Enhanced Multi-Qubit Phase Estimation in Noisy Environments by Local Encoding

TL;DR

This paper demonstrates that local encoding of multi-qubit states enhances robustness against noise, significantly improving phase estimation accuracy in noisy environments, which is crucial for practical quantum metrology and error correction.

Contribution

The study experimentally shows that single-qubit local encoding improves noise resilience of four-qubit graph states for phase estimation, a novel approach for noisy quantum devices.

Findings

Local encoding enhances entanglement robustness against noise.

Significant improvement in phase estimation accuracy with local encoding.

Potential applications in quantum metrology and error correction.

Abstract

The first generation of multi-qubit quantum technologies will consist of noisy, intermediate-scale devices for which active error correction remains out of reach. To exploit such devices, it is thus imperative to use passive error protection that meets a careful trade-off between noise protection and resource overhead. Here, we experimentally demonstrate that single-qubit encoding can significantly enhance the robustness of entanglement and coherence of four-qubit graph states against local noise with a preferred direction. In particular, we explicitly show that local encoding provides a significant practical advantage for phase estimation in noisy environments. This demonstrates the efficacy of local unitary encoding under realistic conditions, with potential applications in multi-qubit quantum technologies for metrology, multi-partite secrecy and error correction.

Figures

Equations

$$\mathcal{D}(\rho )\doteq \left(1-\frac{p}{2}\right)\rho +\frac{p}{2}{\sigma }_z\rho {\sigma }_z,$$

$$|GH{Z}_N⟩\doteq \frac{1}{\sqrt{2}}\left(|0{⟩}^{\otimes N}+|1{⟩}^{\otimes N}\right)$$

$$|{GHZ}_N^{enc}⟩\doteq H^{\otimes N}|{GHZ}_N⟩=\frac{1}{\sqrt{2}}\left(|+{⟩}^{\otimes N}+|-{⟩}^{\otimes N}\right)$$

$$|{\psi }^{-}⟩=\frac{1}{\sqrt{2}}\left(|h⟩|v⟩-|v⟩|h⟩\right),$$

$$\mathrm{Tr}[{\rho }_{\varphi }{\left(|+⟩⟨+|\right)}^{\otimes 4}{]}_{\text{textsc}ghz}=\frac{1}{16}\left((p-1{)}^4\cos (4\varphi )+1\right),$$

$$\mathrm{Tr}[{\rho }_{\varphi }{\left(|+⟩⟨+|\right)}^{\otimes 4}{]}_{\text{textsc}{ghz}^{enc}}=\frac{1}{128}(4\cos (2\varphi )+\cos (4\varphi )+11).$$

$$Var[\varphi ]=\frac{Var[ϵ]}{|\frac{d}{d\varphi }ϵ{|}^2},$$

$$\begin{array}{rl}|C{L}_4⟩\doteq & C{Z}_{1,2}C{Z}_{2,3}C{Z}_{3,4}HHHH|0000⟩=\\ =& \frac{1}{2}(|+00+⟩+|-10+⟩+\\ & +|+01-⟩-|-11-⟩),\end{array}$$

$$|C{L}_4^{enc}⟩=\frac{1}{2}\left(|0000⟩+|1100⟩+|0011⟩-|1111⟩\right)$$

$${\lambda }_0=(1/2)(1-(1-p{)}^N)$$

$${\lambda }_1=1-{\lambda }_0$$

$${\lambda }_k=(1-p/2{)}^{N-k}(p/2{)}^k+(1-p/2{)}^k(p/2{)}^{N-k}$$

$$S(\rho )=\mathrm{Tr}(\rho {\log }_2\rho )=-\sum _k{\lambda }_k{\log }_2{\lambda }_k.$$

$${\mathcal{C}}_k:=\mathrm{conv}\{|\psi ⟩⟨\psi |:r_{\text{textsc}c}(|\psi ⟩)\le k\},$$

