Optimal Non-asymptotic Quantum Metrology with Hierarchical Strategies

One of the main quests in quantum metrology is to attain the ultimate precision limit with given resources, where the resources are not only of the number of queries, but more importantly of the allowed strategies. With the same number of queries, the restrictions on the strategies constrain the achievable precision. In this work, we establish a systematic framework to identify the ultimate precision limit of different families of strategies, including the parallel, the sequential and the indefinite-causal-order strategies, and provide an efficient algorithm that determines an optimal strategy within the family of strategies under consideration. With our framework, we show there exists a strict hierarchy of the precision limits for different families of strategies.

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. 1(a)), 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. 1(b)), 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. 1(c)), 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] (Fig. 1(d) and (e)) 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 non-asymptotic 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 δφ of estimating an unknown parameter φ encoded in a quan-tum state ρ φ , for any unbiased estimatorφ, can be determined via the quantum Cramér-Rao bound (QCRB) as δφ ≥ 1 νJ Q (ρ φ ) [24][25][26], where J Q (ρ φ ) is the quantum Fisher information (QFI) of the state ρ φ and ν is the number of repeated measurements [27]. For singleparameter 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 Ψ φ ⟩ is the purification of ρ φ with an ancillary space H A , Tr A denotes the partial trace over H A , anḋ Ψ ∶= ∂Ψ ∂φ. When the parameter is carried by a quantum channel E φ , i.e., a completely positive tracepreserving (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]: , where S(H) denotes the space of density operators on the Hilbert space H, H S A denotes the Hilbert space of the system/ancillae, and I A is the identity on H A .
We denote by L(H) the set of linear operators on the finite-dimensional Hilbert space H, and L(L(H 1 ), L(H 2 )) denotes the set of linear maps from L(H 1 ) to L(H 2 ). By the Choi-Jamiołkowski (CJ) isomorphism, a parameterized quantum channel E φ ∈ L(L(H 2i−1 ), L(H 2i )) (for 1 ≤ i ≤ N ) can be represented by a positive semidefinite operator (called the CJ operator) E φ = Choi(E φ ) = E φ ⊗ I( I⟫⟪I ), where I⟫ = ∑ i i⟩ i⟩. The CJ operator of N identical quantum channels is i=1 H i . 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 E φ , generates an output quantum state carrying the information about φ. A strategy can be described by a CJ operator on L H F ⊗ 2N i=1 H i , where 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 ∈ L (⊗ a∈A H a ) and B ∈ L (⊗ b∈B H b ) as where T i denotes the partial transpose on H i , and H A B denotes ⊗ i∈A B 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 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: given a strategy set P is where J Q (ρ) is the QFI of the state ρ, and N φ is the CJ operator of N channels.
In general we can write the ensemble decomposition [28] where H r denotes the set of r × 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: withP ∶= P = Tr F P P ∈ P .
To evaluate the QFI, we first exchange maxP and min h without changing the optimal QFI, assured by the minimax theorem [39,40] since the objective function is concave onP and convex on h [41]. Hence, the problem is cast into Then we fix h and formulate the dual problem of maximization over the setP. Finally we further optimize the value of h. To simplify the calculation of QFI we require thatP where Conv{⋅} denotes the convex hull, and each S i for i = 1, . . . , K is an affine space of Hermitian operators. Adopting the above-mentioned method, we get: Given an arbitrary strategy set P such that P given by Eq. (7) satisfies the condition Eq. (9), the QFI of N quantum channels E φ can be expressed as the following optimization problem: where S i ∶= Q is Hermitian Tr(QS) = 1, S ∈ S i is the dual affine space of S i .
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). Algorithm 1. Find an optimal strategy in the set P.
• Given N φ the CJ operator of N channels, solve for an optimal value h = h (opt) in Eq. (10) of Theorem 1 via SDP.
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 Sup later), 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 H i it acts on.
The family of parallel strategies (see Fig. 1(a)) is the first and one of the most successful examples of quantumenhanced 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 A parallel strategy set Par is defined as the collection of P ∈ Strat such that [34] Tr F P = 1 2,4,...,2N ⊗ P (1) , Tr P (1) = 1.
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. 1 (b)). A sequential strategy set Seq is defined as the collection of P ∈ 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 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. 1(c)). 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. 1(d)). 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 Sup is defined as the collection of P ∈ Strat such that where each permutation π is an element of the symmetric group S N of degree N , and each Seq π denotes a sequential strategy set whose execution order of N channels is , having denoted by E k φ the channel from L(H 2k−1 ) to L(H 2k ). Note that SWI is a subset of 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 indefinitecausal-order strategy (see Fig. 1(e)), 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-signalling channels [16,43]. A general indefinite-causal-order strategy set ICO is defined as the collection of P ∈ Strat such that where E j ∈ L(H 2j−1 ⊗ H 2j ⊗ H Aj ) denotes the CJ operator of an arbitrary quantum channel with an arbitrary ancillary space H Aj . We note that, unlike the previous strategies that can always be physically realized, the physical realization of the general ICO is untraceable [50,51]. The optimal value obtained with general 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 (Sup) and J (ICO) are equal or nearly equal. This then shows that the physically realizable strategy obtained from the set Sup is already optimal or nearly optimal among all possible strategies, which we will not be able to tell without J (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 π ∈ S N bijectively maps each affine space S i (in Eq. (9)) to another affine space S j . That is, for any π ∈ S N and any i, there exists a j such that the mapping S ↦ G π SG † π on S i is a bijective function from S i to S j , where G π is a unitary representation of π. Furthermore, if each space S i itself is permutation-invariant, we can restrict each Q i ∈ 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 I compares the number of variables involved in QFI evaluation with/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 (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 φ is a zrotation U z (φ) = e −iφtσz 2 , where t is the evolution time, followed by an amplitude damping channel described by two Kraus operators: K = J (ICO) with an error tolerance of no more than 10 −8 in this case, but the gap between J (Sup) and J (ICO) could exist for larger N or for other types of noise, which can be observed by randomly sampling noise channels.
