{'fraction_non_alphanumeric': 0.0441372475308431, 'fraction_numerical': 0.01912101431940125, 'mean_word_length': 4.446533271288971, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantum-Computing Architecture based on Large-Scale Multi-Dimensional Continuous-Variable Cluster States in a Scalable Photonic Platform

Bo-Han Wu†,1, ∗ Rafael N. Alexander†,2 Shuai Liu,3 and Zheshen Zhang3, 4 1Department of Physics, University of Arizona, Tucson, Arizona 85721, USA 2Center for Quantum Information and Control, University of New Mexico, MSC07-4220, Albuquerque, New Mexico 87131-0001, USA 3Department of Materials Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA 4J. C. Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA (Dated: September 13, 2019)


I. INTRODUCTION
Quantum computing is deemed a disruptive paradigm for solving many classically intractable problems such as factoring big numbers [1], data fitting [2], combinatorial optimization [3], and boson sampling [4]. The development of quantum-computing platforms has significantly progressed over the last decade [5][6][7][8][9], but outstanding challenges associated with the system scalability, fidelity of quantum gates, and controllability of qubits remain. To date, there has not been a single quantum-computing platform that successfully addresses all these challenges.
One-way quantum computing [10] is an intriguing approach to obviate the demanding requirement on quantum-gate fidelity [11]. Unlike quantum-computing schemes based on quantum gates, the quantum logic in one-way quantum computing is implemented via measuring a highly entangled state known as a cluster state [12]. Measurements are implemented sequentially, so that the measurement basis at a given step may be adaptively chosen based on outcomes of prior measure-of the DV cluster state increases. Deterministic means of generating large-scale DV cluster states from single quantum emitters have only appeared in recent theoretical works [19,20].
CV states are encoded into continuous quadratures of bosonic modes. Like DV systems, superdense coding [21], quantum teleportation [22], and quantum cryptography [23] have been demonstrated in CV systems. Moreover, one-way quantum computing can also be generalized to CVs [24]. An appealing feature of this approach is that large-scale entangled states can be deterministically generated at a large scale. Indeed, CVcluster-state sources have been studied in the frequency domain [25,26] and the time domain [27][28][29][30][31][32]. A recent experiment of frequency-multiplexed CV cluster states demonstrated 60 simultaneously accessible spectral modes [25]. In the time domain, temporal modes can be addressed sequentially, enabling demonstrations of cluster states made of 10,000 modes [28] and over one-million modes [31]. Though large in scale, the aforementioned demonstrations all generated onedimensional cluster states, which are insufficient for universal one-way quantum computing. More recently, twodimensional time-multiplexed CV cluster states were generated by Asavanant et al. [32] and Larsen et al. [33]. The utility of such 2D CV cluster states in one-way quantum computing is, however, constrained by the shorter of the two dimensions. Extending this dimension comes at the price of potentially introducing additional losses, limiting the potential scalability of time-multiplexing in more than one dimension. Hybrid time-frequency multiplexed CV cluster states [34,35] would significantly enlarge the size of the shorter dimension, but obtaining phase references to simultaneously access all spectral modes remains an outstanding open problem.
A key factor in assessing the feasibility of faulttolerant measurement-based quantum computation is the amount of squeezing available in a CV cluster state [36,37]. The amount of required squeezing depends on the form of error correction used [38]. Recent work has highlighted the possibility of using a combination of robust bosonic qubits, known as the Gottesman-Kitaev-Preskill (GKP) encoded qubits [39], and 3D entangled structures to implement fault-tolerant quantum computation [40][41][42][43]. While the former have recently been demonstrated experimentally [44,45], the latter still presents a challenge. As such, a platform that generates 3D CV cluster states would be an enabler for faulttolerant quantum computing.
In this article, we bridge two distinct fields, Kerrsoliton microcombs and CV quantum information, to formulate a one-way quantum-computing architecture based on large-scale 3D CV cluster states generated in a scalable quantum-photonic platform. In the proposed architecture, third-order (χ (3) ) Kerr nonlinearity is utilized with both time and frequency multiplexing to produce reconfigurable 1D, 2D, or 3D CV cluster states. Frequency multiplexing can provide access to hundreds of simultaneously accessible, highly-connected spectral modes, whereas the time multiplexing allows for sequential access to an unlimited number of temporal modes. By virtue of large bandwidth (∼GHz) of the spectral modes, the quantum-photonic platform offers the scalability and robustness required to produce large-scale 3D CV cluster states for fault-tolerant quantum computing. A unique advantage of our approach-which uses χ (3) Kerr nonlinearity-is that we can generate a frequencycomb soliton suitable for acting as a local phase reference for all spectral modes, thereby solving a key challenge that faced previous work on frequency-multiplexed CV cluster states. Access to a large number of spectral modes enables us to reach a scale required to see a truly 3D structure, without introducing prohibitively high loss.
In Sec. II, we will first describe the mechanism to obtain classical frequency-comb phase references followed by elaborating the architecture for generating programmable CV cluster states. We proceed in Sec. III to analyze device parameters that determine the scale and quality of the generated CV cluster states. The universal one-way quantum computing framework tailored to the 3D CV cluster state is formulated in Sec. IV.

