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Introduction.—Quantum metrology [1, 2] features a series of promising applications in the near future [3]. In the prototypical setting of quantum metrology, the goal is to estimate an unknown parameter carried by a quantum channel, given $N$ queries to it. A pivotal task is to design a strategy that utilizes these $N$ queries to generate a quantum state with as much information about the unknown parameter as possible. This often involves, for example, preparing a suitable input probe state [4, 5, 6] and applying intermediate quantum control [7, 8, 9, 10] as well as quantum error correction [11, 12, 13, 14].

In reality, the implementation of strategies is subject to physical restrictions. In particular, within the noisy and intermediate-scale quantum (NISQ) era [15], we have to adjust the strategy to accommodate the limitations on the system. For example, for systems with short coherence time it might be favorable to adopt the parallel strategy [Fig. 1LABEL:sub@subfig:parallel_strategy], where multiple queries of the unknown channel are applied simultaneously on a multipartite entangled state [4]. When the system has longer coherence time and can be better controlled, one could choose to query the channel sequentially [Fig. 1LABEL:sub@subfig:sequential_strategy], which may potentially enhance the precision. In addition to the parallel and sequential strategies, it was recently discovered that the quantum SWITCH [16], a primitive where the order of making queries to the unknown channel is in a quantum superposition [Fig. 1LABEL:sub@subfig:quantum_switch_strategy], can be employed to generate new strategies of quantum metrology [17, 18, 19] that may even break the Heisenberg limit [19]. Moreover, indefinite causal structures beyond the quantum SWITCH [16, 20, 21] [Figs. 1LABEL:sub@subfig:causal_superposition_strategy and LABEL:sub@subfig:general_strategy] have recently been shown to further boost the performance of certain information processing tasks [22, 23]. The ultimate performance of these strategies in quantum metrology, however, remains unknown. This is mainly due to the lack of a systematic method that optimizes the probe state, the control and other degrees of freedom in a strategy in a unified fashion which leads to the ultimate precision limit.

In this work, we develop a semidefinite programming (SDP) method of evaluating the optimal precision of single-parameter quantum metrology for finite $N$ (which we call the nonasymptotic regime) over a family of admissible strategies. With this method, we show a strict hierarchy (see Fig. 2) of the optimal performances under different families of strategies, which include the parallel, the sequential, and the indefinite-causal-order [16, 20, 21] ones (see Fig. 1). In conjunction, we design an algorithm to obtain an optimal strategy achieving the highest precision. For the strategy set that admits a symmetric structure, we develop a method of reducing the complexity of our algorithms by an exponential factor.

Quantum Fisher information.—The uncertainty $\delta\hat{\phi}$ of estimating an unknown parameter $\phi$ encoded in a quantum state $\rho_{\phi}$, for any unbiased estimator $\hat{\phi}$, can be determined via the quantum Cramér-Rao bound (QCRB) as $\delta\hat{\phi}\geq 1/\sqrt{\nu J_{Q}(\rho_{\phi})}$[24, 25, 26], where $J_{Q}(\rho_{\phi})$ is the quantum Fisher information (QFI) of the state $\rho_{\phi}$ and $\nu$ is the number of repeated measurements [27]. For single-parameter estimation, the QCRB is achievable, and the QFI thus quantifies the amount of information that can be extracted from the quantum state. One way to compute the QFI is [28, 29]

where $\ket{\Psi_{\phi}}$ is the purification of $\rho_{\phi}$ with an ancillary space $\mathcal{H}_{A}$, $\Tr_{A}$ denotes the partial trace over $\mathcal{H}_{A}$, and $\dot{\Psi}:=\partial\Psi/\partial\phi$. When the parameter is carried by a quantum channel $\mathcal{E}_{\phi}$, i.e., a completely positive trace-preserving (CPTP) map, the channel QFI can be defined as the maximal QFI of output states using the optimal input assisted by arbitrary ancillae [28, 30, 31, 32]: $J_{Q}^{(\mathrm{chan})}(\mathcal{E}_{\phi})=\max_{\rho_{\mathrm{in}}\in\mathcal{S}(\mathcal{H}_{S}\otimes\mathcal{H}_{A})}J_{Q}[(\mathcal{E}_{\phi}\otimes\mathcal{I}_{A})(\rho_{\mathrm{in}})]$, where $\mathcal{S}(\mathcal{H})$ denotes the space of density operators on the Hilbert space $\mathcal{H}$, $\mathcal{H}_{S/A}$ denotes the Hilbert space of the system or ancillae, and $\mathcal{I}_{A}$ is the identity on $\mathcal{H}_{A}$.

