Distributed quantum sensing enhanced by continuous-variable error correction

A distributed sensing protocol uses a network of local sensing nodes to estimate a global feature of the network, such as a weighted average of locally detectable parameters. In the noiseless case, continuous-variable (CV) multipartite entanglement shared by the nodes can improve the precision of parameter estimation relative to the precision attainable by a network without shared entanglement; for an entangled protocol, the root mean square estimation error scales like 1/M with the number M of sensing nodes, the so-called Heisenberg scaling, while for protocols without entanglement, the error scales like 1 / M . However, in the presence of loss and other noise sources, although multipartite entanglement still has some advantages for sensing displacements and phases, the scaling of the precision with M is less favorable. In this paper, we show that using CV error correction codes can enhance the robustness of sensing protocols against imperfections and reinstate Heisenberg scaling up to moderate values of M. Furthermore, while previous distributed sensing protocols could measure only a single quadrature, we construct a protocol in which both quadratures can be sensed simultaneously. Our work demonstrates the value of CV error correction codes in realistic sensing scenarios.


Distributed sensing of real quadrature displacements
For CV sensors, the signal is acquired by measuring the displacement changes in the sensor state-e.g. position and/or momentum change for a mechanical oscillator. The precise measurement of displacements is important for interferometric phase sensing [35], quantum key distribution [49], spin sensing [50], and inertia sensing [51]. Moreover, like the example in RF sensing [52], transducers can transform a even broader class of signals into optical displacements for further sensing purposes. Mathematically, displacements are described by the unitary a a a = - U a a exp ( ) (ˆˆ) † , or equivalently the mode transform a  + a âˆ. Here â is the annihilation operator of the field being sensed. Equivalently, a displacement a Û ( )can also be represented by a quadrature transform a a  + + q p q p , 2R e , 2I m (ˆˆ) (ˆ( )ˆ( )), where = + q a a 2 (ˆˆ) † , = p a a i 2 (ˆˆ) † are the position and momentum quadratures. For simplicity, we will use the notation a a a = Re , Im ( ( ) ( )). In this convention, the quadrature variance is + = + p q n 2 1 2 2ˆˆ, where º n a âˆ † is the number operator. Thus the vacuum noise (á ñ = n 0 ) is á ñ = á ñ = p q 1 2 2 2ˆ.
As shown in figure 1, the original distributed sensing protocol [31] aims to obtain a minimum rms error estimate of a weighted average, a a º å = w To model the imperfections from the distributed sensors, we introduce an independent loss channel h  m with Figure 1. Schematic of a distributed quantum sensing protocol for measuring displacements on a single quadrature. SV: squeezedvacuum state with mean photon number N S and squeezed noise in its real quadrature. h  : pure-loss channel with transmissivity 0<η1. a Û ( ): field-quadrature displacement by real-valued α. homo: homodyne measurement of the real quadrature. 7 Note that negative weights can be merged into the sign of each α m . transmissivity η m on each sensor node, leading to the mode transform m m m mˆˆ, where the environment mode e m is in a vacuum state. To enable performance comparison, we characterize the overall resource for the sensing task by the total mean photon number N S used in modes   a m M , 1 m {ˆ}. This is because for sensing applications like bio-sensing one wants to minimize the light power shining on the fragile samples to avoid any damage.
In an entanglement-enhanced distributed sensing protocol, the modes   a m M , 1 m {ˆ} are in a CV multipartite entangled state, produced by passing a single-mode squeezed vacuum, with mean photon number N S , through a beamsplitter array. Homodyne measurements are applied to obtain the information about the weighted average. To benchmark the performance, we compare the entangled scheme with the optimal separable scheme, where each mode is in a squeezed vacuum state with mean photon number N m , and the total mean photon number å = = N N m M m S 1 for fair comparison. In principle, one can introduce extra ancilla modes; however, this is not necessary in the lossless case-one can show that each scheme is optimal in its own class, given the total mean photon number constraint. In the lossy case, the optimal protocol is still an open question, but [31,36] were able to show that the scheme in figure 1 maximizes the Fisher information among Gaussian states and achieves the best precision when homodyne measurement is applied. Recently [53] proved that this scheme is also the optimal Gaussian protocol for distributed phase sensing.
In the following, we evaluate the performance of the entangled and separable sensing protocols in the presence of loss. We set the beasmplitters such that The optimal separable-state scheme, for the scenario under consideration here, employs a product state, but its precision da h C does not have a closed solution in general. For the simple case of η m =η, w m =1/M, we have [31] da As shown in figure 5, in the lossless case, the separable scheme has its precision obeying the SQL, while the entangled scheme achieves the Heisenberg scaling; even when there is loss, the entanglement enhancement survives, due to the robustness of CV multipartite entanglement. However, the scaling advantage is entirely gone even for small M. To mitigate the loss issue, we consider CV quantum error-correction.