II. THE ARCHITECTURE
The microring resonator (MR) is the workhorse of our photonic platform, providing the means of generating both a classical frequency-comb reference and largescale CV cluster states. In this section, we discuss the physics relevant to utilizing MRs for the generation of both classical and non-classical states of light.

A. Classical frequency-comb phase references
A continuous-wave (c.w.) pump field is sent through a bus waveguide and coupled into the MR, as shown in Fig. 1. The power of the in-coupled field is greatly enhanced by an appropriate quality (Q) factor of the MR. Above the parametric oscillation threshold, side-mode fields (represented as blue bars in Fig. 1) are created via four-wave mixing (FWM). The generated side-mode fields then couple with the pump field to create more side-mode fields via stimulated FWM. Moreover, provided that the power of the generated side-mode fields are above the cavity threshold, they also serve as new pump sources that, in turn, generate other side-mode fields. Ultimately, this cascading FWM process will lead to an extensively-extended spectrum profile, as shown in Fig. 1. Locking the phase of each frequency tooth leads to the generation of a Kerr-soliton [46,47], which can address the corresponding spectral mode of the CV cluster state in coherent quantum measurements.

B. Quantum CV cluster-state sources
We now describe our method for generating zero-, one-, two-, and three-dimensional CV cluster states. This involves sending a c.w. pump field, whose power is below the parametric oscillation threshold, through the configuration described in Sec. II A. Choosing an input pump power level below the cavity oscillator threshold brings multiple benefits. First, operating at a lower power level is more energy efficient; second, this reduces thermallyinduced instabilities; and third, the mean fields of the quantum modes would otherwise be very large above the oscillation threshold, creating a barrier to quantumlimited homodyne detection. We then describe a programmable photonic platform that can switch between generating a variety of different CV cluster states with different dimensions simply by tuning the phase of various MZIs. In our scheme, we pump the MRs at even spectral modes while detecting the output fields at only odd modes.
Throughout this article, we represent the multimode Gaussian states generated using the graphical notation introduced in Ref. [27] and summarized in Appendix A.

Entangled spectral mode pairs: 0D CV cluster states
The FWM process couples different cavity spectral modes, creating side-mode fields in a pair-wise fashion. As shown in Fig. 2, pairs of pump photons at spectral mode l = 0 are converted into signal photons at spectral modes −l = −1, −3, −5, · · · and idler photons at spectral modes l = 1, 3, 5, · · · . These modes become entangled with each other. More specifically, they become two-mode squeezed states, which are equivalent to twomode CV cluster states via application of local phase shifts [27].

1D CV cluster states
The 1D cluster-state source consists of two identical sets of 0D configurations connected by a 50:50 integrated beamsplitter (IBS). This IBS is designed so that it is capable of coupling the fields across a wide-frequency range [48]. The two MRs are pumped at different cavity spectral modes: spatial mode 'a' is at l = 0 and 'b' is at l = 2. The frequency offset of these two pumps results in each frequency being connected by an entangled pair to its neighbor, as shown in stage (i) of Fig. 3. A onedimensional entangled CV cluster state is produced in the frequency domain, as shown in stage (ii) of Fig. 3. This is known as the dual-rail wire and is a resource for single-mode CV one-way quantum computing [37]. States of this type have been successfully generated using bulk-optics setups [25,28,31].