$$R_{{\mathcal{C}}_k}(\rho ):=\underset{\tau \in \mathcal{D}(\mathcal{H})}{\inf }\left\{s\ge 0:\frac{\rho +s\tau }{1+s}\in C_k\right\}.$$

$$\begin{array}[]{lll}R_{\mathcal{C}_{k}}(\rho)\,\,\,=&\min&\operatorname{Tr}(\sum_{I\in\mathcal{P}_{k}}\tilde{\sigma}_{I})-1\\[6.0pt] &\textup{s.t.}&\tilde{\sigma}_{I}\geq 0\quad\forall I\in\mathcal{P}_{k}\\[6.0pt] &&P_{I}\tilde{\sigma}_{I}P_{I}=\tilde{\sigma}_{I}\quad\forall I\in\mathcal{P}_{k}\\[6.0pt] &&\sum_{I\in\mathcal{P}_{k}}\tilde{\sigma}_{I}\geq\rho\,.\\ \end{array}$$

Full text

*Introduction—*Quantum systems are notoriously fragile due to unavoidable interactions with their environment Zurek (2003), resulting in decoherence that grows exponentially with system size. This represents a major roadblock for quantum computing Nielsen and Chuang (2011), quantum communication Aschauer and Briegel (2002) and quantum metrology Giovannetti et al. (2011), rendering noise mitigation Freedman et al. (2003); Lidar et al. (1998); Kim et al. (2012) indispensable. Quantum error correction (QEC) Shor (1995); Calderbank and Shor (1996); Steane (1996) schemes in principle achieve full protection against decoherence. However, daunting experimental requirements on the single qubit noise rate and large resource overheads Aharonov and Ben-Or (2008) make QEC a long-term vision.

A complementary approach, expected to play a central role in near-term quantum technologies Preskill (2018), is to relax the fault-tolerance requirement against arbitrary noise aiming instead at enhanced robustness of quantum systems, under experimentally relevant conditions. One of the dominant types of noise is local dephasing along a privileged direction Noda et al. (1986); Leibfried et al. (2003); Schindler et al. (2013). There, simple single-qubit unitary encoding can drastically improve the resilience of quantum resources Chaves et al. (2012) such as multi-qubit Greenberger-Horne-Zeilinger (GHZ) Greenberger et al. (1990) states, where an exponential decay of entanglement Aolita et al. (2008, 2015) can be turned into a linear decay Chaves et al. (2012). This improvement is crucial for metrology applications such as phase estimation in noisy environments Shaji and Caves (2007); Chaves et al. (2013), whereby the otherwise optimal phase sensitivity of GHZ states Giovannetti et al. (2004) become asymptotically bounded by a constant Demkowicz-Dobrzański et al. (2012).

Here, in a state-of-the-art 4-photon experiment at telecom wavelength, we report an in-depth study of the enhanced noise resilience that can be gained from local encoding Chaves et al. (2012). Using symmetric informationally complete (SIC) Renes et al. (2004) tomographic techniques, we quantify the noise resilience of quantum resources such as coherence and entanglement for all local-unitarily inequivalent classes of 4-qubit graph states HEIN (2005); Hein et al. (2004). Finally, as quantified by the experimental phase variance and the quantum Fisher information Holevo (2011), we observe that our encoding provides a clear improvement of the 4-qubit GHZ states usefulness for noisy quantum phase estimation. Notably, even under full dephasing, where the entanglement is always zero, the states nonetheless remain useful, hinting at a key role played by coherence, shown instead to be independent from the noise when the encoding is applied.

