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A stochastic evaluation of quantum Fisher information matrix with generic Hamiltonians
EPJ Quantum Technology volumeÂ 10, ArticleÂ number:Â 37 (2023)
Abstract
Quantum Fisher information matrix (QFIM) is a fundamental quantity in quantum physics, which closely links to diverse fields such as quantum metrology, phase transitions, entanglement witness, and quantum speed limit. It is crucial in quantum parameter estimation, central to the ultimate CramÃ©rRao bound. Recently, the evaluation of QFIM using quantum circuit algorithms has been proposed for systems with multiplicative parameters Hamiltonian. However, systems with generic Hamiltonians still lack these proposed schemes. This work introduces a quantumcircuitbased approach for evaluating QFIM with generic Hamiltonians. We present a timedependent stochastic parametershift rule for the derivatives of evolved quantum states, whereby the QFIM can be obtained. The scheme can be executed in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we demonstrate the consistency between the results obtained from the stochastic parametershift rule and the exact results, while the results obtained from a standard parametershift rule slightly deviate from the exact ones. Our work sheds new light on studying QFIM with generic Hamiltonians using quantum circuit algorithms.
1 Introduction
The objective of quantum metrology is that using nonclassical quantum resources to enhance the precision in the estimation of unknown parameters [1, 2], including entanglement [3â€“9] and squeezing states [10â€“12]. Its cornerstone is the quantum estimation theory, which imposes the lower bound of precision by the quantum CramÃ©rRao inequality [13]. The bound is associated with quantum Fisher information (QFI) for singleparameter estimation and quantum Fisher information matrix (QFIM) for multiparameter estimation. Beyond the estimation theory, QFI and QFIM also connect to various aspects of quantum physics, including quantum phase transitions, entanglement witness, and the FubiniStudy metric, making them being fundamental quantities with broad applications (See Ref. [14] and references therein). Therefore, the evaluation of QFI and QFIM is crucial for studying these concepts.
Numerous studies on QFI and QFIM mainly focus on multiplicative parameters of Hamiltonians, e.g., a parameter Î¸ in a Hamiltonian Î¸H [15]. However, recent attention was raised to generic parameters of Hamiltonians, such as quantum magnetometry [7, 15, 16], unitary parametrization process [17, 18], and timedependent Hamiltonians [19, 20]. While the estimation with generic Hamiltonians shares some typical properties with the multiplicative case, it likewise indicates other distinct features, such as getting high efficiency with time scaling [15] and quantum control [19, 20]. The study of QFI/QFIM in these generic cases will open a broad range of potential applications in quantum metrology, quantum computing, and others.
On the other side, quantum computers can outperform classical ones and open significant quantum advantages for exponentially speeding up various computational tasks [21, 22]. Specifically, using Noisy IntermediateScale Quantum computers [23] resulted in the brilliant growth of different quantum algorithms (see Refs. [24, 25].) Among them, variational quantum algorithms [24] are the most promising approach for improving the efficiency in noisy and fewqubits devices. These algorithms include variational quantum eigensolvers [26â€“28], quantum approximate optimization algorithms [29], new frontiers in quantum foundations [30â€“32], and so on.
Besides, many computational tools based on variational quantum circuits were developed, including the standard parametershirt rules (Stand.PSR) [33, 34] and quantum natural gradient [35]. The Stand.PSR allows us to get the exact partial derivatives of any function by calculating it with different shifted parameters in the circuits. However, it only applies to cases where the gateâ€™s generators commute. Otherwise, to apply the Stand.PSR, additional treatments are required, such as Hamiltonian simulation techniques [36]. Recently, Banchi and Crooks in their seminal work, have developed a stochastic parametershift rule (Stoc.PSR) for general quantum evolutions, which relies on the stochastic repetitions of quantum measurement [37].