2D CV cluster states
By extending the setup from the 1D case by including an additional unbalanced Mach-Zehnder interferometer FIG. 3. Generation of the 1D cluster state. Top and bottom inserted panels depict the graph-state representations right after stage (i) and (ii). The upper rail is for spatial mode 'a', and the lower rail is for spatial mode 'b'. C = 1 for stage (i) and C = 1/2 for stage (ii).
(UMZI), delay line (DL) and one 50:50 IBS, we are able to generate a 2D universal CV cluster state known as the bilayer square lattice [34]. This approach is equivalent to the bulk-optics scheme proposed in Ref. [34] to produce a (frequency)×(time) 2D CV cluster state, but the large MR bandwidth now allows for a shorter delay line (DL) that can be integrated on a photonic chip.
Stage (i) and (ii) in Fig. 4 are the same as in Fig. 3. After stage (ii), the four ports are processed by two UMZIs. The length difference between the two arms of each UMZI is specially designed so that the spectral modes l = 4n+1 are spatially separated from the spectral modes l = 4n − 1, where n ∈ Z [34,49,50]. The two UMZIs are fine tuned using electrodes via the thermal-optical effect [51,52].
After stage (ii), the field in one arm is temporally delayed. This arrangement extends the spectral entanglement across modes with different temporal indices. The state after stage (iii) is shown in Fig. 4. In stage (iv), the middle two arms are mixed by another 50:50 IBS, generating a 2D CV cluster state. A closely related CV cluster state was recently generated in the time domain [32] using a long DL.

3D CV cluster states
A 3D CV cluster state can be generated using the setup shown in Fig. 5. Part of the setup consists of two copies of the 2D cluster state setup, but all the 50:50 IBSs are replaced by balanced Mach-Zehnder interferometers (BMZIs). The Mach-Zehnder interferometers are tuned to act as 50:50 IBSs. This replacement will FIG. 4. Generation of the 2D cluster state. Gray, green, and magenta shaded areas in the inserted figures denote the temporal modes t 1 , t 2 , t 3 ∈ T , where t 2 = t 1 + δt and t 3 = t 2 + δt, and δt is the time delay due to the DL. For stages (i − iv), (ii), (iii), and (iv), C = 1, 1/2, 1/2, and 1/2 √ 2, respectively. The labels a (4n + 1), a (4n − 1), b (4n + 1) and b (4n − 1) indicate the spatial mode indices ('a' and 'b') followed by the spectral mode index, where n ∈ Z. Electrodes for the UMZI are shown in yellow. be relevant in the next section where we discuss how to tune the Mach-Zehnder inteferometers in order to make CV cluster states with the same chip. These two copies are coupled together with two additional BMZIs at stage (ii) in Fig. 5.
Spatial modes {a, b, c, d} are pumped at spectral modes l = 0, 2, 1 + ∆, 1 − ∆, respectively, where ∆ ∈ {2n + 1, n ∈ N} is a free parameter that sets the length of one lattice direction in frequency, as described below. At stage (i), the state consists of a collection of entangled pairs. At stage (ii), each mode has passed through a BMZI, resulting in a collection of dual-rail wire graphs, just like in the 1D case. At stage (iii), two additional BMZIs stitch these wires together to create a 2D square lattice embedded on a cylinder with circumference ∆ and length set by the overall bandwidth of the experiment. This state is known as the quad-rail lattice [27,53]. In fact, the generation circuit until this point is the same as was proposed in Ref. [53]. At stage (iv), one quarter of the modes are delayed by one time step. This is analogous to the use of DL in 2D case. The result is the 3D CV cluster state shown in Fig. 5 and further elaborated in Fig. 6. Finally, two additional 50:50 IBSs are applied on four of the resulting fields. This is a (frequency)×(frequency) lattice. Single mode input states, such as GKP ancilla states, can be injected into the cluster state by an input port indicated on the chip in Fig. 5. Use of such states for universal fault-tolerant quantum computation will be discussed later in Sec. IV B. The 3D structure of the cluster state becomes apparent when the modes are combined into groups of four, referred to as macronodes.

C. Programming the 3D CV cluster state chip for other lattices
Besides the 3D CV cluster state, the chip proposed in Fig. 5 is able to generate CV cluster states of any dimension from 0D to 3D by controlling phase shifts via the electrodes, E1, E2, and E3 in Fig. 5. These electrode control the relative phases between the two arms of BMZIs such that the splitting ratios are tuned.
First, 0D cluster states, i.e., a collection of pairwise entangled states, can be generated by setting the splitting ratios of all BMZIs to be 100:0. These are shown in To make many copies of 1D entangled states in frequency as shown in Fig. 7(b), one tunes the splitting ratios of the BMZIs at E1, 50:50, and the BMZIs at E2 and E3, 100:0.
In order to make two copies of 2D cluster states shown in Fig. 7(c), one sets the splitting ratios of the BMZIs at E2, 100:0 (rather than 50:50) and modifies the BMZIs at E1, and E3, 50:50. These 2D cluster states are (frequency)×(time) lattices.
The ( f requency) × (time) mode analog of the 2D cluster state proposed in Ref. [33] is shown in Fig. 7 (d). This can be generated by setting the splitting ratio of the BMZIs at E3, 100:0.