Multi-Qubit Robustness by Local Encoding— Consider an ideal quantum system subjected to local dephasing noise, before being used for an information-processing task such as phase estimation. Such is the case, for example, when the system crosses a noisy region before the protocol happens or if its implementation is much faster than the dephasing timescale. We note that the results are the same when the noise acts during the phase estimation task, here however, we keep them separate for simplicity. We aim to encode the system before the noise acts in order to increase its resilience against dephasing with as simple an encoding as possible, and then decode the system before it is being used. Taking the dephasing to act on the state $\rho$ in the computational basis $\{|0\rangle,|1\rangle\}$, the single-qubit dephasing channel is given by

[TABLE]

where $\sigma_{z}=|0\rangle\!\langle 0|-|1\rangle\!\langle 1|$ is the $Z$ Pauli matrix, and $0\leq p\leq 1$ quantifies the noise strength from no noise ($p=0$) to full dephasing ($p=1$). Consider now, for instance, an $N$-qubit GHZ state defined as

[TABLE]

The entanglement of GHZ states under single-qubit dephasing is known to decay exponentially with $N$. More precisely, for the dephased GHZ state $\rho_{N}(p)\doteq\mathcal{D}^{\otimes N}(|GHZ_{N}\rangle\!\langle GHZ_{N}|)$ it holds that $E\left(\rho_{N}(p)\right)\leq(1-p)^{N}E\left(\rho_{N}(0)\right)$ for any convex entanglement quantifier $E$ Aolita et al. (2008, 2015). However, this scaling can be drastically improved Chaves et al. (2012) by encoding the state using local Hadamard gates $H$, defined by $H|0\rangle\doteq|+\rangle$ and $H|1\rangle\doteq|-\rangle$ with $|\pm\rangle\doteq\frac{1}{\sqrt{2}}(|0\rangle\pm|1\rangle)$. The resulting encoded GHZ state:

[TABLE]

has the same entanglement properties, yet its entanglement decay rate becomes independent of $N$ and linear in $p$. Formally, the dephased encoded state $\rho^{enc}_{N}(p)\doteq\mathcal{D}^{\otimes N}(|GHZ_{N}^{enc}\rangle\!\langle GHZ_{N}^{enc}|)$ satisfies the bound $E\left(\rho^{enc}_{N}(p)\right)\geq E\left(\rho_{2}(p)\right)$ for all $N$ and thus possesses at least as much resilience as the two-qubit state $|GHZ_{2}\rangle$. A similar enhancement can be extended for arbitrary graph states Hein et al. (2004), which play a crucial role in measurement-based quantum computing and quantum error correction, see Supplementary Materials (SM) for details.

*Experimental setup—*We now test these passive error protection techniques in a state-of-the-art photonic platform, Fig. 1. Qubits are encoded in the horizontal $|h\rangle=|0\rangle$ and vertical $|v\rangle=|1\rangle$ polarization states of single photons. These are generated at $1550\text{\,}\mathrm{n}\mathrm{m}$ via collinear type-II spontaneous parametric down-conversion in a $22\text{\,}\mathrm{m}\mathrm{m}$ long periodically-poled KTP (PPKTP) crystal, pumped with a $1.6\text{\,}\mathrm{p}\mathrm{s}$ pulsed laser at $775\text{\,}\mathrm{n}\mathrm{m}$ Graffitti et al. (2018). After spectral filtering with a bandwidth of $3\text{\,}\mathrm{n}\mathrm{m}$, the source generates $\sim 3075$ pairs/mW/s with a symmetric heralding efficiency of $\sim 55\%$. Embedding the crystal within a Sagnac interferometer Fedrizzi et al. (2007) enables the generation of high-quality entangled states of the form

[TABLE]

