Optimal approximate quantum error correction for quantum metrology

For a generic set of Markovian noise models, the estimation precision of a parameter associated with the Hamiltonian is limited by the $1/\sqrt{t}$ scaling where $t$ is the total probing time, in which case the maximal possible quantum improvement in the asymptotic limit of large $t$ is restricted to a constant factor. However, situations arise where the constant factor improvement could be significant, yet no effective quantum strategies are known. Here we propose an optimal approximate quantum error correction (AQEC) strategy asymptotically saturating the precision lower bound in the most general adaptive parameter estimation scheme where arbitrary and frequent quantum controls are allowed. We also provide an efficient numerical algorithm finding the optimal code. Finally, we consider highly-biased noise and show that using the optimal AQEC strategy, strong noises are fully corrected, while the estimation precision depends only on the strength of weak noises in the limiting case.

For a generic set of Markovian noise models, the estimation precision of a parameter associated with the Hamiltonian is limited by the 1/ √ t scaling where t is the total probing time, in which case the maximal possible quantum improvement in the asymptotic limit of large t is restricted to a constant factor. However, situations arise where the constant factor improvement could be significant, yet no effective quantum strategies are known.
Here we propose an optimal approximate quantum error correction (AQEC) strategy asymptotically saturating the precision lower bound in the most general adaptive parameter estimation scheme where arbitrary and frequent quantum controls are allowed. We also provide an efficient numerical algorithm finding the optimal code. Finally, we consider highly-biased noise and show that using the optimal AQEC strategy, strong noises are fully corrected, while the estimation precision depends only on the strength of weak noises in the limiting case.
Quantum mechanics places a fundamental limit on estimation precision, the Heisenberg limit (HL), where the estimation precision scales like 1/N for N probes; or equivalently, 1/t for a total probing time t. In the noiseless case, the HL is achievable using the maximally entangled state among probes [1,35]. In practice, decoherence plays an indispensible role. Under many typical noise models, the estimation precision will follow the standard quantum limit (SQL) with scaling 1/ √ N (or 1/ √ t) [29-31, [36][37][38][39][40][41], the same as the central limit theorem scaling using classical strategies. Nevertheless, the superiority of quantum strategies over classical strategies by a constant-factor improvement, as opposed to a scaling improvement, was proven in several cases [11,37,40]. There were also situations where the HL is achievable using quantum strategies even in the presence of noise [16,31].
Due to the difficulty in obtaining the exact precision limits for general noise models using different quantum strategies, several asymptotical lower bounds have been proposed [29-31, [36][37][38][39][40][41][42]. For example, the channel simulation method was used to prove the SQL lower bound for programmable channels [38][39][40]. A necessary and sufficient condition of achieving the HL under Markovian noise was established using the channel extension method [29][30][31]. Although these bounds have been successful at showing the scaling limit of quantum strategies, only in several special cases, the saturability of these lower bounds was established, e.g. for dephasing and erasure noise [40] and for teleportation-covariant channels as a special type of programmable channels [43,44]. A satura-bility statement of the SQL lower bound under general noise models and an efficient algorithm solving the optimal strategy remain missing up to the present day.
We address both of these open questions in this work. Here we consider parameter estimation under general Markovian noise using the most general adaptive sequential strategy (see Fig. 1). We propose an approximate quantum error correction (AQEC) strategy saturating the SQL lower bound of precision (asymptotically) and an efficient numerical algorithm solving the optimal AQEC codes for different noises. The saturability of the precision lower bound we prove here not only answers an important question in quantum metrology theory, but also paves the way for identifying the optimal quantum strategies in future experiments.
