Sufficient condition for universal quantum computation using bosonic circuits

Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest.


I. INTRODUCTION
Despite recent progress in understanding the relationship between genuine quantum properties and quantum computation [1][2][3][4][5], unraveling the origin of quantum computational power remains a challenging task.Adopting insight from the framework of resource theories [4], one approach to develop our understanding consists of breaking down the design of quantum computing architectures into two sub-parts: (i) the implementation of a restricted class of circuits, which can be efficiently simulated with a classical device and therefore are deemed as free or allowed; (ii) the preparation of specific states which are able to promote the restricted class to a universal model [6] and are therefore deemed as resources.The latter identifies key properties that enable quantum advantage -namely the ability to solve certain computational problems exponentially faster than classical computers [7].
The choice of the restricted class depends on the model of quantum computation (QC).In discrete-variable (DV) qubitbased QC, the restricted class most commonly considered is the set of Clifford circuits acting on stabilizer states [8,9].Clifford circuits alone are incapable of achieving universality, and consequently quantum advantage.However certain states, such as the "magic" T -state, are capable of promoting these circuits to universality [6].States that have a fidelity to the T -state beyond a certain threshold also fulfill this scope by means of magic state distillation, whereby a large number of low-quality magic states can be converted to a smaller number of nearly ideal ones [6,[10][11][12][13].Hence the fidelity to the closest ideal magic state yields a sufficient criterion for universality.
In continuous-variable (CV) quantum computing, Gaussian quantum circuits [14][15][16][17] are commonly chosen as the counterpart to Clifford circuits.In fact, it is known that Gaussian circuits are efficiently simulatable and therefore incapable of performing universal QC [18].Adding access to certain CV resource states, such as the cubic phase state [19][20][21][22], Gottesman-Kitaev-Preskill (GKP) states [23] or cat state [24,25] promotes these circuits to universality.More broadly, a significant effort has been devoted to identifying efficiently simulatable circuits [18,[26][27][28], and therefore the requisite properties for a state to act as a resource in the CV setting.In particular, necessary conditions have been provided in terms of the Wigner logarithmic negativity (WLN) [29,30] and the stellar rank [31], which quantify the degree of non-Gaussian features of a state.However, in contrast to the DV case, no sufficient criterion exists which can identify whether an arbitrary CV state is capable of promoting an otherwise simulatable architecture to universality.
In this work, we establish a sufficient criterion for a CV state to promote an otherwise simulatable class of circuits to universality.To accomplish this, we consider a distinct class, different from Gaussian circuits, as resourceless.Specifically, we choose circuits composed of ideal GKP stabilizer states, acted on by Gaussian operations [32] and measured with homodyne detection.These circuits have been shown to be efficiently simulatable [33][34][35][36].As such, throughout this work, we refer to these types of circuits as simulatable GKP (SGKP) circuits.
Leveraging on this criterion, we assess the resourcefulness of generic Gaussian states in this model, thereby extending the set of previously known resourceful Gaussian states which only included the vacuum and thermal states [23].Our approach can be applied to generic states, and in particular we also investigate highly non-Gaussian states, such as realistic GKP states, cat states, and cubic phase states.We identify pa-arXiv:2309.07820v2[quant-ph] 5 Jan 2024 rameter regimes where they can be considered as resources in this framework, and where they exceed the resourcefulness of the Gaussian states.
Our approach comprises two steps.We first map the CV state of interest into a two-dimensional space, effectively associating a qubit state to it.The maps that we define are inspired by some subsystem decompositions (SSDs) recently introduced in order to extract the (qubit-like) logical content of a generic CV state [37][38][39].However, crucially, we identify and focus on maps that can be implemented using solely SGKP circuits.Therefore they are free maps in a rigorous resourcetheoretical sense, ensuring they do not artificially add any resource to the original CV state.In the second step, we apply known results in DV systems to evaluate the resourcefulness of the mapped qubit state, therefore establishing a sufficient condition for the original CV state to be resourceful.
As a byproduct, by establishing which SSD can be obtained using SGKP circuits, we are able to establish whether known SSDs can be grounded in a rigorous resource-theoretic framework.Considering that these SSDs play a pivotal role in extracting logical information from both theoretically proposed [40][41][42] and experimentally generated states [43], we anticipate that this result will be of independent interest.
The subsequent sections are organized as follows.In Sec.II we present an overview of the main results of our work.In Sec.III we present previous methods for understanding the resourcefulness of quantum states for universal QC, along with an overview of existing methods for mapping CV states to DV states.In Sec.IV we introduce a unified approach for mapping CV states to DV states and demonstrate how existing maps can be expressed in this framework.We then introduce a new map which is implementable using SGKP circuits.In Sec.V we present our technique for quantifying the resourcefulness of CV states for quantum advantage by interpreting the CV state as an encoded DV state, and present results quantifying the resourcefulness of a range of different CV states using our technique.In Sec.VI we present the conclusions of our work and provide some open questions.In the Appendix, we provide a physical interpretation of the various maps in terms of circuits, and we demonstrate that the modular subsystem decomposition admits a physical interpretation in terms of SGKP circuits for states symmetric in position.

