Practical Limits of Error Correction for Quantum Metrology

Noise is the greatest obstacle in quantum metrology that limits it achievable precision and sensitivity. There are many techniques to mitigate the effect of noise, but this can never be done completely. One commonly proposed technique is to repeatedly apply quantum error correction. Unfortunately, the required repetition frequency needed to recover the Heisenberg limit is unachievable with the existing quantum technologies. In this article we explore the discrete application of quantum error correction with current technological limitations in mind. We establish that quantum error correction can be beneficial and highlight the factors which need to be improved so one can reliably reach the Heisenberg limit level precision.

Quantum error correction [32][33][34][35][36][37], an essential tool from quantum computation to protect quantum states from errors, is a potential solution to circumvent the noise problem of quantum metrology [38][39][40][41][42][43][44][45]. It has been shown that if the signal and noise occur in orthogonal directions [42,43], then error correction can mitigate the effects of noise without affecting the signal. Furthermore, they showed that the Heisenberg limit is recoverable when the time between error correction applications is infinitesimally small. Unfortunately, this mathematical assumption is impractical with our current quantum technologies: the timescale of current error correction schemes are far from infinitesimally small, instead they are comparable to that of the dephasing times for both spin qubit systems [46,47] and superconducting qubit systems [48,49]. Moreover, a realistic error correction strategy is hindered by other factors such as noisy ancillary qubits and imperfect error correction.
Here we are going to consider a more pragmatic approach to incorporate error correction into quantum metrology -by accounting for the impediments one would face with current quantum technologies. We begin this article by reviewing the fundamentals of phase estimation in a noisy environment without error correction and compare it to a model enhanced with a parity check error correction strategy. We derive the necessary conditions to achieve Heisenberg-like precision, and unsurprisingly, we show that it is not permanently achievable once one discards the assumption of being able to perform arbitrarily fast error correction. Next, we derive the effects of noisy ancillary qubits and imperfect error correction has on the precision. From which, we investigate the limitations of current quantum technologies, determining which factors need to be improved upon to enable Heisenberg-level precision. Even though the results of this study are derived by examining a specific error correction strategy, we conclude by suggesting that they can be generalized to all error correcting codes.

Noisy Quantum Metrology
The canonical example of noisy quantum metrology scheme involves n qubits undergoing the free evolution H 0 = Ω 2 n m=1 Z m , where Ω is the natural frequency of the qubits. The qubits are then governed by two interactions: i) a signal which causes a detuning ω in each of the qubits (represented by H = ω 2 n m=1 Z m ), and ii) an interaction with an environment which causes dephasing (with rate γ) in an orthogonal direction, X. Here (Z m , X m ) are the usual Pauli operators for the mth qubit. In the rotating reference frame, the Lindbladian master equation can be written as [50] The master equation can be further generalized to include a dephasing term which is parallel to the signal if necessary. However, these errors cannot be corrected because one cannot differentiate between the signal and those errors [42,43]. For off-axis noise, we can only correct its perpendicular component. The protocol of the quantum metrology scheme is to let the system evolve for time t, after which an appropriate measurement is performed. The likelihood of the measurement outcomes will be dependent on ω, so by repeating the prepare and measure protocol many times one can precisely establish the value of ω [4]. As the noisy system continues to evolve it becomes significantly more difficult to distinguish the effects of the signal Hamiltonian and the effects of the noise; increasing the uncertainty of the estimate.
The quantum Fisher information (QFI) is a quantity which is inversely proportional to the minimum uncertainty of the estimate [51,52]. If initial quantum state is an n qubit Greenberger-Horne-Zeilinger (GHZ) state, then in a noiseless environment after sensing time t the QFI is given by [2] Q noiseless = n 2 t 2 . ( This quadratic scaling in n is known as the Heisenberg limit, the ultimate precision quantum physics allows. However, in a noisy environment, the achievable precision is ultimately bounded because of said noise. It is straightforward to show (see Appendix A) that in the short sensing time limit the QFI is given by This is not a practical level of precision due to the short time limits. It is important to remark that a more practical figure of merit for quantum metrology is the Fisher information [52] (instead of quantum Fisher information). However, computing the Fisher information is beyond the scope of the study. This is because the QFI is the Fisher Information optimized over all measurement schemes, and the optimal measurement scheme is dependent on the strength of the noise. Nonetheless, we provide sufficient mathematical results in the Appendix to compute the Fisher information for a given measurement strategy.

