A Phase and frequency estimation

# Improved quantum metrology using quantum error-correction

## Abstract

We consider quantum metrology in noisy environments, where the effect of noise and decoherence limits the achievable gain in precision by quantum entanglement. We show that by using tools from quantum error-correction this limitation can be overcome. This is demonstrated in two scenarios, including a many-body Hamiltonian with single-qubit dephasing or depolarizing noise, and a single-body Hamiltonian with transversal noise. In both cases we show that Heisenberg scaling, and hence a quadratic improvement over the classical case, can be retained. Moreover, for the case of frequency estimation we find that the inclusion of error-correction allows, in certain instances, for a finite optimal interrogation time even in the asymptotic limit.

###### pacs:
03.67.-a, 03.65.Ud, 03.65.Yz, 03.65.Ta

Introduction.— Parameter estimation is a problem of fundamental importance in physics, with widespread applications in gravitational-wave detectors Caves (1981); McKenzie et al. (2002), frequency spectroscopy Wineland et al. (1992); Bollinger et al. (1996), interferometry Holland and Burnett (1993); Lee et al. (2002), and atomic clocks Valencia et al. (2004); de Burgh and Bartlett (2005). Quantum metrology offers a significant advantage over classical approaches, where the usage of quantum entanglement leads to an improved scaling in the achievable precision Giovannetti et al. (2004); Huelga et al. (1997). However, noise and decoherence jeopardize this effect, reducing the quadratic improvement with system size to only a constant gain factor in many scenarios Huelga et al. (1997); Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); ?.

General upper bounds on the possible gain have been derived suggesting that no improvement in the scaling of precision is possible in the presence of uncorrelated, Markovian noise including local depolarizing or dephasing noise Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); ?. For non-Markovian noise Matsuzaki et al. (2011); ?, and noise with a preferred direction transversal to the Hamiltonian evolution Chaves et al. (2013), a scaling of and was found respectively, where denotes the number of probes (see also Dorner (2012); ?; ?; ? for results on correlated noise). This is, however, still below the quadratic improvement attainable in the noiseless case. Moreover, for frequency estimation the optimal interrogation time, i.e. the optimal time to perform the measurement, tends to zero for large in both these cases making a physical realization for large impractical.

In this letter we show that, by relaxing the restrictions implicit in standard quantum metrology, namely that the only systems available are the probes, and the unitary dynamics are generated by local Hamiltonians, the no-go results for the case of uncorrelated, Markovian noise Huelga et al. (1997); Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); ?; Chaves et al. (2013) can be circumvented, and Heisenberg scaling can be restored. Specifically, by encoding quantum information into several qubits one can effectively reduce noise arbitrarily at the logical level thereby retaining the Heisenberg limit in achievable precision. The required overhead is only logarithmic, i.e. each qubit is replaced by qubits. Moreover, we show that in the case of frequency estimation the optimal interrogation time in certain scenarios considered here is finite and independent of the system size, in stark contrast to all frequency estimation protocols studied to date. As the methods we employ can be readily implemented experimentally, at least for moderate system sizes, our result paves the way for the first feasible experimental realization of Heisenberg limited frequency estimation.

To be more precise, let us consider a system of qubits which we imagine to be decomposed into blocks of qubits with odd (see Fig. 1). First, we consider a class of many-body Hamiltonians, , acting on each of the blocks, and uncorrelated, single-qubit dephasing or depolarizing noise (scenario I). Here, and in the following, , denote the Pauli operators. We show that, depending on the number of probe systems, , one can choose a sufficiently large (not exceeding ) such that the Heisenberg limit is achieved even in presence of noise and that the optimal measurement time is constant. Furthermore, we generalize this model to arbitrary local noise and show that for short measurement times the Heisenberg limit can be retrieved. Whereas this model may appear somewhat artificial, it nevertheless serves as a good example to illustrate how quantum error-correction can be used to restore the Heisenberg scaling.

The second, and more physically important, scenario we consider is that of a local Hamiltonian, , and local, transversal -noise on all qubits. We show that this scenario can be mapped, for short times, to scenario I, and hence demonstrate how quantum error-correction (and other tools) can be used to arbitrarily suppress noise and restore Heisenberg scaling in precision just as in the noiseless case 1. The key idea of our approach lies in the usage of auxiliary particles to encode and protect quantum information against the influence of noise and decoherence as done in quantum error-correction. In addition, the encoding needs to be chosen in such a way that the Hamiltonian acts non-trivially onto the encoded states, such that the information on the unknown parameter is still imprinted onto the system. As long as is many-body and the noise is local (scenario I), or the Hamiltonian is local and the noise is transversal (scenario II), both conditions can be met simultaneously.