II. MAIN RESULTS
To enhance readability, we report in this section a summary of the main results of this work.Comprehensive details and proofs are deferred to subsequent sections.
As said, we introduce a framework to address the resourcefulness of generic CV states when combined with the otherwise simulatable class of SGKP circuits.The general type of circuits considered is of the form depicted in Fig. 1, where one can also assume to have access to adaptive operations.As proven in Ref. [36], these circuits are efficiently simulatable on a classical device when employing only ideal stabilizer GKP states as input.Drawing on insights from qubit magic state distillation [6] and GKP error correction [19], we will prove that the circuit in Fig. 1 attains instead universality when the input CV states ρ can be mapped into resourceful encoded qubits state via SGKP circuits alone.This approach therefore establishes a sufficient criterion for determining the resourcefulness of ρ.
A circuit diagram displaying the broad class of circuits that we consider in this work.In input, there are m stabilizer ideal GKP states (in the diagram these are indicated as 0-logical states without loss of generality) and n arbitrary CV states ρ.These states are acted on by Gaussian operations and measured with homodyne measurement.When also ρ are stabilizer GKP states, these circuits are efficiently simulatable yielding SGKP circuits [35,36] (see text for details).
In more detail, we first introduce a method to systematically map arbitrary single-mode CV states to qubit states.This method unifies previously defined mappings, specifically subsystem decompositions (SSDs), which have recently been introduced for evaluating the qubit-like logical content of CV states.The approach involves transforming a CV state into an encoded GKP state and subsequently analyzing the resulting encoded qubit state.Depending on the choice of mapping, different qubit states will emerge.Of particular relevance for our objectives are those mappings implementable exclusively using SGKP circuits.This is crucial, since it guarantees that the associated SSDs do not introduce additional resources beyond those present in the original CV state.For this reason, following standard resource theory nomenclature, we term them allowed mappings.
We review the existing mappings of stabilizer SSD [39] and modular SSD [37,38], expressing them within the presented general formalism.By leveraging on the connection between the stabilizer SSD and GKP error correction [39], we show that stabilizer SSD can be constructed in terms of SGKP circuits, therefore it is an allowed mapping.In contrast, the modular SSD lacks this interpretative advantage in general.To address this, we introduce a new map termed Gaussian modular SSD, and prove its equivalence to the modular SSD when the input CV state exhibits symmetry in the position representation.Crucially, like the stabilizer SSD, the Gaussian modular SSD can be understood in terms of allowed maps.This implies that the modular SSD, too, is a resource-theoretically grounded mapping for analyzing relevant position symmetric states such as finitely squeezed GKP states, as well as cat states, among others.However, for non-symmetric states, the equivalence breaks down, and the implementation of modular SSD necessitates operations beyond SGKP circuits, thus losing its interpretative status as an allowed map.These three mappings are summarized in Table I, accompanied by their circuit diagrams.As mentioned earlier, while we have considered these mappings here for instrumental reasons to prove our main results, the revealed connection between SSDs and allowed maps is a result of intrinsic interest.In fact, this connection provides a rigorous resource-theoretical basis for recently introduced SSDs.
Using this framework, we provide our main result.Namely, given a generic CV state ρ, we define a sufficient condition for promoting the circuits presented in Fig. 1 to universality.This condition entails identifying an allowed map that converts ρ into an encoded qubit state sufficiently close to an ideal magic GKP state.In particular, owing to the correspondence between SGKP circuits and Clifford circuits, it sufficies for the mapped qubit state to exhibit fidelity to an ideal magic state surpassing the corresponding known distillation threshold (identified in the context of Clifford quantum computation via state injection).Furthermore, beyond the fidelity, the resourcefulness of qubit states can be quantified by using various magic measures -such as the robustness of magic (ROM) [45], relative entropy of magic [46], GKP magic [47] and stabilizer Rényi entropy [48].For certain measures, such as the ROM, there exists a threshold R * above which a state is guaranteed to have a fidelity greater than the threshold for magic state distillation.Therefore, access to a set of qubit states with a value of ROM beyond this threshold is sufficient to promote Clifford circuits to universality.
In practical terms, given a generic single-mode CV state ρ, our sufficient criterion involves the following steps: (i) employ a chosen allowed mapping to transform ρ into a qubit state ρ(P ) L (details on the nomenclature will be provided later), and (ii) determine its corresponding ROM.If the latter exceeds the threshold R * , than the state ρ is a resource for universal quantum computation.We apply the above criterion by analyzing the ROM of the logical states obtained by the two resource-theoretically motivated mappings, i.e., stabilizer SSD and Gaussian modular SSD.In particular, we analyze the set of Gaussian states and three classes of non-Gaussian CV states, namely finitelysqueezed GKP states, cat states, and cubic phase states.This allows us to identify states able to promote SGKP circuits to universal quantum computation which extend what previously reported in the literature.We stress that all pure Gaussian states are equally resourceful for promoting SGKP circuits to universality, since they can all be generated via SGKP circuits from vacuum.Furthermore, we find that certain non-Gaussian states, albeit not necessarily all, have a value of ROM higher than the set of Gaussian states upon considered allowed mappings.Finally, notice that the fact that Gaussian states can be considered resourceful in this model implies that the resourcefulness for SGKP circuits is independent of the notion of resourcefulness in all-Gaussian circuits.

III. BACKGROUND
A resource is a component of a quantum circuit that promotes an otherwise simulatable model to universality.In this section, we review both DV and CV QC and their existing known measures of resourcefulness.We also introduce the families of quantum states which we analyze in the later Sec.V. Finally, we also recall existing methods to map CV states to DV, before introducing our unified approach for this type of mapping, in the next Section.

A. Universal quantum computation and measures of resourcefulness in discrete variables
Quantum computation over DVs involves quantum states defined over a discrete finite eigenspectrum.For example, qubit-based quantum computation involves qubits that are expressed in terms of the eigenstates of Pauli operators.A complete basis can be defined in terms of the eigenstates of the It is possible to simulate DV quantum circuits under certain conditions.For example, the Gottesman-Knill theorem [49] provides a method to simulate circuits with input Pauli eigenstates, Clifford group operations (i.e., those which map Pauli operators to Pauli operators), and measurements in the Pauli basis.If we introduce access to a distillable magic state, then the circuit can perform universal quantum computation.For example, an ideal magic state such as the T state, defined as [6] |T ⟩ = cos β |0⟩ + e iπ/4 sin β |1⟩ , cos(2β can be combined with Clifford circuits to produce the full span of qubit circuits [9,50].Furthermore, a supply of states sufficiently close to this state can be converted to a smaller number of higher-quality versions of this state via magic state distillation [6].The resourcefulness of a single qubit state ρ can be therefore quantified as the fidelity of the state with its closest T-type magic state, i.e., where the set C is the set of single qubit Clifford operations.

Resourcefulness of DV states: Robustness of magic
Magic measures, such as the robustness of magic (ROM), also provide a method to quantify the resourcefulness of a DV state.First note that by defining S n as the set of all pure stabilizer states over n qubits, any non-stabilizer state can be expressed as a sum of such states -i.e., ρ = i x i σi for σ i ∈ S n .In general, there may be many different choices of {x i } which give the same ρ.The ROM of the qubit state ρ is defined as the minimal 1-norm among all those possible choices of {x i }.Formally, its expression is given by [45] R(ρ) = min If the qubit state ρ is a stabilizer state then the ROM is equal to 1.For non-stabilizer states, the ROM of a single qubit state can be simplified to the convenient expression [51] R (1) where X, Ŷ , Ẑ are the Pauli operators.To avoid confusion, we have denoted the ROM of a single qubit as R (1) (ρ), where ρ must be a single qubit state.Note that we can also express this value in terms of the coefficients of the qubit density matrix ρ, For single qubit states, the ROM is directly related to the fidelity to the closest T -state in Eq. (2) by The proof of this relation is given in Appendix A. It is known [6] that single qubit states are distillable to T -type magic states if they have fidelity We can express this condition in terms of the ROM as Therefore, to perform magic state distillation, a value of ROM greater than R * is sufficient for universality, in combination with Clifford circuits [12,45,51].In addition, the larger the ROM, the more resourceful the state, in the sense that fewer copies of the state are needed for magic state distillation [6].

B. Universal quantum computation and measures of resourcefulness in continuous variables
CV QC involves quantum states defined over a continuous eigenspectrum of relevant observables, such as the position q and momentum p quadratures of the electromagnetic field, satisfying the commutation relations [q, p] = i.A complete basis in CV can be defined in terms of the eigenvectors of the position operator, q |q = s⟩ = s |q = s⟩.
In CV quantum systems -as in the case of DV QCthere exist simulatable models that have no exponential computational advantage over a classical computer.For example, circuits involving all Gaussian input states, Gaussian operations, and Gaussian measurements, such as homodyne measurements, are efficiently simulatable [18,[26][27][28]].Although it is not possible to achieve quantum advantage with this restricted class of circuits, it is known that adding access to specific resource states, such as the cubic phase state or GKP stabilizer states, will promote this model to universality [19,23].

