23K13025 Ho Le Bin issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Quantum sensing utilizes quantum resources like non-classical states, entanglement, and squeezing to improve sensor capabilities beyond classical approaches1. Recent advances in quantum resource theory have been made in quantum-enhanced sensing using non-classical states2–4, entangled cluster and graph states5–8, many-body nonlocality and multiqubit systems9–11, and squeezed resources12–15. Furthermore, various techniques like machine learning algorithms16–20, quantum error correction methods21–24, network sensing25,26, and hybrid algorithms8,27–32, are being explored for enhancing noise resilience and extracting insights from quantum sensing.

In quantum magnetometry, the precise measurement of magnetic fields is crucial in various subjects like fundamental physics research, space exploration, material science, geophysics, and medical biophysics. Recent advances in quantum engineering have led to the development of various quantum magnetometers, such as superconducting quantum interference device (SQUID)33, diamond-based magnetometer34,35, single-spin quantum magnetometer36, submicron-scale NMR spectroscopy37, cold atom magnetometer38, and 2D hexagonal boron nitride magnetic sensor39. These innovations find applications in highly sensitive and broadband magnetic field measurements34,35, scanning gradiometry36, low magnetic fields37, navigation38, and magnetic field imaging39.

Improving the sensitivity of magnetometers is essential for different applications. However, the present methods are insufficient due to high quantum resource efficiency and noise resilience demands. Therefore, there is an urgent requirement for a novel resource that can enable the full potential of quantum magnetometry while being practical for experimental implementation.

Among various candidates, graph states have emerged as a promising avenue in the quest for quantum-enhanced magnetometry. Graph states are particular types of entangled states that can be represented by a graph, where the vertices represent qubits, and the edges represent entangling gates between the qubits40,41. Due to their multipartite entanglement, they have demonstrated great potential in quantum computation42,43, communication42,44, and metrology6–8. Within the context of quantum magnetometry, harnessing the capabilities of graph states introduces a novel dimension to the quest for precision and robustness in the presence of noise.

This work explores symmetric graph state resources for robust quantum magnetometry under time-homogeneous and time-inhomogeneous noises45. Our approach begins by modeling an ensemble of N spin-1/2 particles as a sensor probe for measuring Larmor frequencies of an external magnetic field. Initially, the probe state is set up in a star-graph configuration, where one vertex (spin particle) is connected to the remaining N-1 vertices through CZ gates. We study the influence of noise in the model by analyzing the measurement precision from the perspective of estimation theory and quantum Fisher information.

For uncorrelated probes, the variance of estimating a single phase ϕ follows Δ2ϕ=O(N-1), commonly referred to as the standard quantum limit (SQL), whereas for entangled probes, it is possible to reach the Heisenberg limit (HL), where Δ2ϕ=O(N-2)46–48. However, under time-homogeneous noise, entangled sensors cannot surpass the SQL49,50. In the presence of time-inhomogeneous noise, the variance can reach Δ2ϕ=O(N-1.5)51–53, and similar results have been observed in the context of multiphase sensing8,45.

In our investigation for single Larmor frequency estimation, we observe a transition from SQL to HL behavior for a periodic range of the Larmor frequency. For multiple Larmor frequencies, we find that the variance Δ2ϕ can beat the SQL for both time-homogeneous and time-inhomogeneous noise sources. This marks the initial instance where we observe surpassing the SQL under time-homogeneous noise. Our analysis of quantum magnetometry in these noise factors sheds light on the resilience and potential of graph state-based approaches for conducting highly precise magnetic field measurements in challenging and real-world conditions.

Results

Measurement model and its initialization

Let us consider the measurement of an external magnetic field by employing a spin-1/2 system comprising N particles as the probing mechanism. Each particle interacts with the field and provide information about the field strengths. The coupling Hamiltonian is given by541 H=-∑k=1N(μ(k)·B),

where μ(k)=12γeσ(k) represents the magnetic moment of the kth spin. γe denotes the gyromagnetic ratio, and σ(k)=(σx(k),σy(k),σz(k)) refers to the Pauli matrices. Here, B=(Bx,By,Bz) signifies the external magnetic field. We define ϕj=|γe|Bj for all j∈x,y,z as the Larmor frequency54, and Jj=12∑kσj(k) as the angular momentum. With this, the Hamiltonian recasts as2 H=ϕ·J,

where ϕ=(ϕx,ϕy,ϕz) represents the set of Larmor frequencies requiring estimation, and J=(Jx,Jy,Jz) are three components of the collective angular momentum. Refer to the “Methods” section for a detailed model and its quantum circuit.