Quantum error correction (QEC) [45] was first shown useful in quantum metrology in a typical scenario where the dephasing noise in a qubit probe is corrected by QEC, while the X (the Pauli-X operator) signal remains intact [23][24][25][26]. Later on, the result was generalized to arbitrary Markovian noise [30,31], stating that the HL is achievable using the sequential QEC strategy if and only if the HNLS condition is satisfied, i.e. the signal Hamiltonian is not in the Lindblad span-an operator subspace defined using Lindblad operators [46][47][48]. In practice, however, HNLS is often violated and the estimation precision is limited by the SQL, for example, sensing any single-qubit signal under depolarizing noise. Standard QEC strategies would be useless in this case, as the signal will be completely eliminated if the noise is fully corrected. However, here we show that by performing the QEC in an approximate fashion, the highest possible precision limit is achievable, marking another triumph of the QEC strategy in quantum metrology.
In this Letter, we first review the SQL precision lower bound under Markovian noise using the sequential strategy when HNLS is violated. Then we describe our AQEC strategy consisting of both a two-dimensional AQEC code and an optimal recovery channel. This allows the original quantum channel to be reduced to an effective channel where a Z (the Pauli-Z operator) signal was sensed under dephasing noisea special case where the precision lower bound was known to be saturable [11,37,40]. Finally, we optimize the achievable precision over all possible AQEC codes, which coincides with the precision lower bound, completing the proof.
FIG. 1. The most general adaptive sequential strategy where one probe sequentially senses the parameter for time t, with quantum controls (arbitrary completely positive and trace-preserving (CPTP) maps) applied every dt and an arbitrary number of noiseless ancillas available.
describes the evolution of the probe in an infinitesimally small time interval dt.
Precision lower bound.-We assume the evolution of the quantum system is described by the following quantum master equation [46][47][48]: where ω is the unknown parameter, ρ ∈ H S ⊗ H A , H S is the probe space H and {L i } r i=1 acting on (H, L i are shorthand for H ⊗ 1, L i ⊗ 1, respectively) and H A is the noiseless ancillary space (see Fig. 1). We assume I, {L i } r i=1 are linearly independent, dim H S = d and dim H A = 2d. The Lindblad span associated with Eq. (1) is S = span{1, L i , L † i , L † i L j , ∀i, j}, where span{·} denotes the real linear subspace of Hermitian operators spanned by {·}. According to the quantum Cramér-Rao bound [49][50][51][52], the standard deviation δω of the ω-estimator is bounded by δω ≥ (N expr F (t)) −1/2 , where N expr is the number of experiments and F (t) is the so-called quantum Fisher information (QFI) as a function of the final state ρ(t). The bound is asymptotically saturable using the maximum likelihood estimator as N expr goes to infinity [53,54]. Therefore, finding the optimal sequential strategy boils down to maximizing F (t) over all input states and quantum controls. For an input state |ψ evolving noiselessly under Hamiltonian ωH, F (t) = 4t 2 ( ψ| H 2 |ψ −( ψ| H |ψ ) 2 ) and δω ∝ 1/t follows the HL. In the noisy case, it was proven that the HL is achievable if and only if H / ∈ S (the HNLS condition) and there exists a QEC strategy achieving the HL [30, 31].
The HNLS condition holds usually when the noise has a special structure, e.g. rank-one noise [29] or spacially correlated noise [32,33]. For generic noise, however, the HNLS condition is often violated. In this Letter, we focus on the latter situation where H ∈ S and the QFI follows the SQL [30, 31]: where · is the operator norm of a matrix, h ∈ R, h ∈ C r , h ∈ C r×r is hermitian, where L := (L 1 , L 2 , . . . , L r ) T and h1 := (h 1 1, . . . , h r 1) T .
Here we introduce an AQEC strategy which (asymptotically) saturates the QFI upper bound up to an arbitrarily small error under arbitrary Markovian noise. That is, for any small δ > 0, there exists an AQEC strategy such that where we define the normalized QFI F as the objective function we maximize. The upper bound is saturated asymptotically in the sense that lim t→∞ F (t)/t = F. Approximate quantum error correction.-Here we propose a set of AQEC codes for quantum metrology and show that the effective channel under fast AQEC is an effective qubit dephasing channel in the logical space. In this way, identifying the optimal recovery channel for quantum metrology is equivalent to minimizing the noise rate of the dephasing channel where a closed-form solution exists, as opposed to generic AQEC scenarios where many known AQEC recovery channels are only suboptimal [55][56][57][58][59][60].