In Fig. 2 we plot the QFI versus p for the amplitude damping noise with all 5 strategies for N = 2. A strict hierarchy of Par, Seq and ICO holds if p is neither 1 nor 0, i.e., J (Par) < J (Seq) < J (ICO) . This is in contrast to the asymptotic regime of N → ∞, where the relative difference between J (Seq) and J (Par) vanishes for this channel [45]. Besides, in this case general ICO cannot strictly outperform Sup, implying that causally superposing two sequential strategies is sufficient to achieve the general optimality in this particular scenario. The gap between J (Sup) and J (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 (Par) < J (Seq) < J (Sup) < J (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 non-asymptotic regime, which has seldom been done till this work. Here we take the commonly used, asymptotically tight [45] upper bound for parallel strategies (see [28,Theorem 4] or [30,Eq. (16)]). For p = 0.5, our result shows that the exact parallel QFI J (Par) = 1.795 is 32.7% lower than than the asymptotically tight parallel upper bound 2.667, and even the exact sequential QFI J (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 +⟩ C ∶= ( 0⟩ C + 1⟩ C ) √ 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 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 ( 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 [16][17][18][19][20][21][22][23].
The code accompanying the paper is openly available [57].
Acknowledgement.-We thank Cyril Branciard, Alastair A. Abbott, and Raphael Mothe for helpful discus-sions 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.
The formalism of this proof has been developed for strategies following a definite causal order in [38], and the generalization to indefinite-causal-order strategies considered here is straightforward.
In [28], the QFI of a quantum state is expressed as a minimization problem for any integer q ≥ max φ rank(ρ φ ), where { ψ φ,i ⟩} is a set of unnormalized vectors such that ρ φ = ∑ i ψ φ,i ⟩⟨ψ φ,i 1 . In the main text, the QFI of N quantum channels E φ is defined as the QFI of the output state obtained from the concatenation of the CJ operator N φ of N quantum channels and an optimal strategy P in a given strategy set P: Due to the monotonicity [36] of the QFI under CPTP maps (e.g., the partial trace operation over ancillary space in this case), by choosing a proper global future space H F an optimal P can be taken as a pure process (rank-1 operator) denoted by P ⟩⟨P . For a fixed P = P ⟩⟨P , minimization over decompositions of P * N φ is equivalent to minimization over decompositions of N φ , as rank(P * N φ ) ≤ rank(N φ ). As a positive semidefinite operator, N φ has a decomposition as: Then the QFI of N channels can be expressed as having defined the performance operator where the superscript T denotes the transpose. We can further define an r × r Hermitian matrix h ∶= iV φ V † φ to take care of the freedom of choice for the decompositionÑ φ , by noting thatṄ φṄ † to explicitly manifest its dependence on h. Hence, we finally arrive at Eq. (6) of the main text: whereP ∶= P = Tr F P P ∈ P .
Appendix B: Proof of Theorem 1 Starting from Eq. (6) of the main text, we exchange the order of minimization and maximization thanks to Fan's minimax theorem [39], since Tr P Ω φ (h) is concave onP and convex on h, andP is assumed to be a compact set. Then the problem of QFI evaluation can be rewritten as Reformulating the condition of Theorem 1, we require that each operatorP ∈P can be written as a convex combination of positive semidefinite operators S i , i = 1, . . . , K: where each S i is an affine space of Hermitian operators. Thus Eq. (B1) can be reformulated as For now we fix h and consider the dual problem of maximization overP. For each affine space S i we have defined its dual affine space S i , whose dual affine space in turn is exactly S i [43]. Choose an affine basis {Q i,j } Li j=1 for S i , and the maximization problem is further expressed as Defining P i ∶= q i S i to avoid the product of variables in optimization, we have where the constraints q i ≥ 0 can be safely removed, since Tr S i = ∏ N j=1 d 2j , implying that S i includes a positive operator proportional to identity for any i = 1, . . . , K, having denoted d j ∶= dim(H j ) for simplicity. The Lagrangian of the problem is given by forQ i ≥ 0. Hence, by removingQ i the dual problem is written as min λ, corresponds to a trivial case where the QFI is zero), and clearly Q i is an arbitrary operator in the set S i . Therefore, we cast the dual problem into min λ, Slater's theorem [58] implies that the strong duality holds, since the QFI is finite and the inequality constraints can be strictly satisfied for a positive semidefinite operator Ω φ (h), by choosing λQ i = µ Ω φ (h) 1 1,2,...,2N for µ > 1 and any i = 1, . . . , K, having denoted the operator norm by ⋅ . Finally, by optimizing the choice of h we derive the result of Theorem 1.
Appendix C: Proof of the validity of Algorithm 1 We first recall the minimax theorem: for a function f (x, y) convex on x and concave on y. Assume (x 0 , y 1 ) is a solution for the L.H.S. of Eq. (C1) and (x 1 , y 0 ) is a solution for the R.H.S. of Eq. (C1). It is easy to see that In view of Eq. (C1) both equalities hold. Therefore, (x 0 , y 0 ) is a saddle point of f (x, y), i.e., x 0 = arg min x f (x, y 0 ) and y 0 = arg max y f (x 0 , y). Since the objective function Tr P Ω φ (h) in the primal problem of estimating QFI is convex on h and concave onP , we can substitute x = h and y =P . Obviously, h (opt) is an optimal solution for min h maxP Tr P Ω φ (h) and thus corresponds to a saddle point. Then x 0 = arg min x f (x, y 0 ) can be satisfied by requiring ∂ h f h,P (opt) h=h (opt) = 0, resulting in Eq. (11) in the main text: Meanwhile y 0 = arg max y f (x 0 , y) corresponds to the requirement thatP (opt) is an optimal solution for fixed h = h (opt) . Therefore, h (opt) ,P (opt) is a saddle point and an optimal solution for maxP min h Tr P Ω φ (h) . By definition a purification ofP (opt) is an optimal physically implemented strategy, i.e., we can choose a strategy P (opt) such that Tr F P (opt) =P (opt) . We remark that Eq. (C3) can be reformulated as a set of linear constraints by choosing a basis {H i } r 2 i=1 for the space H r of Hemitian matrices. For example, denoting by E ij the r × r matrix of which only the (i, j)-th element is 1 and all other elements are 0, we can choose H = E jj for j = 1, . . . , r, H = E jk + E kj and H = i (E jk − E kj ) for k = 1, . . . , r and k ≠ j, and obtain a series of constraints, which are equivalent to Eq.(C3).