Resourcefulness of CV states: Wigner logarithmic negativity
Considering states displaying negative regions in their Wigner function as resources, a rigorous monotone has been introduced within a resource-theoretic framework.Such monotone is dubbed Wigner logarithmic negativity (WLN) [29], and it is defined as where the Wigner function W ρ(q, p) is defined as All states ρ with a non-negative Wigner function have W neg (ρ) = 0.Meanwhile, a non-zero value of this quantity is a necessary condition to promote otherwise Gaussian circuits to universality [26,27].However, it is also known that satisfying this criterion is not sufficient for achieving universality.Namely, circuits with input states that do contain Wigner negativity can be simulatable [35,36,52].

C. Families of CV states
Here we provide a short review of some families of CV states experimentally relevant to bosonic quantum computation with continuous variables.We first begin with a quick reminder of Gaussian states [14].Then, we present two types of bosonic code states [53], which encode DV quantum information into CV states.Specifically, we present GKP states [19,23] and cat states [24,25,54].We then recall the cubic phase state [19,55,56].The last three families are known states able to promote Gaussian circuits to QC universality.

Gaussian states
Any pure Gaussian state can be produced via a Gaussian unitary operation Û acting on the vacuum state.A singlemode Gaussian unitary can be decomposed in terms of a rotation where ζ > 0 represents squeezing in the position basis, while ζ < 0 represents squeezing in the momentum basis, and displacement operations [57] V (s) = e iqsp e −i psq , parameterized by s = (s q , s p ) T , where s q ∈ R is the displacement in position while s p ∈ R is the displacement in momentum.Therefore, we can define any pure single-mode Gaussian state in terms of these operations as where |0⟩ is the vacuum state.General Gaussian states can then be constructed out of pure Gaussian states by considering convex mixtures of pure states.

GKP states
The GKP encoding encodes DV quantum information using grid states [19].For qubits, the 0-logical state and the 1-logical state are defined as Using these two basis states, it is possible to define arbitrary qubit states encoded as logical GKP states.For pure singlequbit states, we have However, these ideal states are not normalizable and hence are not physically implementable.By using a wavefunction with Gaussian peaks and a Gaussian envelope parameterized by a squeezing parameter ∆, instead of Dirac delta peaks which extend infinitely in position, it is possible to define realistic GKP states in terms of the unnormalized [58] basis states as [19,38,59] where ϑ(z, τ ) is the Jacobi theta function, Combining these states allows us to encode any pure (and hence also mixed) single qubit state as whereby N GKP is a normalization constant, specific to the squeezing and the parameters of the encoded state.These states are physically implementable, however, the logical basis states are no longer orthogonal.This introduces errors in the encoding that can be interpreted as qubit errors [19].Furthermore, while for large squeezing, i.e., ∆ ≪ 1, the norm of both unnormalized basis states are approximately equal [19], for larger values of ∆ the normalization factors differ and can introduce an asymmetry in the encoded states [38].
GKP states have been physically implemented in a variety of experimental setups [60][61][62][63] and are known to promote all-Gaussian circuits to universality [23].

Cat states
The second type of non-Gaussian states that we analyze in this work are cat states [54,64].Cat states with even symmetry can be used to encode the 0-logical state of a qubit, while cat states with odd symmetry encode the 1-logical state of a qubit.The code space is defined in terms of the unnormalized [65] basis states [54,66,67] where |α⟩ is a coherent state parameterized by the complex number α ∈ C, which can equivalently be expressed as α = re iϕ .The wavefunction of a coherent state |α⟩ in the position basis is given by Any pure (and hence also mixed) qubit state can be encoded using these basis states.In what follows, we do not focus on the code aspect of cat states but rather analyze the ability of the state | 0α cat ⟩ to promote SGKP circuits to universality.Cat states have been successfully experimentally produced in a variety of different CV architectures [68][69][70][71][72][73].These states can also be used to produce GKP states using only Gaussian operations [24,25].Therefore, like GKP states, they can also be considered a resource for quantum advantage in Gaussian circuits.

Cubic phase state
The final type of state that we analyze is the cubic phase state [19].This is defined as where |0⟩ is the vacuum state and the squeezing operator is defined as in Eq. (11).The cubic phase state can be used to produce both a T gate in the GKP encoding [19] and the CV cubic phase gate, which promotes all-Gaussian circuits to universality [74].Cubic phase states have recently been successfully produced in a microwave cavity [62] and in an optical system [75].Theoretical prosposals have been put forward to generate them also in other platforms [76] or by Gaussian conversion from other non-Gaussian states [77,78].

D. Existing methods to map CV states to DV states
There exist different methods [23,37,39] to analyze the logical content of a CV state.GKP states offer a natural analogy to DV quantum states because they specifically encode DV quantum information into a CV state.Furthermore, the logical action of Clifford operations in DV circuits is obtained by Gaussian operations when acting on GKP states [19].
Although the mapping from DV states to CV states through the GKP encoding is clear and well-defined [19,52,53], understanding general CV states in terms of DV states is more challenging.This is due to the fact that the Hilbert space of CV states is infinite and therefore there is an infinite number of possible mappings.However, by grounding our choice of mapping in terms of the information we wish to extract from the CV state, and by using only resourceless states and operations in our mapping, we can define criteria for maps which are appropriate to the situation at hand.Specifically, in this work, we are interested in maps which inform us of the resourcefulness of CV states to promote otherwise resourceless SGKP circuits to universality.
Here we review two existing methods of SSD.Namely, the stabilizer SSD, which effectively implements ideal GKP error correction on the CV state, and modular SSD, which has a convenient mathematical form.Notice that, prior to this work, neither of them had received a resource-theoretical interpretation.

Stabilizer subsystem decomposition
The result of the projection of a CV state ρ into the GKP encoded subspace, due to GKP error correction, gives a state of the form [23] where Π is the GKP projector defined as and V (−t) is the displacement operator in both position and momentum, given in Eq. (12).The circuit for implementing the stabilizer SSD is given in Table I.The output of such a circuit depends on the values t = (t q , t p ).By disposing of these measurement outcomes, after the corrective displacements, we are left with a mixed state.This state is a GKP-encoded qubit state which encodes the result of stabilizer SSD [39].By a slight abuse of notation, we express the result of the stabilizer SSD as Note that the right-hand side of this equation is defined over the continuous-variable Hilbert space, while the left-hand side is defined over the qubit Hilbert space.However, this can be resolved by considering the implicit change of the basis states |l⟩ to |l GKP ⟩.We provide further details on this notation in Appendix B.