The probe is initialized as a graph state, which typically consists of a collection of vertices denoted by V and edges represented by E as3 G(V,E)=∏i,j∈ECZij|+⟩⊗V,

where CZij represents the controlled-Z gate connecting the ith and jth spins, and |+⟩ is an element in the basis of Pauli σx. Graph states serve as valuable assets in quantum metrology6,8, as demonstrated by their application in achieving Heisenberg scaling, as observed with star configurations where the quantum Fisher information (QFI) gives (N-1)2+1, and with local Clifford (LC) operations where the QFI gives N26. Hereafter, we examine the impact of graph-state resources on quantum-enhanced magnetometry within a noisy environment.Figure 1 Quantum Fisher information in a single phase estimation. Plot of QFI vs the noise probability λ and Larmor frequency ϕ for N=5 and t=1. For λ=0, the QFI reaches Q=(N-1)2+1∀ϕ (dashed red line). Increasing λ, the QFI gradually reduces and reaches the minimum at λ=1. Remarkably, this minimum is non-zero for a non-zero ϕ as illustrated by the soiled black curve. For ϕ=π/2, it is given by (N-1)2 (red circle).

Single phase estimation

We examine the estimation of a single Larmor frequency denoted as ϕ=(ϕ,0,0). The coupling Hamiltonian is H=ϕJx, and the corresponding unitary operator is expressed as4 U(ϕ)=exp(-itϕJx).

The initial probe state is prepared in a graph configuration ρ0=|G⟩⟨G|. After the interaction, it evolves to ρ(ϕ)=U(ϕ)ρ0U†(ϕ). During the magnetic field coupling, the probe interacts with its surroundings and decoheres. Our analysis focuses on dephasing noise as a type of phase decoherence that leads to the evolution of the state5 ρ(ϕ,γ)=[∏k=1NeγtL(k)]ρ(ϕ),

where γ is the dephasing rate. Dephasing is often referred to as the spin-spin relaxation process55, which affects the relative phase in the probe’s basis and can be represented by the Pauli operator σz. By employing the Kraus operators to account for dephasing noise as K0=diag(1,1-λ),K1=diag(0,λ), we obtain6 eγtL(k)ρ(ϕ)=K0(k)ρ(ϕ)[K0(k)]†+K1(k)ρ(ϕ)[K1(k)]†,

where λ=1-e-γt∈[0,1] is the dephasing probability. For other noisy scenarios, please see the “Methods” section.

The final state ρ(ϕ,γ) contains detailed information about the unknown Larmor frequency ϕ. To evaluate the precision of the estimation, we examine the QFI Q. By decomposing ρ(ϕ,γ)=∑kℓk|ℓk⟩⟨ℓk|, the QFI yields567 Q=2∑i,j,ℓi+ℓj≠0|⟨ℓi|∂ϕρ(ϕ,γ)|ℓj⟩|2ℓi+ℓj.

For numerical calculation, let us fix the sensing time t=1 in an arbitrary unit. The results are presented in Fig. 1, focusing on the star-graph configuration and using N=5 as an illustrative example. In the absence of noise, i.e., λ=0, the QFI yields68 Q=4[⟨G|Jx2|G⟩-(⟨G|Jx|G⟩)2]=(N-1)2+1,

which does not depend on ϕ as shown in the dashed red line. See the “Methods” section for detailed calculation.

In the presence of noise, the QFI depends on both λ and ϕ. As λ increases, the QFI gradually decreases, reaching its minimum value at λ=1. Interestingly, this minimum value remains nonzero for ϕ≠0, which is demonstrated by the solid black curve. Specifically, for ϕ=π/2, the QFI is even by (N-1)2 (red circle). See the “Methods” section for detailed calculation.