Appendix D: Symmetry reduced optimization
This section demonstrates how to reduce the size of the optimization problems concerned in Theorem 1 and Algorithm 1, by exploiting the permutation symmetry as applicable.
Let us begin with some notations. We consider the action of S N , the symmetric group of degree N , on the finite- π ∈ X for any X ∈ X and any π ∈ S N . We now explicitly express the components of the performance operator Ω φ (h). Given N identical quantum channels

Symmetry reduced QFI evaluation
With these notations, we first consider the optimization in QFI evaluation and have the following: Lemma 1. In the optimization problem of Theorem 1, if, for any π ∈ S N and any i, there exists a j such that the mapping S ↦ G π SG † π on S i is a bijective function from S i to S j , then there must exist a permutation-invariant h as a feasible solution.
Proof. Without loss of generality we assume all S i are distinct spaces; otherwise, we just remove the duplicate ones. We first prove that, under any permutation operation Q ↦ G † π QG π , for any i ∈ {1, . . . , K} there exists a unique j ∈ {1, . . . , K} such that the dual affine space S j is bijectively mapped to S i . The condition of the lemma implies that, for any G π and any i we can find j i such that G π S i G † π ∈ S ji for all S i ∈ S i . Due to the bijectivity, S i and the corresponding S ji are isomorphic, with different j i for different i. Apparently S i and S ji are also isomorphic. If we We then prove that there exists a permutation-invariant performance operator Ω φ (h) as a feasible solution. The group action is characterized by the permutation operator G π on the representation space Suppose h (opt) is an optimal solution, i.e., for any i ∈ {1, . . . , K}, there exist optimal values of λ and Q i such that Then for any permutation π ∈ S N we have which implies that G π Ω φ (h (opt) )G † π is also a feasible solution for any π. Furthermore, the permutation-invariant solution of the performance operator 1 For distinction we denote the group action on h by G ′ π hG ′ † π and the group action on where π(i) ∶= (i π(1) , . . . , i π(N ) ). Note that G π N φ,i ⟩ = N φ,π(i) ⟩ and which results in where in the last equation we have used Eq. (D3). Therefore, by choosing the permutation-invariant solution )G † π , and this permutation-invariant choice of h is also a feasible solution.
If a stronger condition holds, then not only can we choose a permutation-invariant h, but we can also restrict Q i ∈ S i to be permutation-invariant. In this case all matrix variables concerned in the optimization are permutation-invariant.