So far, different variational quantum algorithms for quantum metrology were developed, which open a new way to achieve quantumenhanced precision [31, 32, 38, 39]. Moreover, the Stand.PSR was widely used in various aspects, including finding the QFI with multiplicative Hamiltonians [40, 41]. However, it is lacking in the study of generic Hamiltonians. In reality, many systems are governed by generic Hamiltonians. Therefore, studying these cases using quantum algorithms is urgent.
This paper introduces a general timedependent Stoc.PSR and applies it to evaluate QFI/QFIM. We utilize the proposed Stoc.PSR for the derivatives of evolved quantum states, then compute the QFI/QFIM and examine the estimation precision in quantum metrology. Our scheme can execute in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we show an excellent agreement between the results obtained from the Stoc.PSR and the exact results while the Stand.PSRâ€™s results deviate from the exact values. This observation suggests the significance of the Stoc.PSR for studying QFI/QFIM with generic Hamiltonians and its applicability to variational quantum metrology. Furthermore, we extend our approach to examine the precision in manybody Hamiltonian tomography, such as estimating unknown coupling constants in the Hamiltonian.
2 Results and discussion
2.1 Quantum Fisher information for generic Hamiltonians
Estimation is a measurement process that uses a probe to extract information from an interesting system with d unknown parameters in a field \(\boldsymbol{B} = \phi _{1}\boldsymbol{e}_{1} + \cdots + \phi _{d}\boldsymbol{e}_{d}\), where \(\{\boldsymbol{e}_{j}\}\) are unit vectors in \(\{j\}\) directions. The probe interacts with the system through a generic Hamiltonian \(H(\boldsymbol{\phi} ) = \boldsymbol{B} \cdot \boldsymbol{H} = \sum_{j=1}^{d} \phi _{j}H_{j}\), where the \(\{H_{j}\}\) do not necessarily commute. The task of quantum parameters estimation is to evaluate these unknown coefficients by measuring the probe.
Let \(\rho _{0}\) be the initial probe state, it evolves to \(\rho (\boldsymbol{\phi} ) = U(\boldsymbol{\phi} )\rho _{0}U^{\dagger}(\boldsymbol{\phi} )\) after the interaction, where \(U(\boldsymbol{\phi} ) = e^{itH(\boldsymbol{\phi} )}\) is the unitary evolution during the interaction time t. Note that \(H(\boldsymbol{\phi} )\) is a general Hamiltonian, therefore \(U(\boldsymbol{\phi} )\) cannot be expanded in terms of multiplicative. By measuring the probe state in a general basis set, such as the positive operatorvalued measure (POVM) \({E_{x}}\) for the outcome x, one can obtain the corresponding probability distribution \(p(x\boldsymbol{\phi} ) = \operatorname{tr} [\rho (\boldsymbol{\phi} )E_{x} ]\), which can be used to estimate the unknown parametersÂ Ï•.
In the estimation theory, different estimators can be used to obtain the estimated value \(\boldsymbol{\check{\phi}}(x)\) of the unknown parameters Ï•, each yielding different precisions. The precision is characterized by the covariance matrix \(C(\boldsymbol{\phi} ) = {\mathrm{E}} [(\boldsymbol{\phi}  {\mathrm{E}}[\boldsymbol{\check{\phi}}(x)]) ( \boldsymbol{\phi}  {\mathrm{E}}[\boldsymbol{\check{\phi}}(x)])^{\intercal} ]\) [8], where \(\mathrm{E}[\boldsymbol{\check{X}}]=\int p(x\boldsymbol{X}) \boldsymbol{\check{X}}(x) \,\mathrm{d}x\) is the expectation value of the estimator \(\boldsymbol{\check{X}}(x)\). The diagonal term \(C_{k,k} \equiv \Delta ^{2}\phi _{k} = {\mathrm{E}}[\phi _{k}^{2}]{\mathrm{E}}^{2}[ \phi _{k}]\) is the variance for estimating \(\phi _{k}\), and the offdiagonal term \(C_{k,l}\) is the covariance between \(\phi _{k}\) and \(\phi _{l}\). An estimator is unbiased when \(\mathrm{E}[\check{\phi}_{k}(x)] = \phi _{k}\), \(\forall k \in \{ 1,\ldots ,d\}\). The precision obeys classical and quantum CramÃ©rRao bounds (CRBs) [13]
where M is the number of repeated measurements, \(F(\boldsymbol{\phi} )\) is the classical Fisher information matrix (CFIM) defined by