A specific case of star graph is a GHZ state up to a local unitary (LU) transformation41,57. Let us consider the initial probe state to be a GHZ state9 |ψ⟩GHZ=12(|νmax⟩+|νmin⟩),

where |νmax⟩ and |νmin⟩ are eigenstates of Jx corresponding to the maximum and minimum eigenvalues νmax and νmin, respectively. Particularly, this state can be prepared by applying a Hadamard gate to the first spin particle of the star graph state in Eq. (25). The QFI gives (see the “Methods” section)10 QGHZ=4[⟨ψGHZ|Jx2|ψGHZ⟩-(⟨ψGHZ|Jx|ψGHZ⟩)2]=N2.

Here, the QFI remains independent of ϕ. Upon closer examination, it becomes evident that the QFI attains the Heisenberg limit of N2 in the absence of noise. In the presence of dephasing noise, the QFI is invariant, preserving its N2 as detailed in the “Methods” section. This result is trivial as noise primarily affects the phase or coherence of quantum states along the z-axis, while GHZ here points toward the x-axis.

Next, we examine the quantum Cramér-Rao bound (QCRB) for various values of N. It is the ultimate bound that imposes the precision achievable in the estimation process, i.e, M·Δ2ϕ≥CF≥CQ, where Δ2ϕ=⟨(ϕ-ϕ^)2⟩-⟨(ϕ-ϕ^)⟩2 is the variance of ϕ, which indicates the difference between the true value ϕ and its estimated counterpart ϕ^, M is the repeated experiments. Here, CF and CQ are classical and quantum Cramér-Rao bound, respectively. The QCRB is determined through the inversion of the QFI as11 CQ=Q-1,

which can be achieved in the single-phase estimation, such as using a Bayesian estimator or neural network technique (see20 and Refs therein).Figure 2 Quantum Cramér-Rao bound in a single phase estimation. The plot of QCRB as a function of the number of spins N for various λ and ϕ. Additionally, SQL and HL are displayed for comparative purposes. The plot is presented with the star-graph configuration.

The numerical results are showcased in Fig. 2. For ϕ=0, the QCRB can beat the SQL event for large noise. Here we illustrate for λ=0.8 () and 0.9 (), and the fitting curves are proportional to (N-1)-1.583 (blue dashed line) and (N-1)-1.607 (orange dotted line), respectively. Throughout the paper, we use the fitting function as f(N)=a(N-1)b∀a,b∈R, which is inspired by the exact result when λ=0. For ϕ=π/2, the analytical findings in Fig. 1 suggest that the QCRB fluctuates between 1(N-1)2+1 and 1(N-1)2 for λ∈[0,1]. Comparatively, the cases of λ=0.8 () and 0.9 () align closely with the λ=0 scenario (the black line), exhibiting a remarkable match. Notably, they attain the Heisenberg scaling. For comparison, we show the standard quantum limit SQL = N-1 and the Heisenberg limit HL = N-2. This result represents an advanced approach in leveraging graph states for robust sensing in noisy environments, indicating a transition from the SQL to the HL within a periodic range of the Larmor frequency.Figure 3 Quantum Cramér–Rao bound in a multiphase estimation. The plot of QCRB as a function of sensing time for time-homogeneous (HO) and time-inhomogeneous (INHO) noises. Here we fixed the Larmor frequencies ϕx=ϕy=ϕz≡φ=π/6 in (a) and fixed γ=0.5 in (b). The QCRB initially decreases, reaches a minimum, and then increases with increasing sensing time.

Multiple phases estimation

We consider the estimation of Larmor frequencies as ϕ=(ϕx,ϕy,ϕz) with the coupling Hamiltonian being given in Eq. (2). The unitary evolution yields12 U(ϕ)=exp(-itϕ·J).