with typical fidelities $F(\rho_{e},\rho_{t})=(Tr[\sqrt{\sqrt{\rho_{t}}\rho_{e}\sqrt{\rho_{t}}}])^{2}=99.62^{+0.01}_{-0.04}\%$ where $\rho_{e}$ and $\rho_{t}$ are the experimental and target state respectively. The measured purity is $P=99.34^{+0.01}_{-0.09}\%$ and entanglement as measured by the concurrence Hill and Wootters (1997) is $\mathcal{C}=99.38^{+0.02}_{-0.10}\%$. The photons are detected using superconducting nano-wire single-photon detectors (SNSPDs) with an efficiency of $\sim 80\%$ and processed using a time-tagging module with a resolution of $156\text{\,}\mathrm{p}\mathrm{s}$. Using two such photon-pair sources in the setup of Fig. 1, we can prepare the 4-qubit GHZ state $|GHZ_{4}\rangle$ of Eq. (2) by subjecting one photon of each entangled pair to nonclassical interference on a polarizing beam splitter (PBS), which transmits horizontal and reflects vertically polarized photons. This implements a so-called type-I fusion gate Browne and Rudolph (2005) for which we achieved a visibility of $91.80^{+1.73}_{-1.73}\%$, translating into a purity of $P=87.09^{+1.15}_{-2.18\%}$ and fidelity of $F=92.53^{+0.63}_{-1.23}\%$ for the 4-qubit GHZ state. The states are generated at a measured rate of $47.6\text{\,}\mathrm{H}\mathrm{z}$ using $60\text{\,}\mathrm{m}\mathrm{W}$ pump power.

Single qubit dephasing of Eq. (1) is experimentally implemented in a controllable manner by applying individual phase flips, using HWPs, to a fraction of the runs weighted by the dephasing strength $p$. The density matrices of the experimentally generated states are then reconstructed using maximum-likelihood quantum state tomography. The tomography was performed using the set of symmetric informationally complete (SIC) measurements Renes et al. (2004), which reduces the number of measurements compared to the standard Pauli basis by a factor $(2/3)^{N}$, leading to improved precision at equal acquisition time.

*Experimental noise protection—*In this section we present the experimental results attesting the capability of the method proposed, whose application in phase estimation is shown in the next section. We investigate the effect of the encoding proposed on two paradigmatic quantum resources: quantum entanglement and coherence. The former is quantified by the negativity Vidal and Werner (2002), the latter instead using the recently developed resource theory of multilevel coherence Napoli et al. (2016); Ringbauer et al. (2018), see SM for details. The results for both these figures of merit are shown in Fig. 2. The top panel shows that the negativity of the encoded $|GHZ_{4}^{enc}\rangle$ states is significantly more resilient against dephasing than the bare states. Moreover, the inset in Fig. 2 shows how the enhancement for a fixed amount of dephasing becomes more significant as the number of qubits increases, instead of the exponential decay observed for the bare states Chaves et al. (2012). As shown in the SM, such noise resilience is notably achieved by increasing the amount of entanglement with the environment. In fact, the encoded states experience a higher loss of purity than the non-encoded ones.

The bottom panel of Fig. 2, shows how coherence of the encoded states not only is protected from the action of noise but distinctly manifests independence from it. Non-encoded states, on the other hand, show an exponential decay. Intuitively, this may be understood based on the distribution of coherence within the state. Concentrating all coherence on two terms (coherence rank 2), such as the bare GHZ4 state leaves the state vulnerable to dephasing. In contrast, maximally spreading it out (coherence rank $2^{N}$), as in the encoded state, achieves increased resilience. In the latter case indeed, the decoding map (a non-free operation in the resource theory of coherence) can under certain conditions recover a significant amount of coherence, see SM for details. We remark that, multi-level coherence is independent of entanglement measures and can unlock information on the encoding effects otherwise inaccessible. This, as we will see in the next section, provides useful insights on phase estimation in noisy environment. Finally, noise protection, is also achieved for the linear cluster state, see SM.

*Enhanced phase estimation—*We now exploit our passive error correction for quantum metrology, by performing a 4-qubit phase estimation task Tóth and Apellaniz (2014) in a noisy environment. The goal is to estimate an unknown phase $\phi$ imparted on a probe state $\rho$ by the unitary $U_{\phi}=e^{-\frac{i}{2}\phi\sigma_{z}}$ by measuring the evolved state $\rho_{\phi}\doteq U_{\phi}^{\otimes N}\rho{U_{\phi}^{\dagger}}^{\otimes N}$. It is well known that GHZ states are optimal for phase estimation Giovannetti et al. (2011). In the presence of dephasing noise, however, this task becomes much more challenging, and different inequivalent strategies can be devised Demkowicz-Dobrzański and Maccone (2014); Pirandola and Lupo (2017). We now show how our local encoding can significantly enhance the metrology performance of 4-qubit GHZ state under such conditions. To assess the performance of phase estimation we study the expectation value:

