Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behavior of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for noisy intermediate-scale quantum devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.

Figure 1

Various ways to obtain physical states starting from ideal grid states. From left to right, the approximations in green clouds correspond to discussions around Eq. (20), (12), (16), (15), and (8).

Figure 2

A schematic of how to obtain the normalizable GKP state $\left|{\psi }_{\epsilon }\right.〉\equiv e^{-\epsilon \widehat{n}}\left|{\psi }_I\right.〉$ through an optical circuit, as noted in [13] and shown in Appendix pp2-s1. An ideal GKP state and the vacuum state pass through the first and second modes of a beam splitter $B(\theta ,\varphi )$ with transmissivity $-ln\epsilon$ [see Eq. (62)]; the second mode is measured and postselected on a vacuum state.

Figure 3

The Wigner logarithmic negativity $W_N$ (defined in Appendix pp2-s4) of the normalizable GKP state $\left|0_{\epsilon }\right.〉$ from Eq. (12) as a function of $\epsilon$ shown as solid (blue) line. Insets : Highlighted from left to right are wave functions corresponding to $\epsilon$ values of 0.158, 0.398, and 1 ($\mathrm{\Delta }$ values of 8, 4, and 0 dB). The overall Gaussian envelope of width $\kappa ={\mathrm{\Delta }}^{-1}$ is also drawn. The Gaussian in the third inset noticeably does not perfectly envelop the $\epsilon =1$ state because here there is an error of order 0.04 in the approximation between $\mathrm{\Delta }$ and $\epsilon$ [as shown in Eq. (13) and Table 4]. $W_N$ decreases with increasing $\epsilon$, i.e., as the number of peaks in our GKP state decreases, the width of each one increases, and the state becomes more Gaussian.

Figure 4

Effects of the normalization envelope $E\left(\epsilon \right)=e^{-\epsilon \widehat{n}}$ on the logical content of GKP states. In (a), trajectories of $\left|0_{\epsilon }\right.〉,\left|1_{\epsilon }\right.〉,\left|{\pm }_{\epsilon }\right.〉$, and $\left|{H_{+}}_{\epsilon }\right.〉$ as a function of $\epsilon$ confined to the $X-Z$ plane of the Bloch sphere [here $H_{+}$ is the $+1$ eigenstate of the Hadamard operator; see (31)]. For large $\epsilon$, all Fock states but $\left|0\right.〉$ are exponentially suppressed, causing all the trajectories to converge at the logical subsystem point corresponding to the physical vacuum state. We additionally show locations for reasonable values of $\epsilon$: $\epsilon =0.0625\approx \mathrm{\Delta }=12$ dB and $\epsilon =0.25\approx \mathrm{\Delta }=6$ dB are denoted by a star and triangle, respectively [see Eq. (13) and Appendix pp1 for conversions). We see that the required $\epsilon$ (dB) values to achieve higher purity states are lower (higher) toward the equator of the Bloch sphere. In (b), the modification of the $X-Z$ plane of the ideal Bloch sphere as a function of $\epsilon$. We see that points toward the equator are more distorted.

Figure 5

(a) The ideal logical circuit: a Clifford unitary $\overline{U}$ is applied to a qubit $\left|\overline{\psi }\right.〉$, followed by Pauli $Z$ measurements resulting in a bit $x$, with probability $p_{\overline{U}}\left(x|\overline{\psi }\right)$. (b) The ideal encoded physical circuit: a Gaussian unitary $U$ which implements $\overline{U}$ is applied to an ideal GKP state, followed by (binned) homodyne measurements to produce a bit with probability $p_U\left(x\right|{\psi }_I)$. (c) The circuit in (b) redrawn with a modular subsystem decomposition (note that we are only able to decompose the $U$ in this way for an input ideal GKP state). The probability distributions in (a), (c), and (b) are identical: $p_{\overline{U}}\left(x\right|\overline{\psi })=p_U\left(x\right|{\psi }_I)$. (d) The realistic GKP circuit, where the initial states are subject to a finite-energy envelope $G$. The unitary remains the same and a homodyne measurement produces a bit with probability $p_U\left(x|{\psi }_G\right)$. (e) The circuit in (d) but we highlight that the envelope can be conjugated with the Gaussian with appropriate modifications. (f) A general GKP circuit: the same as (e), but now we include a unitary recovery operation on the data states and ancillae $\left|{\psi }^A\right.〉$. The measurement operator, which could be homodyne or PNR, is generalized to be on all modes, now producing a bit string of some composite variable $y$. See the remark in the main text for a generalization of this schematic to a multimode circuit.