We consider the Ornstein–Uhlenbeck noise model, originating from the stochastic fluctuations of the external magnetic field58. The noise is characterized by Kraus operators5913 K0(t)=diag(1,1-q(t)),K1(t)=diag(0,q(t)),

where q(t)=1-e-f(t) and f(t)=γ[t+τc(e-t/τc-1)]. Here, τc is the memory time of the environment. In the limit of time-homogeneous behavior or white noise limit (τc→0), we have f(t)=γt, which corresponds to the previous dephasing case. In the time-inhomogeneous case, the time τc is large, and thus t/τc≪1 (short-time limit). In this case, the expression becomes f(t)=γt22τc. The function q(t) is defined as14 q(t)=1-exp(-γt)time-homogeneous,1-exp(-γt22τc)time-inhomogeneous.

In the numerical simulation, τc is fixed at τc=20 for the time-inhomogeneous case.

Similar as the single phase case, we first calculate ρ(ϕ)=U(ϕ)|G⟩⟨G|U†(ϕ), and then derive ρ(ϕ,γ) by applying the Kraus operators in Eq. (13) for all qubits. Next, given the decomposed form as ρ(ϕ,γ)=∑kℓk|ℓk⟩⟨ℓk|, the quantum Fisher information matrix (QFIM) gives15 Qαβ=2∑i,j,ℓi+ℓj≠0⟨ℓi|∂ϕαρ(ϕ,γ)|ℓj⟩⟨ℓj|∂ϕβρ(ϕ,γ)|ℓi⟩ℓi+ℓj,

and the QCRB in the multiphase case is given by CQ=Tr[Q-1]. See detailed calculations in the “Methods” section.

Figure 3 illustrates the QCRB concerning time-homogeneous and time-inhomogeneous noises as a function of time t. Consistent with findings reported in8,45, a pivotal insight surfaces: an optimal sensing time emerges leads to minimized CRBs across the examined scenarios. For time-homogeneous (HO) noise, the optimal sensing time tends to be shorter, whereas for time-inhomogeneous (INHO) noise, an extended sensing time is favored. Remarkably, the presence of time-inhomogeneous dephasing yields lower metrological bounds compared to the time-homogeneous counterpart.

In Fig. 4, we can observe the minimum QCRB for various values of N. The results demonstrate that as N increases, time-inhomogeneous noise consistently outperforms the SQL across all levels of noise (represented by open triangles). The relationship follows a fitted function that scales as ∝(N-1)-1.071. Similarly, in the case of time-homogeneous noise, the bounds tend to surpass the SQL as N grows larger. This is the first instance we observe exceeding the SQL under time-homogeneous noise.Figure 4 Quantum Cramér–Rao bound in a multiphase estimation. The plot of QCRB vs N at fixed Larmor frequencies ϕx=ϕy=ϕz≡φ=π/6 for two cases of time-homogeneous and time-inhomogeneous noises.

Discussion

We discuss the Bayesian inference in estimating the single Larmor frequency. In this approach, the process begins with defining a likelihood function that expresses the probability of the observed data set {D} concerning the parameter of interest ϕ. In our scenario, the likelihood function is calculated as the product of probabilities while measuring the final state ρ(ϕ,γ) in Eq. (5) with the computational bases {|k⟩} as16 P(D|ϕ)=∏k=12NTr[ρ(ϕ,γ)|k⟩⟨k|].

The Bayesian theorem is then applied to compute the posterior distribution, which describes the uncertainty associated with the parameter as17 P(ϕ|D)=P(D|ϕ)∫P(D|ϕ)dϕ.

Finally, the estimated phase is given by18 ϕestimated=∫ϕP(ϕ|D)dϕ.

Practically, sampling techniques such as Markov Chain Monte Carlo (MCMC) or Nested Sampling (NS) are used to generate samples from the posterior distribution20. These samples are then used for estimates, typically as the posterior mean or credible intervals. These estimates convey not only the point estimate but also the related uncertainty.

To illustrate, we focus on the case where λ=1 and proceed to theoretically derive the likelihood function (16) using ρN(ϕ) from Eq. (35). It gives19 P(D|ϕ)=∏k=12NTr[ρN(ϕ)|k⟩⟨k|]=12N·(2N)∑k=02N-12N-1k(-1)ksin2k(ϕ)sin2k[(N-1)ϕ],

Then P(ϕ|D) yeilds20 P(ϕ|D)=12N·(2N)∑k=02N-12N-1k(-1)k∫sin2k(ϕ)sin2k[(N-1)ϕ]dϕ.