D. Nullifiers
An N-mode Gaussian pure state |ψ with zero mean and complex graph Z can be efficiently specified by a list of N linear combinations of the quadrature operators that satisfy the nullifier relation: where operators that satisfy this relation are referred to as nullifiers [27], andq = (q 1 , . . . ,q N ) T ,p = (p 1 , . . . ,p N ) T , Measuring expectation values of nullifiers plays a key role in verifying Gaussian pure states and genuine multi-partite inseparability, e.g., via the van Loock-Furusawa criterion [54]. Particularly convenient are states which have nullifiers that can be re-expressed such that each only consists of either position or momentum operators. These enable particularly efficient state verification since they can be measured by setting all homodyne detectors to measure either the local position or momentum operator.
Any state prepared from two-mode squeezed states and beamsplitters that do not mix position and momentum quadratures in the Heisenberg picture has nullifiers of this type. Explicit formula were given in Ref. [29]: where I is the identity operator, and V is the infinite squeezing limit of Z and is a real symmetric matrix.

A. Material considerations
Bulk quantum-optics platforms have successfully demonstrated the generation of large-scale CV cluster states. For next-generation quantum information processing, however, issues arising from long-term stability, cost, portability, and mass productivity need be accounted for. Silicon photonics, in this regard, is a promising scalable platform as mass integration of hundreds of devices on a single chip for classical optical communication has already been accomplished [55]. With respect to quantum information processing, silicon photonic implementations of integrated on-chip DV nonclassical sources [56][57][58][59], single-photon detectors [60], and DV logic gates [61] have already been demonstrated. More recently, DV high-dimensional entangled states were demonstrated in a silicon-photonics platform with 500+ waveguide and interferometer components [62]. Critically, quantum information processing in silicon photonics is carried out in the telecommunication band, and is thereby compatible with mature modulation, transmission, and detection technologies. Silicon, however, is not an ideal material for quantum information processing based on CVs due to its strong two-photon absorption in the telecommunication band, which precludes the generation of, e.g., highly squeezed light. Indeed, optical parametric oscillation, a key ingredient for the generation of large squeezing, has only been observed in silicon at mid-infrared [63], where efficient photo detectors and processing units have not yet been fully developed. Lithium niobate has been a widely used photonic material by virtue of its large nonlinearity and low absorption in the telecommunication window. Lithium niobate was recently employed in quantum information processing [64,65], but the development of quantuminformation-processing platforms based on lithium niobate has been highly challenging and cost ineffective due to a lack of fabrication recipe for large-scale devices composed of hundreds to thousands of elements.
Silicon nitride (Si 3 N 4 ), in this regard, shows its superiority in this exciting area. As a well-developed commercially-available material, Si 3 N 4 has been widely used in both microelectronic and optical integrated circuits. The compatibility with the mature CMOS fabrication technology makes the Si 3 N 4 platform stable, high performance, and cost effective. Unlike silicon, Si 3 N 4 's ultrabroad transparency window spanning from the visible to the mid-infrared makes it immune to two-photon absorption in the telecommunication band [66]. In addition, the Si 3 N 4 platform enjoys three key features that render it ideal for CV quantum information processing. First, the nonlinearity of Si 3 N 4 is about 20 times lower than that of silicon but the nonlinear interactions can be enhanced in ring resonators, as demonstrated in the generation of twin beams [67,68] and entangled states [69]. Very recently, ∼1-dB quadrature squeezing was observed in Si 3 N 4 -based devices [70,71], opening the door to a scalable CV quantum information processing platform. Second, the Si 3 N 4 platform enjoys an additional advantage in measuring frequency-multiplexed CV cluster states over the bulk quantum-optics platform based on the second-order nonlinearity: a phase-coherent soliton frequency comb produced [46,[72][73][74][75] via the thirdorder Kerr nonlinearity of Si 3 N 4 allows for simultaneous addressing of all spectral modes of the CV cluster state. Such a capability is demonstrated in the generation of octave-spanning Kerr-soliton frequency combs in Si 3 N 4 [76] and is unmatched by conventional bulk quantum-optics platforms in which the number of accessible spectral modes is fundamentally limited by the bandwidth of the electro-optic modulator used to produce the phase references for each spectral mode. Third, as a critical ingredient for time-multiplexed CV cluster states, long DLs of a few meters and an ultra-low loss level (0.1 dB/m) have been demonstrated in the Si 3 N 4 platform [77], representing a nearly two orders of magnitude improvement over that of silicon-based DLs.