Error Correction Enhanced Quantum Metrology
With the issue of noise in mind, let us now incorporate error correction protocols into the quantum metrology scheme as a potential solution. Current literature suggests that noise which is perpendicular to the signal can be mitigated by performing error correction in rapid succession [42,43], where the time between applications of error correction is taken to be arbitrarily small, such that non-linear terms in the Taylor expansion of the QFI are negligible. Further, they include the assumption of perfect error correction and access to noiseless ancillary qubits. These assumptions are implausible for near term implementations of repeated error correction [46][47][48][49].
We overcome these restrictions by finding an exact solution of a specific error correction strategy: a parity check error correction code [53][54][55] involving a single The input state ρ in , is composed of n sensing qubits and one ancillary qubit. The sensing qubits evolve according to the signal ω and noise γ. The parity check code, denoted by C, is repeatedly applied after a given time τ . The final quantum state used for parameter estimation, ρ out , undergoes t/τ rounds of error correction. The scheme can easily be generalized; allowing for arbitrary input states, error correction strategies and more ancillary qubits. ancillary qubit, depicted in Fig. 1. This scheme could be implemented for instance in a hybrid quantum system involving an electron/nuclear spin system. Here the electron spin could be used a sensing qubit while the nuclear spin would act as an ancillary qubit [46,47,56]. The parity check code is composed of two components. The first is n non-destructive parity measurements between individual sensing qubits and the ancillary qubit. The second is the correction to any qubits in which an error is detected [57].
In our model, the quantum state is initialized as an n+1 qubit GHZ where n qubits are used for sensing while the remaining one is used as an ancilla for error correction. The sensing qubits are influenced by a signal ω, as well as the noise γ. The parity check code is applied after time τ to mitigate the effects of the noise; the procedure is then repeated t/τ times where t is the total sensing. Let us now determine the criteria to achieve the Heisenberg limit. Initially we assume the ancillary qubit is noiseless and error correction is perfect, after which we consider the scenarios without these assumptions.

Noiseless Ancilla and Perfect Error Correction
In the ideal error correction scenario (noiseless ancilla and perfect error correction), after t/τ rounds of error correction, the quantum state can be expressed as a bipartite mixed state Figure 2: Plot of Q/τ 2 for a n = 25 qubit GHZ state after undergoing repeated error correction with (a) a noiseless ancilla and perfect error correction, (b) a noisy ancilla (ξ/γ = 10 −4 ) and imperfect error correction, and (c) a noiseless ancilla and imperfect error correction (p = 0.01), with total sensing times t/τ = 10 3 , 10 6 . The characteristics of a noisy state without the inclusion of a quantum error correction code (QECC) after sensing time τ is also displayed. Note that these curves are cutoff when Q/τ 2 = 1 for clarity purposes. Additionally, we illustrate the corresponding normalized QFI curves, Q/(nt) 2 , in plots (d), (e) and (f) respectfully, to emphasize the deviation from the Heisenberg limit. where and with ∆ = ω 2 − γ 2 . It is important to note that r 2 < 1 if γτ > 0. Consequently, the quantum state becomes more mixed (and less useful for quantum metrology) once the quantity nt/τ becomes very large. In Appendix B, we show that the QFI of such a quantum state can be written in the form where for small times τ , We immediately observe that that a Heisenberg level of precision is obtained if two conditions are met. The first condition is that 2γτ 1; it was derived in [42,43] and is synonymous to the noisy scenario without error correction given in Eq. (3). The second condition is that r 2nt/τ ≈ 1, which suggests that the Heisenberg limit cannot be maintained indefinitely in a noisy environment (r 2 < 1) and that the QFI will eventually tend to zero. For small τ we have meaning we can re-write the second condition as 4 3 nω 2 τ 2 γt 1. This second condition is of second order with respect to τ , which is the a possible reason it was overlooked in previous studies [42,43].
Both conditions are clearly illustrated in Fig. 2a and Fig. 2d, where we have two different family of curves plotted for a 25 qubit GHZ state. The first family of curves is with ω/γ = 20, and shows the Heisenberg limit level of precision is lost once r 2nt/τ begins to tend to zero. The second family of curves is for ω/γ = 1/20, and the Heisenberg limit level of precision is lost once γτ ≈ 10 −2 , regardless of if t = 10 3 τ or t = 10 6 τ . The reason for the stark contrast in the two families of curves (ω 2 γ 2 versus ω 2 γ 2 ), is due to larger deviations from the ideal case when ω 2 γ 2 . Information about ω is stored in the relative phase, nφt/τ , and if an error does occur between applications of error correction, the phase will deviate further from the ideal case. Thus, each round of error correction introduces a small amount of variance to the phase, and this variance is larger for larger values of ω.

