Dynamical decoupling leads to improved scaling in noisy quantum metrology

We consider the usage of dynamical decoupling in quantum metrology, where the joint evolution of system plus environment is described by a Hamiltonian. We demonstrate that by ultra-fast unitary control operations acting locally only on system qubits, essentially all kinds of noise can be eliminated. This is done in such a way that the desired evolution is reduced by at most a constant factor, leading to Heisenberg scaling. The only exception is noise that is generated by the Hamiltonian to be estimated itself. However, even for such parallel noise, one can achieve an improved scaling as compared to the standard quantum limit for any local noise by means of symmetrization.

We consider the usage of dynamical decoupling in quantum metrology, where the joint evolution of system plus environment is described by a Hamiltonian. We demonstrate that by ultra-fast unitary control operations acting locally only on system qubits, essentially all kinds of noise can be eliminated. This is done in such a way that the desired evolution is reduced by at most a constant factor, leading to Heisenberg scaling. The only exception is noise that is generated by the Hamiltonian to be estimated itself. However, even for such parallel noise, one can achieve an improved scaling as compared to the standard quantum limit for any local noise by means of symmetrization. Introduction.-Quantum mechanics offers the promise to significantly enhance the precision of estimating unknown parameters as compared to any classical approach [1]. Such high-precision measurements are of central importance in physics and other areas of science, and include possible applications in frequency standards [2][3][4], atomic clocks [5][6][7], or gravitational wave detectors [8,9]. However, this quantum advantage seems to be rather fragile and can in general not be maintained in the presence of incoherent noise processes [10][11][12][13][14][15].
Identifying schemes that realize this advantage in practice are of high theoretical and practical relevance. The usage of quantum error correction was suggested in this context, which is however restricted to certain specific noise processes [16,17]. Limited control over the environment also allows for an improved scaling in certain situations [18,19], but may be difficult to realize.
Here we provide a practical scheme to maintain quantum advantage based on the usage of dynamical decoupling techniques [20,21], which have been shown to be applicable for storage and for the realization of quantum gates [22]. These techniques are nowadays widely used in various experimental settings [23][24][25][26][27][28][29][30][31]. With the help of ultra-fast control operations that act locally on the system qubits, we show that one can effectively decouple the system from its environment and fully protect it against decoherence effects, while at the same time maintaining its sensing capabilities. System and environment are described by a (pure) state, and interact via a coherent evolution governed by some Hamiltonian H SE . In addition, the sensing system is effected by a Hamiltonian H S that includes an unknown parameter that should be estimated. We assume that a coherent description is appropriate at all times, and that the evolution can be interrupted by (ultra)-fast control operations. This is similar to what is done in dynamical decoupling [20,22,32], or digital quantum simulation [33,34]. In practice, these intermediate pulses will not be infinitely fast, and only noise up to a certain frequency can be treated and eliminated this way. However, in what follows we will assume that the accessible time is much faster than any other timescale in the problem.
We show that for any local system-environment interaction of rank one or two one can completely eliminate the noise process at the cost of reducing the sensing ca-pabilities of the system by a constant factor leading to Heiseneberg scaling in precision. Only interactions that are of rank three-and hence necessarily contain a component that is generated by the system Hamiltonian that should be estimated-cannot be fully corrected. Still one can achieve that only this parallel noise part remains, and, even in this case one can still maintain a quantum advantage. In particular, for any local noise process, we show that one can achieve a super-standard quantum scaling in precision by preparing N qubit probes in the Greenberger-Horne-Zeilinger (GHZ) state and randomizing the qubits via fast, intermediate swap gates. For correlated noise processes we provide a general lower bound on the achievable precision, applicable even if one assumes perfect quantum control and auxiliary resources, demonstrating that one is in general restricted to the standard quantum limit. Even in this case, one may still improve the precision by a constant factor, however the exact effect depends on the details of the systemenvironment interactions and the type of fluctuations.
Background.-Quantum metrology is the science of optimally measuring and estimating an unknown parameter such as the frequency in an atomic clock or the strength of a magnetic field. In a typical metrology scenario, a system of N probes is subjected to an evolution for a certain time t with respect to a Hamiltonian where ω is the parameter to be estimated, and is subsequently measured. This process is repeated ν times and ω is estimated from the measurement statistics. The Cramér-Rao bound then provides a limitation on the estimation precision δω ≥ , which is attainable for large enough number of repetitions ν. Here F is the quantum Fisher information (QFI) [35]. For classical strategies (separable probe states) the ultimate precision is given by the standard quantum limit (SQL), F ∝ N . While using entangled states, one can achieve the so-called Heisenberg limit, F ∝ N 2 , i.e., a quadratic improvement as compared to any classical strategy [1].