[TABLE]

for noise of strength $p$, showing that for maximal dephasing i.e. $p=1$, no phase information can be recovered. Conversely, if the encoding of Eq. (3) is used, we find improved performance for all $p$. Most strikingly, the encoding preserves phase sensitivity even under full dephasing:

[TABLE]

Although entanglement is recognized as a fundamental resource for phase estimation Tóth (2012); Hyllus et al. (2012), is remarkable to note that phase sensitivity is observed even in the full dephasing regime, whereby the entanglement is always zero, even for the encoded states. It follows that the phase sensitivity observed is instead provided by the coherence only, which as we have previously seen is left untouched by the dephasing. Only when both coherence and entanglement are zero (as for the non-encoded states) the phase sensitivity is completely suppressed. This suggests, at least for this instance, the coherence to be a useful resource whereby the entanglement is not.

Experimentally, we applied a phase-shift $\phi\in[0,\pi]$ to each qubit, by shifting the measurement’s waveplates accordingly and reconstructed the expectation values $\operatorname{Tr}[\rho_{\phi}\left(|+\rangle\langle+|\right)^{\otimes 4}]$ as a function of $\phi$ for a range of $p$, see Fig. 3a. The results clearly show a steeper slope of $\operatorname{Tr}[\rho_{\phi}\left(|+\rangle\langle+|\right)^{\otimes 4}]$ for the encoded state compared to the bare state for all non-zero values of $\phi$. This directly translates into a more sensitive phase estimator in the encoded case. Moreover, we emphasize that the encoded fringes preserve at least half the visibility of the $p=0$ case, even for $p=1$ where instead, without our encoding, the bare fringes flatten to a constant value. In other words, whereby phase estimation would be normally impossible, our encoding makes it feasible again.

This qualitative behavior is turned into a quantitative result by measuring the experimental variance of the estimated phase in the point where the fringes are the steepest for different values of $p$, according to

[TABLE]

where $\epsilon$ is the measured average value of our estimator $\epsilon\equiv\left\langle+_{1}+_{2}+_{3}+_{4}\right\rangle$. The results are shown in Fig. 3b.

More in general however, the primary figure of merit in quantum metrology is the so called quantum Fisher information (QFI) Holevo (2011). Indeed, in the noiseless case, the statistical deviation $\delta\phi$ in the estimation of $\phi$, is bounded as $\delta\phi\geq 1/\left(\sqrt{\nu\mathcal{F}(\rho_{\phi})}\right)$ Braunstein and Caves (1994), where $\nu$ is the number of repetition runs in the estimation and $\mathcal{F}(\rho_{\phi})$ is the QFI, measuring the maximum amount of information about $\phi$ that can be extracted from $\rho_{\phi}$. For separable states $\mathcal{F}(\rho_{\phi})\leq N$, namely the QFI is bounded by the shot-noise limit (SNL), while for GHZ states the QFI attains the optimal value $\mathcal{F}_{\text{max}}=N^{2}$, known as the Heisenberg limit. On the other hand, the QFI of a locally dephased GHZ state $\mathcal{F}(\rho_{N}(p))=N^{2}(1-p)^{2N}$, indicates that for fixed noise strength $p$ the precision of the estimate of $\phi$ decreases exponentially with $N$. Such drastic decay is turned into a quadratic one for the encoded GHZ state, \mathcal{F}(\rho_{N}^{T}(p))=N^{2}(1-p)^{2}+4N\big{(}1-\frac{p}{2}\big{)}\frac{p}{2}. The Fisher information is measured experimentally for both the encoded and non-encoded density matrices Liu et al. (2014), see Fig. 3c. In absence of noise we experimentally observe a value close to the Heisenberg limit $N^{2}=16$ exponentially turned to [math] when the encoding is not applied. On the other hand, encoded GHZ states preserve their quantum advantage for significantly high noise strengths.