Lemma 2.
In the optimization problem of Theorem 1, if each affine space S i is permutation-invariant for any i = 1, . . . , K, then there must exist a permutation-invariant Q i ∈ S i as a feasible solution for each i.
Proof. By Lemma 1 we can restrict h to be permutation-invariant in optimization, and therefore the performance operator Ω φ (h) is permutation-invariant. It is easy to see that each dual affine space S i is also permutation-invariant.
Then for any i = 1, . . . , K, for Q i,(opt) ∈ S i satisfying the constraint λQ i ≥ Ω φ (h), we have the permutation-invariant i which also satisfies the same constraint. Hence, this permutation-invariant choice of Q i is also a feasible solution.
Now since the optimization is restricted to the invariant subspace, we can reduce the matrix sizes by block diagonalization. Generally, for a group-invariant space of complex matrices C n×n,(inv) , there exists an isomorphism ϕ preserving positive semidefiniteness between C n×n,(inv) and a direct sum of complex matrix spaces [53, Theorem 9.1]: where m k is the multiplicity of the k-th inequivalent irreducible representation of the group, and I is the number of inequivalent irreducible representations. To be more specific, for the symmetric group S N , the representation space where each partition µ ∶= (µ 1 , . . . , µ d ) (with nonnegative integers µ i ≥ µ j for any i < j) corresponds to a Young diagram, and µ ∶= ∑ i µ i . Each isotypic component W µ can be further decomposed into a direct sum of equivalent irreducible subspaces: We define the dimension of the irreducible representation d µ ∶= dim W µ,i . It is worth mentioning that the first decomposition into isotypic components is unique while the second decomposition into equivalent irreducible representation spaces is not (see, e.g., [59]). In terms of the unitary group action G π on W, there exists a unitary transformation of basis U such that for any π we have where G µ π is a unitary operator on the irreducible representation space associated with the Young diagram label µ, m µ is the corresponding multiplicity, and 1 (m µ ) is an m µ × m µ identity matrix acting on the multiplicity subspace. By Schur's lemma, for any group-invariant operator X on W, i.e., X commuting with all G π , we have for any π, where X µ is an m µ × m µ matrix.
where the upper bound is obtained straightforwardly from the definition of the binomial coefficient. Specifically, if X is further restricted to be a Hermitian matrix variable, then the number of real scalar variables contained in all elements of X is reduced from d 2N to N +d 2 −1 d 2 −1 . Now let us turn to the optimization problem of QFI evaluation. If the Hermitian matrix h is taken to be permutationinvariant by Lemma 1, the number of variables concerned in h is reduced from s 2N to N +s 2 −1 s 2 −1 . Similarly, if further by Lemma 2 each Hermitian matrix Q i is permutation-invariant, the number of variables in Q i is also reduced from . By this reduction the complexity gets polynomial rather than exponential, with respect to N .
We can then reformulate the optimization in Theorem 1 with the reduced form. We consider two cases relevant to the strategy sets mentioned in the main text. First, under certain circumstances we can reduce the number of constraints in optimization as follows: Theorem 2 (Symmetry reduced Theorem 1, first case). If, for any π ∈ S N and any i, there exists a j such that the mapping S ↦ G π SG † π on S i is a bijective function from S i to S j , and meanwhile for any i, j there exists some permutation operation such that S i is bijectively mapped to S j , then the QFI of N quantum channels E φ can be expressed as: where h = U ′ ⊕ µ =N 1 (d µ ) ⊗ h µ U ′ † with each h µ as an m ′ µ × m ′ µ matrix variable and U ′ as a unitary transformation of basis.
Proof. We first prove that, for any i, j there exists some permutation operation Q ↦ G π QG † π such that S i is bijectively mapped to S j . Given any i and j, if we choose an arbitrary Q i ∈ S i , i.e., Tr Q i S i = 1 for all S i ∈ S i , then the condition of the theorem implies that there exists G π such that G π S i G † π ∈ S j . As the map is bijective, in fact G † π S j G π ∈ S i for all S j ∈ S j . Then we have Tr G π Q i G † π S j = Tr Q i G † π S j G π = 1 for all S j ∈ S j . Therefore, following the same argument in the proof of Lemma 1, under Q ↦ G π QG † π , S i is bijectively mapped to S j .
By Lemma 1 we can take h to be permutation-invariant and apply the block diagonalization to h given by Eq. (D8). Then for any Q 1 ∈ S 1 satisfying the constraint for any i = 2, . . . , K. Therefore, all the constraints except for λQ 1 ≥ Ω φ (h) are redundant and can be removed. Now we consider the second case. If by Lemmas 1 and 2 h and each Q i are permutation-invariant, we can then reformulate the constraints in optimization with reduced matrix dimensions. To relate the group representation on µ i is the label of the i-th Young diagram, and U ′,µ i has d µ i m ′ µ i columns. Thus by defining the matriẋ we have the permutation-invariant performance operator Analogously, we apply the block diagonalization using a change of basis U to where Q i,µ is an m µ × m µ matrix variable. The unitary transformation U = U µ 1 , . . . , U µ I is first divided into blocks, and then each U µ i = U µ i ,1 , . . . , U µ i ,d µ i is divided into blocks for i = 1, . . . , I, where µ i is the label of the i-th Young diagram, and U µ i ,j has m µ i columns for j = 1, . . . , d µ i . Note that U also gives the block diagonalization of Ω φ (h).
Before proceeding further we prove that the range ofṄ µ * φ is exactly contained in the irreducible representation space corresponding to µ: Proof. To characterize the representation space W µ we introduce the key notion of Young symmetrizer as well as other related concepts very briefly (see, e.g., [59] for mathematical details). A Young tableau is a filling into the boxes of the Young diagram with positive integers weakly increasing along each row and strictly increasing along each column. For a standard Young tableau labeled by ν, i.e., a Young tableau filled with the entries 1, . . . , N , we define two permutation subgroups P ν ∶= {σ ∈ S N σ preserves each row} (D14) and Q ν ∶= {σ ∈ S N σ preserves each column} .
In the group algebra CS N we define two elements a ν ∶= ∑ σ∈Pν e σ and b ν ∶= ∑ σ∈Qν sgn(σ)e σ , where e σ is the unit vector corresponding to σ and sgn(⋅) denotes the parity of the permutation. Then the Young symmetrizer is defined by It is known that a Young diagram of µ corresponds to d µ standard Young tableaux, with each Young tableau of ν characterizing an irreducible representation space, given by the image of c ν on ⊗ i W i under the natural group algebra representation CS N → End (⊗ i W i ), where End(V ) denotes the set of endomorphisms on V . With these notions we now have an explicit characterization of the representation space. Note that we can prove Range Ṅµ * φ ⊆ W µ if Range Ṅ * φ U ′,µ * ⊆ W µ and Range N * φ U ′,µ * ⊆ W µ . In the proof of Lemma 1 we have seen G π Ṅ φ,i ⟩ = Ṅ φ,π(i) ⟩ and G π N φ,i ⟩ = N φ,π(i) ⟩, from which it follows that and G π N φ = N φ G ′ π . Now we prove that Range Ṅ * φ U ′,µ * ⊆ W µ . From the discussions above we know that for all Young tableau labels ν corresponding to the Young diagram of µ. Explicitly, we have Note that there always exists a unitary transformation of basis V such that we can obtain a real matrix U ′,µ (real) = U ′,µ V † , which leads to for all Young tableau labels ν corresponding to the Young diagram of µ. To show Eq. (D21), note that by Eq. (D17) we have Since the R.H.S. of Eq. (D22) is the image of the Young symmetrizer c ν on Range Ṅ * φ , it is a subset of W µ . Therefore, Range Ṅ * φ U ′,µ * ⊆ W µ . In the same way we can also show that Range N * φ U ′,µ * ⊆ W µ and thus complete the proof. Therefore, U † Ω φ (h)U is block diagonal with each block on the space Range Ṅµ φ . This results in: Theorem 3 (Symmetry reduced Theorem 1, second case). If each affine space S i is permutation-invariant for any i = 1, . . . , K, then the QFI of N quantum channels E φ can be expressed as: where Q i,µ is given by Eq. (D13).
Proof. By Lemmas 1 and 2, h, Ω φ (h) and all Q i are taken to be permutation-invariant. The original constraint λQ i ≥ Ω φ (h) on the permutation-invariant space is reformulated as Since by Lemma 3 Range Ṅµ * φ ⊆ W µ = Range (U µ ) for any µ, we have U µ † ∑ ν =NṄ ν * φṄ ν * † φ U µ = U µ †Ṅµ * φṄ µ * † φ U µ . Then Eq. (D24) can be reformulated as Furthermore, both sides of the inequality in Eq. (D25) are block diagonal with the same repeating blocks and we only need to compare one of the blocks. Without loss of generality, we choose the first block for comparison and obtain which holds for all i = 1, . . . , K and µ = N .
We have not characterized each dual affine space S i for i = 1, . . . , K yet. If we choose an affine basis {S i,j } Mi j=1 for each S i , then the constraint Q i ∈ S i can be reformulated as a set of linear constraints Tr Q i S i,j = 1 for all j = 1, . . . , M i . If each affine space S i is permutation-invariant, then by the proof of Lemma 2 we have, for a feasible solution of Q i ∈ S i , G π Q i G † π is also a feasible solution for any π ∈ S N . Then by defining each linear constraint Tr Q i S i,j = 1 can be replaced by Tr Q iSi,j = 1 without changing the problem. SinceS i,j is permutation-invariant, similar to Eq. (D13) it can be decomposed as Finally, it remains to be seen how to find the unitary transformation U and U ′ for block diagonalization. Known rigorous numerical algorithms for identifying the transformation are fairly expensive [61]. Fortunately, RepLAB [62], a numerical approach to decomposing representations based on randomized heuristics works very well in practice and is thus adopted here.