Modular bosonic subsystem decomposition
The logical content of a general CV state can also be identified using modular analysis.Modular analysis of CV states has a long history in quantum information [79,80].Notably, it was used to first test the Bell inequalities [81][82][83], which enabled much higher detection efficiency in comparison with using DV systems.Furthermore, it has recently been realized that modular analysis can be used to reconstruct the logical content of realistic GKP states [37,38].
The modular SSD has been introduced in Ref. [38] in an abstract context, without reference to a specific circuit.Its primary feature is to decompose a CV state into a logical component and a gauge part.As in Ref. [38], we begin by providing an example of the decomposition for a real number, s ∈ R. It is always possible to write the number in terms of an integer part ⌊s⌋ and its remainder s−⌊s⌋, where ⌊•⌋ is the integer floor function which rounds the number down to the nearest integer.We can consider this decomposition as splitting the number s into different bins on the real number line, whereby each bin has width 1.Similarly, we can find a different decomposition of the number s by using a different bin width α ∈ R. We can then decompose the number s into the closest integer multiple of α using the centered floor function ⌊s⌉ α = α⌊ s α + 1 2 ⌋ and its remainder {s} α = s − ⌊s⌉ α .
The position quadrature q can be similarly decomposed.The position eigenstates |q = s⟩ of the position quadrature operator have eigenvalues over the real numbers.The operator can be written as q = α m + û where α m = ⌊q⌉ α is the integer part of the operator and {û} is the fractional part.This provides a method of writing the position eigenstates as simultaneous eigenstates of α m and û.We can express the position eigenstate as |q = s⟩ = |α m + û = s⟩ or |q = s⟩ = | m = m, û = u⟩, with αm + u = s.Furthermore, by separating the odd and even integers m we can define a logical subsystem.This can be achieved by expressing q = α l + 2α mG + ûG where l = m mod 2, ûG = û and mG = 1 2 ( m − l).We can then write the position basis states in terms of the logical part and gauge parts We can therefore describe the complete Hilbert space of a CV state in terms of a logical qubit and a gauge mode, i.e., The identity operator 1 CV can be expressed as It is possible to calculate the logical component of the density matrix by tracing out the gauge part of the state.The logical density matrix can be expressed as While this method has a clear and robust mathematical definition, it was previously unknown whether this partial trace corresponds to implementable operations using physical circuits.
In the next section, specifically Sec.IV B 2, we demonstrate that in the analysis of the logical content of GKP states, the modular SSD is in fact a well-motivated mapping that can be implemented with SGKP circuits.

IV. UNIFIED APPROACH FOR MAPPING CV STATES TO QUBITS
In this section, we establish a general mapping from CV states to logically encoded GKP states using continuousvariable operations.The modular SSD and the stabilizer SSD, introduced in Sec.III D, fall within this broad category.However, this general class of maps lacks a clear interpretation in terms of quantum computational resources.It may encompass operations that could potentially artificially enhance the computational capabilities of the original CV state.To address this, we further narrow down the scope to the class of maps implementable solely using SGKP circuits.This ensures that no artificial computational power is introduced during the mapping process.We will demonstrate that this class includes the stabilizer SSD but not the modular SSD.Additionally, we introduce a new map, the Gaussian modular SSD, inspired by the modular SSD but exclusively relying on components from SGKP circuits.Consequently, it also falls within the restricted class of maps.

A. Mapping CV states to DV
We begin defining a general map M P from an arbitrary CV state ρ to an encoded qubit GKP state as where Pi (s) are Kraus operators which -according to some parameters s that may depend on measurement results -consist of CV operations, and R is some integrable region of the space of the measurement outcomes s.We denote the set of these Kraus operators as P , i.e., P = { P1 (s), . . ., Pk (s)}.These Kraus operators must include the GKP projector such that the state is mapped to a perfectly encoded ideal GKP state; i.e., the Kraus operators Pi = Π P ′′ i are expressed as an arbitrary CV operation P ′′ i followed by the projection Π onto the GKP code space.The encoded qubit state achieved as a result of applying the set of Kraus operators P is denoted The state that arises from the mapping M P depends on the choice of the Kraus operators P .As said, the crucial point to notice is that, depending on the choice of Kraus operators, this general map may introduce additional resourcefulness to the original CV state.Therefore, in order to quantify the resourcefulness of CV states for SGKP circuits, we must restrict the Kraus operators to be chosen from the set of SGKP-type Kraus operators, which we label P SGKP .The corresponding restricted set of maps, i.e., the set of M P such that P ∈ P SGKP , are hence all maps that can be implemented using resourceless operations.
As said, each Kraus operator in any set P must project onto the GKP basis using the operator Π.However, this operator is not, by itself, a valid operation in SGKP circuits.Despite this apparent contradiction, it remains possible to perform GKP error correction using SGKP circuits, which effectively introduces a random displacement and projects the CV state onto the GKP code basis, and which can instead be expressed using the Kraus operator [23] Therefore, we identify a class of allowed Kraus operators, which are both implementable with SGKP circuits and also project onto the GKP basis as where P ′ i (s) is selected from the set of Kraus operators implementable by probabilistic GKP-encoded Clifford operations and Ûi is any unitary Gaussian operation (encompassed in Ref. [36]), which occurs prior to the GKP error correction routine, and therefore does not depend on the measurement outcomes.For simplicity, we choose Ûi = 1 in our analysis of CV states.A complete characterization of the class of maps M P such that P ∈ P SGKP is lacking and we leave it for further investigation.

B. Considered maps in terms of the general map
The two maps introduced in Sec.III D can all be expressed in the form given in Eq. ( 31).As we will now see for the stabilizer subsystem decomposition, as well as for the Gaussian modular subsystem decomposition that we will introduce below, the Kraus operators can be further expressed as in Eq. (33), implying that these maps can be implemented by means of SGKP circuits.However, the Kraus operators implementing modular SSDs are not in the set P SGKP .

Stabilizer subsystem decomposition
If we consider the set of Kraus operators P in Eq. ( 31) to consist of a single operator P = K = { K(s)}, where K(s) is defined in Eq. ( 32) and R is the interval [− √ π/2, √ π/2) over both s q and s p , then we recover the stabilizer SSD [39] as defined in Eq. ( 26), i.e., ρ(K) L = ρΠ .This map can be implemented by performing GKP error correction according to the original proposal provided by Ref. [19].In turn, it is easy to see that GKP error correction is a SGKP circuit, namely an allowed map.In fact, from the circuit diagram in Table I, it consists of measuring the two GKP stabilizers and displacing the mode in both position and momentum, whereby the corrective displacements are performed modulo √ π over the interval Equivalently, this can be implemented by performing the corrective displacements t q , t p directly but only accepting the state when the values of the measurement results t q , t p , modulo 2 √ π, are within the acceptable interval (− √ π/2, √ π/2]; otherwise, the state is discarded [23].In any case, all these elements belong to the class of SGKP circuits, therefore ensuring that the stabilizer SSD map is an allowed map from a resource theory viewpoint, in that it does not add any computational power to the original state ρ.This provides a resource-theoretic foundation to the statiblizer SSD therefore strongly grounding its use when one wants to associate a binary (qubit-like) logical content to a generic CV state ρ.
In Appendix B 1, we also provide a new alternative form of the stabilizer SSD, namely expressing it in the position basis.This alternative form is useful for comparing the effect of the stabilizer SSD with the modular SSD and also provides a convenient method to calculate the stabilizer SSD of a general CV state.As mentioned, in Appendix B 2 we derive the circuit implementation of the stabilizer SSD, also reproduced in Table I.