We denote by $\mathcal{L}(\mathcal{H})$ the set of linear operators on the finite-dimensional Hilbert space $\mathcal{H}$, and $\mathcal{L}[\mathcal{L}(\mathcal{H}_{1}),\mathcal{L}(\mathcal{H}_{2})]$ denotes the set of linear maps from $\mathcal{L}(\mathcal{H}_{1})$ to $\mathcal{L}(\mathcal{H}_{2})$. By the Choi-Jamiołkowski (CJ) isomorphism, a parametrized quantum channel $\mathcal{E}_{\phi}\in\mathcal{L}[\mathcal{L}(\mathcal{H}_{2i-1}),\mathcal{L}(\mathcal{H}_{2i})]$ (for $1\leq i\leq N$) can be represented by a positive semidefinite operator (called the CJ operator) $E_{\phi}=\mathsf{Choi}(\mathcal{E}_{\phi})=\mathcal{E}_{\phi}\otimes\mathcal{I}(\lvert I\rrangle\llangle I\rvert)$, where $\lvert I\rrangle=\sum_{i}\ket{i}\ket{i}$. The CJ operator of $N$ identical quantum channels is $N_{\phi}=E_{\phi}^{\otimes N}\in\mathcal{L}\left(\otimes_{i=1}^{2N}\mathcal{H}_{i}\right)$.

Strategy set in quantum metrology.—A strategy is an arrangement of physical processes (the blue shaded area in Fig. 1) which, when concatenated with given queries to $\mathcal{E}_{\phi}$, generates an output quantum state carrying the information about $\phi$. A strategy can be described by a CJ operator on $\mathcal{L}\left(\mathcal{H}_{F}\otimes_{i=1}^{2N}\mathcal{H}_{i}\right)$, where $\mathcal{H}_{F}$ denotes the output Hilbert space of the concatenation, referred to as the global future space. The concatenation of two processes is characterized by the link product [33, 34] of two corresponding CJ operators $A\in\mathcal{L}\left(\otimes_{a\in\mathcal{A}}\mathcal{H}_{a}\right)$ and $B\in\mathcal{L}\left(\otimes_{b\in\mathcal{B}}\mathcal{H}_{b}\right)$ as

where $T_{i}$ denotes the partial transpose on $\mathcal{H}_{i}$, and $\mathcal{H}_{\mathcal{A}/\mathcal{B}}$ denotes $\otimes_{i\in\mathcal{A}/\mathcal{B}}\mathcal{H}_{i}$. The output state lies in the global future $F$, which should not affect any state in the past.

Following the above formalism, given a sufficiently large ancillary Hilbert space, a strategy set determined by the relevant causal constraints is described by a subset $\mathsf{P}$ of

Here, without loss of generality [35], we have restricted $P$ to pure processes (rank-1 operators) due to the monotonicity of QFI [36]. Our goal is to identify the ultimate precision limit of parameter estimation characterized by the QFI within such constraints:

In general we can write the ensemble decomposition [28] of the CJ operator $N_{\phi}$ as $N_{\phi}=\sum_{i=1}^{r}\outerproduct{N_{\phi,i}}{N_{\phi,i}}=\mathbf{N}_{\phi}\mathbf{N}_{\phi}^{\dagger}$, where $\mathbf{N}_{\phi}:=\left(\ket{N_{\phi,1}},\dots,\ket{N_{\phi,r}}\right)$ and $r:=\max_{\phi}\mathsf{rank}({N_{\phi}})$. We also define $\dot{\tilde{\mathbf{N}}}_{\phi}:=\dot{\mathbf{N}}_{\phi}-\mathrm{i}\mathbf{N}_{\phi}h$ for $h\in\mathbb{H}_{r}$, where $\mathbb{H}_{r}$ denotes the set of $r\times r$ Hermitian matrices, and the performance operator [38]

With these notions, we can show (see Appendix A, which is analogous to the approach in [38]) that the QFI admits the form

with

To evaluate the QFI, we first exchange $\max_{\tilde{P}}$ and $\min_{h}$ without changing the optimal QFI, assured by the minimax theorem [39, 40] since the objective function is concave on $\tilde{P}$ and convex on $h$ [41]. Hence, the problem is cast into