Symmetry reduced algorithm for optimal strategies
The idea is similar to the symmetry reduced evaluation of QFI. Since the first step of Algorithm 1 is simply solving the optimization problem in Theorem 1, now we only consider its second step, where we need to solve for the optimal value ofP in We have the following: Lemma 4. If, for any π ∈ S N and any i, there exists a j such that the mapping S ↦ G π SG † π on S i is a bijective function from S i to S j , then there must exist a permutation-invariantP as an optimal solution in Algorithm 1.
Proof. By definitionP is permutation-invariant. By Lemma 1 we can choose a permutation-invariant h (opt) and thus Ω φ h (opt) is also permutation-invariant. IfP (opt) is an optimal solution ofP , then G πP (opt) G † π for any π ∈ S N is also an optimal solution, since and for all H ∈ H r having used G † π N φ = N φ G ′ † π and G † πṄφ =Ṅ φ G ′ † π for any π ∈ S N in the second equality of Eq. (D33). Therefore, there exists a permutation-invariant solutionP Now by following the same line of arguments as used from Eqs. (D27) to (D29), we define Similar to the arguments in Appendix C, by choosing a basis H i J i=1 for the permutation-invariant subspace of H r×r , where J = N +s 2 −1 s 2 −1 , Eq. (C3) can be reformulated as a set of J linear constraints. We further define ⊗ Ω µ φ h (opt) U † , and then reformulate the optimization problem as the following reduced form: By choosing an affine basis {Q i,j } Li j=1 for S i we can also characterize S i ∈ S i by a set of linear constraints Tr S i Q i,j = 1 for i = 1, . . . , K and j = 1, . . . , L i , and follow the same routine to tackle the constraints on the permutation-invariant subspace. Thus both the number of variables and the number of constraints are polynomial with respect to N .

Appendix E: Evaluation of QFI using different strategies
In this section we provide explicit formulas of the QFI for all strategy sets considered in the main text, in the forms which can be numerically solved by SDP. Without the positivity constraints, parallel, sequential and general indefinite-causal-order strategy sets are affine spaces themselves, while quantum SWITCH and causal superposition strategy sets are convex hulls of affine spaces. In some cases the result of Theorem 1 can be simplified a bit, as it is possible to trace over certain subspace while formulating the primal problem at the beginning.

Parallel strategies
When definite causal order is obeyed, a strategy can be described by a quantum comb [33,34,63]. The dual affine space is the set of dual combs without the positivity constraint [43]. For parallel strategies the primal problem can be written as s.t.P ≥ 0, P = 1 2,4,...,2N ⊗P (1) , Equivalently, the problem can be formulated as

(E2)
The dual problem is given by which simplifies the result directly obtained from Theorem 1 a bit. To solve the problem via SDP, we define a block matrix wherein 1(d) denotes a d-dimensional identity matrix, and The problem can be solved by SDP since h is incorporated linearly in the blocks of A.