CV error correction
Quantum error correction [54] codes are originally developed for protecting DV quantum information for scalable quantum computing, sometimes even with the aid of CV systems [46,55,56]. However, various quantum sensing applications require CV quantum information processing. To facilitate these applications, the question to be addressed in this section is: can we protect CV quantum information against noise?
The general idea to correct a CV mode is to encode a single mode into multiple modes. Indeed, previous such proposals can correct single-mode errors [57,58]. However, a key difference of CV systems is that errors (e.g. thermal noises and excitation loss) happen with unity probability on all modes. Thermal noise can be described by an additive white Gaussian noise channel (AWGN) F s 2 , which applies a Gaussian distributed complexvalued random displacement ¢ = + +   a a i 2 q pˆ( ) on the input mode. Here ò p , ò q are real Gaussian distributed with standard deviation σ. In fact, it suffices to consider AWGN channels for all Gaussian noise models, due to channel reduction relations [59][60][61][62][63]. As an example, the excitation loss channel h  can be combined with an amplification channel  G in front, described by the mode transform ¢ = -a G a G e 1ˆ † joint on the vacuum environment mode eˆ. Choosing the gain G=1/η, we obtain the composite channel and 1−η1/η−1, it is always beneficial to apply amplification before the loss.
To correct the AWGN noise, the new idea in [38] is to use GKP grid states to encode CV information. This builds on an observation emphasized in [47,64]-that with GKP grid states we can simultaneously measure two quadratures with high precision, as long as we're promised that the displacement of both quadratures is small. The GKP grid state has wave function When Δ=1, its Wigner function is peaked around a square grid of spacing p 2 in the phase space. The overall variance á ñ á ñ D q p 1 2  equals the mean photon number N S ; however, if we consider only the phase space region close to a single peak, the variances in position and momentum are Δ 2 /2;1/4N S =1, only twice the squeezed-vacuum variance.
Below we will recall two codes introduced in [38], the GKP-two-mode-squeezing code and the GKPstabilizer code. To understand the error correction mechanism, consider two input modes to an encoding circuit. Mode 1 will be used to detect the signal, and mode 2 is an ancilla which has been prepared in a GKP state. The encoding circuit applies a Gaussian unitary operator U Ŝ to this pair of modes, where S is a symplectic tranformation; its inverse U Ŝ † can be used to decode the state. Between encoding and decoding, the two modes are subjected to additive noise-the noise operation is a displacement Thus, after decoding, the noise is transformed to a modified displacement [65]. By choosing a proper entangling transform U Ŝ , one induces a correlation of the effective displacements ¢  1 and ¢  2 of the two modes. Then by measuring the displacement ¢  2 of the GKP ancilla, one can infer a displacement  U ĉ ( ) which corrects the additive error ¢  1 on the signal mode. In this scheme, while all operations are Gaussian, the input ancilla is a non-Gaussian GKP grid states, so the effectiveness of error correction is compatible with the no-go theorem for Gaussian error correction in [66].

GKP-two-mode-squeezing code
As illustrated in figure 2, the GKP-two-mode-squeezing code uses a two-mode squeezing operation T G T 12 ( )to entangle the input state with an ancilla initialized in the GKP grid state. After both modes go through the noise channel Φ σ , another conjugate two-mode squeezing operation T G T 12 ( ) † is performed. Finally, both quadratures of the ancilla are measured modulo p 2 to diagnose the displacement error on the input state. After conjugation of the displacement error by the two-mode squeezing operator, we obtain the the effective displacements  T G U ) we see that when G T is large, the effective displacements are highly correlated. Thus measuring the displacement noise of the ancilla provides a good estimate of the displacement error on the signal, and therefore enables approximate correction of the error through a counter-displacement. Although the uncertainty principle forbids the simultaneous precise measurement of displacements on both quadratures, an ancilla in the GKP grid state allows the precise measurement of both quadrature displacements modulo p 2 . Reference [38] has given a detailed analysis on the amount of noise reduction in this scheme. We plot the rms logical noises σ p , σ q given by equation (24) in [38] on each quadrature with the physical noise σ in figure 3. The code helps when σ0.558, which corresponds to loss η0.689.