B. Classical frequency-comb phase references
The generation of Kerr-soliton frequency combs has been studied extensively both in theory [46,74,75] and in experiments [47,72,73,76]. Here, we provide a brief review on the generation mechanism for Kerr-soliton frequency combs, which will be subsequently used as phase references to address each spectral mode of the CV cluster state.

Microring resonators
We consider a MR with circumference L. In the absence of optical nonlinearities and dispersion, the resonant frequency of the cavity eigenmodes are equally spaced across the whole spectrum as shown in Fig. 8 (a). The spectral linewidth, κ, is determined by the loaded Qfactor, Q (L) , as κ = ω 0 /Q (L) with ω 0 = 2πc/λ 0 , where λ 0 is the pump wavelength. The free spectral range (FSR) is ∆ω/2π = c/n g L, where n g is the group-velocity refractive index.

Kerr nonlinearity
The Kerr effect is a third-order nonlinear phenomenon that manifests itself as a intensity-dependent refractive index n = n 0 +n 2 I, where n 0 denotes the original material refractive index, I is the intensity of the field propagating in the material, and n 2 is the nonlinear refractive index.
To study the nonlinear interactions in a MR, it is more convenient to define an n 2 -related nonlinear coefficient where = h/2π is the reduced Planck constant, ω 0 is the angular frequency of the pump, c is the speed of light, and V 0 is the mode volume of the MR [78]. Physically, g 0 quantifies the shift of the resonant frequency induced by a single pump Photonics Since n 2 > 0 in Si 3 N 4 , the resonant frequency for the pump will be red shifted relative to a cold cavity by self-phase modulation. The presence of intracavity pump power also shifts the resonant frequencies of other cavity-resonant modes via cross-phase modulation. The magnitude of cross-phase modulation is twice that of self-phase modulation, thereby leading to a doubled shift for other resonant frequencies aside from the pump, as illustrated in Fig. 8 (b).

Dispersion
To employ MRs in broadband applications such as the generation of Kerr-soliton frequency combs or largescale frequency-multiplexed CV cluster states, the frequency dependence of refractive index n(ω) must be accounted for. Let us first expand the frequency-dependent wavevector, β(ω) ≡ n(ω)ω/c, at ω 0 as The group velocity is v g = 1/β 1 and β 2 is the groupvelocity dispersion (GVD) parameter. Non-zero β s 's for s ≥ 2 lead to dispersion-induced resonant-frequency shifts. For the cavity mode indexed by l ∈ Z, the shifted resonant frequency becomes where ζ s can be derived from β s 's and ∆ω, as shown in Appendix B. The shifted resonant frequencies are illustrated in Fig. 8 (c) for ζ 2 > 0, i.e., anomalous dispersion and Fig. 8 (d) for ζ 2 < 0, i.e., normal dispersion. A strong pump red-shifts the resonant frequencies, while the anomalous dispersion blue-shifts the resonant frequencies.
The overall frequency shift is balanced to ensure that a large number of spectral modes reside approximately on the cavity resonances, as shown in Fig. 8 (e). This is a key to achieving efficient Kerr-soliton and CV cluster-state generation [46,47,[72][73][74][75][76].