Sequential strategies
For sequential strategies the problem can be written as (having traced over H 2N ) from which it follows that the dual problem is where Q (N −1) is Hermitian. Similarly, in order to solve the problem via SDP we rewrite it as having defined where Ṅ φ,j ⟩ = Ṅ φ,j ⟩ − i ∑ k N φ,k ⟩ h kj and { j⟩ , j = 1, . . . , d 2N } forms an orthonormal basis of H 2N .

Quantum SWITCH strategies
We first formally define a quantum SWITCH strategy set SWI as the collection of P ∈ Strat such that P = (ρ T,A,C ) * P (SW) ⟩⟨P (SW) , ρ T,A,C ≥ 0, Tr ρ T,A,C = 1, where ,F T π⟩ F C corresponds to a (generalized) quantum SWITCH for N operations, each permutation π is an element of the symmetric group S N whose order is N !, and { π⟩ C } forms an orthonormal basis. We suppose each H i for i = 1, . . . , 2N has the same dimension d 1 . H T ≃ H i denotes the input space of the target system, H A the ancillary space, and H C the space of the control system. Correspondingly, H F T , H F A and H F C denote the future output spaces of each part. The global future space Using the quantum SWITCH strategy set, after tracing over the global future space H F the QFI evaluation problem is written as where the superscript π of an operator denotes a permutation label, and the subscript denotes the subspace it lies in. Note that the primal set ofP is a convex hull of affine spaces. Equivalently the problem can be rewritten as Following the method in the proof of Theorem 1, the dual problem is given by min λ, Equivalently in an SDP form the problem is written as having defined for where Ṅ φ,j ⟩ = Ṅ φ,j ⟩ − i ∑ k N φ,k ⟩ h kj and { j π ⟩} forms an orthonormal basis of H 2π(N ) .
Symmetry reduction.-For SWI, by Theorem 2, h can be taken to be permutation-invariant and the constraint corresponding to one permutation π (e.g., the identity element of S N ) is sufficient. As explained in Appendix D we can decompose the permutation where { j 2N ⟩} forms an orthonormal basis of H 2N andṄ µ φ is given by Eq. (D11). If we define then by Theorem 2 we can reformulate the optimization problem as (E24)

(E25)
For each causal order in the superposition the dual affine space is the set of dual combs without the positivity constraint. Thus the dual problem is given by min λ, where the constraints hold for any π ∈ S N . To solve the problem via SDP we can formulate it as min λ,h λ, for where Ṅ φ,j ⟩ = Ṅ φ,j ⟩ − i ∑ k N φ,k ⟩ h kj and { j π ⟩} forms an orthonormal basis of H 2π(N ) . Symmetry reduction.-Similar to SWI, by Theorem 2, for Sup we take a permutation-invariant h and only need the constraint corresponding to the identity element of S N . As explained in Appendix D we can decompose the where { j 2N ⟩} forms an orthonormal basis of H 2N andṄ µ φ is given by Eq. (D11). If we define then by Theorem 2 we can reformulate the optimization problem as (E32)

General indefinite-causal-order strategies
In this case the explicit linear constraints on strategies have been derived in [21], and the dual affine space turns out to be the set of CJ operators of N -partite no-signalling quantum channels without the positivity constraint [16,43], mathematically defined by The intuitive interpretation for no-signaling channels is that locally the input of each channel only affects the output of this single channel, but cannot transmit any information to N − 1 other channels. To solve the QFI evaluation problem via SDP we can write it in the form min λ,Q,h λ, having defined Symmetry reduction.-For ICO, by Lemmas 1 and 2, both h and Q can be taken to be permutation-invariant. As explained in Appendix D we can decompose the permutation whereṄ µ φ is given by Eq. (D11), then by Theorem 3 we can reformulate the optimization problem as having removed the redundant constraints on Q by permutation symmetry. Similar to the case of Par, we only need to consider the constraint on Q on the permutation-invariant subspace, since both Q and Tr 2N −1,2N Q are permutationinvariant. Therefore, both the number of scalar variables and the number of constraints in terms of scalar variables are polynomial with respect to N .

Appendix F: Complexity analysis
Here we refer to the number of real scalar variables concerned in optimization as the complexity. In Appendix E we have presented both the original and the symmetry reduced QFI evaluation for all families of strategies considered in this work as applicable, from which we can obtain Table I in the main text. Now we consider the algorithm for identifying an optimal strategy. Since the algorithm is based on Theorem 1, its complexity is no less than that of the QFI evaluation. For the second step of the algorithm, by Lemma 4 there exists a permutation-invariant optimal strategy for all families of strategies except for the sequential one, and thus we can apply the group-invariant SDP to achieve the permutation-invariant solution. Recall thatP = ∑ K i=1 q i S i for S i ∈ S i . In fact, for SWI and Sup, we only need to characterize S 1 ∈ S 1 and obtainP = ∑ K i=1 1 K S 1 due to the permutation invariance. In summary, taking both steps of the algorithm into account, we obtain Table II:  N ). The asymptotic numbers of variables in optimization are compared between the original (Ori.) and group-invariant (Inv.) SDP. We denote d ∶= d1d2 for di ∶= dim(Hi) and s ∶= max φ rank(E φ ) ≤ d.
As a concrete example, we apply the group-invariant SDP to the evaluation of the QFI J (ICO) for the general indefinite-causal-order strategies up to N = 5. We consider the amplitude damping noise and take φ = 1.0, t = 1.0 and p = 0.5. As illustrated in Fig. 3, the growth of J (ICO) is faster than linear growth but slower than quadratic growth. Table III compares the complexity between the original and the group-invariant SDP and indicates that the symmetry reduced approach can save the computational resources dramatically.  Numerical results in this work are obtained by implementing SDP using the open source Python package CVXPY [65,66] with the solver MOSEK [67]. We plot the QFI for N = 3, amplitude damping noise in Fig. 4 and observe a similar hierarchy of the estimation performance using parallel, sequential and indefinite-causal-order strategies. Different from the N = 2 case presented in the main text, general indefinite-causal-order strategies indeed provide a small advantage over causal superposition strategies, which is presented in Table. IV.
On the other hand, analogous to the N = 2 case, simple quantum SWITCH strategies without any additional intermediate control operations could have advantage over any definite-causal-order strategies when the decay parameter p is small, but this advantage becomes more insignificant. This should not be surprising since the control can make a bigger difference as N grows. Hierarchy of QFI using parallel, sequential and indefinite-causal-order strategies for N = 3, the amplitude damping noise. We take φ = 1.0, fix the evolution time t = 1.0 and vary the decay parameter p. To see the gaps between different strategies more clearly we insert an inset plot where the decay parameter p ranges from 0.1 to 0.2.