Let P be the projection on to the code space |0 L 0 L | + |1 L 1 L |, where |0 L and |1 L are the logical zero and one states. Applying the AQEC quantum operation P + R • P ⊥ infinitely fast, the effective evolution would be (up to the first order of dt [31, 33]) where P ⊥ = 1 − P , P(·) = P (·)P , P ⊥ (·) = P ⊥ (·)P ⊥ and R is a CPTP map describing the AQEC recovery channel. We define the following class of AQEC codes where A 0 , A 1 ∈ C d×d and A 0/1,ij = C ij ± εD ij satisfy Tr(A 0 A † 0 ) = Tr(A 1 A † 1 ) = 1 and Tr(C † D) = 0. Here C describes the part of the code which |0 L and |1 L have in common and D describes the part distinguishing |0 L from |1 L which generates non-zero signal and noise. In the special case where ε = 0, the effective signal and noise are zero. Let H A = H A ⊗ H 2 where dim H A = d and dim H 2 = 2, the last ancillary qubit in H 2 makes the signal and noises both diagonal in the code space, i.e. 0 L |H|1 L = 0 L |S|1 L = 0 for all S ∈ S. Later on, we will assume ε is a small parameter and consider the perturbation expansion of the effective dynamics around ε = 0. We consider the recovery channel restricted to the structure (we will show that this type of recovery channels is sufficient for our purpose) where {|R m }, {|S m } ⊂ H S ⊗H A are two sets of orthonormal basis and R is CPTP. A few lines of calculation shows the effective channel (Eq. (6)) under the AQEC code (Eq. (7)) and the recovery channel (Eq. (8)) is We can remove the term H S in Eq. (9) by applying a reverse Hamiltonian constantly [29]. For dephasing channels, the optimal F is reached using a special type of spin-squeezed state as the input [9-11, 37, 40], where we have To simulate the evolution of multipartite spin-squeezed states using the sequential strategy where we have only a single probe, one could first prepare the desired spin-squeezed state in For simplicity in furture calculation, we perform a two-step gauge transformation on the Lindblad operators is a diagonal matrix. Note that above transformations only induce another parameterindependent shift H S in the Hamiltonian which could be eliminated by a reverse Hamiltonian. Now we have a new set of and we replace in Eq. (10). First, we maximize F over the recovery R, which is equivalent to minimizing γ(R) over R. We claim that the minimum noise rate γ = min R γ(R) is where we have used max U :U † U =1 Tr(M U + M † U † ) = 2 M 1 for arbitrary square matrices M and U , where · 1 is the trace norm (see details in Appx. A of [61]). Next, we could like to maximize F over all possible AQEC codes of the form Eq. (7). It is not clear yet how that could be done mathematically with the presence of trace norm in the denominator. To arrive at an expression of γ free of the trace norm, we further sacrifice the generality of our AQEC code and assume ε 1. We call it the "perturbation" code in the sense that the signal and the noise are both infinitesimally small when ε → 0. Under the limit ε → 0, we have Tr(HZ L ) = 2εTr(HC) + O(ε 2 ), wherẽ and the noise rate is (ignoring all o(ε 2 ) terms) For a detailed derivation of the noise rate, see Appx. B of [61] and [62]. Finally, we have the following expression of the normalized QFI (up to the lowest order of ε) as a function ofC and C (implicitly through the choice of . The effective dynamics of the perturbation code has the feature that both the signal and the noises are equally weak and only the ratio between them matters. Therefore the exact value of ε will not influence the normalized QFI F as long as it is sufficiently small. On the other hand, it does influence how fast F (t)/t reaches its optimum F, characterized by a coherence time O(1/ε 2 ).