and the maximum over all possible measurements \(\{E_{x}\}\) yields the quantum Fisher information matrix (QFIM) \(Q(\boldsymbol{\phi} )\) with elements
where \(L_{k}\) is the symmetric logarithmic derivative (SLD) that obeys \(2\partial _{\phi _{k}}\rho (\boldsymbol{\phi} ) = L_{k}\rho (\boldsymbol{\phi} ) + \rho ( \boldsymbol{\phi} )L_{k}\) [13]. For a single parameter estimation (such as Ï•), the CRBs simplify to \(\Delta ^{2}\phi \ge 1/F(\phi ) \ge 1/Q(\phi )\), where \(F(\phi ) = \int p(x\phi ) [\partial _{\phi}\ln p(x\phi ) ]^{2}\,\mathrm{d}x\) and \(Q(\phi ) = \operatorname{tr} [L^{2} \rho (\phi ) ]\) are the classical and quantum Fisher information, respectively. Note that both CFIM and QFIM may depend on the parameters Ï• regardless of the unitary process.
The QFI and QFIM set ultimate bounds for the estimation precision of any estimator. Therefore, it is crucial to derive these QFI and QFIM for the estimation theory with generic Hamiltonians. Let us start with the derivative of the unitary evolution [16, 42]
where \(Y_{j} = \int _{0}^{t} e^{isH(\boldsymbol{\phi} )} [\partial _{\phi _{j}} H( \boldsymbol{\phi} ) ]e^{isH(\boldsymbol{\phi} )} \,\mathrm{d}s \) is a Hermitian operator [16]. Then, we obtain
The QFIM (3) straightforwardly yields
for \(\rho (\boldsymbol{\phi} ) = \sum_{\lambda }p_{\lambda}\lambda \rangle \langle \lambda \), and \(\partial _{\phi _{k}}\rho (\boldsymbol{\phi} )\) is given from Eq.Â (5). For pure quantum states, i.e., \(\rho _{0} = \psi _{0} \rangle \langle \psi _{0}\), the QFIM is defined by [13]
where \(\psi (\boldsymbol{\phi} )\rangle = U(\boldsymbol{\phi} )\psi _{0}\rangle \) is the evolved probe state. Substituting Eq.Â (4) into Eq.Â (7), it yields [7, 16]
Computing QFI and QFIM requires the derivatives of the probe state, i.e., \(\partial _{\phi _{j}}\rho (\boldsymbol{\phi} )\), \(\forall j\in \{1,\ldots ,d\}\). Hereafter, we introduce a stochastic parametershift rule (Stoc.PSR) to compute these derivatives on quantum circuits, allowing for precision evaluation in different quantum computing platforms.
2.2 Stochastic parametershift rule
In this section, we present a timedependent stochastic parametershift rule (Stoc.PSR) for evaluating QFI/QFIM with generic Hamiltonians, where we particularly calculate \(\partial _{\phi _{j}}\rho (\boldsymbol{\phi} )\) using quantum circuits. This method is thus helpful for studying different variational quantum algorithms [24], including variational quantum metrology [31, 32, 39] and evaluating FubiniStudy metric tensor in quantum natural gradient [35].
We first recast Eq.Â (5) in the following form
where \(O_{j} = e^{isH(\boldsymbol{\phi} )} [\partial _{\phi _{j}} H(\boldsymbol{\phi} ) ]e^{isH(\boldsymbol{\phi} )} \). Referring to [37] and using the BakerCampbellHausdorff formula [43] for \(H_{j}^{2} = I\), we derive
for all \(t\mu \notin \frac{\pi}{2} \mathbb{Z}\). See Methods section for the detailed proof. Recall that Ref. [37] fixes \(\mu = \pi /4\) and \(t = 1\). Here, we consider any time t and introduce Î¼ as an arbitrary parameter shift, which makes our scheme more general, especially in timedependent and noisy metrology. For \(t\mu = \pi /4\), it reduces to Ref. [37] and maximizes the accuracy for parametershift approaches (se also Ref. [44].) Finally, using \(e^{it\mu O_{j}} = e^{isH(\boldsymbol{\phi} )} e^{it\mu [\partial _{\phi _{j}} H(\boldsymbol{\phi} )]} e^{isH(\boldsymbol{\phi} )} \) [42], and substituting Eq.Â (10) into Eq.Â (9), we obtain (see the Methods section):
which is the timedependent stochastic parametershift rule (Stoc.PSR), where