Background.—We begin by describing the standard scenario in quantum metrology. A probe is prepared in a possibly entangled state of particles and subsequently undergoes an evolution that depends on some parameter, , after which it is measured. This process is repeated times and is estimated from the statistics of the measurement outcomes. The achievable precision is lower-bounded by the quantum Cramér-Rao bound Braunstein and Caves (1994), with the quantum Fisher information (QFI). For local Hamiltonians and uncorrelated (classical) probe states, , leading to the so-called standard quantum limit. Entangled probe states, such as the GHZ state, lead to , i.e. a quadratic improvement in precision, the so-called Heisenberg limit. In frequency estimation, time is also a variable that can be optimized, and the quantity of interest in this case is given by . We refer the reader to Appendix A for details.

In the presence of noise, however, a number of no-go results show that for many uncorrelated noise models, including dephasing and depolarizing noise, the possible quantum enhancement is limited to a constant factor rather than a different scaling with  Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); ?. To be more specific, we describe the time evolution of the state by a master equation of Lindblad form

 ˙ρ(t)=−iλ[H,ρ]+N∑j=1Lj(ρ), (1)

where the action of the single qubit map is given by

 Ljρ=γ2(−ρ+μxσ(j)xρσ(j)x+μyσ(j)yρσ(j)y+μzσ(j)zρσ(j)z), (2)

and denotes the strength of the noise. The choice and corresponds to local unitary evolution and local, uncorrelated, and commuting dephasing noise scenario considered in Huelga et al. (1997), whereas for the same Hamiltonian the choice corresponds to transversal noise considered in Chaves et al. (2013). The choice corresponds to local depolarizing noise. We remark that this approach includes phase estimation for fixed , and frequency estimation when can be optimized.

For any such scenario investigated so far the attainable precession scales worse than , and the optimal interrogation time tends to zero whenever the noise is not vanishing (see Appendix B for details).

Quantum metrology with error-correction.—We now demonstrate that error-correction can be used to recover the Heisenberg limit in the presence of noise in the two scenarios (scenario I and II) mentioned above. For the case of frequency estimation we show that, in certain scenarios, our technique asymptotically allows for a finite, non-zero optimal time to perform measurements in contrast to all current metrological protocols.

Scenario I.—The evolution of the qubits is governed by the class of Hamiltonians (see Fig. 1) , where acts on block . We assume locality with respect to the blocks, i.e. this situation is equivalent to having , d-level systems with . We describe the overall dynamics by Eq. (1), where the decoherence mechanism is modeled by Eq. (2). In the noiseless case (), the maximal attainable QFI is given by and is obtained by a GHZ-type state, , with and .

Let us now consider the standard metrological scenario in the presence of local dephasing noise, acting on all qubits, where the noise operators commute with the Hamiltonian evolution. In this case Eq. (1) can be solved analytically and the resulting state is given by , where and , with , are acting on all physical qubits. Phase estimation corresponds to the case where , for some fixed time , and the parameter to be estimated is resulting from the unitary evolution for time . Note that in this case one can start directly with the equation for , with being time independent, and a time independent gate (see Appendix B). As the subsequent discussion is independent of whether is time dependent or not, we simply write in the following whenever it does not lead to any confusion.

We now encode each logical qubit in physical qubits. On each block of qubits we make use of an error-correction code, similar to the repetition code, capable of correcting up to phase-flip errors (recall that we chose to be odd), with code words , , where . The error-correction procedure consists of projecting onto subspaces, , spanned by , where with . Here, denotes the qubit local operator, . After obtaining outcome the correction operation is applied. As long as fewer than errors occur we obtain no error at the logical level. Otherwise, a logical error occurs. Hence, the noise at the logical level can again be described as logical phase-flip noise, , with

 pL=m−12∑k=0(mk)pm−k(1−p)k, (3)

where for . For small errors, i.e. , the Taylor expansion of can be approximated by , to leading order in . That is, noise at the logical level is exponentially suppressed.

We now consider a logical GHZ state, , as input state 2. At the logical level, acts as a logical operation, , , and the (time) evolved state, , remains within the logical subspace. The state is then subjected to phase noise acting on each of the qubits. After correcting errors within each block of qubits, phase noise at the logical level is reduced (see above). The state after error-correction is given by . As a result, the situation is equivalent to the standard phase estimation scenario with a single-qubit, Hamiltonian and local phase noise, where the error probability is, however, exponentially suppressed.