Finally, we obtain21 ϕestimated=∫ϕP(ϕ|D)dϕ.

The detailed calculation for obtaining the estimated ϕestimated is provided in the “Methods” section. We present our findings in Fig. 5. In (a), a comparison is made between the true values and the estimated values. The true values are represented by the blue line when ϕtrue=ϕestimated. The average estimated values are indicated by orange dots with error bars, obtained through the Bayesian inference method from M=100 repeated experiments in a quantum circuit. In (b), we plot the squared error Δ2ϕ as a function of ϕ and compare it with the inverse QFI extracted from the black curve in Fig. 1 after re-scaling, i.e., 2.5×1MQ. It indicates the QCRB relation as Δ2ϕ≥1MQ.Figure 5 Bayesian inference. (a) A comparison between true and estimated values. The true values are represented by the blue line where ϕtrue=ϕestimated. The estimated values are indicated by orange dots along with error bars, derived from 100 repeated experiments using Bayesian inference. (b) Plot of squared errors as a function of ϕ and their comparison with the inverse QFI extracted from the black curve in Fig. 1. These results are presented for λ=1.

Methods

Qubits model

We introduce a measurement model using qubits system as shown in Fig. 6. In metrology schemes, we measure a system by using a probe that couples to it. After the interaction, we measure the probe to estimate the system’s information. In our model, the system is the magnetic field, and the probe is an ensemble of spins. When considering noise, it couples to the probe, and this coupling is different from the system-probe interaction. In our case, we consider dephasing and Ornstein-Uhlenbeck noise models. Dephasing is often referred to as the spin-spin relaxation process. It affects the relative phase in the probe’s basis, which can be represented by the Pauli operator σz . The noise occurs during the interaction process between the system and the probe. Initially, at t=0, we turn on the coupling between the probe and the system. At time t=tf , we turn off the interaction and measure the probe. Assuming the preparation and measurement times are very short, the probe does not evolve before time t=0 and after t=tf. Thus, noise only affects the probe during the interaction time. The measurement scheme is given in Fig. 6a.Figure 6 (a) Quantum magnetometry scheme with noise model. The probe is an ensemble of spins prepared in a graph state. The probe interacts with an external filed during the time t=0 to t=tf. The noise also appears during the interaction time. (b) The quantum circuit designed for quantum magnetometry. The first block is a star configuration, applying Hadamard gates to |0⟩ qubits to transform them into |+⟩ states and connecting them as a star, with the first qubit at the center linked to the surrounding qubits via CZ gates. The second block involves phase and noise encoding using U1(ϕ), followed by Kraus operators K=(K0,K1) apply to all qubits.

For the interaction, we first rewrite Eq. (12) as22 U(ϕ)=exp[-it(ϕxJx+ϕyJy+ϕzJz)]=exp[-it∑k=1N(ϕx2σx(k)+ϕy2σy(k)+ϕz2σz(k))]=∏k=1Nexp[-it(ϕx2σx(k)+ϕy2σy(k)+ϕz2σz(k))].

We set a single-qubit unitary as23 U1(ϕ)=exp[-it(ϕx2σx+ϕy2σy+ϕz2σz)],

and apply it to all qubits in a quantum circuit. For a single-phase estimation, it becomes a rotation gate, i.e. U1(ϕ)=exp[-itϕx2σx].

The quantum circuit is given in Fig. 6b. The first block is the graph state generation with star configuration, wherein a Hadamard gate is applied to each qubit initially set in the state |0⟩, transforming them into |+⟩ states. These qubits are connected as a star, with the first qubit at the center and connects to the surrounding qubits through CZ gates. The second block is phase and noise encoding, given by applying U1(ϕ) and followed by the Kraus operators K=(K0,K1) apply to all qubits. The final state is used to calculate quantum Fisher information (QFI) for single phase estimation and quantum Fisher information matrix (QFIM) for multiphase estimation. From these, we derive the corresponding QCRB60.