Estimation of randomly sampled channels
To demonstrate the universality of the hierarchy of different families of strategies considered in the main text, we randomly sample noise channels drawn from an ensemble of CPTP maps defined by Bruzda et al. in [68]. In this work we only sample rank-2 qubit channels for N = 2, which is enough to show the hierarchy. The sampling process is implemented via an open source Python package QuTiP [69,70]. We set an error tolerance of 10 −8 , i.e., we claim J 1 > J 2 only if the gap is no smaller than 10 −8 . We find that for 984 of 1000 random channels, a strict hierarchy J (Par) < J (Seq) < J (Sup) < J (ICO) holds, implying that general indefinite-causal-order strategies can provide advantage over causal superposition strategies. In addition, we find that of the same 1000 channels J (Par) < J (SWI) for 34 channels and J (Seq) < J (SWI) only for 1 channel, so with a high probability quantum SWITCH strategies cannot outperform strategies following definite causal order for a random noise channel, which highlights the estimation enhancement from intermediate control in the general case.

Elusive advantage of sequential strategies in the asymptotic limit
We observe a gap between parallel and sequential strategies for amplitude damping channels and SWAP-type interactions for N = 2 and 3. Now we show that for both examples there is no advantage of sequential strategies asymptotically since there exists an h such that β = 0.
Appendix I: Implementation of optimal strategies with universal quantum gates In this section we apply Algorithm 1 in the main text to numerically solve for an optimal strategy in the set of sequential and causal superposition ones respectively. The CJ operator of an optimal sequential strategy corresponds to a sequence of isometries with a minimal ancilla-space implementation provided by [46], and can then be decomposed into single-qubit gates and CNOT gates [71]. By taking advantage of the freedom of choosing a parameter-independent unitary on the final output state, we can adjust the strategy to reduce the CNOT count without affecting the QFI. In terms of an optimal causal superposition strategy we follow the same routine for each sequential strategy branch respectively in the superposition.