Let us now bound the precision for both phase and frequency estimation. As is of rank 2 the Fisher information can be easily calculated Demkowicz-Dobrzański et al. (2012); ? (see Appendix B), and for phase estimation one finds . In contrast to the standard scenario, where the strength of the noise is independent of , here can be made arbitrarily close to 1. Hence, one encounters a quadratic scaling and thus recovers the Heisenberg limit. For any fixed value of and , we have Heisenberg scaling up to a certain, finite-system size, . For example, for we find with for respectively. Hence, , i.e. a constant close to 1, as long as . Thus, for up to our error-correction technique would yield Heisenberg scaling in precision. More importantly, if , and using the approximation , it can be shown that and for as long as . Thus, the QFI can be stabilized, and the Heisenberg limit is attained, with only a logarithmic overhead  3.

If instead of phase estimation we consider frequency estimation, i.e. , we obtain (see Appendix B) , where , and is the noise parameter at the logical level. Assuming that the optimization of over can be easily performed. Assuming that the optimal interrogation time and the bound on precision for an arbitrary number of are presented in Appendix B . We find that the optimal interrogation time decreases for larger system sizes . However, gets larger with increasing , and can hence be much more feasible in practice. Assuming that and , can be approximated using Stirling’s formula and we find . Thus the optimal measurement in our scenario can be performed at a finite time for large . This is to be contrasted with the optimal times for previously considered frequency estimation scenarios, based on GHZ and other entangled states, where for large  Huelga et al. (1997); Chaves et al. (2013). The maximum QFI per unit time is then given by , and the Heisenberg limit is approached for .

In Appendix C we show that any kind of local error can be treated in this way. This is done by using an error-correction code that corrects for arbitrary single-qubit errors rather than just bit-flip errors, and where the Hamiltonian still acts as a logical operator on the codewords. We find that one obtains Heisenberg scaling for short measurement times, .

Scenario II.—Let us now consider the physically more relevant scenario where the Hamiltonian is given by , and transversal noise 4.

We now show that the Heisenberg limit is attainable also in this case. To this aim, we attach to each of the system qubits ancilla qubits, not affected by the Hamiltonian, that may also be subjected to (directed) local noise (see Fig. 1). In practice, this may be achieved using qubits associated with different degrees of freedom (e.g. other levels in an atom), or another type of physical system. The situation is hence similar to scenario I, i.e. we have qubits that are decomposed into blocks of size . The Hamiltonian is given by . and we consider transversal noise acting on each of the qubits, see Eqs. (1,2).

In the following we show that the above situation can indeed by mapped precisely to the situation considered in scenario I. To this end, imagine that after preparing the entangled (encoded) resource state (i.e. a logical GHZ state ), we apply an entangling unitary operation to all qubits, allow them to freely evolve according to Eq. (1), and apply before the final measurement. The result is that the evolution takes place with respect to a unitarily transformed master equation , where , and . Here, with , where acts on a single block, and with the controlled phase gate, and the Hadamard operation. The action of such a transformation has been studied and applied in the context of simulating many-body Hamiltonians Dür et al. (2008). It is straightforward to verify that Dür et al. (2008) , , where the transformed Hamiltonian, , acts within a block. Up to Hadamard operations on particles , this corresponds to the situation described in scenario I, i.e. an -qubit Hamiltonian, , and local, single-qubit noise ( noise on particle 1 and noise on all ancilla particles). As shown in Appendix C one can achieve Heisenberg scaling for any local noise model using logical GHZ states as input states. This implies that we also achieve Heisenberg scaling—at least for short measurement times,  5—for transversal local noise, where the required block size is again .

Experimental realization.— We now consider a simplified version of scenario II, where only particles that are affected by the Hamiltonian are affected by noise, i.e. noise is part of the coupling process, involving a two-qubit error correction code which can be easily demonstrated experimentally. The error correction code with , as codewords, is capable of correcting arbitrary errors occurring on the first qubit, while the Hamiltonian still acts as a logical after the transformation . This opens the way for simple proof-of-principle experiments in various set-ups, including trapped ions or photonic systems, where a total of qubits prepared in a GHZ-type states suffices to obtain a precision .