However, in practice the system is not isolated but also interacts with its environment. In general, this leads to a noisy evolution, where it was shown that in case of incoherent noise described by a master equation, the quantum advantage is limited to a constant factor rather than a different scaling [10][11][12][13][14][15]. The way to describe noise processes crucially depends on the timescales of the prob-lem. Here we assume that we have ultra-fast access to the system, and can interject the evolution by fast control operations, much faster than the relaxation time of the environment. In this case it is appropriate to describe the dynamics of system plus environment in a coherent way, i.e., by means of an overall Hamiltonian that governs the unitary evolution of system plus environment. As we will show shortly, the fact that system and environment evolve coherently grants us with additional freedom that allows to maintain Heisenberg scaling even in the presence of uncontrolled interaction with some environment.
General setting.
-We now specify the exact form of the overall Hamiltonian describing the coherent evolution of both the system and environment. We begin by first considering the case of a single qubit. The most general evolution of a single qubit plus environment is described by the Hamiltonian where H S = ωσ 3 ⊗ ½ describes the evolution of the system, H SE describes the system-environment interaction with A j arbitrary environment operators and c j the coupling strengths. Here and throughout the remainder of this work σ j , j ∈ {1, 2, 3} denote the Pauli matrices, σ denotes the vector of Pauli matrices and σ n = n · σ.
In the case where we have N probe systems, the exact form of the Hamiltonian governing their coherent evolution with the environment depends on whether the N probes couple to individual environments, or to a common environment. In the former case the Hamiltonian is given by where A (a) j act on different Hilbert spaces. If the N probes couple to a common environment then the operators A (a) j are entirely unspecified allowing for both temporal and spatial correlations. Before proceeding to the results let us outline the decoupling procedure.
Decoupling strategy.-The most general dynamical decoupling strategy consists of applying an arbitrarily fast time sequence of unitary gates, i.e., intersecting the system evolution with fast pulses. Without loss of generality any strategy corresponds to a time ordered sequence of gates {u i } n 0 applied at times {0, t 1 , . . . , t n }. If this is done fast enough, one can use a first order Trotter expansion to describe the effective evolution as being generated by the Hamiltonian modulo an irrelevant final unitary, where we redefine the gates as U 1 = u 0 , U i+1 = U i u i and the probabilities are obtained from the time sequence p i = ti−ti−1 tn . This is similar to what is done in optimal control theory or bangbang control [20], with the requirement that the desired evolution H S is not completely eliminated.
Single qubit.-Notice that in the single qubit case the parametrization of completely positive trace preserving (CPTP) maps in Eq. (3) spans exactly all the unital maps, i.e. such that E(½) = ½ [36], and an arbitrary unital map can always be constructed from the set of generators {σ i } 3 i=0 . As all single qubit CPTP maps corresponds to affine transformations on the Bloch sphere, a unital CPTP map is uniquely specified by a real matrix A = RDR, where R,R ∈ SO(3), D = diag(η 1 , η 2 , η 3 ) [37], and the action of such map on any Pauli matrix is given by E(σ n ) = An·σ. The second rotationR corresponds to an inconsequential change of basis, so we assumeR = R T .
Noting that a noise term can be eliminated if and only if it belongs to the kernel of DR, on the other hand D = 0 since some part of the system evolution has to survive. Therefore in order to identify all the noises that can be removed it is sufficient to consider rank one projectors A = Π r in a general direction r = (r 1 , r 2 , r 3 ) T . It follows that the action of the corresponding map E = π r on Pauli matrices is given by π r (σ n ) = (r · n) σ r . Notice that this remains true even if one allows for auxiliary systems and intermediate unitary operations in Eq. (3) to act on the enlarged system. As the argument is a purely geometrical embedding everything in a larger dimensional space does not change the conclusions. Finally note that such a unital map π r can be easily implemented by applying U = σ r at regular intervals, so that the effective Hamiltonian is H eff = π r (H) = (H + σ r Hσ r )/2. Now for the single qubit we identify all the noises that can be removed by dynamical decoupling. As the noise term ½ ⊗ A 0 in Eq. (1) only acts on the environment and does not affect the system evolution, it cannot be altered by any control operations performed on the system alone.