The algorithm for timedependent Stoc.PSR is described in Algorithm 1, which is an extended version of the original (without timedependent) in Ref. [37]. FigureÂ 1 depicts a quantum circuit for the Stoc.PSR. To obtain \(\partial _{\phi _{j}}\rho (\boldsymbol{\phi} )\) for a given time t, we perform the following steps: (s1) generate a random number s from a normal distribution within the interval \([0,t]\); (s2) initialize the circuit with \(\rho _{0}\); (s3) apply the quantum gates \(e^{isH(\boldsymbol{\phi} )}\), \(e^{it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\), and \(e^{i(ts)H(\boldsymbol{\phi} )}\); (s4) extract the final state \(\rho ^{+}\) from the circuit; (s5) repeat steps s2s4, replacing \(e^{it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\) with \(e^{it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\), and assign the quantum state to \(\rho ^{}\); (s6) repeat steps s1s5 N times and compute the derivative via \(\frac{t}{N*\sin (2t\mu )}\sum_{n=1}^{N}(\rho ^{+}\rho ^{})\). The term \(t/N\) comes from MonteCarlo sampling, i.e., \(\int _{a}^{b} f(x) \,dx \approx \frac{ba}{N}\sum_{i=1}^{N}f(x_{i})\). Apply the procedure for all \(j\in {1,\ldots , d}\) and use Eqs.Â (6)(7) we can compute the QFIM. Finally, we repeat the scheme for other time instances.
Note that the scheme can be implemented in universal quantum computers. Assuming a programmable quantum computer that can execute a family of native quantum gates \(U(t,\boldsymbol{\phi} ) = e^{itH(\boldsymbol{\phi} )}\), where \(H(\boldsymbol{\phi} ) = \sum_{j}\phi _{j}H_{j}\), the evolution terms \(e^{i(ts)H(\boldsymbol{\phi} )}\) and \(e^{isH(\boldsymbol{\phi} )}\) in step 3 can be implemented by using the quantum gates \(U(ts,\boldsymbol{\phi} )\) and \(U(s,\boldsymbol{\phi} )\), respectively. The remaining term \(e^{it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\) in step 3 yields \(e^{it\mu H_{j}}\), which can be implemented by the quantum gate \(U (t\mu ,\boldsymbol{e}_{j} )\), where \(\boldsymbol{e}_{j}\) is a unit vector with 1 at the \(j^{\mathrm{th}}\) element and zeros for the others. Therefore, all the evolution terms can be implemented by the device. The density states \(\rho ^{+}\) and \(\rho ^{}\) can be extracted and subtracted from each other using classical computers or quantum subtraction technology in real hardware, e.g., see Ref. [45].
So far, the accuracy of an approach (such as finite difference, Stand.PSR, and Stoc.PSR) is determined by its variance, which is a statistical error raising from a finite number of measurements. The variance of the Stoc. PSR is comparable with that of the Stand. PSR when an infinite number of measurements are taken. [37]
Finally, the algorithmâ€™s complexity is calculated using the Big O notation (\(\mathcal{O}\)). Following Ref. [46], the complexity of the evolution gate \(e^{it H(\phi )}\) is \(\mathcal{O}(N^{3})\). When considering the set of three evolution terms shown in Fig.Â 1, the total complexity is \(\mathcal{O}(3N^{3})\), which can be simplified to \(\mathcal{O}(N^{3})\) by removing the constant term.
2.3 Applications
To demonstrate advantaged features of the Stoc.PSR method for evaluating QFIM, we scrutinize quantum metrology in two cases of single and multiple magnetic fields. We further discuss its application to Hamiltonian tomography in manybody systems, which involves determining unknown coupling constants in the Hamiltonian.
2.3.1 Single parameter estimation
Let us consider a magnetic field \(\boldsymbol{B} = \cos (\phi )\boldsymbol{e}_{x} + \sin (\phi )\boldsymbol{e}_{z}\), and our goal is to estimate the angle Ï• between the fieldâ€™s direction and the z axis [15]. The field interacts with an exposed qubit probe and imprints its information into the probe via the interaction Hamiltonian
where \(\boldsymbol{\sigma} = (\sigma _{x}, \sigma _{y}, \sigma _{z})\) are the Pauli matrices. The unitary evolution is given by \(U(t,\phi ) = e^{itH(\phi )}\). Applying this transformation, an initial probe state, i.e., \(\psi _{0}\rangle = (0\rangle + 1\rangle )/\sqrt{2}\) evolves to \(\psi (\phi )\rangle = U(t,\phi )\psi _{0}\rangle \). The evolved probe state \(\psi (\phi )\rangle \) provides the best quantum strategy for the estimation of Ï•, which can be evaluated via the QFI, similar to Eq.Â (8)
where \(Y_{\phi }= \int _{0}^{t} e^{isH(\phi )} [\partial _{\phi} H( \phi ) ]e^{isH(\phi )} \,\mathrm{d}s \) (see detailed in the Methods section). The QFI \(Q(\phi )\) is timedependent and achieves a maximum value of 4 at \(t = \pi /2\), as shown by the solid curves in Fig.Â 2. This behavior is caused by the rotation of the probe state under magnetic field. Furthermore, the QFI depends on the true parameter value, it thus becomes a function of Ï•. In the limit \(\phi \to 0\), the QFI yields \(Q(\phi ) = Q_{\mathrm{max}} = 4\sin ^{2}(t)\) [15].