GKP-stabilizer codes
Reference [38] has also proposed a more general GKP-stabilizer code. In the GKP stabilizer code, a hierarchical structure of squeezing and GKP encoding is used, as shown in figure 2. We begin by analyzing the lowest level (n=2).
The encoding is achieved by a sequence of Gaussian operations-a two-mode SUM gate and single-mode squeezing operations. Recall that a SUM gate  SUM 1 2 acts on a pair of modes according to and that a squeezing operation acts on mode m according to To protect mode 1, we make use of the GKP ancilla mode 2 via the encoding circuit Here the order of operations is read from right to left-that is, d S 1 ( ) acts first, followed by the sum gate and then g b S S 1 2 ( )ˆ( ). The overall 4×4 symplectic matrix applied to the four quadratures can be calculated to be dg The decoding circuit is the encoder run in reverse and implements the unitary U ( )and then the decoder is applied. If there is no noise, the decoder perfectly restores the input signal. But when there is noise the decoder distorts the noise, yielding noise in the output signal Because mode 2 was initially encoded as an ideal GKP grid state, it is possible to simultaneously measure the offset of both quadratures in mode 2, assuming the (distorted) noise is sufficiently weak. Once we know the offset in mode 2, we can approximately diagonose the additive shift in the (unmeasured) mode 1. Specifically, after measuring we apply a corrective displacement ), obtaining the partially corrected noise in mode 1: Suppose for example that δ=1, β=γ −1 =λ>1, and that  q are all comparable; then the noise in both quadratures of mode 1 is suppressed by a factor λ −1 relative to an unprotected mode. In practice the noise suppression is limited because, for a given noise strength there is a limit to how much we can squeeze the noise and still read out both quadratures of the GKP grid state unambiguously. Furthermore, if the GKP grid states themselves are only finitely squeezed, further squeezing during the protocol may compromise their errorcorrecting power.
We can go further in a protocol that makes use of multiple GKP-encoded modes. To see how that works, we consider the case where the noise acting on the two modes is asymmetric, so that  q 2 ≈ p 2 ≈ <   2 1 ≈ q 1 ≈ p 1 , and we adjust the protocol so that the output noise on mode 1 after decoding and recovery is balanced between q and p; hence There is a further constraint-we do not want the shift error in mode 2 after decoding to be too large, which would compromise our ability to measure both quadratures accurately. To ensure that the distorted error is comparable in both quadratures in mode 2, we impose To summarize, if the noise in mode 2 is weaker than the noise in mode 1 by the factor κ<1, and if we want the error-corrected noise in mode 1 to be balanced between the q and p quadratures, we use the encoder with where λ>1; then the error corrected noise in mode 1 is ò 2 /λ in both quadratures.
Because the scheme works for asymmetric noise, it can be used iteratively. For example, with three modes, where mode 1 is the sensing mode and modes 2 and 3 are GKP-encoded ancillas, we can use mode 3 to reduce the additive noise in mode 2 by a factor of 1/λ, and then use mode 2 to reduce the noise in mode 1 by a further factor of 1/λ, achieving all together a reduction by 1/λ 2 Due to the imperfect measurement of quadratures that a GKP grid state offers, after the error correction the probability density function (PDF) of the logical noise P Z x n , (·), x=p, q is not Gaussian. It can be obtained through the following recursion relation where n ± =n±1/2. Note that P Z x n   , which indeed agrees with the intuitive understanding. To summarize, if there is one input signal model we wish to protect in a sensing experiment, we may introduce n−1 ancilla GKP grid states and then iterate the protocol n−1 times, thereby reducing the noise strength to σ n−1 ;σ/ λ n−1 in the error-corrected signal state. To precisely evaluate the error correction performance under moderate noise, we perform numerical integration of equation (20) repeatedly and obtain the standard deviation σ q,n , σ p,n . The results, for the case n=7, are in figure 3, where different levels of the squeeze parameter λ are chosen, indicated by the color; we did not perform the computations for larger values of n due to limitations on numerical precision. When the initial noise σ is large, the modulo p 2 property of the measurement leads to excess noise and thus hinders the error correction performance. At certain critical noise level, the code ceases to reduce the noise, as indicated by the termination of the plots on the right-hand-side in figure 3.

Improved distributed sensing
Now we apply the CV error correction codes in the distributed sensing protocol introduced in section 1. We will evaluate the standard deviations in parameter estimation given the loss and error correction. Although the GKP based error correction codes lead to non-Gaussian random errors in the parameter estimation, the standard deviation of noise is still a good characterization of the measurement precision. This is because in a parameter estimation scenario, one can average multiple independent repetitions of the same measurement, even when each measurement is already multi-mode. When the number of repetitions is large, the central limit theorem guarantees that the averaged measurement error is Gaussian distributed and thus can be characterized entirely by its standard deviation.