Conclusion and outlook.—We have demonstrated that quantum error-correction can be applied in the context of quantum metrology and allows one to restore Heisenberg scaling in several scenarios. This includes the estimation of the strength of a multi-qubit Hamiltonian in the presence of arbitrary independent local noise, as well as a single-body Hamiltonian in the presence of transversal noise. In the latter case, an improvement in the precision from , previously shown in Chaves et al. (2013), to is demonstrated. Furthermore, for frequency estimation we have shown that the interrogation time can be finite and independent of in contrast to all previously known parameter estimation protocols. This demonstrates that, even though recent general bounds suggest a limitation of the possible gain in noisy quantum metrology to a constant factor for dephasing or depolarizing noise, this is actually not the case in general. It remains an open question whether tools from quantum error-correction can also be applied in other metrology scenarios, most importantly in the context of estimating local Hamiltonians in the presence of parallel (phase) or depolarizing noise 6.

Acknowledgements.—This work was supported by the Austrian Science Fund (FWF): P24273-N16, Y535-N16, SFB F40-FoQus F4012-N16, J3462.

Note added.—After completing this work we learned about independent work using similar approaches Arad et al. (2013); Kessler et al. (2013); Ozeri (2013).

## Appendices

In the following appendices we provide detailed calculations for the main results in the paper. Specifically, Sec. A includes a brief review of phase and frequency estimation. In Sec. B we discuss the quantum Fisher information (QFI), and provide a proof of finite, non-zero optimal time and Heisenberg scaling in precision for scenario I. In Sec. C we show how our error-correcting scheme is capable of dealing with arbitrary local noise provided we consider short measurement times.

## Appendix A Phase and frequency estimation

We start by describing the standard scenario in quantum metrology. A probe is prepared in a possibly entangled state of particles. It undergoes an evolution that depends on some parameter, , and the probe is measured afterwards. The overall process is repeated times and is estimated from the statistics of the measurement outcomes. The achievable precision in the estimation of , , which measures the statistical deviation of the estimator from the actual parameter, is lower-bounded by the quantum Cramér-Rao bound Braunstein and Caves (1994),

 δλ≥1√νF(ρλ), (4)

where denotes the quantum Fisher information of the state resulting from the evolution of the initial state of the probes Braunstein and Caves (1994). Note that the bound can be reached asymptotically, i.e. for .

In the noiseless case we have , where for some Hamiltonian . In the literature one distinguishes between phase estimation, where is the parameter to be estimated, and frequency estimation, where and the frequency has to be estimated. In the later case not only the number of particles, , counts as a resource but the additional resource of the total running time, , has to be taken into account. The QFI for pure input states, , is then given by , where denotes the variance of with respect to the state . If the aim is to estimate frequency the bound in precision, Eq. (4), can be written as in order to account for the total running time . Here, the QFI obtained per unit time, , has to be optimized over time leading to an optimal interrogation time .

## Appendix B Fisher Information

In this section we briefly recall the definition and some properties of the quantum Fisher information, . The latter is defined as Braunstein and Caves (1994)

 F(ρ)=tr(ρ′Lρ)=tr(ρL2ρ), (5)

where the Hermitian operator is the symmetric logarithmic derivative of and is defined via the equation

 dρdλ=ρ′≡12(ρLρ+Lρρ). (6)

Writing in its spectral decomposition as , it can be easily seen that

 Missing or unrecognized delimiter for \left (7)

 F(ρ)=2∑j,k:pj+pk≠01pj+pk|⟨Ψj∣∣ρ′|Ψk⟩|2. (8)

The computation of the QFI is in general hard since the diagonalization of is required. However, there exist several upper bounds on the Fisher information in the literature Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); ?.

Throughout the paper we consider the situation where , with for some Hamiltonian, , and initial state, . Here, denotes a completely positive, trace-preserving map that is independent of the parameter to be estimated. Such a map could result, for example, from solving the master equation, in case the unitary and dissipative evolution are commuting, from approximating the solution of the master equation for short times, or from a time-independent evolution which the system is subject to.

In the case of phase estimation, i.e. , and one obtains for the QFI

 Missing or unrecognized delimiter for \right (9)

For frequency estimation, where , one obtains

 F(ρλ)=2t2∑j,k:pj+pk≠0(pj−pk)2pj+pk|⟨Ψj∣∣H|Ψk⟩|2. (10)

Note that the sums in Eqs. (9,10) run over terms. Furthermore, if , for some single qubit state, , (which is the case for local Hamiltonians and local noise acting on a product state as input state) it can be shown that , and the Fisher information scales linearly in .