We now apply the Stoc.PSR to a singlequbit quantum circuit. The circuit is initially prepared in \(0\rangle \), and it becomes \(\psi _{0}\rangle \) after applying a Hadamard gate. Using the definition \(\partial _{\phi }\psi (\phi )\rangle = [\partial _{\phi }U(t, \phi ) ]\psi _{0}\rangle \), and the first line in Eq.Â (4), we have
where \(O_{\phi }= e^{isH(\phi )}[\partial _{\phi }H(\phi )] e^{isH(\phi )}\). Similar as above, we have
where \(t\mu \notin \pi \mathbb{Z}\). Using \(e^{it\mu O_{\phi}} = e^{isH(\phi )}e^{it\mu [\partial _{\phi }H( \phi )]} e^{isH(\phi )}\), we derive Eq.Â (16) as
where \(\psi ^{\pm}\rangle \) are given by
In the numerical calculation, we derive \(\partial _{\phi}\psi (\phi )\rangle =\frac{t}{N*2\sin (t\mu )} \sum_{n=1}^{N} [\psi ^{+}\rangle  \psi ^{}\rangle ]\) with N samplings of \(s\in [0,t]\). This is a simplified version of Algorithm 1 for pure states. We set \(N = 1000\) and obtain the QFI \(Q(\phi )\) which is of the form (7)
where \(\Psi \rangle = \sum_{n=1}^{N} [\psi ^{+}\rangle \psi ^{} \rangle ]\).
To implement the Stoc.PSR in quantum computers, we assume there exists a universal quantum hardware that allows for executing the quantum gate \(U(t,\phi )\). Changing the variables in \(U(t,\phi )\) by \(U(x,z) = e^{it(x\sigma _{x} + z\sigma _{z})}\) where \(x = \cos (\phi )\) and \(z = \sin (\phi )\), it yields \(\partial _{\phi }U(x,z) = \partial _{x} U(x,z)\partial _{\phi }x + \partial _{z} U(x,z)\partial _{\phi }z\). This is a universal quantum device because all the evolution terms in Eq.Â (19) can be implemented via this quantum gate in the device.
Finally, let us compare the results with the Stand.PSR. To apply the Stand.PSR, we first decompose the evolution \(U(t,\phi )\) into a sequence of subevolutions through TrotterSuzuki transformation [47]
where these subevolutions can be executed in quantum circuits through rotation gates, specifically Rx and Rz. The derivative \(\partial _{\phi}\psi (\phi )\rangle \) now can be implemented by using the Stand.PSR. See detailed calculation in the Method section.
FigureÂ 2 shows a comparison between the performance of Stand.PSR and Stoc.PSR with the exact theoretical result. The Stoc.PSR consistently demonstrates a good agreement with the exact results all the time while the Stand.PSR deviates from the exact results as time increases. It implies that using Stoc.PSR in quantum circuits for studying quantum systems with generic Hamiltonian is essential and cannot be replaced by similar approximation methods. This is further supported by considering the meansquare error (MSE), defined as \((1/M)\sum_{i} [y_{i}(t)  f(t)]^{2}\), where M denotes the number of data points, \(y_{i}(t)\) represents the data obtained using the Stand.PSR or Stoc.PSR and \(f(t)\) represents the exact results given by Eq.Â (15). We emphasize that the MSE here plays no role with the error of the estimated parameter, it is rather a systematic error caused by different methods when comparing with the exact theoretical result. The MSEs are shown in the figure as the error bars and error areas. As we can see, the MSE for Stoc.PSR remains small throughout the duration, while that one for the Stand.PSR divers for large sensing time t.
2.3.2 Multiple parameters estimation
Next, we apply the Stoc.PSR scheme to estimate the components of a magnetic field pointing in an arbitrary direction. Consider the probe state initially prepared in nqubit GHZ state \(\psi _{0}\rangle = (00\cdots 0\rangle + 11\cdots 1\rangle )/\sqrt{2}\), such that allows for obtaining the maximum QFIM [6]. The interaction Hamiltonian is given by
where \(\boldsymbol{\phi} = (\phi _{x}, \phi _{y}, \phi _{z})\) are three components of the given magnetic field that we want to estimate, and \(J_{j} = \sum_{k=1}^{n}\sigma _{j}^{(k)}\) is a collective Pauli matrix. Potential platforms for the probe include spin1/2 ensemble semiconductors, ions traps, NMR systems, and NV centers. In these systems, such as spin1/2 ensemble, \(J_{j}\) becomes the collective angular momentum operator [7].