Optimal sequential strategy
Let H 0 = C and H 2N +1 = H F , and we have the CJ operator of a sequential strategy P ∈ L H F ⊗ 2N i=0 H i (an (N + 1)-step quantum comb), as illustrated in Fig. 6. In this way of relabeling Hilbert spaces we have Concatenation of a sequential strategy and N quantum channels. P is the CJ operator of a sequential strategy and E φ is the CJ operator of a parameterized quantum channel. H0 = C is a trivial one-dimensional Hilbert space, and therefore the first step of the strategy is the process of state preparation.
According to Theorems 1 and 2 in [46], P corresponds to a sequence of isometries {V (k) } for k = 1, . . . , N + 1 by Stinespring dilation: for any input state ρ ∈ L ⊗ N i=0 H 2i , where the whole process corresponding to P is described by an isometry P ∈ L ⊗ N i=0 H 2i , ⊗ N i=0 H 2i+1 , and in each step a choice of isometry V (k) ∈ L (H 2k−2 ⊗ H A k−1 , H 2k−1 ⊗ H A k ) with minimal ancilla space is given by where H A k = Supp(P (k) * ) is an ancillary space given by the support of P (k) with H A0 = C, H i ′ is a copy of the Hilbert is an identity map from H 2k−2 to H (2k−2) ′ , and P (k−1) * − 1 2 denotes the Moore-Penrose pseudoinverse of P (k−1) * 1 2 with its support on H A k−1 . As the last isometry V (N +1) preserves the QFI, it is therefore only necessary to consider the implementation of P (N ) instead of the full strategy P (N +1) . From this explicit construction it follows that the minimal dimension of the ancilla space for implementing the sequential strategy P is dim(H A N ) = rank(P (N ) ). In the case of N = 2 for the amplitude damping noise considered in the main text, it is easy to see that dim(H A1 ) ≤ 2 and dim(H A2 ) ≤ 8, so V (1) is an isometry from 0 to (at most) 2 qubits and V (2) is an isometry from 2 to (at most) 4 qubits, as illustrated in Fig.  7. 7. A sequence of isometries corresponding to a sequential strategy for N = 2. The first qubit is the system qubit going through the channel E φ twice, while the three other qubits are ancillary.
Next, we apply a circuit decomposition of each isometry into single-qubit gates and CNOT gates. In practice it is often desirable to achieve a CNOT count as low as possible. Note that V (1) actually corresponds to the preparation of a 2-qubit state, which in general requires only one CNOT gate [72]. In terms of V (2) , an isometry from 2 to 4 qubits, the state-of-the-art decomposition scheme is the column-by-column approach which requires at most 54 CNOT gates [71]. However, as an arbitrary parameter-independent unitary on 3 ancillae can always be deferred and regrouped into V (3) and therefore does not affect the QFI, in fact we can choose a proper V (2) to further reduce the worst CNOT count to 47 without changing the QFI. To see this we need to briefly introduce the main ideas behind the column-by-column decomposition scheme.
An isometry V from m to n qubits (m ≤ n) can be represented in the matrix form by V = U † 1(2 n × 2 m ), where U † is a 2 n × 2 n unitary matrix and 1(2 n × 2 m ) is the first 2 m columns of the 2 n × 2 n identity matrix. If we obtain a decomposition of U † , then we can simply initialize the state of the first n − m qubits to 0⟩ to implement V . Equivalently we can find a decomposition of U such that U V = 1(2 n × 2 m ) and then inverse the circuit representing U . The idea is to find a sequence of unitary operations such that U = U 2 m −1 ⋯U 0 transforms V into 1(2 n × 2 m ) column by column. More specifically, we first choose a proper U 0 to map the first column of V to the first column of 1(2 n × 2 m ), i.e., U 0 V 0⟩ m = 0⟩ n , then choose U 1 satisfying U 1 U 0 V 1⟩ m = 1⟩ n as well as U 1 U 0 V 0⟩ m = 0⟩ n . . . until we determine U 2 m −1 .
Here we only focus on U 0 , the inverse of which can be seen as a process preparing a state V 0⟩ m from 0⟩ n . In terms of decomposing V (2) from m = 2 to n = 4 qubits, preparing a 4-qubit state in general requires 8 CNOT gates [73]. Fortunately, without changing the QFI, we have the freedom to choose a unitary U anc on the ancillae after applying V (2) such that the state V (2)′ 0⟩ 2 = U anc V (2) 0⟩ 2 can be prepared using only one CNOT gate. This can be seen by dividing the 4 qubits into two parties, including the single system qubit (in the space H S ) and the three ancillary qubits (in the space H A ), and taking the Schimidt decomposition of the 4-qubit state V (2) 0⟩ 2 where { e i f i ⟩ S A } forms an orthonormal basis of H S A , and {λ i } is a set of non-negative real numbers satisfying ∑ i λ 2 i = 1. Therefore, to prepare V (2) 0⟩ 2 , we only need a local unitary on H S to generate ∑ 1 i=0 λ i i⟩ S 0⟩ A , then apply one CNOT gate taking the system qubit as the control to obtain ∑ 1 i=0 λ i i⟩ S i⟩ A , and finally apply local unitary operations U S = ∑ i e i ⟩ S ⟨i S on the system and U A = ∑ i f i ⟩ A ⟨i A on the ancillae respectively. If we take U anc = U † A , then it is easy to see that V (2)′ 0⟩ 2 = U anc V (2) 0⟩ 2 = ∑ 1 i=0 λ i e i ⟩ S i⟩ A can thus be prepared using one CNOT gate. This choice of V (2)′ saves 7 CNOT gates compared to the general state preparation scheme, and leads to a worst CNOT count of 47 in total. Now we present numerical results of the circuit implementation of an optimal sequential strategy. The decomposition of ismometries is implemented using the Mathematica package UniversalQCompiler [74] based on the method described above. As in the main text, we consider the amplitude damping noise and take N = 2, φ = 1.0, p = 0.5 and t = 1.0. The circuits implementing V (1) and V (2)′ are illustrated in Fig. 8. The state preparation V (1) requires 1 CNOT gate and the intermediate control operation V (2)′ requires 33 CNOT gates.

Optimal causal superposition strategy
A causal superposition strategy for estimating N channels can be implemented by an N !-dim quantum control system entangled with N ! sequential strategies of applying the channels: where { π⟩ C } forms an orthonormal basis of the Hilbert space H C of the control system, and each P π = P π ⟩⟨P π is a sequential strategy. Once we obtain an optimal causal superposition strategy by applying Algorithm 1, we can apply the circuit decomposition for each sequential strategy in the superposition, following the method described in Appendix I 1. Taking account of the permutation symmetry, we can choose an optimal causal superposition strategy such that apart from the execution order of two channels, sequential strategies in the decomposition of the strategy are the same, containing the same state preparation and intermediate control.
As a concrete example, we again take N = 2, φ = 1.0, p = 0.5 and t = 1.0 for the amplitude damping noise and present numerical results of the circuit implementation of an optimal causal superposition strategy. As illustrated in Fig. 9, we use the qubit ψ⟩ C to coherently control which sequential order is executed. Due to the permutation invariance of the optimal strategy, we can simply control the query order of the identical channels while fixing V (1) and V (2) for all sequential orders. In view of this, generally we can use a (2N − 1)-quantum SWITCH to control the order of N channels E φ and N − 1 intermediate control operations V (i) .
FIG. 9. Sequences of isometries corresponding to each sequential order in the causal superposition for N = 2.
The first qubit of the circuit is the system qubit, and the query order of two identical channels E (1) φ and E (2) φ is entangled with the state of the control qubit ψ⟩ C . When ψ⟩ C is a superposition of the two states shown in the figure, the causal order is also in a superposition given by Eq. (I5).
Further decomposition shows that each sequential branch requires one CNOT gate for state preparation V (1) and 36 CNOT gates for intermediate control V (2) , as illustrated in Fig. 10.