Saturating the bound.-Now we maximize the normalized QFI (up to the lowest order of ε) over C andC and show that the optimal F is exactly equal to its upper bound in Eq. (2). The domain of C is all complex matrices satisfying Tr(C † C) = 1. We assume the domain ofC is all traceless Hermitian matrices satisfying Tr(J † i J jC ) = 0 for all i, j ∈ n := {i|λ i = 0}. When C is full-rank, n is empty and for arbitrary tracelessC, we could always take 14) is satisfied. When C is singular, we could replace it with an approximate full-rank version (e.g. C ← C + δ1). In this case, F will only be decreased by an infinitesimal small amount when ε = o(δ 2 ) because the numerator in Eq. (16) is only slightly perturbed after the replacement.
Consider the following optimization problem over h, h, h and C, Fixing C, we introduce a Hermitian matrixC as the Lagrange multiplier associated with the constraint β = 0 [63]. Strong duality implies Eq. (17) has the same solution as the following dual program (see Appx. C of [61]) max C,C F(C,C), subject to Tr(C † C) = 1, Tr(C) = 0, and Tr(J † i J jC ) = 0, ∀i, j ∈ n.
(18) whose optimal value could be achieved using the perturbation code up to an infinitesimal small error according to the discussion above. On the other hand, thanks to Sion's minimax theorem [64,65], we can exchange the order of the maximization and minimization in Eq. (17) because we could always confine (h, h) in a convex and compact set (see Appx. D of [61]) such that the solution of Eq. (17) is not altered and the objective function 4Tr(C † αC) is concave (linear) with respect to CC † and convex (quadratic) with respect to (h, h). Therefore, the optimal value of Eq. (18) is also equal to 4 min h,h,h|β=0 α , the upper bound of the normalized QFI. Numerical algorithm.-It is known that the upper bound in Eq.
(2) could be calculated via a semidefinite program (SDP) [30,42], where and " 0" means positive semidefinite. However, the minimax theorem does not guaranteed an efficient algorithm to solve Eq. (18) after exchanging the order of the maximization and minimization in Eq. (17). Now we provide an efficient numerical algorithm obtaining an optimal (C ,C ) in three steps. The validity of this algorithm is proven in Appx. E of [61]. The algorithm runs as follows: (a) Solving min h,h,h|β=0 α using the SDP gives us an optimal α (and corresponding h , h , h ) satisfying α = min h,h,h|β=0 α . (b) Suppose Π is the projection onto the subspace spanned by all eigenstates corresponding to the largest eigenvalue of α , we find an optimal C C † satisfying Π C C † Π = C C † and for all (∆h, ∆h) such that ∆h1+∆h † L+L † ∆h+L † ∆hL = 0 for some ∆h. Note that this step is simply solving a system of linear equations. (c) Find are Hermitian, M h,ah 0 ∈ S 0 and M h,ah ⊥ S 0 (in terms of the Hilbert-Schmidt norm). Using the vectorization of matrices |· = jk j| (·) |k |j |k , let .
According to the Cauchy-Schwarz inequality, and the optimal |C = B −1 |H h . Here −1 means the Moore-Penrose pseudoinverse.
Highly-biased noise.-We consider a special case where noises are separated into two groups -strong ones and weak ones [31][32][33]. To be specific, we consider the following quantum master equation where the indices of Lindblad operators {L i } r i=1 are separated intos and s, representing weak and strong noises respectively. η 1 is a small parameter characterizing the relative strength of the weak noises. Moreover, we assume that H / ∈ span{1, L i , L † i , L † i L j , i, j ∈ s} so that it is possible to fully correct all strong noises and also preserve a non-trivial signal in the code space. Taking η → 0, it is easy to show that (see Appx. F of [61]), the optimal F in this case is equal to whereᾱ = (h1 + hL) † Πs(h1 + hL) and Πs is a diagonal matrix whose i-th diagonal element is one when i ∈s and zero when i ∈ s. This reduces the running time of the SDP in Eq. (19) by reducing A from a d(r + 1) × d(r + 1) matrix to a d(|s| + 1) × d(|s| + 1) matrix. Using the optimal AQEC strategy, F is boosted by a factor of O(1/η), compared to the case where no QEC is performed. To find the optimal AQEC code, we can solve the dual program of a modified version of Eq. (17) where α is replaced withᾱ: Conclusions and outlook.-In this Letter, we proposed an AQEC strategy such that the optimal SQL in Hamiltonian parameter estimation could be achieved asymptotically. An interesting open question is whether the perturbation code we introduced here could be turned into non-pertubative ones. We provide an example in Appx. G of [61], where by slightly modifying the ancilla-free QEC code (non-perturbation) proposed in Ref.