The QFIM can be obtained theoretically from Eq.Â (8), and the total variance yields \(\Delta ^{2}\boldsymbol{\phi} = \operatorname{tr}[Q^{1}]\). Concretely, with \(n = 3\) qubits and \(\phi _{x} = \phi _{y} = \phi _{z} = \varphi \), we obtain
We show the exact theoretical results by the solid curves for various Ï† in Fig.Â 3a. For each Ï†, there is a minimum variance at a certain time t, which is caused by the rotation of the probe state under magnetic field. In the limit of small phase, i.e., \(\varphi \to 0\), the total variance is \(\operatorname{tr}[Q^{1}] = \frac{7}{108t^{2}}\), which results in the minimum of total variance.
In the Stoc.PSR method, we model the probe in an nqubit quantum circuit initially prepared in the GHZ state. The circuit can be implemented in the existing noisy intermediatescale quantum computers [23]. Its state evolves under the transformation \(U(t,\boldsymbol{\phi} )= e^{itH(\boldsymbol{\phi} )}\), and results in the evolved state \(\psi ({\boldsymbol{\phi}})\rangle = U(t,\boldsymbol{\phi} )\psi _{0}\rangle \). As discussed above, this unitary evolution can be implemented in a universal quantum computer. Therefore, we employ the Stoc.PSR using Algorithm 1 to obtain \(\partial _{\phi _{j}}\psi (\boldsymbol{\phi} )\rangle \) for all j and get the QFIM as in Eq.Â (7). The \(\operatorname{tr}[Q^{1}]\) is shown in Fig.Â 3a (dotted curves) for the number of sampling \(N = 1000\). The Stoc.PSRâ€™s results agree with the exact results.
We further apply the scheme to noisy cases, where the probe is described by mixed states. We consider timedependent dephasing, which is given by a quantum channel \(\mathcal{E}\) that acts on a single qubit as
where we used the Kraus representation for the dephasing channel [31]
The timedependent probability is \(p(t) = e^{\gamma t}\) for the Markovian noise, where Î³ is the decay rate [31].
We apply the quantum channel \(\mathcal{E}\) to all qubits in the probe during the interaction time and use Algorithm 1 to derive the QFIM. The results for the total variance versus the interaction time t are shown in Fig.Â 3b. We plot the results for several decay rates Î³ and compare the Stoc.PSR approach with the theoretical analysis. Again, they match excellently.
2.3.3 Hamiltonian tomography
We additionally discuss the application to Hamiltonian tomography in manybody systems, which involves determining unknown coupling constants in the Hamiltonian. Hamiltonian tomography aims to reconstruct a generic manybody Hamiltonian by measuring multiple pairs of the initial and timeevolving states. It is a challenging task due to the complexity of the manybody dynamics. So far, the progress is limited to particular Hamiltonians and smallsize systems [48â€“51]. For example, a simple task is to identify the Hamiltonian in an Ising model of a spin1/2 chain placed under an external field. AÂ generic Hamiltonian is given by \(H = \sum_{j} c_{j,j+1}\sigma _{z}^{(j)} \sigma _{z}^{(j+1)} + \sum_{j} h_{j} \sigma _{x}^{(j)} \), where the coupling constants \(\{c_{j,j+1}\}\) and the external field strengths \(\{h_{j}\}\) are unknown factors, j stands for the site \(j^{\mathrm{th}}\) in the chain.
Recently, Li et al. introduced a quantum quench approach for the Hamiltonian tomography that can apply to both analog and digital quantum simulators [52]. Hereafter, we evaluate the quantum quench precision by using Stoc.PSR to calculate the classical CramÃ©rRao bound.
A generic Hamiltonian of a manybody system can be decomposed into dinteraction terms as
where \(\{x_{j}\}\) are unknown coupling constants that need to be determined, and \(\{H_{j}\}\) are Hermitian operators. An initial state \(\rho _{0}\) evolves to \(\rho (\boldsymbol{x}) = e^{iHt}\rho _{0}e^{iHt}\) after time t, for \(\boldsymbol{x} = (x_{1}, \ldots , x_{d} )^{\intercal}\). The system obeys a conservation law [52]
for every pair of given \(\rho _{0}\) and \(\rho (\boldsymbol{x})\). To determine d coefficients \(\{x_{j}\}\), we need to solve at least \(p \ge d1\) linear equations which form a matrix equation as \({\boldsymbol{X}} \boldsymbol{x} = \boldsymbol{0}\), where X is a \(p\times d\) matrix with the elements