Modular subsystem decomposition
The modular SSD is calculated by tracing out the gauge part of a bosonic state, i.e., Eq. (29).For a single mode, this can be expressed in a convenient form using the density matrix of the state in the position basis, as we show in Appendix C 1.However, we can also interpret this operationally as performing GKP error correction, followed by a logical Ẑ rotation acting on the logical qubit state, as we explicitly show in Appendix C 2. The resulting interpretation in terms of a quantum circuit is reproduced in Table I and makes explicit the connection between modular SSD with GKP error correction that was implicitly established in Ref. [84].Our analysis allows us to express the modular SSD as where the logical Ẑ rotation is given by and ρΠ (t) is given in Eq. ( 24).The set of Kraus operators defining the modular SSD, in terms of the general map defined in Eq. ( 31), therefore consists of a single Kraus operator, i.e., P = { RZ (t p √ π) K(−t)}.It is relevant to notice that the logical Ẑ rotation is, for general θ, a non-Clifford operation in the qubit framework and its GKP-encoded operation is accordingly non-Gaussian.
By inserting the rotation operator given in Eq. ( 36) into Eq.( 35), we show that the expression can be interpreted as a summation of a Gaussian ρG L term and a non-Gaussian ρNG L term, i.e., ρL = ρG L + ρNG L .As we explicitly derive in Appendix C 3, these terms are given by and In general, since the logical Ẑ rotation corresponds to a non-Gaussian operation, it is not implementable via an SGKP circuit and therefore it could add computational power to the original state ρ, as it could increase the magic content of the corresponding qubit state.However, for certain states ρ, the state ρL is equivalent to ρG L and can therefore be prepared with only SGKP circuits.In fact, as we explicitly demonstrate in Appendix E, when the input state is symmetric in position, the non-Gaussian part of the density matrix, Eq. ( 38), evaluates to zero, i.e., For the purpose of analyzing realistic GKP states, as given in Eq. (19), which are symmetric in position, we therefore find that the modular SSD can, in fact, be implemented using only components selected from the class of SGKP circuits.This implies that the modular SSD is also endowed with a resourcetheoretic fundation, as the stabilizer SSD, when it is applied to the analysis of the logical content of realistic GKP states.

Gaussian modular subsystem decomposition
We now introduce a new map that can be performed using only the set of simulatable SGKP circuits.I.e., the set of Kraus operators P is contained within P SGKP .This map is the result of performing only the Gaussian part of the modular SSD and, therefore, the resulting state is given by ρG L .To operationally produce this state from the state ρ with the otherwise free resources of SGKP circuits, we perform GKP error correction which gives measurement outcomes t q , t p and then randomly apply a logical Z gate with probability The measurement results should then be discarded to produce the statistical mixture over the possible values of t q , t p .Further details, and a circuit diagram of this procedure, are presented in Appendix D, see also Table I.
This mapping has the benefit of being implementable with the resourceless SGKP operations, while also maintaining part of the structure of the modular SSD.In fact, as a result of Eq. ( 39), this map is equivalent to the modular SSD when the input state is symmetric in position.

V. RESOURCEFULNESS OF CV STATES FOR SGKP CIRCUITS
We now use the maps described in the previous subsections to analyze the resourcefulness of arbitrary CV states to promote the otherwise simulatable model of SGKP circuits to universality.

A. Resourcefulness of DV state resulting from general mapping
In order to quantify the resourcefulness of generic CV states, we calculate the ROM of its associated qubit:

R(M P (ρ)).
( As said, for this quantity to have a grounded resourcetheoretic meaning, we restrict the allowed Kraus operators to those which are included in the simulatable model of simulatable GKP circuits, i.e., P ∈ P SGKP .
By this logic, we can quantify the resourcefulness of an arbitrary single-mode CV state to promote SGKP circuits to universality by means of the functional Although a full search over all possible mappings is challenging, for the purpose of a sufficient condition of universality it is only required that there exists some map such that the ROM is above the threshold of distillability R * .This is because SGKP circuits contain stabilizer GKP states and Gaussian operations, yielding encoded Clifford circuits.The addition of a supply of GKP-encoded magic states, above the distillation threshold, promotes these circuits to universal QC.
We can therefore inspect the quantity given in Eq. ( 41) for different choices of mappings M P , all with P ∈ P SGKP .If, for one of these mappings, the ROM is greater than R * , then the CV state can clearly be converted to a GKP-encoded distillable magic state by some allowed mapping.Hence, the ROM of the logical state found via a specific mapping M P gives a lower bound of R SGKP .
In other words, given access to a supply of the CV state ρ, if the ROM of a logical state found via a specific mapping M P is above the distillation threshold, R(M P (ρ)) > R * , then it is possible to produce a supply of GKP-encoded magic states above the distillation threshold from ρ using only resourceless SGKP operations.Furthermore, since the operations required for magic state distillation consist of only encoded GKP-Clifford operations and adaptive homodyne measurements, it is possible to produce a supply of T states with arbitrarily high quality using a polynomial number of operations [6], given access to a supply of the CV state ρ.In this sense, the value of ROM after an allowed mapping yields an upper bound to the number of copies to be used in the magic state distillation procedure.Therefore, for a given mapping, the larger the ROM, the more resourceful the state.Note however that different mappings can yield different hierarchies between states, as we show in Appendix F.
Finally, we note that the value of ROM of a logical state found via a mapping of the form given in Eqs.(30,31) is convex.Specifically, when considering a mixed CV state ρ = k p k ρk with k p k = 1 consisting of a weighted sum of pure CV states, the corresponding logical state is equal to a weighted sum of the set of logical states found from each of the corresponding CV pure states ρk , i.e., ρL = k p k M P (ρ k ).Since ROM is convex for qubit states [45], we therefore must have that

B. Analysis of CV states
We use the methods described in Sec.V A to analyze the ROM of the mapped CV states selected from the set of Gaussian states and three families of non-Gaussian states introduced in Sec.III C.Here we present the values of ROM for the resource-theoretically motivated SSDs, i.e., the stabilizer SSD and the Gaussian modular SSD.For the symmetric cat and GKP states, the Gaussian modular SSD is equivalent to the modular SSD, and therefore the values of the ROM for the two decompositions are equal.