Conclusions— We have shown that, given knowledge of the dominant noise sources in the experiment, protection of quantum resources without complex encoding and additional overhead in the number of physical qubits, is feasible in practice and can enable significant performance improvements. We revealed different behaviors of graph states for all the quantum figure of merits under study, highlighting the importance of a deep understanding of state dynamics in the multi-qubit scenario. In particular, we exploited our method in one of the most common quantum information tasks i.e. phase estimation. Adding dephasing noise, we simulated a realistic implementation of such protocol where without any action no quantum advantage would be observed. With our encoding, we instead observe phase-sensitivity up to the full-noise regime where only coherence and not entanglement might be the useful resource for such task. In conclusion, we successfully proposed an alternative route for noise-protection which might be further exploited as far as active multi-qubit quantum error correction remains out of reach for near-term technology.

SI Supplementary Material

*Extension to linear cluster graph state—*Graph states are a sub-class of multi-qubit states which can be expressed by means of a graph, where vertices represent qubits and links entangling interactions. These play a fundamental role from quantum computation to quantum error correction. For $N=4$ qubits there exist only two classes of connected graph states in-equivalent under local unitary transformation Hein et al. (2004) i.e. a state in one class can not be mapped to a state in the other by means of any local unitary operation. The paradigmatic representatives of these two families are the GHZ state $|GHZ_{4}\rangle$ and the linear cluster state $|CL_{4}\rangle$, with the latter commonly defined as

[TABLE]

where the tensor product is omitted and $CZ_{i,j}$ is a controlled-Z gate Nielsen and Chuang (2011) on qubits $i$ and $j$. This expression corresponds to the graph state represented as a linear chain with qubits $1,4$ as external vertices. By investigating with our protocol both the GHZ and the linear cluster quantum states, we could in essence covers all 4-qubit graph states. The optimal local-unitary encoding for $|CL_{4}\rangle$ turns out to be $H\otimes\openone\otimes\openone\otimes H$, with $\openone$ the single-qubit identity operator, resulting in the state

[TABLE]

This state can be generated experimentally by subjecting one photon of an entangled pair to two sequential fusion gates with uncorrelated single photons in the state $|+\rangle$ and with a Hadamard gate in between, see Fig. S1. The purity and fidelity of the obtained state are $\mathcal{P}=81.77^{+0.65}_{-0.85\%}$ and $\mathcal{F}=89.03^{+0.38}_{-0.60}\%$ respectively. As for the GHZ case we compute the negativity in the partition $(1|234)$ and the robustness of multilevel coherence. As shown in Fig. S2 the encoding provided enhances the resilience of linear cluster states against the action of dephasing. Moreover, it is even possible to qualitatively change the behavior from finite-time disentanglement to infinite-time disentanglement. Regarding the coherence, contrary to the GHZ case, a constant behavior cannot be achieved, due to subtle differences in the structure of the states. Nonetheless protection is observed for all the values of dephasing. We note that, since contrary to the GHZ state the linear cluster state is known to be non-optimal for phase estimation, we limited our study to the case of entanglement and coherence enhancement of locally encoded linear cluster states.

Finally, for both the GHZ state and Linear Cluster, we consider all the 1-vs-rest partitions $(i|jkl)$ and 2-vs-2 bipartitions : $(12|34),(13|24),(14|23)$. The encoded 4-GHZ state is optimally protected in both all the 1-vs-rest and 2-vs-2 partitions. The scenario is more complex when instead we study the linear cluster, in fact there always exists at least one 1-vs-rest partition where the protection is optimal. However, the partition depends from the encoding chosen. Nevertheless, encoding providing protection for all the 1-vs-rest partitions exist, but the protection is not optimal.

*Evolution of the purity and entanglement entropy with the environment—*Intuitively, one would assume that the noise resilience is achieved by reducing the amount of entanglement with the environment that is generated (thus increasing the state’s purity). Surprisingly, however, the results of Fig. S3 show that, for the GHZ case, the opposite is true and the encoded states experience a higher loss of purity than the non-encoded ones.