[33], we show that the optimal F could be achieved in the correlated dephasing noise model. However, it is unclear how to generalize the result to generic noise models.
Another two interesting open questions are (1) how to characterize the power of QEC in improving quantum metroloy for parameters encoded in generic quantum channels [39,40], for example when the rate of quantum controls is constant, rather than infinitely fast; (2) how to optimize the QEC strategy when considering a constant probing time, rather than an infinitely long probing time.
Appendix B: Perturbation expansion of the noise rate γ In this appendix we expand the minimum noise rate γ around ε = 0 using the perturbation code. For simplicity, the equal sign "=" in this appendix means approximate equality up to the second order of ε (ignoring all o(ε 2 ) terms). We also state a useful lemma here: Lemma 1 ( [68]). X + εY 1 = X 1 + O(ε) for arbitrary X and Y .
To calculate Eq. (13), we first consider the terms independent of R, − Re The remaining term is equal to (thanks to Lemma 1) minus where Λ ∈ R r×r is a diagonal matrix whose k-th diagonal element is λ k andΛ ∈ R r×r is a diagonal matrix whose k-th diagonal element is λ k if λ k > 0 and 1 if λ k = 0. Assume {λ k } r k=1 is arranged in a non-ascending order and r 0 is the largest integer such that λ r0 is positive. X 1 , X 1 ∈ C r×r satisfy for 1 ≤ j ≤ r 0 and for r 0 + 1 ≤ j ≤ r. X 2 , X 2 ∈ C (d 2 −r)×r satisfy for r + 1 ≤ j ≤ d 2 − 1 and Here are two sets of orthonormal basis of H S ⊗ H A .
To calculate the first and second order expansion of Eq. (B2), we consider the singular value decompositions Then where and Π Λ is the projector onto the support of Λ. Using Theorem 2 in Ref. [62], we have Note that therefore Here we show the Lagrange dual program of Eq. (17) is Eq. (18). From the definition of α (Eq. (3)) and β (Eq. (4)), we see that the upper bound in Eq. (2) is invariant under the transformation L → J, that is, after the transformation L → J there is always another set of (h, h, h) such that β = 0 and α is the same. Therefore we let where J = (J 1 , J 2 , . . . , J r ) T . To proceed, we simplify the notations by letting . (C3) Note that the r-dimensional vector j is to be distinguished from the index j, then we have and 4Tr(C † αC) = 4(h † h + Tr(Λh 2 )).

Appendix D: Confining (h, h) in a compact set
The minimax theorem [65] states that for convex compact sets P ⊂ R m and Q ⊂ R n and f : P × Q → R such that f (x, y) is a continuous convex (concave) function in x (y) for every fixed y (x), then In Eq. (17), the operator CC † satisfying Tr(CC † ) = 1 is contained in a convex compact set, but the domain of (h, h, h) is not compact. Here we show that we could always confine (h, h) in a convex and compact set such that the solution of Eq. (17) is not altered. First we note that min h,h,h|β=0 α = a < ∞ when H ∈ S. Note that It is clear that there exists some b > 0 such that for all (h, h) 2 > b ( · 2 is the Euclidean norm), we have α > a. Therefore it is easy to find some b > 0 such that and has the same optimal value equal to 4a, and there exists a saddle point (h * , h * , C * ) such that for all (∆h, ∆h) satisfying ∆h1 + ∆h † L + L † ∆h + L † ∆hL = 0 (D7) for some ∆h. Therefore, (h * , h * , C * ) is also a saddle point of Eq. (17): subject to ∃h, β = 0, Tr(C † C) = 1, proving that the optimal value of Eq. (17) must also be equal to 4a.