Gaussian states
We begin with an analysis of pure Gaussian states.We then consider the case of mixed Gaussian states.
We recalled in Sec.III C 1, specifically in Eq. ( 13) that any pure Gaussian state can be defined via a Gaussian unitary operation -parameterized in terms of the squeezing parameter ζ, a rotation angle Θ and a displacement vector s -acting on the vacuum state.In Fig. 2 we plot the value of ROM of the qubit state arising from the different choices of SSD of a pure Gaussian state, parameterized by the squeezing parameter ζ and rotation angle Θ.Specifically, in Fig. 2a we plot the ROM of the stabilizer SSD of a Gaussian state and in Fig. 2b we plot the modular SSD of a Gaussian state.Note that in the figures we choose the value of the displacement vector s to be zero in both position and momentum, however when we later optimize to identify the maximally resourceful states, we also optimize over the choice of s.Also, note that Fig. 2b equivalently shows the ROM of the Gaussian modular SSD because the wavefunction of a pure Gaussian state centered in phase space is symmetric in position.
We start by inspecting Fig. 2a, which shows the ROM of the stabilizer SSD for different pure Gaussian states.As a reminder, values of Θ = 0 and ζ = 0 correspond to the vacuum state, while non-zero values correspond to a rotated and squeezed state.We find that the stabilizer SSD of the vacuum state has a value of ROM of R(ρ Π ) ≈ 1.160, which is greater than the threshold for T -magic state distillation.This result is in line with what reported in Ref. [23], where it was shown that a supply of vacuum states allows to distill 0-logical GKP states into magic states with Gaussian operations alone, although in that work distillation towards H states was rather considered.However, the vacuum is not the optimal Gaussian state to achieve a high value of ROM [85].Instead, we find that the value of ROM (and hence, fidelity to T ) is greater when using a rotated squeezed state.Specifically, by numerically optimizing over all the parameters of the Gaussian unitary we find that by choosing rotation angle Θ = π/4, squeezing parameter ζ ≈ 0.26, and displacement s = (0, 0), a ROM of 1.303 can be achieved.Note that the ROM of the stabilizer SSD is symmetric in both ζ and Θ.
Next, inspecting Fig. 2b, we see that the ROM of the modular SSD of the vacuum state is 1.This is because the modular SSD of the vacuum state evaluates to the maximally mixed state.To achieve a value of ROM above the distillation threshold for the modular SSD, it is necessary to instead use a Gaussian state with both non-zero squeezing and rotation.
Note that, since it is possible to convert between any Gaussian state with only Gaussian operations, in virtue of the possibility of optimizing over Gaussian unitaries in Eq. ( 33), all pure Gaussian states should be considered equally resourceful for SGKP circuits.Furthermore, given that the value of ROM of the SSD of a CV state is convex, as shown in Eq. ( 43), the value of ROM of the SSD of a mixed Gaussian state can only be less than or equal to the value of ROM of the SSD of a pure Gaussian state.This implies that the optimal values of ROM for the pure single-mode Gaussian states are also optimal over all single-mode Gaussian states, including thermal states.

GKP states
We start by numerically calculating the ROM of an encoded realistic GKP state of the form given in Eq. ( 19), where we fix ϕ = π/4.The state is given by to identify a selection of insightful logical states with only varying θ.For θ = 0, this state is simply the 0-logical GKP state.For θ = π, this state is the 1-logical GKP state.For θ = arccos 1/ √ 3 and θ = π − arccos 1/ √ 3 , the state is an encoded magic |T ⟩ state and its orthogonal magic state, respectively.Each of these states is an encoded finitely squeezed state, parameterized by ∆.From these states, we evaluate the resulting qubit state from each SSD introduced in the previous Sec.IV and plot the ROM of each state in Fig. 3.The red vertical lines correspond to the encoded T states and have the highest ROM for any choice of ∆, in fact reaching the maximal achievable ROM (i.e, R = √ 3) for ∆ = 0.The plot shows ample regions of distillability in terms of the parameters θ and ∆, i.e., regions where R(M P (ρ)) > R * , identified by the shaded white contour.These results are in line with and generalize what is reported in Ref. [36], where it was shown that a supply of realistic GKP states allows to distill magic states from 0-logical GKP states with Gaussian operations alone, although that result referred to distillation towards H states.
Note that there is an asymmetry in the shape of the contour levels of the ROM in Fig. 3 which arises as the level of squeezing is decreased, i.e., ∆ is increased.This asymmetry can be interpreted as arising from the fact that the norm of the unnormalized 0-logical state is in general larger than the norm of the unnormalized 1-logical state.Therefore, when the states are combined in superposition and normalized together, the 0-logical component contributes more than the 1-logical component.For values of ∆ ≪ 1, the norm of each state is approximately equal and hence this asymmetry is no longer present [19,38].In Fig. 4 we plot the wavefunction of this GKP state for various levels of squeezing.This provides a visual explanation as to why the ROM of the stabilizer SSD of the encoded GKP state is asymmetric.We observe that for ∆ = 1 and θ = 0, the state approximates the vacuum state, which is a known resource [23,36].However, at θ = π the state retains two peaks.This difference affects the logical content of the decomposed state.Note that the definition of the wavefunction of a GKP state can affect the logical content of the encoded state and hence also the resulting ROM of the encoded state.We use the same definition of the GKP state analyzed in Ref. [38] rather than that of Ref. [39].We provide a more detailed discussion in Appendix G.
We also note that the stabilizer SSD state has higher values of ROM for all ∆ and θ, as compared to the ROM of the modular SSD.
Furthermore, we stress that the threshold of the ROM, R * , is not a necessary condition for achieving quantum advantage.For example, the 0-logical state with squeezing ∆ = 1, which approximates the vacuum state, has a value of ROM below the distillation threshold, meaning it cannot be distilled to the T state.Despite this fact, the state can be distilled to the H state and can therefore still be considered a resource for quantum advantage [23].
We leave further discussion of these results for specific GKP states to Appendix H, whereby we also provide a comparison of the ROM of the stabilizer SSD of the GKP states with the WLN of the same states.
Finally, we note that the maximal achievable ROM using GKP states is significantly higher than that which is possible using Gaussian states.