Deriving QFI for single parameter estimation

For star graph state

We first calculate the QFI with the initial star graph state in the case of without noise. We express the QFI in terms of its generator as24 Q=4⟨ΔHϕ2⟩=4[⟨G|Hϕ2|G⟩-(⟨G|Hϕ|G⟩)2],

where Hϕ=iU†(ϕ)∂ϕU(ϕ). For single-phase estimation, it gives Hϕ=Jx=12∑k=1Nσx(k).

To calculate Eq. (24), we first expand the graph state to a star configuration, with one central qubit (qubit 1) connects to the remaining surrounding N-1 qubits:25 |G⟩=∏k=2NCZ1,k|+⟩⊗N=(CZ1,NCZ1,N-1⋯CZ1,2)|+⟩⊗N=12(|0⟩|+⟩⊗(N-1)+|1⟩|-⟩⊗(N-1)).

Next, we compute Hϕ|G⟩:26 Hϕ|G⟩=[12∑k=1Nσx(k)]|G⟩=122[(|1⟩|+⟩⊗(N-1)+|0⟩|-⟩⊗(N-1))+(N-1)(|0⟩|+⟩⊗(N-1)-|1⟩|-⟩⊗(N-1))].

Using Eq. (26), the first term in Eq. (24) gives:27 ⟨G|Hϕ2|G⟩=⟨G|HϕHϕ|G⟩=14(1+(N-1)2).

Using Eqs. (25) and (26), the second term in Eq. (24) yields28 ⟨G|Hϕ|G⟩2=0.

As a result, the QFI in (24) gives29 Q=1+(N-1)2,

and the corresponding QCRB is CQ=1Q=11+(N-1)2.

Now, we calculate the QFI under dephasing noise. We first recast the unitary Eq. (4) as30 U(ϕ)=[exp(-iϕ2σx)]⊗N=[cos(ϕ′)I-isin(ϕ′)σx]⊗N,

where we fixed t=1 and set ϕ′=ϕ/2. The initial graph state |G⟩ evolves to31 |G(ϕ)⟩=[cos(ϕ′)I-isin(ϕ′)σx]⊗N·12(|0⟩|+⟩⊗(N-1)+|1⟩|-⟩⊗(N-1))=12{(cos(ϕ′)|0⟩-isin(ϕ′)|1⟩)⊗e-iϕ′(N-1)|+⟩⊗(N-1)+(cos(ϕ′)|1⟩-isin(ϕ′)|0⟩)⊗eiϕ′(N-1)|-⟩⊗(N-1)}=12{e-iϕ′(N-1)cosϕ′-isinϕ′⊗|+⟩⊗(N-1)+eiϕ′(N-1)-isinϕ′cosϕ′⊗|-⟩⊗(N-1)}.

The single-qubit dephasing is represented by Kraus operators K0=diag(1,1-λ),K1=diag(0,λ). They act on a qubit i as K0(i)=I⊗⋯K0⋯⊗I and K1(i)=I⊗⋯K1⋯⊗I. Under dephasing, the quantum state (6) explicitly gives32 ρk(ϕ)=K0(k)ρ(k-1)(ϕ)[K0(k)]†+K1(k)ρ(k-1)(ϕ)[K1(k)]†,

for k=1,⋯,N with ρ0(ϕ)=|G(ϕ)⟩⟨G(ϕ)|.