In the noiseless case, where , it can easily be seen that for pure input states Eqs. (9,10) reduce to

 F(ρλ)=4Var(H)ρλ (11) F(ρλ)=t24Var(H)ρλ (12)

respectively, where denotes the variance of with respect to the state .

It follows that for uncorrelated (classical) input states, the precision of phase and frequency estimation is bounded by and respectively, as the QFI can only scale as for such states. This is also known as the standard quantum limit. In contrast, a scaling of for the QFI is possible for entangled probe states, leading to the so-called Heisenberg limit with an attainable precision of and respectively. The bound is achieved by preparing the probe in the Greenberger-Horne-Zeilinger (GHZ) state, .

When taking noise into account, Heisenberg scaling can however no longer be achieved. For instance, as shown in Escher et al. (2011), if we consider noise described by Eq. (2) in the main text, where and , the ultimate precision in frequency estimation is given by . In contrast the best classical strategy yields a bound , i.e. only a gain by a constant factor is found. Notice that the GHZ state—which is optimal in the noiseless case—has an optimal interrogation time , but does not provide any gain in the noisy case. For the case of transversal noise the achievable precision and corresponding interrogation time were shown to be , and respectively Chaves et al. (2013). Note that in both cases, the interrogation time tends to zero as gets large, making a physical realization of the optimal protocol very challenging. In fact, for larger measurement times it has been shown that the scaling cannot be achieved Chaves et al. (2013).

We now compute the QFI, in the case of phase estimation, for scenario I where . The only two non-vanishing eigenvalues of are

 p0,1=12(1±(2p−1)2N), (13)

and the corresponding eigenstates are . All other eigenvalues are zero and do not contribute to the QFI. This can be seen by considering the kernel of which is given by the span of . As for and , we obtain for the QFI

 F(ρλ)=4(p0−p1)2|⟨Ψ0|H|Ψ1⟩|2=(2p−1)2NN2.

Similarly, for frequency estimation we have

 F(ρλ)=t2(2p(t)−1)2NN2.

We now consider the process at the logical level, i.e. where error-correction has been employed and we obtain with given by Eq. (3) in the main text. The optimal interrogation time and QFI can be straightforwardly determined. Using the approximation as indicated in the main text), together with , and assuming that and is small, optimization of over yields for the optimal interrogation time and precision bound:

 topt = ⎛⎜ ⎜ ⎜ ⎜⎝12(mm+12)(γ2)m+12(3+Nm)⎞⎟ ⎟ ⎟ ⎟⎠2m+2 (14) (Ft)opt = N2(2(mm+12)(γ2)m+12(3+Nm))2m+2(Nm+2Nm+3)2N.

Using Stirling’s approximation we obtain and as stated in the main text. Notice that above equations are only valid for sufficiently large , , and we have used to arrive at the final result.

As a second example let us compute the QFI for the standard metrology scenario with a local Hamiltonian, , and depolarizing noise described by (see Sec. C). As in this case the local noise commutes with the local Hamiltonian we have . If the initial state, , is the GHZ state the eigenbasis, , of is given by , where , and the two states . This can be easily verified as

 D(p)⊗N(ρ0)=(1−p2)Nρ0+ N−1∑k=0pk(1−p2)N−k∑PP[1l⊗1l…⊗tr1,…N−k(ρ0)]P,

where the sum runs over all possible permutations, and denotes the reduced state of qubits . Thus, the eigenstates of are the states . Since commutes with , we need to determine the overlaps . As is diagonal in the computation basis this overlap vanishes for unless =. Thus, the QFI is given by

 F=4(p0−p1)2p0+p1|⟨Ψ0|H|Ψ1⟩|2=p2N(1+p2)N+(1−p2)NN2, (15)

where denote the eigenvalues of respectively.

## Appendix C Local noise

Here we show that the error-correction method presented in scenario I, with given by , apply to any kind of local noise if we consider short measurement times. We first consider local depolarizing noise, and then demonstrate that the results also hold for arbitrary local noise. Depolarizing noise is described by the completely positive map

 E(ρ)=pρ+(1−p)43∑i=0σiρσi=pρ+(1−p)21l. (16)