Cat states
After having analysed the two most natural classes of states for our framework -namely, Gaussian and (realistic) GKP states -we now move to a class of states with no specific relation to SGKP circuits.In particular, we analyze the even cat state as defined in Eq. (20).We parameterize the cat state using the complex number α by separating its magnitude r and phase Φ, i.e., α = re iΦ .
Due to the fact that these states have a wavefunction that is symmetric in position, the modular SSD is equivalent to the Gaussian modular SSD.The ROM of the stabilizer SSD and the Gaussian modular SSD (equivalently, the modular SSD) of the state, for different values of r and Φ, are plotted in Fig. 5, where the regions above the distillation threshold are enclosed by the dashed white lines.We find that for most choices of α, the ROM of the stabilizer SSD is greater than the ROM of the Gaussian modular SSD.
Note that the value of Φ corresponds to a rotation in phase space and can be implemented using Gaussian unitary operations, which are included in the set of SGKP circuits.Therefore, we should consider the lower bound of the maximum ROM, as defined in Eq. (42), to be the maximum of all angles Φ for a given r.
We also observe that both the values of the stabilizer SSD ROM and the values of the modular SSD ROM each display symmetry.Specifically, when the state is rotated by π/2, the values of each respective ROM are equal.This can be seen from the equal values of ROM in each of Figs.5a and 5b at values of Φ and Φ + π/2.This angle corresponds to a Fourier transform which can equivalently be considered a change of basis of the quadratures, q, p.The stabilizer SSD is known to be symmetric in q, p, so this symmetry is to be expected for the stabilizer SSD ROM [39].However, the modular SSD is not symmetric in general.Instead, this symmetry arises from the definition of the cat state.First note, that the cat state is symmetric under rotations around π, i.e., We also see that the wavefunction of a coherent state with angle Φ is the complex conjugate of the wavefunction of a state with angle −Φ.This also implies that the wavefunction of the cat state with angle Φ is equal to the complex conjugate of the wavefunction with angle −Φ, i.e.
Given that the state is also symmetric under rotations by π, we see that the wavefunction of the cat state with angle Φ is equal to the complex conjugate of the cat state with angle π − Φ.This also means for a density matrix ρ of a cat state with angle Φ, the corresponding density matrix of a cat state with angle π − Φ can be considered to be ρ * .In terms of the logical density matrix of a cat state ρL,Φ with angle Φ, the corresponding density matrix with angle π − Φ is given by ρL,π−Φ = ρ * L,Φ , which is equivalent to applying a phase gate to the density matrix ρL,Φ .The phase gate is Clifford and therefore the ROM of each state is equal.
We also note the decreasing values of ROM for higher values of r.This effect can be understood by considering the wavefunction of the state.The value of r corresponds to the distance between the peaks of the wavefunction, and also the width of the individual peaks.As the peaks become further apart, each peak can be binned inside one type of region, rather than across two or more regions.To illustrate this, we have provided plots of the wavefunction of the cat states for some selected values of r, Φ in Fig. 6.In the limit of large r, the state consists of two peaks, both contained entirely in either the region corresponding to the 0-logical state or the region corresponding to the 1-logical state.This implies that the SSD of the state will be a logical basis state.
Finally, we note that the maximal ROM of the stabilizer SSD of an even cat state is 1.39, which is higher than the maximal achievable with Gaussian states alone.

Cubic-phase state
The ROM of the stabilizer SSD and the Gaussian modular SSD of the cubic-phase state is plotted in Fig. 7.We find that, counter-intuitively, the ROM of the stabilizer SSD of the state is maximum when both the cubicity and squeezing are zero, i.e., γ = 0, ζ = 0, which corresponds to the vacuum state.Note that this is somewhat surprising since the maximally resourceful state among this family of states is the one for which the state is Gaussian and the WLN is zero.However, given that for SGKP circuits we consider the already highly Wigner negative stabilizer GKP states to be resourceless, we know that negativity is not necessary for the promotion of these circuits to universality.
Note that unlike the other states considered in this work, the cubic-phase state is not symmetric in the position basis.Therefore, the modular SSD of this state is not equivalent to the Gaussian modular SSD.Hence, evaluating the ROM of the modular SSD does not provide a resource-theoretically meaningful quantifier of the resourcefulness of the cubicphase state -as in this case the modular SSD requires non-Gaussian operations, in addition to GKP states, to be implemented.For completeness we provide a plot of the ROM of the modular SSD for the cubic phase state in Appendix I.

VI. CONCLUSION
In quantum computation over DV systems, the fidelity to a target magic state is a well-established criterion for determining whether a state can promote otherwise simulatable Clifford circuits to universality, potentially leading to quantum computational advantage.In CV systems, while the presence of negativities in the Wigner function of a given circuit serves as a necessary condition for universality, it falls short of providing a sufficient criterion.To bridge this gap, we have introduced a resource-theoretically motivated framework, enabling the formulation of a sufficient criterion to assess the resourcefulness of a generic CV state for quantum computation.
Specifically, we have introduced such a criterion in the framework of SGKP circuits.Our criterion is based on the evaluation of a measure of magic on the encoded logical state associated to a generic CV state ρ, upon mapping it to the computational subspace of the GKP code.For resourcetheoretically grounded mappings -such as the stabilizer SSD and the Gaussian modular SSD -this quantity can be understood as the resourcefulness of the state ρ to promote otherwise simulatable GKP circuits to universality.Applying such a criterion we find that all pure Gaussian states are equally resourceful for promoting SGKP circuits to universality.Moreover, we found that certain non-Gaussian states, albeit not necessarily all, have a value of ROM higher than the set of Gaussian states.
Furthermore, our work provides a rigorous and resourcetheoretically grounded interpretation of recently introduced methodologies aimed at extracting the binary logical content of generic CV states.In particular, we have elucidated that the mapping established by the stabilizer SSD [39] can be understood in terms of resourceless operations in the context of SGKP circuits, for any state.This interpretation also holds for the modular SSD [37], albeit exclusively for states symmetric in the position representation.Considering the relevant role of SSDs in extracting the logical content of states in the emerging field of quantum computation over CV systems, we expect that this result will hold independent interest.
We conclude by recalling that the ideal GKP states considered in the SGKP framework are infinitely squeezed.Therefore, in order to provide a conclusive validation of the result presented here from a practical and operational viewpointincluding the interpretation of SSDs in term of resourceless mapping -it would be necessary for our findings to hold also in the presence finite squeezing -and for the SSDs to be implemented with finite squeezing.We leave this analysis for future work.