Appendix E: The validity of the numerical algorithm
Here we prove the validity of the three-step algorithm introduced in the main text. Let (h * , h * , C * ) be the saddle point of Eq. (17). The first inequality in Eq. (D5) implies which means that Π * C * = C * where Π * is the projection onto the subspace spanned by all eigenstates corresponding to the largest eigenvalue of α * . Now assume we have a solution (h , h ) of Eq.

Appendix F: Highly-biased noise
We derived the optimal F and the corresponding optimal AQEC code under the highly-biased noise model, taking the limit η → 0. Using the highly-biased model (Eq. (23)), we need to replace L by (Π s + Πs √ η)L in Eq.
(2), where Πs (or s) is an r-by-r diagonal matrix whose i-th diagonal element is one when i ∈s(or s) and zero when i ∈ s(ors). After the following parameter transformation in Eq.
(2), we have the optimal QFI is equal to F = min h,h,h|β=0 α , where α = (h1 + hL) † (Π s + Πs/η)(h1 + hL), Lettingᾱ = (h1 + hL) † Πs(h1 + hL), we have where min h,h,h|β=0 ᾱ > 0 as long as H / ∈ span{1, L i , L † i , L † i L j , i, j ∈ s}. Now consider the dual program of the modified version of Eq. (17) with α replaced byᾱ. We first simplify the calculation by performing a gauge transformation such that the new set of Lindblad operators J satisfies Tr(C † J i C) = 0, J ss is diagonal with the i-th diagonal element equal to λ i when i ∈ s and zero when i ∈s, and its Schur complement Jss − Js s J −1 ss J ss is diagonal with the i-th diagonal element equal to λ i when i ∈s and zero when i ∈ s. Here −1 means the Moore-Penrose pseudoinverse and we use the notations (·) = Π (·)Π for , =s, s and J ij = Tr(C † J † i J j C). Note that the gauge transfromation here is divided into two steps (1) L i ← L i − Tr(C † L i C)1 and (2) L ← (uΠ s + vΠs)L where u = Π s uΠ s and v = ΠsvΠs are some unitary operators within the subspaces defined by Π s and Πs. In this way, the solution is invariant. Again, we introduce a Hermitian matrixC as a Lagrange multiplier, the Lagrange function is L(C, h, h, h) = 4Tr(C † (h1 + hJ) † Πs(h1 + hJ)C) + Tr(C(H + h1 + J † h + h † J + J † hJ)). (F5) under the constraints that Π ns j ss = 0, Π ns J ss = 0, Π ns jss − Js s J −1 ss j ss − js s J −1 ss J ss Π ns = 0, where Π ns,ns are projection operators defined by n s = {i ∈ s|λ i = 0}, ns = {i ∈s|λ i = 0}. Otherwise, ( * ) = −∞. Note that the second constraint Π ns J ss = 0 is automatically satisfied by definition.
To conclude, the dual program after replacing α byᾱ is equal to where −1 means the Moore-Penrose pseudoinverse, and (G3) Now we introduce a QEC code where θ = 1 2 arccos χu, defined element-wise, satisfying and P (v · Z)(v · Z)P ∝ P for any v and v . χ is a tunable parameter ∈ (0, u −1 ∞ ] where · ∞ is the infinity norm. The QEC code is designed to correct every mode v perpendicular to u. Using the recovery channel introduced in Appx. F of [33], we would have an effective channel Using spin-squeezed states as input states, we could achieve the optimal F because F = 4χ 2 (u T w) 2 2χ 2 (u T Γu) ≤ 2wΓ −1 w, where the second equality holds when (u T Γu)(w T Γ −1 w) = (u T w) 2 ⇔ u ∝ Γ −1 w.
Note that F = 2wΓ −1 w could be very large if there exist an i such that µ i (v i · w) 2 .