Classical dynamics
We now formulate the generation of Kerr-soliton frequency combs that will serve as phase references for the CV cluster states. We consider an MR pumped by a single mode situated at l = 0 with amplitude A in (in units of photons/s). Let the intracavity field of spectral mode l be A l (in units of photon number). The evolution of the intracavity modes is governed by the coupled-mode equations [75,[78][79][80][81]: (6) Here, κ = κ (i) + κ (o) is the spectral linewidth, which describes the total power decay rate. It accounts for the intrinsic cavity loss κ (i) and the out-coupling loss κ (o) ; σ is the pump detuning away from pump frequency ω 0 . On the right-hand side of Eq. (6), the first term corresponds to intracavity power decay, the second term relates to the resonant frequency shift due to dispersion, the third term describes the nonlinear Kerr interactions between different cavity modes, including self-phase modulation, cross-phase modulation, and FWM, and the last term links the extracavity pump with the intracavity field. The coupled-mode equations represent a frequency-domain approach in which the evolution of each spectral mode is derived. Alternatively, the classical dynamics can be studied in the time domain by the Lugiato-Lefever equa-FIG. 8. Shifting of cavity modes. δω X and δω S denote the frequency shifts from cross-phase modulation and self-phase modulation. δω l is the dispersion frequency shift of spectral mode l. (a) Zero-detuned (σ = 0) cold cavity without considering dispersion effect (ζ 2 = 0). (b) Zero-detuned hot cavity without considering dispersion effect. (c) Zero-detuned cold cavity in the case of anomalous dispersion (ζ 2 > 0). (d) Zerodetuned cold cavity in the case of normal dispersion (ζ 2 < 0). (e) Balancing between dispersion and Kerr nonlinearity (e.g. σ < 0, ζ 2 > 0). tion (LLE) [47,73,82]: Here, t R = 2π/∆ω is the round trip time, E(t) describes the intracavity field involving all cavity modes (in units of √ W). t and τ denote the slow time and fast time of the fields, α = ω 0 t R /Q (i) is the normalized intrinsic cavity loss, where Q (i) is the intrinsic Q-factor. θ is the normalized out-coupling loss. The normalized total cavity loss encompassing both the intrinsic and out-coupling contributions is α + θ = ω 0 t R /Q (L) . δ is the normalized pump detuning, γ = n 2 ω 0 L/cV 0 is the effective nonlinear coefficient, and E in is the input pump field.
We simulated the formation of Kerr-soliton frequency combs in the overcoupling regime using the LLE. In our simulation, we consider a MR circumference L = 15.7 mm, Q (L) = 2 × 10 6 , and Q (i) = 2.22 × 10 7 [83,84]. The pump wavelength is chosen to be λ 0 = 1549.6 nm. To design the MR waveguide with the desired dispersion properties, we utilized the simulation environment COMSOL supplied with Sellmeier equations for Si 3 N 4 reported in Ref. [85]. The designed MR waveguide has a rectangular cross-section with width W C = 2.1 µm and height H C = 0.82 µm. The simulation result gives the effective refractive indices n 0 = 1.85 and n g = 2.05 and dispersion coefficients, as shown in Table I. The nonlinear index is n 2 = 2.5×10 −19 m 2 /W [86], corresponding to an effective nonlinear coefficient of γ = 0.59 (1/Wm). Also, the simulated effective refractive index determines an FSR of ∆ω/2π = 9.32 GHz.
To produce Kerr-soliton frequency combs, the MR is pumped above its oscillating threshold P th = 51.53 mW by a c.w. pump with a power level of P in = 1.2 W and an initial normalized pump detuning of δ = 0. Subsequently, the pump detuning is adjusted to 0.21, 0.42 and 0.75 at, respectively, 25 ns, 50 ns and 75 ns, when a stable Kerr-soliton is observed, as plotted in Fig. 9 its spectrum.

C. Quantum dynamics
To study the quantum dynamics, in particular, the formation of entanglement between different spectral modes, the classical coupled-mode equations need to be augmented with quantum field operators. Specifically, the quantum field operatorâ l (t) for the l-th intracavity Each frequency tooth can serve as a pump or a phase reference that addresses a corresponding spectral mode in the CV cluster state.
field can be decomposed into a classical mean field A l (t) and a quantum fluctuation operator δâ l (t) [75]: The evolution of A l (t) is derived using the classical coupled-mode equations in Eq. 6, while the dynamics of δâ l (t) is governed by quantum coupled-mode equations: where κ = ω 0 /Q (L) , A 0 is the stabilized classical field at the pump mode. Here, without loss of generality, we assume the pump mode is situated at l = 0.V (i) l (t) andV (o) l (t) are the vacuum noise field operators at the spectral mode l, induced by the cavity intrinsic loss and the out-coupling loss, respectively. The introduction of the vacuum noise operators is required to preserve the Heisenberg uncertainty principle [87]. Both noise operators satisfy the commutation relations: To study the quantum dynamics, it is convenient to derive the spectrum of the quantum field operators. To do so, we take the Fourier transform on both sides of Eq. (9) and obtain The intracavity field is coupled out to the bus waveguide to form the out-coupling field residing in the bus waveguide, which can be directly measured and characterized. Here, the out-coupling field is represented as δâ (out) , which relates to the intracavity field, δâ l , via From Eq. (11) and Eq. (12), δâ (out) l (ω) and δâ †(out) −l (ω), are written in a matrix-form representation, Here, The introduction of IBSs, DLs and waveguide crossings causes attenuation on the power of the extracavity quantum fields by a factor of 1 − η. To account for the power attenuation, we introdue an attenuated quantum-field operator, δâ (att) l , modeled by where η ∈ [0, 1], andV (a) l (ω) is the vacuum noise operator associated with the power attenuation.V (a) l (ω) and its corresponding noise operator in the time domain, V (a) l (t), satisfy the commutation relations similar to that of Eq. (10).
To derive the nullifiers, recall the definition of position and momentum operators (q andp): whereâ andâ † are the annihilation and creation operators, respectively, applied to any dimension from zero to three. Similarly, we can define the rotated quadrature operators asq (θ) =q cos θ −p sin θ, p(θ) =q sin θ +p cos θ.
This θ parameter arises due to the phase difference between the local oscillator and the quantum fields. By tuning θ ∈ [0, 2π), we search for the angle that results in the maximum squeezing of the nullifier variance. Fig. 10 shows the simulation result for nullifier variances, under the physical parameters β 2 , L, λ 0 , Q (L) , Q (i) , n 2 , n 0 and V 0 specified in Sec. III E.