Fidelity to H state
For completeness, we also now provide an explicit relation between the robustness of magic and the threshold of the fidelity to the H state for magic state distillation.We begin with an explicit derivation of Theorem 2 of Ref. [12].We now consider the fidelity of an arbitrary Bloch vector with the Bloch vector of the H state. I.e., we choose a (1) = a H in Eq. (A1), where which can also be transformed under single-qubit Clifford operations as The fidelity to an arbitrary state ρ with Bloch vector a is therefore given by which can be evaluated as The maximum value of the fidelity to any H state is thus given by Given that the distillation threshold for the H state is tight [86], we know that meaning the threshold for H state distillation can be expressed as [86] max(|a Unlike the case of the T state, the robustness of magic is not directly related to this quantity.Instead, we must consider the minimum robustness of magic of an arbitrary state required to satisfy this inequality.
Formally, we need to identify R * H such that .This is significantly higher than the bound found in terms of the fidelity to the nearest T state.Also note, that, unlike the previous bound, this does not identify all qubits states that have a value of fidelity to the closest H state above the distillation threshold. .GKP error correction as a circuit.We draw the operation e iq 3 p1 using the symbol ⊖, which can be considered the inverse of the SUM gate [88].
We begin by evaluating the first operator by inserting the wavefunction of the 0-logical GKP state Kp (t q ) =e i p1{tq} √ π ⟨p 2 = t q | e iq1 q2 where we have dropped the index in the last line as the effect of the Kraus operator only applies to the mode being error corrected.Note that we can also use the fact that the wavefunction of the 0-logical state in the momentum basis ψ(x) can be expressed as [23] ψ0,L (x) = n e 2in to simplify the expression further.We find that Kp (t q ) ∝e i p{tq} √ π ψ0,L (q − t q ) =e i p{tq} √ π e −itq p ψ0,L (q)e itq p =e −i p(tq−{tq} √ π ) ψ0,L (q)e i(⌊tq⌉ √ π +{tq} √ π ) p =e −i p⌊tq⌉ √ π ψ0,L (q)e i(⌊tq⌉ √ π +{tq} √ π ) p = ψ0,L (q − ⌊t q ⌉ √ π )e i{tq} √ π p = ψ0,L (q)e i{tq} √ π p (B12) where in the final line we have used the fact that ⌊t q ⌉ √ π is an integer multiple of √ π and the wavefunction in the momentum basis is periodic in √ π.The second Kraus operator also simplifies using the same methods, Combining these two operators allows us to find an expression for the combined Kraus operator as K(t) = ψ0,L (p)e −i{tp} √ π q ψ0,L (q)e i{tq} √ π p = ψ0,L (p) ψ0,L (q)e −i{tp} √ π q e i{tq} √ π p = Π V (−{t} √ π ), (B14) where we have used that the GKP projector Π, defined in Eq. ( 25), is equivalent to [23] ψ0,L (p) ψ0,L (q) = n,n and the displacement operator implements a displacement whereby both elements of the vector are taken modulo √ π on the interval (− √ π/2, √ π/2], i.e., {t} √ π = ({t q } √ π , {t p } √ π ).The statistical mixture of the output state after a round of GKP error correction, whereby the measured values are ignored, can be evaluated as (B16) However, due to the fact that the Kraus operator is periodic in both elements of t with a period of √ π, centered around the origin, we can evaluate the output state by integrating over a single period, dt p K † (t)ρ K(t). (B17) This also allows us to simplify the expression of the Kraus operator to K(t) = Π V (−t), where K is now only defined over this interval.This is precisely the same expression (up to normalization) as the density matrix we identified at the beginning of this appendix, in Eq. (B1). the connection to the Gaussian modular SSD due to the fact that the state does not have a density matrix that is symmetric in position.Note that the plot includes squeezed vacuum states along the axis γ = 0.These states are symmetric in position and therefore their modular SSD decomposition is equivalent to the Gaussian modular SSD in Fig. 7b along the same axis.This axis, in turn, also includes the vacuum state at γ = 0, ζ = 0.As is the case for the Gaussian modular SSD ROM, the state prepared using the modular SSD from the vacuum state is not above the distillation threshold.However, two new distillation regions appear, characterized by low squeezing and moderate cubicity.
FIG. 14. ROM of the modular SSD of the cubic phase state.The white dashed lines show the distillation threshold of the qubit state prepared from the modular SSD.However, this threshold is not a criterion for universality.
FIG. 2. The ROM of a decomposed rotated and squeezed Gaussian state, as defined in Eq. (44), for different values of squeezing ζ and rotation angles Θ.The regions inside the dashed white boundaries in each plot indicate the regions of distillability.
FIG. 3. The ROM of a decomposed encoded qubit GKP state, as defined in Eq. (44), for different values of squeezing ∆ and rotation angles θ and a fixed phase of ϕ = π/4.The red dashed lines indicate the values of θ for which the state is an encoded T -state, i.e., θ = arccos ±1/ √ 3 .The large regions inside the dashed white boundaries in each plot indicate the regions of distillability.The inset plots show a subset of the same data plotted with θ on the x-axis and the value of ROM on the y-axis.The solid blue, red, green and purple lines correspond to ∆ = 0, 1/2, 3/4, 1, respectively.Equivalently, the lines for each increasing ∆ have decreasing maxima.Note that the value of ROM in the main figures and the insets is always greater than or equal to 1.

FIG. 5 .
FIG. 5. Plot a) shows the ROM of the cat states decomposed using stabilizer SSD, while plot b) shows the ROM of the same class of states decomposed with the (Gaussian) modular SSD.The white dashed lines show the regions where the SSD ROM is above the threshold for distillability and hence the states are resourceful for quantum advantage with SGKP circuits.The wavefunctions of the states labeled with a cross are plotted in Fig. 6.

FIG. 6 .
FIG. 6. Wavefunctions of the cat state for different choices of α = re iΦ , corresponding to points specified in Fig 5.The regions with a white and blue background represent areas that contribute to the 0-logical and 1-logical components of each SSD, respectively.
FIG. 7. The ROM of a decomposed cubic phase state for different values of squeezing ζ and cubicity γ.Plot a) shows the ROM of the stabilizer SSD of the state, while plot b) shows the ROM of the Gaussian modular SSD of the state.Note that because the wavefunction of the cubic phase state is not symmetric in position, plot b) is not equivalent to the ROM of the modular SSD of the cubic phase state.The white region in plot a) shows the region of states that are resourceful for SGKP circuits.Note that no states are above the distillation threshold for the Gaussian modular SSD ROM.

FIG. 11 .FIG. 13 .
FIG.11.Comparison of ROM of the resulting logical state found by the modular SSD (blue, solid lines) and the stabilizer SSD (red, dashed lines) of a GKP state, Eq. (44), with Bloch angles ϕ = π/4 and θ = π/25 and varying squeezing ∆.The inset plot shows the gradient of the ROM of each decomposition, for each value of ∆.

TABLE I .
[44]mmary of the three types of maps considered in this work.The operator K(t) = Π V (−t) is the Kraus operator which is implemented by GKP error correction[23], where t = (tq, tp) contains the measurement results.The modular SSD and Gaussian modular SSD are equivalent for CV states that are symmetric in position.Note that the implementation of all of these maps requires access to ideal GKP stabilizer states.The operation RZ (θ) is a GKP-encoded rotation around the Z-axis on the Bloch sphere, it is therefore non-Gaussian.The functions Γ(t) and Γ(t) = 1 − Γ(t), defined later in Eq. (40), correspond to the probability of implementing each operation.Finally, ẐL is the GKP-encoded Pauli Ẑ operator.The probabilistic implementation of the ẐL operator can equivalently be expressed as a Gaussian channel εt p (ρ) = Γ(t)ρ + Γ(t) Ẑ ρ Ẑ † .The controlled gate with the symbol ⊖, shown in each circuit, denotes the inverse of the SUM gate, namely e iq 3 p1[44].Each SSD can be implemented by the circuit shown in this table, whereby the outcome over the measurement results are averaged to produce a mixed state. L