On each block, one uses an error-correction code corresponding to graph states Hein et al. (2006), e.g. a 5-qubit code corresponding to a ring graph, that can correct an arbitrary error on one qubit Gottesman (1997); Grassl et al. (2002); Schlingemann and Werner (2001); Schlingemann (2002). Using such a code in a concatenated fashion allows one to reduce noise at the logical level to an arbitrary degree as long as . In fact, one finds that the noise at the logical level is logical depolarizing noise Kesting et al. (2013). Let be a graph state, , where is a phase gate acting on qubits , and the graph is described by edges . Defining the logical states

 |0L⟩ =(|G⟩+σ⊗mz|G⟩)/√2, |1L⟩ =(|G⟩−σ⊗mz|G⟩)/√2, (17)

the action of on these logical states is given by and . That is acts as a logical phase flip, . If we only consider the noisy part of the evolution, which on each block is given by , this leads to depolarizing noise acting on each of the qubits, with

 D(p)ρ=pρ+1−p43∑j=0σ(k)jρσ(k)j=pρ+1−p21l (18)

and .

As the noise and unitary evolution do not commute the master equation can not be easily solved as in the case of dephasing noise. However, we might approximate the solution for short evolution times using the Trotter expansion. For times the output state is well approximated by

 ρ(δt)=[D(p)]⊗N(Uδt|ψ⟩⟨ψ|U†δt). (19)

If we apply error-correction before performing the final measurement, the noise for each block acts as depolarizing noise at the logical level with parameter for sufficiently large Kesting et al. (2013). That is, the situation at the logical level is equivalent to a standard metrology scenario with local Hamiltonian, , and depolarizing noise described by . The QFI in this case is given by (see Sec. B)

 F=p2NL(1+pL2)N+(1−pL2)NN2, (20)

and can be approximated, for sufficiently close to 1, as . Note that this QFI would be obtained whenever the state is described by Eq. (19).

Noise at the logical level can be exponentially reduced when using a concatenated error-correction code Hein et al. (2005); Kesting et al. (2013). For the concatenated 5-qubit code with concatenation levels the block size is . For one finds that the probability, , to have no error at the logical level is well approximated by Hein et al. (2005); Kesting et al. (2013)

 qL=q5+5q4(1−q), (21)

where , and for depolarizing noise. That is all events that correspond to zero error (probability ) or one error at one of the qubits (5 instances, each with probability ) can be corrected by the code leading to no error at the logical level. A simple concatenation of Eq. (21) leads to the logical error probability when using a concatenated code Nielsen and Chuang (2000). One finds that the effective noise parameter, , is exponentially suppressed Hein et al. (2005). Similar to dephasing noise, for we again recover a quadratic scaling of the QFI and hence of the achievable precision.

A generalization to arbitrary local noise is straightforward. The reason is that quantum error-correction codes can deal with any kind of local noise, as long as the probability for no error is sufficiently large. In fact, as shown in Kesting et al. (2013), Pauli noise acting on the individual qubits is mapped to (logical) Pauli noise at the logical level. The probability to have no error at the logical level is given by Eq. (21) and the above approximations still hold when dealing with concatenated codes. Alternatively, one can actually bring arbitrary local noise process described by a completely positive map, or noise in a master equation described by a local Liovillian, to a standard form corresponding to local depolarizing noise. This is done by means of depolarization, i.e. by applying certain local unitary operations randomly, and might increase the noise level by a constant factor Dür et al. (2005).

### Footnotes

1. Note that the reason for obtaining Heisenberg scaling lies in the usage of error-correction, and not in the (logarithmic) increase of system size (which could only lead to a logarithmic improvement). In fact, the Hamiltonians we consider are such that the achievable precession in the noiseless case is independent of . Moreover, it only depends linearly on , which is also in contrast to the non-linear metrology scheme studied in Boixo et al. (2007); ?; ?. We show in both scenarios that we can obtain the same (optimal) precession as in the noiseless case.
2. See also Refs. Fröwis and Dür (2011); ? for studies on the stability of this state under noise.
3. By logarithmic overhead we mean that each particle is replaced by particles. Note that in practice there is no need for a separate error-correction step followed by measurements to determine the parameter, but a single measurement with proper re-interpretation suffices.
4. We remark that in practical situations, parallel noise will often be dominant. The optimal measurement is typically transversal to the Hamiltonian, and imperfections in the measurement lead to parallel noise.
5. If , then we have to take higher order terms in the solution of the master equation into account. This leads to parallel noise of and limits the maximal until which Heisenberg scaling can be achieved Chaves et al. (2013).
6. Note that our results from scenario II can not be directly applied in the case of parallel or depolarizing noise. Using a unitary transformation to obtain a many-body Hamiltonian also transforms parallel noise to correlated noise, that can not be corrected by the error-correction code used here.

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