Noisy Ancilla and Perfect Error Correction
To further augment the reality of our quantum metrology scheme we impede the error correction by adding a dephasing rate, ξ, to the ancillary qubit. This modification results in a QFI of the form where g is bounded by (see Appendix B) which we can interpret as a third condition that is needed to be satisfied to obtain Heisenberg-like scaling: ξt 1. This is not very surprising; once ξt becomes significantly large, the ancillary qubit will be too noisy for practical error correction. This additional condition is displayed in Fig. 2b and Fig. 2e, where we set the ancillary qubit to have a dephasing rate of ξ/γ = 10 −4 . As expected, the noisy ancilla causes the Heisenberg limit to be lost sooner when compared to the case with a noiseless ancilla. The impact is much more prominent for the curve with ω/γ = 1/20 and t = 10 6 τ , where the loss of the Heisenberg limit is due to ξt becoming too large instead of γτ . This loss can be overcome by occasionally re-initializing the ancillary qubit.

Noiseless Ancilla and Imperfect Error Correction
The second hindrance we explore is the inclusion of imperfect error correction. We simulate this by adding a probability p that a parity check outputs the wrong outcome, which results in an unnecessary bit-flip correction (or lack there of). In this scenario, the QFI can be written as with and Notice that the inclusion of imperfect error correction makes the Heisenberg limit unattainable; Q → n 2 t 2 (1 − 2p) 2 as τ → 0. The multiplicative factor (1 − 2p) 2 is is due to uncertainty in the error correction propagating to the uncertainty in the parameter estimation.
The additional conditions are illustrated in Fig. 2c and Fig. 2f, where we have set p = 0.01. Firstly, as γτ → 0, Q/(nt) 2 → (1 − 2p) 2 ≈ 0.96. Secondly, the additional condition q 2nt/τ ≈ 1 is again associated with additional variance being added to the relative phase because of the imperfect error correction. The impact of which is again more influential when ω/γ = 20, as opposed to when ω/γ = 1/20.

Limitations of Current Quantum Technologies
In [46,47], the electron spin of a nitrogen-vacancy center is entangled to carbon-13 nuclear spins, the nuclear spins act as ancillary qubits, and error correction is performed on the electorn spin. The reported dephasing rates are γ −1 ∼ 10 −6 s and ξ −1 ∼ 5×10 −4 s. The error correction is being performed on a similar timescale of τ ∼ 10 −6 s, with infidelity reported at p = 0.06 [47]. In Fig. 3, we benchmark the sensing capability of a single qubit systems when ω 2 γ 2 , using the listed values. As expected, the Heisenberg limit is unattainable. Moreover, it is still unattainable if one performs near perfect error correction (p = 0.001) and uses noiseless ancillary qubits (ξ = 0) which can be approximately achieved by frequently re-initializing the ancilla before it becomes too noisy. Notably though, when the quantity γτ decreases by a factor of ten, one attains a QFI of ∼ 80% of the Heisenberg limit for a total sensing time t = 10 5 τ ; greatly outclassing the precision achieved in current experiments [9][10][11] using spin systems. Of course, this result should be used with caution. A realistic metrology scheme is hindered  [46,47] ( ) a QFI of ∼ 20% of the Heisenberg limit can be achieved for sensing times t = 10 1 τ . With improved error correction fidelity and a noiseless ancilla ( ) this can be sustained for a sensing time t = 10 3 τ . The QFI is significantly improved when γτ = 0.1 ( ).
by other factors not discussed in this study, such as imperfect resources and noise parallel to the signal, which cannot be suppressed with error correction.