Error corrected real quadrature sensing
We apply the GKP-two-mode-squeezing code and the GKP-stabilizer code in a real-quadrature distributed sensing protocol. As shown in figure 4, to perform distributed sensing on different nodes, one first locally generates signal modes   a m M , 1 m {ˆ} in the same CV multipartite entangled state as in section 1. After the beamsplitter array, each mode a m in the multipartite entangled state is immediately encoded (with additional ancilla) to protect against independent loss errors; to facilitate error correction, amplifiers h  1 transform the loss channels h  to AWGN channels that can be corrected with the standard GKP decoder. Before the sensing process, decoding is applied to the received signal modes and ancillae, and then the error-corrected signal inputs, ¢ a m {ˆ}, are injected to sense displacements. Note that, while a total of ∝M ancilla modes are used in the Multiple anculla modes can be utilized, however, only a single ancilla mode is shown for simplicity. entanglement distribution process, the sensing process occurs after the GKP-assisted decoding, and only a single error-corrected signal mode interacts with the sample at each sensing node. As a demonstration, we consider the case of equal weights and equal displacements; similar advantages are expected for more generic cases. Suppose the error correction code reduces the original noise h -1 to σ EC (η); as in equation (2), one can then obtain the precision Note that, due to the amplification, compared with equation (2) the error is larger by a factor h , and also the second term inside the square root is a factor of two larger in addition to the change from 1−η to s h EC 2 ( ). Using the results in section 2, we can evaluate the performance of both error correction codes. As shown in figure 5, we see that the GKP-two-mode-squeezing code (orange) only has a small advantage over the scheme without error correction (blue) in the low loss region (η0.95). The GKP-stabilizer code with n=7 (red) gives a much better performance improvement. In the low loss region , the Heisenberg scaling of of precision can be reinstated up to M∼10 2 modes. Moreover, when η=0.85 there is still appreciable advantage over the scheme without error correction (blue).
A few comments are worthy of mention here. First, the above performance is valid for arbitrary displacement values, but we can do better the if displacement at each sensing node is guaranteed to be smaller than p 2 . A GKP-decoding error could result in a displacement of ¢ â by an unknown integer multiple of p 2 , but this error has no damaging effect if we decode the result of the homodyne measurement of each mode by evaluating it modulo p 2 . Second, in a fair comparison between sensing schemes, we usually fix the mean photon number of the source that interacts with the samples. In the above comparison, we have not quite done ). QEC1: GKP-two-mode-squeezing code. QEC2: GKP-stabilizer code with n=7. Note that in (c) and (d) the performance of QEC 1 is worse than non-corrected case, due to the extra amplification required to reduce loss to additive thermal noise. that, because in the case without error correction the mean photon number at the sensing nodes has been attenuated by the loss factor η, while in the case with error correction we have compensated for the loss channel h  with the amplification channel h  1 , which transforms the loss channel into an AWGN channel. However, as one can see in figure 5, we have plotted the separable scheme in the lossless case (gray dashed) for comparisonit has the same input mean photon number ∼N S /M, while its performance is limited by the SQL. Also, the dominant noise in the entangled scheme without error correction (blue) comes from loss, and further increasing the initial mean photon number N S barely changes the performance. Finally, we address the necessity of the GKP grid states. For single-quadrature measurement, one might think that a CV repetition code [57] will also be able to suppress the noise due to an effective squeezing. However, in that case while the noise in one quadrature decreases, the noise in the other quadrature increases, leading to an overall increase in the mean photon number. Only a code with non-Gaussian resources such as GKP grid states can suppress noise in both quadratures.
With the above equivalence relation, afterwards an equal-weight addition of the measurement results will give the correct weighted average of displacements, up to some overall noise with variance å

Discussion
In principle, CV error correction codes such as GKP-stabilizer codes may be used to enhance the reliability of any protocol that makes use of CV quantum information. In this paper, we focus on the enhancement of distributed sensing tasks that can be achieved with CV error correction, providing a detailed evaluation of the effectiveness of GKP-stabilizer codes used for this purpose. When used for distributed sensing of CV displacements, the GKPstabilizer code with six iterations (level n=7 code) reinstates Heisenberg-scaling of precision up to about 10 2 nodes for transmissivity η0.95, while the Heisenberg scaling is destroyed entirely when no error correction is used. Since the GKP grid state enables simultaneous precise measurements of small displacements on both quadratures, we also use it to extend the distributed sensing protocol from single-quadrature displacements to displacements of both quadratures. When good prior information is available, simultaneous Heisenberg scaling of rms estimation errors on both quadratures can be achieved for the equal-weight case. Three future directions are worth pointing out. First, there is room for improvement on the conversion from a pure loss channel to an AWGN channel in section 2. During the amplification, part of the information about the input state is stored in the environment mode, and it might improve performance if the environment mode is used in the decoding process as well. It will also be worthwhile to investigate whether using CV error correction in quantum repeaters will improve their performance against loss. Finally, our methods can be applied to sensing other parameters, such as a weighted average of phases, where we expect similar enhancements of performance.