Figure 6

The normalizable GKP state $\left|0_{\epsilon }\right.〉$ (blue, solid line) and the same state after two applications of the $X\left(2\sqrt{\pi }\right)$ gate (red, dashed line) for $\epsilon =0.063(\mathrm{\Delta }\approx 12\text{dB})$. Because finite-energy GKP states have finite support in position space and hence lack a translational symmetry, the physical fidelity between the displaced state and the original state decreases with each application of $X\left(2\sqrt{\pi }\right)$ even though this gate preserves the logical content.

Figure 7

Averaged GKP gate fidelity ${\overline{F}}_{U,G}^{\mathcal{L}}\left(\overline{U}\right)$ from (50) evaluated for the logical and physical identity {$\overline{U}=\overline{\mathbb{1}}, U=\mathbb{1}$} (blue), and logical and physical phase gate {$\overline{U}=\overline{P}, U=P$} (orange) assuming a normalizable GKP state with $G=E\left(\epsilon \right)$ as a function of $\epsilon$. This plot shows that even without the application of any gate, the error envelope $E$ creates a deviation, on average, from unity fidelity with the ideal data qubit state. The deviation is even greater in the case of a phase gate, the reasons for which are given in Sec. 2d4. In both cases, the fidelity becomes worse as $\epsilon$ becomes bigger, as expected.

Figure 8

Physical fidelity of the normalizable GKP state $\left|0_{\epsilon }\right.〉$ after sequential applications of the $X\left(2\sqrt{\pi }\right)$ gate, which in the ideal case implements a logical identity. The lower the $\epsilon$, the better the approximation to the ideal GKP state, the better the translational symmetry of the state, and the slower the physical fidelity falls as a function of the number of gate applications.

Figure 9

The energy cost mean ${\mathcal{D}}_U^{\overline{n}}$ and standard deviation ${\mathcal{D}}_U^{\sigma }$ of multiple applications of the $X\left(\sqrt{\pi }\right)$ gate to the state $\left|0_{\epsilon }\right.〉$ for $\epsilon =0.063$ (corresponding to a $\mathrm{\Delta }\approx$ 12 dB). Each subsequent displacement comes at an increased energy cost, and the spread increases more slowly than the mean. Note that an analytic expression for ${\mathcal{D}}_U^{\overline{n}}$ for an arbitrary state $\left|\psi \right.〉$ yields $\frac{\pi }{2}{k}^2-〈\psi |\widehat{q}|\psi 〉k$, where $k$ is the number of gate applications. Therefore, applying $X$ gates to zero-mean states will result in the same behavior.

Figure 10

Physical fidelity of normalizable GKP state $\left|0_{\epsilon }\right.〉$ after sequential applications of $P^2$ gates for various values of $\epsilon$. In the ideal case, this gate implements the logical $\overline{Z}$, i.e., an identity on the $\left|0_I\right.〉$ state. The physical fidelity falls more slowly when $\epsilon$ is higher, that is, when $\left|0_{\epsilon }\right.〉$ is further from the ideal state. In this regime, the state has a lower squeezing and hence a wider peak at $q=0$, where the phase gate introduces a close-to-trivial phase $e^{i\frac{1}{2}}{q}^2$.

Figure 11

Warping of the logical Bloch sphere after application of the physical phase gate $P$ to the normalizable GKP states $E\left(\epsilon \right)\left|0_I\right.〉$, as compared to the action of a true logical phase gate $\overline{P}$. We show the result for various values of $\epsilon$ (for conversion to dB see Appendix pp1). We apply the gate to states in the $X-Z$ plane, so the $Y-Z$ plane is depicted since the ideal $\overline{P}$ gate rotates the sphere about the $Z$ axis. Whereas $\overline{P}$ simply applies the rotation to an already distorted Bloch sphere, $P$ additionally warps the Bloch sphere, notably near the equator where states lose more purity.