To simplify the calculation, we focus on the maximum noise probability, λ=1. We first calculate33 ρ1(ϕ)=K0(1)|G(ϕ)⟩⟨G(ϕ)|K0(1)+K1(1)|G(ϕ)⟩⟨G(ϕ)|K1(1)=12{cos2ϕ′00sin2ϕ′⊗(|+⟩⟨+|)⊗(N-1)+e-2iϕ′(N-1)i2sin(2ϕ′)σz⊗(|+⟩⟨-|)⊗(N-1)+e2iϕ′(N-1)-i2sin(2ϕ′)σz⊗(|-⟩⟨+|)⊗(N-1)+sin2ϕ′00cos2ϕ′⊗(|-⟩⟨-|)⊗(N-1)},

Continuously we calculate ρ2(ϕ) as:34 ρ2(ϕ)=K0(2)ρ1(ϕ)K0(2)+K1(2)ρ1(ϕ)K1(2)=14{cos2ϕ′00sin2ϕ′⊗I⊗(|+⟩⟨+|)⊗(N-2)+e-2iϕ′(N-1)i2sin(2ϕ′)σz⊗σz⊗(|+⟩⟨-|)⊗(N-2)+e2iϕ′(N-1)-i2sin(2ϕ′)σz⊗σz⊗(|-⟩⟨+|)⊗(N-2)+sin2ϕ′00cos2ϕ′⊗I⊗(|-⟩⟨-|)⊗(N-2)},

and so on. Finally, we get35 ρN(ϕ)=12N(I⊗N+sin(ϕ)sin[(N-1)ϕ]σz⊗N).

To calculate the QFI, we use Eq. (7) in the main text with ρN(ϕ)=∑kℓk|ℓk⟩⟨ℓk|, and get36 Q=2∑i,j,ℓi+ℓj≠0|⟨ℓi|∂ϕρN(ϕ)|ℓj⟩|2ℓi+ℓj=(N-1)2,

where we first calculated ∂ϕρN(ϕ) from Eq. (35), and then used ϕ=π/2.

For GHZ state

We consider the case where the initial probe state is a GHZ state. For the generator Hϕ=Jx=12∑k=1Nσx(k), the defined GHZ state in Eq. (9) explicitly gives:37 |ψGHZ⟩=12(|+⟩⊗N+|-⟩⊗N),

where |±⟩⊗N are eigenstates of Hϕ corresponds to the maximum and minimum eigenvalues. This state can be prepared from a star-graph state by adding a Hadamard gate into the first qubits of Eq. (25).

We first compute two terms Hϕ|ψGHZ⟩ and Hϕ2|ψGHZ⟩, where38 Hϕ|ψGHZ⟩=[12∑k=1Nσx(k)]12(|+⟩⊗N+|-⟩⊗N)=N22(|+⟩⊗N-|-⟩⊗N),

and39 Hϕ2|ψGHZ⟩=N24|ψGHZ⟩.

Then, we obtain ⟨ψGHZHϕ|ψGHZ⟩=0 and ⟨ψGHZ|Hϕ2|ψGHZ⟩=N2/4. Finally, the QFIM yields40 QGHZ=4[⟨ψGHZ|Hϕ2|ψGHZ⟩-(⟨ψGHZ|Hϕ|ψGHZ⟩)2]=N2.

Similarly, for noisy cases, we have QGHZ=N2.

Deriving QFIM for multiparameter estimation

The QFIM for a pure star graph state is given by41 Qαβ=12⟨G(ϕ)|(LαLβ+LβLα)|G(ϕ)⟩,

where L is given in the symmetric logarithmic derivative (SLD) as42 Lα=2(|∂ϕαG(ϕ)⟩⟨G(ϕ)|+|G(ϕ)⟩⟨∂ϕαG(ϕ)|).

For concreteness, we first derive43 |∂ϕαG(ϕ)⟩=∂ϕαU(ϕ)|G⟩=∂ϕαe-itH|G⟩=-i∫0tdue-i(1-u)H[∂ϕαH]e-iuH|G⟩=-ie-iH∫0tdueiuHJαe-iuH|G⟩=-iU(ϕ)Aα|G⟩,

where H is given in Eq. (2), and44 Aα=∫0tdueiuHJαe-iuH,

is a Hermitian operator45,61.

Then, the SLD (42) and QFIM (41) are explicitly given as45 Lα=2iU(ϕ)[|G⟩⟨G|,Aα]U†(ϕ),

46 Qαβ=4Re[⟨G|AαAβ|G⟩-⟨G|Aα|G⟩⟨G|Aβ|G⟩].

In quantum circuits, the QFIM can be calculated using a stochastic method60.