Generation of 0D CV cluster states
In the 0D case, the output quantum fields form pairwise two-mode squeezed states between mode l and −l with the nullifiers The mode configuration is illustrated in Fig. 2. We consider l ≤ l 3dB , where ±l 3dB index the entangled spectral modes at which the squeezing level is right above 3 dB. The optimal squeezing spectrum is displayed in Fig. 10 (a) and shows a 10.18 dB squeezing level.

Generation of 1D CV cluster states
To generate 1D cluster states, we prepare two 0D cluster states produced at spatial modes 'a' and 'b'. The two 0D cluster states are subsequently mixed through a 50:50 IBS shown explicitly in Fig. 3. In passing through the 50:50 IBS, quantum fields are linearly processed, lead-ing to the nullifiers, where the sub-indices denote "spatial mode" and "spectral mode". IBS can operate across a wide frequency range from 1500 nm to 1580 nm, while introducing an estimated insertion loss of 0.28 dB/IBS [48]. The output squeezing spectrum is displayed in Fig. 10 (b) with a squeezing level of 8.17 dB.

Generation of 2D CV cluster states
To generate 2D CV cluster states, 1D cluster states are processed by linear optics [89], as shown in Fig. 4. With the introduction of the DL, the quantum-field operators acquire an additional index, time. We denote the set of temporal labels as T = {t j | t j = t 0 + j δt, j ∈ Z + 0 }, where t 0 is the starting time. The number of temporal modes is determined by the amount of time over which the experiment is executed. The DL spreads the momentary entanglement to a series of entangled temporal modes with spacing δt. The nullifiers can be derived as the linear combinations of the fields, δâ s, l, t n , where s ∈ {a, b}, l ≤ l 3dB , and t n ∈ T , are fully described by Eq. 2. The corresponding graph state is further displayed in the stage (iv) of Fig. 4.
The generation of 2D cluster states requires two 50:50 IBSs, one UMZI, and one DL. Overall, the insertion loss of the IBSs and the propagation loss of DL need be considered. Reported propagation loss of DLs was as low as 0.1 dB/m [77]. Taking into account all losses, the output squeezing level becomes 6.25 dB, as shown in Fig.10 (d).

Generation of 3D CV cluster states
To generate 3D CV cluster states, we replicate two sets of the 2D cluster state setup and process the output fields based on the structure illustrated in Fig. 5. At each time step, the state consists of a 2D entangled structure, with axes f 1 and f 2 made by appropriate frequency multiplexing. To achieve this, we choose our four input pump spectral modes to be l = 0, 2, 1 − ∆, 1 + ∆, where the total number of spectral modes is determined by the phase-matching bandwidth, and ∆ specifies how the total number of frequency indices are split between the two lattice axes f 1 and f 2 [53]. The nullifiers of 3D cluster state are shown in stage (v) of Fig. 5 and described by Eq. 2.
Making the structure depicted in Fig. 5 in a small chip requires some of the waveguides to cross others. To examine the squeezing level in the 3D cluster case, we need to account for the loss arising from waveguide crossing, aside from insertion loss and propagation loss. The experimentally achievable crossing loss can be as low as 0.015 dB/cross [90]. Thus, the overall attenuation results in an output squeezing level of 4.03 dB, as shown in Fig. 10 (d).