Then we fix $h$ and formulate the dual problem of maximization over the set $\tilde{\mathsf{P}}$. Finally we further optimize the value of $h$. To simplify the calculation of QFI we require that

where $\mathsf{Conv}\{\cdot\}$ denotes the convex hull, and each $\mathsf{S}^{i}$ for $i=1,\dots,K$ is an affine space of Hermitian operators. Adopting the above-mentioned method, we get

The proof can be found in Appendix B. We remark that similar optimization ideas have been applied to other tasks, such as quantum Bayesian estimation [42], quantum network optimization [43], non-Markovian quantum metrology [38], and quantum channel discrimination [23]. The minimization problem in Theorem 1 can be further written in the form of SDP and solved efficiently, with detailed numerically solvable forms given in Appendix E, where the constraints in Eq. (10) can be further simplified in some cases.

Optimal strategies.—By itself, the QFI does not reveal how to implement the optimal strategy achieving the highest precision. Here, in addition to Theorem 1, we design an algorithm that yields a strategy attaining the optimal QFI for any strategy set satisfying Eq. (9). The method, which generalizes the method of finding an optimal probe state for a single channel [44, 45], is summarized as Algorithm 1 (see Appendix C for its derivation).

By Algorithm 1 we obtain the CJ operator of a strategy that attains the optimal QFI. For strategies following definite causal order, there exists an operational method of mapping the CJ operator of the strategy to a probe state and a sequence of in-between control operations with minimal memory space [46]. For causal order superposition strategies (see the strategy set $\mathsf{Sup}$), we show that they can always be implemented by controlling the order of operations in a circuit with a quantum SWITCH (see Appendix I). In this way, we obtain a systematic method to identify optimal sequential and causal superposition strategies, one of the key problems in quantum metrology.

Strategy sets.—We consider the evaluation of QFI for five different families of strategies. In all the following definitions the subscript $i$ of an operator denotes the Hilbert space $\mathcal{H}_{i}$ it acts on.

The family of parallel strategies [see Fig. 1LABEL:sub@subfig:parallel_strategy] is the first and one of the most successful examples of quantum-enhanced metrology, featuring the usage of entanglement to achieve precision beyond the classical limit [47]. By making parallel use of $N$ quantum channels together with ancillae, we can regard these $N$ channels as one single channel from $\mathcal{L}\left(\otimes_{i=1}^{N}\mathcal{H}_{2i-1}\right)$ to $\mathcal{L}\left(\otimes_{i=1}^{N}\mathcal{H}_{2i}\right)$. A parallel strategy set $\mathsf{Par}$ is defined as the collection of $P\in\mathsf{Strat}$ such that [34]

Note that the optimal QFI of parallel strategies can also be evaluated using the method in [28, 30].

A more general protocol is to allow for sequential use of $N$ channels assisted by ancillae, where only the output of the former channel can affect the input of the latter channel, and any control gates can be inserted between channels [see Fig. 1LABEL:sub@subfig:sequential_strategy]. A sequential strategy set $\mathsf{Seq}$ is defined as the collection of $P\in\mathsf{Strat}$ such that [34]

Unlike the case of parallel strategies, there is no existing way of evaluating the exact QFI using sequential strategies.

We also consider families of strategies involving indefinite causal order. The first one, denoted by $\mathsf{SWI}$, takes advantage of the (generalized) quantum SWITCH [48, 49], where the execution order of $N$ channels is entangled with the state of an $N!$-dimensional control system [see Fig. 1LABEL:sub@subfig:quantum_switch_strategy]. See Appendix E for the formal definition.

More generally, we consider the quantum superposition of multiple sequential orders, each with a unique order of querying the $N$ channels [see Fig. 1LABEL:sub@subfig:causal_superposition_strategy]. This can be implemented by entangling $N!$ definite causal orders with a quantum control system [50]. If $N=2$ and the control system is traced out, this notion is equivalent to causal separability [20, 21]. A causal superposition strategy set $\mathsf{Sup}$ is defined as the collection of $P\in\mathsf{Strat}$ such that

where each permutation $\pi$ is an element of the symmetric group $S_{N}$ of degree $N$, and each $\mathsf{Seq}^{\pi}$ denotes a sequential strategy set whose execution order of $N$ channels is $\mathcal{E}_{\phi}^{\pi(1)}\rightarrow\mathcal{E}_{\phi}^{\pi(2)}\rightarrow\cdots\rightarrow\mathcal{E}_{\phi}^{\pi(N)}$, having denoted by $\mathcal{E}_{\phi}^{k}$ the channel from $\mathcal{L}(\mathcal{H}_{2k-1})$ to $\mathcal{L}(\mathcal{H}_{2k})$. Note that $\mathsf{SWI}$ is a subset of $\mathsf{Sup}$, where the intermediate control is trivial. There are other strategies, such as quantum circuits with quantum controlled casual order (QC-QCs) and probabilistic QC-QCs [50, 51], which we will not discuss here.