Other Error Correction Strategies
Until now, we have only discussed one possible error correcting strategy, whereas [42,43] make no assumptions regarding the error correction strategy. Although a completely general result is more satisfying, it is unfeasible with our solution method. Nonetheless, one can imagine that regardless of the error correction strategy (assuming noiseless ancilla and perfect error correction), one will obtain similar conditions imposed by Eq. (7). This is because correcting the deviation in the relative phase is impossible for an error correction strategy without first knowing the value of ω; but this would defeat the purpose of quantum metrology. For example, if one instead utilizes the n qubit bit flip code [36] then the details of the proof can be found in Appendix C. Hence, for large n the QFI using the bit flip code is effectively the same as if one utilizes the parity check code. It is even more unfeasible to generalize the results for error correction with noisy ancillary qubits or imperfect error correction to other error correction strategies. However, we still expect the results to be similar to Eq. (10) and Eq. (12) respectfully. This is because any noisy ancilla will eventually become obsolete for error correction, and uncertainty from the imperfect error correction will propagate to the QFI, as well as increase the amount of variance added to the relative phase after each round of error correction.

Remarks
In this article we analysed the ability of an error correction protocol to recover the Heisenberg limit in noisy quantum metrology. As expected from previous results [42,43], the Heisenberg limit can be achieved for longer duration's when compared to a similar scheme without error correction and is conditioned on the fact that 2γτ 1. Notably though, our results suggest that the Heisenberg limit is not permanently achievable, and is lost once the quantity nt/τ becomes too large. This previously undiscussed requirement is due to the fact that a small amount of variance is introduced to the relative phase after each round of error correction; eventually the total variance is too large to overcome. We also showed that any uncertainty in the error correction will propagate to the QFI, making the actual Heisenberg limit unachievable, regardless of the speed of the frequency of error correction. Moreover, the uncertainty of error correction increases the added variance to the relative phase.
We chose to specifically analyse the effects of repeated error correction in the scenario when the quantum state used for sensing is initialized in an n qubit GHZ state. The logical generalization would be to expand the results to a broader scope of initial states; such as squeezed states [58], symmetric states [59], or graph states [60]. It is possible that these quantum states (which do not achieve the Heisenberg limit, but do achieve a quantum advantage) are more robust to the effects of noise and can maintain a quantum advantage for a longer total sensing time.

Acknowledgements
Nathan Shettell and Damian Markham acknowledge support of the ANR through the ANR-17-CE24-0035 VanQuTe. Nathan Shettell would also like to acknowledge the support of the NII international internship program.

Appendix A. The QFI of a Noisy GHZ without Error Correction
We wish to solve the matrix equation where j, k ∈ {0, 1} ⊗n are bit strings of length n. To obtain a solution, we instead consider Eq. (A.1) as a set of linear differential equations and solve for the amplitudes α j,k . Since we are considering the scenario where the quantum state is initialized in a GHZ state, the only non-zero of the amplitudes are those of the form α j,j or α j,j (where |j = X ⊗n |j . We set a to be the vector of size 2 n containing all amplitudes of the form α j,j . Similarly, we set b to be the vector of size 2 n containing all amplitudes of the form α j,j . Both vectors are ordered with respect to ascending values of j. This framework transforms Eq. (A.1) into two separate matrix differential equations, i) d a dt = A a, and ii) The solutions of the differential matrix equations are a = e At a 0 = e −nγt cosh(γt) sinh(γt) sinh(γt) cosh(γt) where a 0 and b 0 are the initial amplitude vectors and ∆ = ω 2 − γ 2 . The symbolic representation of the entries of the matrix B (x ± and y) is to add clarity in latter computations.
The QFI of a quantum state σ = j λ j |ψ j ψ j |, where {|ψ j } form an orthonormal basis and j λ j = 1, is where we have used the notation˙ = ∂ ω for clarity. Combining this with the final quantum state ρ, one can obtain the QFI of a noisy GHZ without error correction. We again float a factor of 1/2 in front of the sum to avoid double counting. In doing so one obtains (A.13)