Figure 12

The energy cost mean and standard deviation of successive applications of the $P$ gate to the state $\left|0_{\epsilon }\right.〉$ for $\epsilon =0.063$ ($\mathrm{\Delta }\approx 12$ dB). In contrast with the $X$ gate in Fig. 9, the spread grows faster than the mean. The analytic expression for ${\mathcal{D}}_U^{\overline{n}}$ after $k$ applications of the phase gate on the state $\left|0_{\epsilon }\right.〉$ is $\frac{1}{2}\left(k^2〈0_{\epsilon }\left|{\widehat{q}}^2\right|0_{\epsilon }〉+k〈0_{\epsilon }\left|\{\widehat{q},\widehat{p}\}\right|0_{\epsilon }〉\right)$. Insets: highlighted are the wave functions after one (left) and three (right) applications of the gate to visualize the damage the gate inflicts on the state in the form of shearing, that is, increasing oscillatory behavior near the peaks.

Figure 13

(a) Entanglement negativity (defined in main text) of a two-qubit system initialized in $\left|{+}_{\epsilon }\right.〉\left|0_{\epsilon }\right.〉$ and then subjected to the application of a cnot [sum(1)] gate, as compared to the ideal (maximal) negativity for perfect qubits. We see that to generate entanglement in the logical subsystem, we require $\epsilon \approx 0.36$, which corresponds to roughly $\mathrm{\Delta }\approx 4.5$ dB of squeezing; however, to generate maximal entanglement, significantly more stringent thresholds are required, with $\epsilon \approx$ 0.05 only generating 80% of the maximal entanglement negativity. (b) The logical subsystem trace distance between cnot$\left[\left|{+}_{\epsilon }\right.〉\left|0_{\epsilon }\right.〉\right]$ and cnot$\left[\left|{+}_I\right.〉\left|0_I\right.〉\right]$. Again, we see that to achieve the ideal distance of 0, smaller values of $\epsilon$ are required.

Figure 14

Error syndrome measurement with normalizable GKP states following the Steane approach. First, shifts in $q$ are corrected: an encoded data qubit $\left|{\psi }_G\right.〉$ and an ancilla $\left|{+}_G\right.〉$ are sent through a sum gate, and $\left|{\psi }_G\right.〉$ is displaced according to the result of a homodyne $q$ measurement on the ancilla. A similar procedure follows for shifts in $p$.

Figure 15

The circuit in Fig. 14 but with the measurements pushed to the end. Now, a unitary is applied that encompasses both the correcting position and momentum shifts determined by a new function $\tilde{f}$ of the homodyne measurement results $q_0$ and $p_0$. This unitary shifts the projected state back onto the GKP grid. When the error is too big, these shifts are mistakenly performed in the wrong direction, and a logical error results.

Figure 16

GKP error correction using the Knill approach. Here, the ancillary states $\left|{+}_G\right.〉$ and $\left|0_G\right.〉$ are entangled and then the data state $\left|{\psi }_G\right.〉$ is sent through a sum gate with $\left|{+}_G\right.〉$ as the target. Homodyne position and momentum measurements are conducted on $\left|{\psi }_G\right.〉$ and $\left|{+}_G\right.〉$, respectively; this teleports the logical data to $\left|0_G\right.〉$ up to a unitary, applied to $\left|0_G\right.〉$ depending on the measurement results. Notice that, unlike in the Steane approach (Fig. 15), the output state after the projection is already on the GKP grid. The purpose of this unitary, therefore, is to correct the logical-Pauli by-product operators that result from the teleportation. When the error is too big, these by-product operators are misidentified, and a logical error results.