Finally, we introduce the family of general indefinite-causal-order strategy [see Fig. 1LABEL:sub@subfig:general_strategy], which is the most general strategy set considered in this work. Here the only requirement is that the concatenation of the strategy $P$ with $N$ arbitrary channels results in a legitimate quantum state. The causal relations in this case [21] are a bit cumbersome, but for our purpose what matters is the dual affine space (see Theorem 1), which is simply the space of no-signaling channels [16, 43]. A general indefinite-causal-order strategy set $\mathsf{ICO}$ is defined as the collection of $P\in\mathsf{Strat}$ such that

for any $E^{j}\in\mathcal{L}(\mathcal{H}_{2j-1}\otimes\mathcal{H}_{2j}\otimes\mathcal{H}_{A_{j}})$ that denotes the CJ operator of an arbitrary quantum channel with an arbitrary ancillary space $\mathcal{H}_{A_{j}}$.

We note that, unlike the previous strategies that can always be physically realized, the physical realization of the general $\mathsf{ICO}$ is untraceable [51, 50]. The optimal value obtained with general $\mathsf{ICO}$ nevertheless serves as a useful tool that can gauge the performances of different strategies. For example, as we will show, in some cases the optimal QFI $J^{(\mathsf{Sup})}$ and $J^{(\mathsf{ICO})}$ are equal or nearly equal. This then shows that the physically realizable strategy obtained from the set $\mathsf{Sup}$ is already optimal or nearly optimal among all possible strategies, which we will not be able to tell without $J^{(\mathsf{ICO})}$.

Symmetry reduced programs for optimal metrology.—The complexity of the original optimization problems in Theorem 1 and Algorithm 1 can be reduced by exploiting the permutation symmetry. In Appendix D, we prove that we can choose a permutation-invariant matrix $h$ for Theorem 1 and solve for a permutation-invariant optimal strategy [52] by Algorithm 1 based on this choice, if any permutation $\pi\in S_{N}$ bijectively maps each affine space $\mathsf{S}^{i}$ [in Eq. (9)] to some affine space $\mathsf{S}^{j}$. That is, for any $\pi\in S_{N}$ and any $i$, there exists a $j$ such that the mapping $S\mapsto G_{\pi}SG_{\pi}^{\dagger}$ on $\mathsf{S}^{i}$ is a bijective function from $\mathsf{S}^{i}$ to $\mathsf{S}^{j}$, where $G_{\pi}$ is a unitary representation of $\pi$. Furthermore, if each space $\mathsf{S}^{i}$ itself is permutation invariant, we can restrict each $Q^{i}\in\overline{\mathsf{S}}^{i}$ to be permutation invariant, further reducing the complexity of optimization. For both optimization problems we can apply the technique of group-invariant SDP to reduce the size as there exists an isomorphism which preserves positive semidefiniteness, from the permutation-invariant subspace to the space of block-diagonal matrices [53, Theorem 9.1]. Table 1 compares the number of variables involved in QFI evaluation with and without exploiting the symmetry (see Appendix E for its derivation as well as Appendix F for the complexity of Algorithm 1, where by group-invariant SDP we also numerically evaluate the growth of QFI $J^{(\mathsf{ICO})}$ up to $N=5$).

Hierarchy of strategies.—By substituting the definitions of different strategy sets into Theorem 1, we obtain the exact values of the optimal QFI. We find that a strict hierarchy of QFI exists quite prevalently. For demonstration purposes, here we show only the result for the amplitude damping channel for $N=2$ and supplement our findings with bountiful numerical results in Appendix G. In this case, the process encoding $\phi$ is a $z$ rotation $U_{z}(\phi)=e^{-\mathrm{i}\phi t\sigma_{z}/2}$, where $t$ is the evolution time, followed by an amplitude damping channel described by two Kraus operators: $K_{1}^{(\mathrm{AD})}=\outerproduct{0}{0}+\sqrt{1-p}\outerproduct{1}{1}$ and $K_{2}^{(\mathrm{AD})}=\sqrt{p}\outerproduct{0}{1}$, with the decay parameter $p$.