In what follows we will ignore this term, but remark that it generally plays a role for the overall evolution unless all noise components can be canceled. The action of the decoupling strategy π r on the rest of the Hamiltonian in Eq. (1) leads to Consequently the noise can be effectively decoupled if and in which case the effective system evolution is slowed down by the factor r 3 = (1 + α 2 1 + α 2 2 ) −1/2 . In the case of bounded operators A j the Hamiltonian Eq. (1) can be put in the standard from where trB j B k = δ jk , {n 1 , n 2 , n 3 } is an orthonormal frame and b 1 ≥ b 2 ≥ b 3 are the ordered Schmidt coefficients, see supplementary material. This allows a more intuitive geometrical picture of dynamical decoupling. For any rank one or two noise, i.e. b 3 = 0, we can choose r = n 1 × n 2 , orthogonal to the plane where noise acts. The fact that the desired evolution σ 3 ⊗ ½ is not completely canceled requires that r · z = 0, which is equivalent to n 1 , n 2 = z. In this case, the noise is completely eliminated by dynamical decoupling and,as above, the system evolution H S = r 3 ω σ r ⊗ ½ is slowed down by a factor r 3 = z · (n 1 × n 2 ). The reduction of the coupling strength leads to a constant reduction of the achievable accuracy by (r 3 ) 2 but, as all noise is completely eliminated, we still obtain Heisenberg scaling precision. Notice that noise that is perpendicular to the system Hamiltonian, i.e., any combination of σ nx and σ ny noise, can be eliminated without altering the evolution, i.e. r 3 = 1. This is done by using fast intermediate σ nz pulses.
Single qubit and full rank noise.-From the above geometrical argument, it follows that one cannot eliminate rank three noise as such noise spans the whole three-dimensional space, and we can only eliminate a two-dimensional plane. To see this note that the effect of the decoupling procedure on any rank-three noise model is to project both the system Hamiltonian and noise onto direction r, so one obtains an effective Hamiltonian (4) for system plus environment which is unitarily equivalent to a σ nz evolution and parallel noise, with j r j c j A j = j (n j · r) b j B j for the standard form. As noise generated by the same operator as the system Hamiltonian cannot be eliminated without eliminating the system evolution as well, the best one can hope for in this case is to reduce any rank-three noise to noise parallel to the system evolution.
The choice of direction r in Eq. (4) determines both the effective coupling strength of the system Hamiltonian r 3 and the strength of the noise. One can then optimize r to optimize the ratio between the modified coupling strength r 3 and the variance of noise fluctuations after projection [38]. Later on we will show what this optimal ratio is for the case of local Gaussian noise.
-We now turn to the case of N two level systems. Consider first the case where each qubit encounters an independent environment which corresponds to a local noise process. The total Hamiltonian describing the evolution of all N qubits plus environment is given by Eq. (2), with the system evolution H S = ω a σ (a) 3 ≡ ωS 3 , and the system-environment interac- . One can use the above dynamical decoupling strategy independently on each of the systems so that the results of the previous section directly apply. For each qubit, noise of rank one or two can be eliminated, while full rank noise can be reduced to parallel noise H SE = ac In addition, one can randomize the system particles by means of fast intermediate permutations, where each permutation can be efficiently realized by O(N ) two-qubit swap gates. Random permutations leave H S unchanged, but project out all asymmetric noise terms onto their symmetric contribution [39]. Hence, the only remaining noise term is given by Notice that in generalĀ depends on the individual coupling strengthsc (a) unless allÃ (a) are identical. As we show later symmetrization of all system qubits can, in the presence of independent couplings or fluctuating coupling strengths, help boost precision to super-classical scaling. We remark that if the noise has no symmetric contributions thenc 3 = 0, and even locally full rank noise can be eliminated by symmetrization.
We now consider the case where the N qubits couple to a common environment, which may possess both temporal and spatial correlations. In this case the environment operators A (a) in Eq. (2) are unspecified. Let us first suppose that the system-environment interactions are such that each system qubit interacts individually with the environment. In principle, a similar strategy as illustrated in the single-qubit case can be applied, where one eliminates all noise except the one generated by the (symmetrized) system Hamiltonian itself by appropriate fast control operations. By way of example consider the following local decoupling strategy where one applies fast local σ (a) z on each of the qubits. This allows to eliminate all noise terms including σ (a) x , σ (a) y without altering H S , and together with fast random permutations reduces all noise to one generated by the system Hamiltonian itself, see Eq. (6). The only difference as compared to the case of independent environments treated above is now that the operatorsÃ (a) may act on the same environment. In general a more involved decoupling strategy requiring non-local operations may be needed in order to partially or fully remove the noise. However, it is not clear if all noise operators except those parallel to H S can be completely removed in this case as not all unital maps can be expressed as convex combinations of unitary operations [40]. Moreover, whatever the dynamical decoupling procedure, the condition that H S has a non-zero overlap with the kernel of the unital map must hold in order to be able to estimate ω.