For a general mixed state, such as a star graph under noise, i.e., ρ(ϕ,γ)=∑kℓk|ℓk⟩⟨ℓk|, the QFIM gives47 Qαβ=2∑i,j,ℓi+ℓj≠0⟨ℓi|∂ϕαρ(ϕ,γ)|ℓj⟩⟨ℓj|∂ϕβρ(ϕ,γ)|ℓi⟩ℓi+ℓj.

The QCRB in this case is given by CQ=Tr[Q-1].Figure 7 QFI under different noises.

Noisy QFI of graph states

In this section, we examine how different types of noise impact the QFI. In addition to dephasing, we also encounter bit flip, phase flip, and depolarizing. The relevant Kraus operators are as follows: K0=1-γ1001;K1=γ0110,for bit flip, K0=1-γ1001;K1=γ100-1,for phase flip, K0=1-γ1001;K1=γ/30110;K2=γ/30-ii0;K3=γ/3100-1,for depolarizing, where γ∈[0,1] the noise probability. The QFI is shown in Fig. 7, with different noises.

Bayes inference

We start from the final state48 ρN(ϕ)=12N(I⊗N+sin(ϕ)sin[(N-1)ϕ]σz⊗N),

when λ=1. Because σz⊗N has the same numbers of component +1 and -1, the state would contain the same number of components 1+sin(ϕ)sin[(N-1)ϕ] and 1-sin(ϕ)sin[(N-1)ϕ] on the diagonal. Therefore49 P(D|ϕ)=∏k=12NTr[ρN(ϕ)|k⟩⟨k|]=12N·(2N)[(1+sin(ϕ)sin[(N-1)ϕ])(N-1)(1-sin(ϕ)sin[(N-1)ϕ])(N-1)]=12N·(2N)[1-sin2(ϕ)sin2[(N-1)ϕ]](N-1)=12N·(2N)∑k=02N-12N-1k(-1)ksin2k(ϕ)sin2k[(N-1)ϕ],

And then P(ϕ|D) is50 P(ϕ|D)=∫P(D|ϕ)dϕ=12N·(2N)∫∑k=02N-12N-1k(-1)ksin2k(ϕ)sin2k[(N-1)ϕ]dϕ=12N·(2N)∑k=02N-12N-1k(-1)k∫sin2k(ϕ)sin2k[(N-1)ϕ]dϕ.

Finally, we obtain51 ϕestimated=∫ϕP(ϕ|D)dϕ.

The integrals in Eqs. (50) and (51) are derived numerically.

Conclusion

This study focuses on precise measurement techniques in noisy quantum systems. By using graph-state resources, we developed a method that enhances measurement accuracy and resilience against noise. Particularly, we used graph-state resources for robust quantum magnetometry under noise and showed promise for overcoming practical measurement challenges. Our demonstrations highlight significant advancements in accurately measuring both single and multiple Larmor frequencies. These outcomes showcase a spectrum ranging from surpassing the standard quantum limit to achieving Heisenberg scaling, marking significant progress.

These advancements are crucial across quantum computing, communication, and sensing, where precise measurements are indispensable. The capability to achieve high-precision measurements despite noise underscores the reliability and efficiency of quantum technologies. Further research in graph state-based quantum metrology could revolutionize various fields and pave the way for practical quantum technologies despite noise. We remark that with advancements in experimental generation of arbitrary photonic graph states via atomic62 sources, and quantum-error-correcting code based on graph states63, this line of research will provide a theoretical basis for quantum-metrological advantages through experiments with graph states.

Acknowledgements

This work is supported by JSPS KAKENHI Grant Number 23K13025.

Author contributions

P.T.N and T.K.L. wrote the initial code and implemented the numerical simulation. L.B.H. derived the theoretical framework, implemented the numerical simulation, and analyzed the results. L.B.H. and H.Q.N supervised the work. All authors discussed and wrote the manuscript.

Data availability

Data are available from the corresponding authors upon reasonable request.

Code availability

All codes used to produce the findings of this study are incorporated into tqix64,65 and available at: https://github.com/echkon/tqix-developers.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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