In Fig. 2 we plot the QFI versus $p$ for the amplitude damping noise with all 5 strategy sets for $N=2$. A strict hierarchy of $\mathsf{Par}$, $\mathsf{Seq}$ and $\mathsf{ICO}$ holds if $p$ is neither 1 nor 0, i.e., $J^{(\mathsf{Par})}<J^{(\mathsf{Seq})}<J^{(\mathsf{ICO})}$. This is in contrast to the asymptotic regime of $N\to\infty$, where the relative difference between $J^{(\mathsf{Seq})}$ and $J^{(\mathsf{Par})}$ vanishes for this channel [45]. Besides, in this case general $\mathsf{ICO}$ cannot strictly outperform $\mathsf{Sup}$, implying that causally superposing two sequential strategies is sufficient to achieve the general optimality in this particular scenario. The gap between $J^{(\mathsf{Sup})}$ and $J^{(\mathsf{ICO})}$, however, could be observed for the same channel with larger $N$ or for other channels when $N=2$ (see Appendix G). In fact, by randomly sampling noise channels from CPTP channel ensembles, we find that for 984 of 1000 random channels, a strict hierarchy $J^{(\mathsf{Par})}<J^{(\mathsf{Seq})}<J^{(\mathsf{Sup})}<J^{(\mathsf{ICO})}$ holds for $N=2$, implying that there exist more powerful strategies than causal superposition strategies in these cases. We note that a strict hierarchy of strategies has been found for channel discrimination in [23], but much less is known in quantum metrology until our work.

Our method can also test the tightness of existing QFI bounds in the nonasymptotic regime, which has seldom been done until this work. Here we take the commonly used, asymptotically tight [45] upper bound for parallel strategies [see [28, Theorem 4 and Eq. (17)] or [30, Eq. (16)]]. For $p=0.5$, our result shows that the exact parallel QFI $J^{(\mathsf{Par})}=1.795$ is $32.7\%$ lower than the asymptotically tight parallel upper bound $2.667$, and even the exact sequential QFI $J^{(\mathsf{Seq})}=2.179$ is $18.3\%$ lower than this parallel upper bound [54]. Similar phenomena are observed in other noise models and for different $N$ (see Appendix H).

With Algorithm 1 we can also construct strategies to achieve the optimal QFI. Remarkably, we find that a simple strategy of applying a quantum SWITCH using a control qubit $\ket{\plus}_{C}:=\left(\ket{0}_{C}+\ket{1}_{C}\right)/\sqrt{2}$ (without any additional control operations on the probe) beats any sequential strategies (which can involve complex control) in certain cases (e.g. $p<0.5$). To our best knowledge, this is the only instance of noisy quantum metrology so far, where the advantage of indefinite causal orders is established rigorously. In Appendix I we also present two explicit examples of implementing optimal sequential and causal superposition strategies, obtained by first applying Algorithm 1 and converting the CJ operators into quantum circuit consisting of single-qubit rotations and CNOT gates (as well as a quantum SWITCH for the case of $\mathsf{Sup}$). For optimal causal superposition strategies, the permutation symmetry allows us to only control the execution order of channels while fixing state preparation and intermediate control [$\rho_{\uparrow}=\rho_{\downarrow}$ and $U_{\uparrow}=U_{\downarrow}$ in Fig. 1LABEL:sub@subfig:causal_superposition_strategy], which can be implemented by a $(2N-1)$-quantum SWITCH of $N$ channels $\mathcal{E}_{\phi}$ and $N-1$ intermediate operations.

Our result serves as a versatile tool for the demonstration of optimal quantum metrology and the design of optimal quantum sensors, especially in the context of control optimization [55, 56] and indefinite causal orders [20, 16, 21, 18, 22, 19, 17, 23].

The code accompanying the paper is openly available [57].

We thank Cyril Branciard, Alastair A. Abbott, and Raphael Mothe for helpful discussions and comments on our first manuscript. This work is supported by Guangdong Natural Science Fund—General Programme via Project 2022A1515010340, by HKU Seed Fund for Basic Research for New Staff via Project 202107185045, and by the Research Grants Council of Hong Kong through the Grant No. 14307420.

Note added.—Recently, it has been shown that neither sequential nor causal superposition strategies provide any advantage over parallel strategies asymptotically [58].

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