Figure 17

Homodyne measurement probabilities in a successful position error-correction step with data state $\left|0_{\epsilon }\right.〉$ and ancilla $\left|{+}_{\epsilon }\right.〉$ with $\epsilon =0.063$ ($\mathrm{\Delta }\approx 12\text{dB}$). (a) The probability of the integer part of the measurement outcome equaling $n_a+n_d$, where $n_a\sqrt{\pi }$ and $2n_d\sqrt{\pi }$ are peak locations in the ancilla and data state, respectively, and (b) the probability of the data state having a peak at $q=2n_d\sqrt{\pi }$ given that the integer part of the measurement outcome was $n_a+n_d$. After several rounds of error correction, we may be able to use data like this to conclude, with some probability, how far the mean of our data state is from 0 so that we may recalibrate it.

Figure 18

The model for the GBS-like device used for state preparation. Gaussian states consisting of squeezed, displaced vacuum states $|z_i,{\alpha }_i\rangle$ are sent through an $N$-mode interferometer parametrized by $U\left(\overline{\mathrm{\Theta }}\right)$, followed by a PNR measurement on all but one of the modes. Given a PNR outcome of $\overline{\mathbf{n}}$, the desired output state $\left|{\psi }_{\text{out}}\right.〉$ is produced in the unmeasured mode. Our task is to optimize the circuit components ($\mathit{\alpha }, \mathit{z}$, and $\overline{\mathrm{\Theta }}$) for a given $\overline{\mathbf{n}}$ to produce a desired approximate GKP state. We loop over $\overline{\mathbf{n}}$ (subject to constraints) to find the best $N$-mode circuit for the task.

Figure 19

Physical fidelity of our approximate states $\left|0_A\right.〉$ to the target state $\left|0_{\mathrm{\Delta }}\right.〉$ as $\mathrm{\Delta }$ (dB) is varied [see Eq. (8) for the definition of $\left|0_{\mathrm{\Delta }}\right.〉$ and Appendix pp1 for conversion of dB to other conventions]. The $\left|0_A\right.〉$ states are constructed by applying squeezing to a core superposition of Fock states, so different line colors and styles correspond to different values of $n_{max}$ for the core state. Note that, for different $n_{max}$, the optimal squeezing parameter and Fock coefficients are generally different. The optimal $\left|0_A\right.〉$ states for different $n_{max}$ are found by maximizing the fidelity to $\left|0_{\mathrm{\Delta }}\right.〉$.

Figure 20

Average photon number of the approximate states $\left|0_A\right.〉$ as a function of $\mathrm{\Delta }$ (dB), the parameter that characterizes the target state $\left|0_{\mathrm{\Delta }}\right.〉$ that $\left|0_A\right.〉$ is designed to approximate. Line colors (styles) reflect different values of $n_{max}$ for the core state used to construct $\left|0_A\right.〉$. As expected, higher-quality states, those with larger $\mathrm{\Delta }$ in dB, require more energy.

Figure 21

Overlap $〈1_A|0_A〉$ between the approximate logical states as a function of $\mathrm{\Delta }$ (dB), the parameter that characterizes the GKP states that the approximate states are targeting. Line colors (styles) reflect different values of $n_{max}$ for the core state used to construct $\left|0_A\right.〉$. For comparison, we also plot the overlap $〈1_{\mathrm{\Delta }}|0_{\mathrm{\Delta }}〉$ of the target states [gray/(dashed) line]. A small overlap is a minimal condition for being able to distinguish logical states.

Figure 22

The Glancy-Knill property $P_{\text{no}}\text{err}$, which characterizes the probability a state would not yield a logical error if used as an ancilla for Steane error correction with GKP states (see Sec. 2e for definition and details). The line colors (styles) reflect $P_{\text{no}}\text{err}$ for the various $\left|0_A\right.〉$, each constructed with a different $n_{max}$ in the core state. We vary $\left|0_A\right.〉$ with $\mathrm{\Delta }$, which parametrizes $\left|0_{\mathrm{\Delta }}\right.〉$, the state $\left|0_A\right.〉$ approximates. Additionally, we provide $P_{\text{no}}\text{err}$ for $\left|0_{\mathrm{\Delta }}\right.〉$ [gray (dashed) line]. For $\mathrm{\Delta }=11$ dB and $n_{max}=12$, we see that, even though $\left|0_A\right.〉$ can have 93% fidelity to $\left|0_{\mathrm{\Delta }}\right.〉$ (see Fig. 19), $P_{\text{no}}\text{err}$ drops by 16%.