where \(k\in \{1,\ldots , p\}\) and \(l\in \{1, \ldots ,d\}\) for different pairs of \(\rho _{0}^{(k)}\), \(\rho ^{(k)}(\boldsymbol{x})\). Here, \(\{\rho _{0}^{(k)} \}\) is a set of (random) initial states and \(\{\rho ^{(k)}(\boldsymbol{x}) \}\) is a set of evolved states.
For \(\{H_{j}\}\) are measured observables, such as Pauli matrices, SICPOVM, and polarization bases [53], the matrix elements \(\{X_{k,l}\}\) become measured probabilities under the eigenbases of these observables. Thus, to evaluate the best estimation of \(\{x_{j}\}\), we examine the classical bound, i.e., via the CFIM Eq.Â (2). Firstly, from Eq.Â (28), we derive
where \(\frac{\partial \rho ^{(k)}(\boldsymbol{x})}{ \partial x_{j}}\) is given by Stoc.PSR Eq.Â (11). We later define the CFIM as
and hence obtain the classical CramÃ©rRao bound, i.e., \(\Delta ^{2}\boldsymbol{x} \ge \operatorname{tr}[F^{1}]\). The equality can be achieved by an appropriate estimator.
For numerical demonstration, we consider a singlequbit system whose Hamiltonian is given by [53]
where \(\{x_{i}\}\) are unknown coefficients, \(\psi _{1}\rangle = 0\rangle , \psi _{2}\rangle = (0\rangle + 1 \rangle )/\sqrt{2}\), and \(\psi _{3}\rangle = (0\rangle + i1\rangle )/\sqrt{2}\). We apply the quantum quench method to find \(\{x_{i}\}\) and analyze the variance \(\Delta ^{2}\boldsymbol{x}\). It is given by the classical CramÃ©rRao bound, i.e., \(\Delta ^{2}\boldsymbol{x} = \operatorname{tr}[F^{1}]\). The derivative \(\frac{\partial \rho ^{(k)}(\boldsymbol{x})}{ \partial x_{j}}\) in Eq.Â (29) is given by either Stoc.PSR or finite difference approach for comparison. For the Stoc.PSR, we run 1000 random samples of \(s \in [0,1]\), \(t = 1\), \(\mu = \pi /4\). For the finite difference method, we use \(\partial _{x}\rho (x) = \frac{ \rho (x+\epsilon )  \rho (x\epsilon )}{2\epsilon}\), where Ïµ is a small step size. The variance \(\Delta ^{2}\boldsymbol{x}\) is averaged after 10 repetitions.
The results are shown in Fig.Â 4 as functions of p. In principle, \(p = d1\) is sufficient to estimate d parameters. However, \(p > d1\) gives better statistical results [52]. In this context, we compare the Stoc.PSR and finite difference approach, and find that they tend to converge when increasing p. While the finite difference method consistently produces better results, it also has a larger bias due to the computational challenges of computing \(\rho (x+\epsilon )\) and \(\rho (x\epsilon )\) in quantum circuits for \(\epsilon \ll 1\). Furthermore, since p is equivalent with the number of repeated measurements, we can define the standard quantum limit (SQL) as \(\propto 1/p\) and Heisenberg limit (HL) by \(\propto 1/p^{2}\). We compare the bound in quantum quench with these limits and find that it scales slightly worse than the SQL, opening further exploring to improve the limit in quantum quench approaches.
3 Conclusions
We proposed a timedependent stochastic parametershift rule (Stoc.PSR) framework for deriving the differential in studying quantum Fisher information matrix with generic Hamiltonian generators. This method allows for obtaining the exact derivative using universal quantum circuits. Different from the standard parametershift rule (Stand.PSR), which particularly replies on commuting Hamiltonians, here, the Stoc. PSR applies to generic Hamiltonians.
Using the Stoc.PSR approach, the evaluation of QFI/QFIM can be practically executed in quantum circuits. In all cases, the Stoc.PSR results have agreed with the exact theoretical analysis, while the Stand.PSR results have deviated from the exact values. This framework can also be used for Hamiltonian tomography in manybody systems.
While variational quantum algorithms are extensively developing, our framework uses these advantages for the development of quantum metrology and quantum measurement with generic Hamiltonian and makes them reliable for studying in the area of quantum computers.
4 Methods
4.1 Proof of timedependent stochastic parametershift rule