One may also consider noise where several systems are affected simultaneously. From a physical standpoint such many-body noise processes are less important as they usually correspond to higher order processes. Nevertheless, these correlated noise processes can be eliminated by means of dynamical decoupling, and for any quasi-local noise process one still recovers Heisenberg scaling in the absence of noise generated by S 3 , see supplementary material.
Parallel noise.-Hitherto, we have seen how to eliminate all kinds of noise, except noise generated by S 3 . The latter is indistinguishable from the desired evolution, and can not be eliminated. However, we will now show that even such parallel noise does not automatically imply the SQL. In fact, the scaling of the QFI depends on the particular situation considered. For instance, if the noise is due to uncorrelated fluctuations of single-qubit noise terms, then a super-SQL scaling O(N 3/2 ) of the QFI can be achieved.
Consider the effect of the system plus environment evolution described by Eq. (6) on the system alone. Tracing out the environment in the eigenbasis {|ℓ } ofĀ 3 one can always represent the noise by the CPTP map where p(c 3 ) corresponds to fluctuations of the interaction strength between experimental runs, and f (ℓ) = ℓ| ρ E |ℓ depends on the initial state of the environment [41]. The effect of the system-environment coupling, when the environment is not in an eigenstate ofĀ 3 , is similar to a fluctuating interaction strength. In both cases, one has to average over evolutions governed by the same Hamiltonian as H S with a fluctuating parameter, where the latter is described by a suitable probability distribution. These fluctuations are what ultimately limit the achievable accuracy in parameter estimation, as they directly correspond to fluctuations of the parameter ω to be estimated. However, the resulting scaling strongly depends on the details of the situation, such as the spectrum of environment and whether these fluctuations are correlated or uncorrelated. We now consider some of these different cases.
The worst case is when the interaction strength,c 3 , is fixed but unknown (within a certain range). This type of noise leads to a systematic error on the estimated value of ω and there is no way to decrease the error below a certain value set by the initial knowledge of the interaction strength and the state of the environment (except the trivial case where the environment is in the zero eigenstate ofĀ 3 ).
We now turn to the case where the mean interaction strengthc 3 is known but fluctuates around the mean value between experimental runs following a smooth distribution p(c 3 ) andĀ 3 = ½. This is is equivalent to the case of a fixedc 3 but a continuous spectrum ofĀ 3 with smooth f (ℓ). We show in the supplementary material that for any p(c 3 ) the optimal QFI per unit time is upper bounded by where F cl (p(c 3 )) = p ′ (c3) 2 p(c3) dc 3 remains finite for every smooth noise distribution p(c 3 ) enforcing the SQL in this case. If p(c 3 ) is normally distributed with width σ the bound takes the simple form F /t ≤ N/σ, whereas a strategy utilizing an N qubit GHZ state 1 gives a maximal QFI per unit time F /t ≈ 0.43 N/σ for the optimal choice of t. Next consider local parallel noise, where each c in Eq. (6) is an independent and normally distributed random variable with width σ. After randomly permuting the probes one finds thatc 3 is also a normally distributed random variable whose widthσ is reduced by a factor √ N ,σ = σ √ N . Consequently, preparing the probes in the GHZ state yields a super-SQL precision in estimating ω where t opt = 1/ √ N σ 2 . Consequently, the Cramér-Rao bound δω ≥ (νF ) −1/2 = (T F /t opt ) −1/2 is attainable for large total running time T = νt opt . This demonstrates that the use of symmetrization of the noise operators allows one to significantly reduce the overall effects of noise (a fact that was also noted in [42,43]), and restore super-SQL scaling.
Finally, in the case wherec 3 is fixed andĀ 3 has a discrete spectrum, the effective noise distribution f (ℓ) is discrete and F cl (p(c 3 )) is unbounded. Consequently, the bound of Eq. (8) is trivial and no general statements can be made with regards to the optimal QFI per unit time. For example, ifĀ 3 has an equally gapped spectrum with gap ∆, then at time t = 2π ∆c3 the noise completely cancels. This final example, though artificial, demonstrates that one cannot provide general statements on achievable scaling without specifying further details of the type of fluctuations, interaction, spectrum, and initial state of the environment.