Figure 23

Examples of different normalizable and approximate GKP wave functions and their performance under the Glancy-Knill property (71): $\left|0_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=11$ dB, which yields $P_{\text{no}}\text{err}=74\%$ [gray (solid) line]; $\left|0_A\right.〉$ with $n_{max}=12$ that achieves $P_{\text{no}}\text{err}=57\%$ by maximizing fidelity to $\left|0_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=11$ dB [blue (dashed) line]; and the state obtained from optimizing directly with respect to $P_{\text{no}}\text{err}$ with core states of $n_{max}=12$, giving $P_{\text{no}}\text{err}=61\%$ [red (dotted) line]. Although the red (dotted) wave function has higher $P_{\text{no}}\text{err}$, the blue (dashed) wave function is better confined to the bin structure of the GKP states, meaning it has better logical encoding when using the modular subsystem decomposition which only depends on the bin structure of the wave function.

Figure 24

Location of the logical subsystem of the approximate states $\left|{\mu }_A\right.〉$, where $\mu =0,1,+,H_{+}$, on the $X-Z$ slice of the Bloch sphere. The trajectories are formed as the $\left|{\mu }_A\right.〉$ vary with $\mathrm{\Delta }$, which parametrizes the states $\left|{\mu }_{\mathrm{\Delta }}\right.〉$ that the $\left|{\mu }_A\right.〉$ approximate. We denote where the trajectories begin and end using a point for $\mathrm{\Delta }=3$ dB and a star for $\mathrm{\Delta }=11$ dB. We repeat the plot for different choices of $n_{max}$ for the core states of $\left|{\mu }_A\right.〉$. While in certain cases $\left|0_A\right.〉$ and $\left|{+}_A\right.〉$ can be mapped correctly on the Bloch sphere even with $n_{max}=2$, if we want all $\mu$ to be well mapped, we need core states with higher $n_{max}$.

Figure 25

Wigner functions for (a) $\left|0_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=10\mathrm{dB}$, as well as for the optimal states output by the three-mode GBS devices designed to produce $\left|0_A\right.〉$ with $n_{max}=$ (b) 12, (c) 8, and (d) 4 photons. These correspond to the starred results for $N=3$ in Table 2. We see the peak structure gets better with increasing $n_{max}$, but differences to $\left|0_{\mathrm{\Delta }}\right.〉$ are still apparent further from the origin in phase space.

Figure 26

Stability analysis for the optimal three-mode circuit designed to produce $\left|0_A\right.〉$ with a core state of $n_{max}=12$ targeting $\left|0_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=10$ dB. We find the worst-case fidelity between the circuit output and $\left|0_A\right.〉$ as a function of the circuit beam-splitter and squeezer stability parameters, which are defined as the relative error to the ideal parameters.

Figure 27

The effect of lossy optical components on the optimal GBS devices for GKP state preparation. We examine how the optimal circuits for $N=3$ modes from Table 2 for core states of $n_{max}=4,8,12$ perform as a function of a single loss parameter $\eta$. Our loss model consists of applying a loss channel parametrized by $\eta$ to each optical component, as well as to the mode outcoupling and PNR detectors. We plot how the fidelity, probability, and Wigner log negativity change as loss increases. Additionally, for a few values of loss, we plot reoptimized results for lossy circuits designed to produce $n_{max}=12$ (green stars with arrows indicating which values were reoptimized). We see an increase in fidelity as a result. We stop reoptimization once fidelity is on the same order as that of the Gaussian state with highest fidelity to the target, a threshold we plot with a dashed black line.