We consider the derivative of a mixed quantum state as in Eq.Â (9) in the main text
where \(O_{j} = e^{isH(\boldsymbol{\phi} )} [\partial _{\phi _{j}} H(\boldsymbol{\phi} ) ]e^{isH(\boldsymbol{\phi} )} \). We have
Proof
Using the BakerCampbellHausdorff formula [43], we derive
and
Subtracting Eq.Â (35) from Eq.Â (34) yields
where using the algebraic expansion with the condition \(O_{j}^{2} = \boldsymbol{I}\), we have \([O_{j}, [O_{j}, [O_{j},\rho _{0} ] ] ] = \frac {2^{3}}{2} [O_{j},\rho _{0} ]\), and so on. Finally, Eq.Â (36) becomes
Multiplying two sides of Eq.Â (37) by \(\frac{i}{\sin (2t\mu )}\) we arrive at Eq.Â (33)
Now, substituting Eq.Â (33) into Eq.Â (32), we have
Using \(e^{it\mu O_{j}} = e^{isH(\boldsymbol{\phi} )} e^{it\mu [\partial _{ \phi _{j}} H(\boldsymbol{\phi} ) ]} e^{isH(\boldsymbol{\phi} )} \) and \(U(\boldsymbol{\phi} ) = e^{itH(\boldsymbol{\phi} )}\), we set
Substituting Eq.Â (39) into Eq.Â (38), we obtain
where we used \(\rho _{j}^{\pm}(\boldsymbol{\phi} ,s) = U_{j}^{\pm}(\boldsymbol{\phi} ,s)\rho _{0} [U_{j}^{ \pm}(\boldsymbol{\phi} ) ]^{\dagger} \).â€ƒâ–¡
4.2 Theoretical analysis for singleparameter estimation
Firstly, let us discuss the exact calculation method for quantum Fisher information in single parameter estimation. Starting from \(H(\phi ) = \cos (\phi )\sigma _{x} + \sin (\phi )\sigma _{z}\), we derive \(\partial _{\phi }H(\phi ) = \sin (\phi )\sigma _{x} + \cos (\phi ) \sigma _{z}\). Substituting it into \(Y_{j}\) for \(j = \phi \), we obtain
Finally, we derive the quantum Fisher information as in Eq.Â (7):
which results in Eq.Â (15).
4.3 TrotterSuzuki transformation and Stand.PSR
From now on, let us show the detailed calculation for the TrotterSuzuki transformation and Stand.PSR for single parameter estimation. From the evolution (21), we set
Then, Eq.Â (21) is recast as
and thus
Concretely, we have
Note that \(U(\pi ,0) = \lim_{m\to \infty}(i\sigma _{x})^{m}\). For \(m = 4k+1\ \forall k\in \mathbb{N}\), we have \(U(\pi ,0) = i\sigma _{x}\), from which the Pauli matrix \(\sigma _{x}\) can be implemented by the unitary (quantum gate) \(U(\pi ,0)\). Likewise, \(U(0,z+\pi ) = i\sigma _{z}\). Now, Eqs.Â (45), (46) are recast as
Here, m obeys the periodic property, therefore its choice will not affect the results. Hence, these derivatives (47), (48) can be obtained in quantum circuits by modifying the Stand.PRS. Substituting Eqs.Â (47), (48) and Eq.Â (42) into Eq.Â (44), we derive
where \(\psi _{0}\rangle \) is the initial probe state. In this form, the QFI is given as
The procedure for calculating the quantum Fisher information is shown in Algorithm 2.
4.4 Multiple parameters estimation
Hereafter, we derive the multiple parameters estimation. For \(n = 3\), we first calculate \(J_{j}\) for \(j = \{x, y, z\}\) as
where \(I_{2}\) is the \(2\times 2\) identity matrix. The Hamiltonian \(H(\boldsymbol{\phi} )\) is given by Eq.Â (22), and its derivative yields \(\partial _{\phi _{j}}H(\boldsymbol{\phi} ) = J_{j}\). Similar to the above, we derive \(Y_{j}\)
and obtain the quantum Fisher information matrix from Eq.Â (8).
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Abbreviations
 Stand.PSR:

Standard parametershift rule
 Stoc.PSR:

Stochastic parametershift rule
 QFI:

Quantum Fisher information
 QFIM:

Quantum Fisher information matrix
 CFI:

Classical Fisher information
 CFIM:

Classical Fisher information matrix
 POVM:

Positive operatorvalued measure
 CRB:

CramÃ©rRao bound
 SLD:

Symmetric logarithmic derivative
 MSE:

Meansquare error
 NMR:

Nuclear magnetic resonance
 NV:

Nitrogenvacancy
 SICPOVM:

Symmetric, informationally complete, positive operatorvalued measure
 SQL:

Standard quantum limit
 HL:

Heisenberg limit
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This work is supported by JSPS KAKENHI Grant Number 23K13025.
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Ho, L.B. A stochastic evaluation of quantum Fisher information matrix with generic Hamiltonians. EPJ Quantum Technol. 10, 37 (2023). https://doi.org/10.1140/epjqt/s4050702300195w
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DOI: https://doi.org/10.1140/epjqt/s4050702300195w