Appendix B. The QFI of a Noisy GHZ Enhanced with Repeated Error Correction
Recall in the scenario enhanced with error correction, we let the system evolve for a total time τ , after which error correction is performed. This processes is repeated until the system has evolved for total time t (without loss of generality we assume that t/τ is a positive integer). Additionally, an ancillary qubit is included into the dynamics, which we set to have an index m = n + 1. We begin by solving a completely general scenario in which the ancilla is noisy and the error correction is imperfect, from which we can obtain varying results for all the combinations of noiseless/noisy ancilla and perfect/imperfect error correction by taking the appropriate limits.
We model the noisy ancilla by subjecting it to dephasing in the X direction with rate ξ, this changes the matrix equations to be Here we have recycled the same notation which we used in Appendix A. To be concise we set c γ = cosh(γτ ), s γ = sinh(γτ ), c ξ = cosh(ξτ ) and s ξ = sinh(ξτ ).
Recall that in the parity check code, a correction is made on a sensing qubit if it has a different parity to the ancillary qubit. We simulate imperfect error correction by adding a probability that the syndrome measurement outputs an incorrect result with probability p, which would result in the incorrect correction being applied. This operation can be represented as a the matrix Combining the evolution matrices, e Aτ and e Bτ , and the error correction matrix, E, we obtain an expression for the amplitudes of the final quantum state where we define re ±iφ = e −γτ (x ± + y) = e −γτ cos(∆τ ) + γ ± iω ∆ sin(∆τ ) . . Recall that we have set the final qubit to be the ancillary qubit, and the first n to be the sensing qubits. Therefore, in this scenario h j is the Hamming weight of the bit string of the sensing qubits. The solution to Eq. (B.5) is more complex. However, after the first round of error-correction (and each subsequent round), the amplitude corresponding to the outer product |j0 j 1| is of the form (1−p) n−h j p h j Re −iθ 2 , and the amplitude corresponding to the outer product |j1 j0| is of the form (1−p) n−h j p h j Re iθ 2 . Therefore, one can simplify the problem of solving for the final values of Re ±iθ . This can be done by constructing a recurrence relation from Eq. (B.5). By setting N = n(t/τ − 1), we obtain and υ ± = c ξ e ±inφ + s ξ e ∓inφ . (B.10) Therefore, in the general case with a noisy ancilla and imperfect error correction, the quantum state after t/τ rounds of error correction can be written as and From which, we compute the QFI to be (B.14) We can now determine the properties of the QFI in the varying scenarios.

Case 1: Noiseless Ancilla and Perfect Error Correction
In the simplest case with a noiseless ancilla (ξ = 0) and perfect error correction (p = 0), we obtain that Re ±iθ = re ±iφ nt/τ . Therefore, we can write that and It is easy to verify that r 2 < 1 when γτ > 0 (assuming ω = 0), implying that the QFI will eventually tend to zero in a noisy environment. Note that we can write Suppose that ω 2 − γ 2 ≥ 0, then we have that ∆ = |∆| is real, as well as sin(∆τ ) ≤ ∆τ and sin(2∆τ ) ≤ 2∆τ . Thus we have the inequality where on the right hand side of the above inequality is strictly decreasing (with respect to γτ ) and equal to one when γτ = 0. The other possible case is that ω 2 − γ 2 < 0, then we have that ∆ = i|∆|, as well as From which we obtain a second inequality r 2 < e −2γτ 1 + 2 sinh(γτ ) + 2 sinh 2 (γτ ) = 1. (B.21) Combining both results, assuming ω = 0, we obtain that r 2 < 1 if γτ > 0.

Case 3: Noiseless Ancilla and Imperfect Error Correction
In the third case we have a noiseless ancilla (ξ = 0) and imperfect error correction (p = 0). It is straightforward to obtain Re ±iθ = re ±iφ nt/τ q ± e ∓iφ n(t/τ −1) . (B.29) We compute the QFI to be of the form where we define q 2 = q + q − , which when raised to the exponent nt/τ satisfies Appendix C. The QFI when using the Bit Flip Code Just as in Appendix A and B we solve for the final quantum state via matrix equations. Note that the n qubit bit flip code [36] does not have an ancilla, thus the whole system is n qubits. The bit flip code maps the outer product |j j| to |0 0| ⊗n if h j < n/2 and |1 1| ⊗n if h j > n/2 (we assume that n is odd to avoid complications when h j = n/2). Similarly, the outer product |j j | is mapped to |0 1| ⊗n if h j < n/2 and |1 0| ⊗n if h j > n/2. We again define E to be the error correction matrix, define E (j,k) to be the entry of E in the jth row and kth column, it follows from the above that Notice that ζ ± ∈ O τ n+1 2 , therefore Additionally, we have that re ±iφ n = η ± + ζ ± . So it follows that for large n the final quantum state in this scenario is very similar to the final quantum state using the parity check code (with a noiseless ancilla and perfect error correction). Mathematically, it is equivalent up to order τ n−1 2 , and therefore the QFI is similarly equivalent up to the same order.