Figure 28

Probabilities of different even Fock states in $|0_{\epsilon }\rangle$ up to $n=12$ as a function of $\epsilon \approx {\mathrm{\Delta }}^2$. We see that the probability of measuring 8 photons is higher than 6; thus, when constructing the GKP states from squeezed core states, increasing $n_{max}$ of the core state from 4 to 6 provides little to no advantage compared to increasing $n_{max}$ to 8.

Figure 29

Wave functions produced from summing the first eight Fock states in the distribution for $|0_{\epsilon }\rangle$ (using $\epsilon \approx {\mathrm{\Delta }}^2$), as well as the wave functions for the weighted Fock states in the superposition. We see the important contribution of the 8-photon state to the outer set of peaks for the normalizable GKP state. Thus, without the 8-photon state in the core of $\left|0_A\right.〉$, the $n=4$ or 6 states are needed to build the outer peaks, leading to worse fidelities.

Figure 30

Fidelity of approximate states $\left|{\mu }_A\right.〉$ to normalizable states $\left|{\mu }_{\mathrm{\Delta }}\right.〉$; and photon number vs $\mathrm{\Delta }$ for (by column) $\mu$ = (a) 1, (b) $+$, and (c) $H_{+}$. Line colors (styles) in each plot represent the results for different $n_{max}$ in the core states of $\left|{\mu }_A\right.〉$. The trend in all cases is that a greater $n_{max}$ leads to better fidelity to the target state and higher energy; if the quality of the target state is improved by increasing the dB value of $\mathrm{\Delta }$, for a fixed $n_{max}$, the fidelity gets worse, and the required energy gets higher. We expect comparable circuit resources (see Sec. 3d) are required to construct approximate states for any point on the Bloch sphere.

Figure 31

Single-qubit code-space projectors $P(q,q^{\prime })$ in $q$ basis (in units of $\sqrt{\pi }$) for three cases: (a) The normalizable GKP states $\left|0_{\mathrm{\Delta }}\right.〉\left.〈0_{\mathrm{\Delta }}\right|+\left|1_{\mathrm{\Delta }}\right.〉\left.〈1_{\mathrm{\Delta }}\right|$, with $\mathrm{\Delta }=10$ dB. (b) Our approximate GKP states $\left|0_A\right.〉\left.〈0_A\right|+\left|1_A\right.〉\left.〈1_A\right|$, designed to target the normalizable states with $\mathrm{\Delta }=10$ dB, constructed using squeezing applied to a core state with $n_{max}=12$. Here, we assume we prepare both $\left|0_A\right.〉$ and $\left|1_A\right.〉$. (c) Our approximate GKP states $\left|0_A\right.〉\left.〈0_A\right|+X\left(\sqrt{\pi }\right)\left|0_A\right.〉\left.〈0_A\right|X^{†}\left(\sqrt{\pi }\right)$ with the same parameters as (b), except here we assume we prepare only $\left|0_A\right.〉$ and create the logical 1 state by displacing $\left|0_A\right.〉$. We see that the symmetries of the target projector are best preserved when preparing both $\left|0_A\right.〉$ and $\left|1_A\right.〉$, although that symmetry may not be required for encoding.

Figure 32

Visualization of the quantum error-correction (QEC) matrices [Eq. (E1)] of normalizable and approximate GKP states $\left|{\psi }_{\mathrm{\Delta }}\right.〉$ and $\left|{\psi }_A\right.〉$, subject to displacement errors. Each row of subfigures corresponds to a different choice of code states. In each subfigure, the $y$ ($x$) axis corresponds to a value of displacement along $q$ ($p$) in phase space, so that a single point corresponds to a net displacement error. The QEC matrix associated with this displacement error can then be decomposed in terms of the identity and Pauli operators. Each column corresponds to the magnitude of the coefficients in the decomposition, which can be interpreted as no-error, bit-flip, bit-phase-flip, and phase-flip probabilities. The first row corresponds to $\left|{\psi }_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=10$ dB. The second through fourth rows correspond to $\left|{\psi }_A\right.〉$ meant to approximate $\left|{\psi }_{\mathrm{\Delta }}\right.〉$ with $\mathrm{\Delta }=10$ dB with core states of $n_{max}$ = 12, 8, and 4 photons.