Summary.-We have shown that when an overall Hamiltonian description of the system plus environment is appropriate, ultra fast control allow one to alleviate a large class of noise processes, and recover Heisenberg scaling. We remark that the dynamical procedure outlined here can also be experimentally realized with finite duration control pulses as was shown in [44]. Ultimately, the only noise processes that forbid Heisenberg scaling precision are those generated by the system Hamiltonian to be estimated itself. The effect of such parallel noise strongly depends on the details of interactions, the spectrum of the environment and the type of fluctuation of the coupling parameter.
Our results are in stark contrast to situations where a master equation description of the system environment interaction is required. There, it has been shown that with the help of auxiliary systems and fast error correction only rank one Pauli noise processes can be eliminated, while even full quantum control including ultrafast pulses and quantum error correction do not allow one to go beyond SQL scaling [17]. Hence, our results provide a big promise for practical applications of quantum metrology in various contexts, opening the way towards ultrasensitive devices with widespread potential application in all branches of science.
which is real and symmetricÕ =Õ T (as imposed by the hermiticity of the HamiltonianC j =C † j ). Expressing the Pauli operators in a rotated frameσ = R σ allows one to rewrite the Hamiltonian as with C k = 3 j=1 R jkCk . Accordingly the overlap matrix for the operators O ik = trC i C k is given by Choosing the rotation that diagonalizes the symmetric matrixÕ = R diag(λ 1 , λ 2 , λ 3 ) R T leads to Which also shows that λ j ≥ 0, being the trace of the square of an Hermitian operator. Finally, denoting B j = 1 √ λj C j and b j = λ j allows one to rewrite the Hamiltonian as with trB j B k = δ jk .

Correlated noise
We now consider correlated noise processes where several systems are affected jointly. In general, the systemenvironment Hamiltonian of a N -qubit system is given by denotes a tensor product of Pauli operators. Using fast intermediate σ 3 operations on all qubits allows one to eliminate all terms containing σ 1 , σ 2 somewhere. We are then left with a Hamiltonian where j k ∈ {0, 3} and noise is solely diagonal. In case of localized noise, i.e., where there is a certain spatial structure and only qubits that are spatially close are jointly affected by noise, one can use fast intermediate σ 1 operations acting sparsely to eliminate noise terms of range k. For instance, performing such an action on every second qubit eliminates all nearest neighbor two-qubit noise terms in a 1-D setting. However, this also eliminates the desired evolution for half of the particles, and these particles no longer contribute to the sensing process. As long as the number of systems to be decoupled is given by αN with α being some constant-which is the case for any finite range k noise operators-we still obtain Heisenberg scaling O(α 2 N 2 ).

Parallel noise upper-bound QFI
In this section we derive a limitation on the maximally achievable QFI in presence of the parallel noise, i.e., noise that is described by the same generator as the Hamiltonian H that governs the evolution. Such parallel noise results in the channel E(ρ) = p(λ)e −itλH ρ e itλH dλ, (16) where λ is a random variable and p(λ) is a probability distribution with standard deviation σ characterizing the strength of the noise. As already mentioned such type of noise cannot be ameliorated using error correction as the operator generating it is identical to the Hamiltonian generating the desired evolution. The noise process in Eq. (16) can be viewed as describing classical noise applied directly on the estimated parameter ω, i.e., in every run of the experiment the observed parameter fluctuates by an amount λ, with λ being a random variable with corresponding probability distribution p(λ).
As the Uhlmann fidelity is strongly concave it follows that Eq. (18) is lower bounded by the fidelity of the probability distributions p(λ) and p(λ + dω). Consequently the QFI in the presence of parallel noise is bounded by F E(ρ) ≤ (p ′ (λ)) 2 p(λ) dλ = F cl p(λ) .
On the other hand we know that the QFI in the noisy case is lower that the noiseless QFI (atteined by the GHZ state), therefore F E(|ξ ) ≤ t 2 N 2 . Combining the two bounds one gets for the QFI per unit time It remains to find the time t that maximizes the r.h.s. Trivially the maximum is attained when tN 2 = F cl (p(λ)) t , which yields For any smooth distribution p(λ) the classical Fisher information F cl (p(λ)) is finite, and therefore SQL scaling for the QFI per unit time is enforced. In particular for a Gaussian noise with p(λ) = 1 √ 2πσ 2 e −λ 2 /2σ 2 this bound implies