Purity of a density matrix $\rho$ is given by $Tr[\rho^{2}]$ and can be expressed in terms of the eigenvalues $\lambda_{k}$ as $\mathcal{P}=\sum_{k}\lambda^{2}_{k}$. The evolved dephased bare GHZ state has eigenvalues

[TABLE]

leading to $P(\rho^{N}(p))=\frac{1}{2}\left(1+(1-p)^{2N}\right)$. On the other hand, the encoded state has eigenvalues given by

[TABLE]

with $0\leq k\leq N-1$ (each with a degeneracy of $\binom{N-1}{k}$) leading to a purity $P(\rho^{N}_{T}(p))=\left(p^{N}(1-\frac{p}{2})^{N}+(1-p+\frac{p^{2}}{2})^{N}\right)$. Although both purities decay exponentially with $N$, the purity of the bare state tends to $1/2$, while the purity of encoded state tends to $2^{1-N}$. The entanglement between system and environment can also be quantified via the von Neumann entropy of the system

[TABLE]

For this quantity, closed formula expressions are not any longer possible but one can easily see some interesting properties. For the bare GHZ state the entanglement entropy tends to $S(\rho_{N}(p))=1$ while for the encoded state $S(\rho_{N}^{T}(p))=N-1$ as $p\rightarrow 1$. Moreover, for any $p>0$ it follows that $S(\rho_{N}^{T}(p))>S(\rho_{N}(p))$, that is, at all times of the noisy evolution the encoded and more robust state is surprisingly more entangled with the environment.

Intuitively, this can be understood as a consequence of the special structure of the GHZ state, which even after full dephasing retains classical correlations that manifest in relatively high residual purity. In the encoded case, the coherence is more distributed, thereby reducing the resilience of the state’s purity. In the case of the linear cluster, on the other hand, the bare state features uniformly distributed populations, while the encoded state is sparser. As a consequence, the optimal encoding protects both entanglement and purity.

*Robustness of multilevel coherence—*Following Ref. Ringbauer et al. (2018), we now review the concept of multilevel quantum coherence as a fine-grained quantifier for the amount of coherence present in a given quantum state. We consider composite $N$-qubit systems and measure coherence with respect to the computational basis $\{|0\rangle,|1\rangle\}^{\otimes N}$. In order to capture the structure of coherence in such a system, one first defines the following sets of states

[TABLE]

where $\operatorname{conv}$ stands for convex hull and $r_{\textsc{c}}$ is the coherence rank of $|\psi\rangle$, given by the number of non-zero coefficients in the basis-decomposition of $|\psi\rangle$. $\mathcal{C}_{1}$ is the set of fully incoherent states, given by density matrices that are diagonal in the computational basis, while $\mathcal{C}_{d}\equiv\mathscr{D(\mathscr{H})}$ is the set of all states in the $d$-dimensional Hilbert space $\mathscr{H}$. It was shown in Ref. Ringbauer et al. (2018) that these sets obey a strict hierarchy and that the amount of $k$-level coherence can be quantified by the robustness of multilevel coherence

[TABLE]

Here $\tau$ is any density matrix. A state has coherence number $k$, if it can be decomposed into pure states which are superpositions of at most $k$ basis elements, while every decomposition must contain at least one such state. For $k=1$, this measure simply quantifies the total amount of coherence in the system and it can be computed efficiently for all $k$, given the density matrix. For $k=2$ and $k=3$ instead the results are shown in Fig. S4. Experimentally, we can quantify the multilevel coherence for a given density matrix using the following semi-definite program, see Ref. Ringbauer et al. (2018) for details.

[TABLE]

Here $\mathcal{P}_{k}$ is the set of all the $k$-element subsets of $\{1,2,\ldots,d\}$, and $P_{I}:=\sum_{i\in I}|i\rangle\langle i|$