D. Effect of dispersion
To take into account the effects of dispersion, we model the cross-section of each quantum MR as a rectangle, with width W Q and height H Q , similar to that of the classical MR in Sec. III B 4. Tuning the aspect ratio W Q /H Q allows us to obtain different dispersion parameters, which in turn determine the dimensions of the cluster state.
Given W Q , H Q , and the pump wavelength λ 0 = 1549.6 nm, dispersion parameters β 1 , β 2 , and β 3 can be calculated by COMSOL. Four combinations of W Q and H Q are listed in Table II. The 0D-cluster-state squeezing levels as a function of available mode pairs for the four settings are shown in Fig. 11. The wavevector mismatch at spectral modes l and −l, ∆β l;−l = β l + β −l − 2β 0 , is written as Since ∆β l;−l ties to the amount of squeezing, Eq. (20) indicates that the size of the cluster state is determined by β 2 . A lower β 2 results in greater number of entangled spectral modes, i.e., a larger l 3dB , as depicted in Fig. 11.

E. Experimental realization
In this section we summarize the parameters for the CV cluster-state platform, estimate the size and squeezing of the resultant state, and provide further details about the experimental realization.
Given the dispersion parameters in Table II, a quantum MR with W Q = 2.7 µm and H Q = 0.81 µm is chosen so that the dispersion parameters, β 1 , β 2 , and β 3 , result in a large number of entangled spectral modes. In the following, we focus on improving the squeezing level, by optimizing other physical parameters of the quantum MR.
The pump detuning is selected to be σ/2π = −82 MHz apart from the pump at ∼193 THz. The input pump power is P in = 55.63 mW, below the threshold P th = 65.45 mW. The length of the DLs is designed to be L DL ≥ v g ∆T = 151.27 cm, where ∆T = 2π/κ = 10.34 ns. In our designed quantum MR, we set its loaded Q-factor Q (L) = 2 × 10 6 , intrinsic Q-factor Q (i) = 2.22 × 10 7 , and FSR ∆ω/2π = 9.32 GHz, so that these parameters match those of the classical MR described in Sec. III B 4. By doing so, we can utilize the frequency comb, from the classical MR, to address the CV cluster state, from quantum MR, in homodyne detection.
The above physical parameters lead to maximum squeezing levels of 10.18 dB, 8.17 dB, 6.25 dB and 4.03 dB for 0D, 1D, 2D, and 3D cluster states, respectively. We estimate the size of CV cluster states for each dimension assuming a 3 dB cutoff for modes in the frequency direction. For the states displayed in Figs. 2, 3, 4, and 5, these are respectively, which follow the multiplication forms as In the 3D case, we choose ∆ ≈ O( N (3D) /4τ) so that the number of spectral modes in f 1 agrees with that of f 2 . Overall, our scheme does not set an upper bound for the temporal mode index τ, and, therefore, N (2D) and N (3D) can, in principle, be extended to infinity.
Thus, our approach should provide an experimentally feasible scheme to generate time-frequency multiplexed cluster states in a photonic circuit. Nevertheless, in a real experiment, there are still some challenges to be overcome. In the following, we list one primary challenge along with its solution.
To generate large-scale CV cluster states, we need to prepare several identical MRs-2 MRs for the 1D and 2D cases, and 4 MRs for the 3D case. However, fabricating several effectively identical MRs poses an engineering challenge. Fabrication errors may, for example, result in variations in the FSR of each MR and ultimately a reduction in the quality of the output CV cluster states. One way to overcome this problem is to sandwich each MR by two parallel bus waveguides. Then, we send a pair of pump fields from each bus waveguides in counterpropagating directions so that they are into the same MR base. This allows for the FSRs of the MRs to be matched by the thermo-optical fine tuning.

IV. QUANTUM COMPUTING WITH THE CV CLUSTER STATE
In previous sections, we provided details for how to generate 0D, 1D, 2D, and 3D CV cluster states. Here we describe how such states can be used for one-way quantum computing.
Implementing one-way quantum computing requires homodyne measurements of each spectral-temporal mode of the CV cluster state. The local oscillators re-quired for homodyne detection can be generated via classical frequency combs from a supplementary MR system. We pump the supplementary MR above threshold to experimentally realize optical soliton generation by choosing the physical parameters in Sec. III B 4. The frequency teeth of the generated optical are coherent, nearly equidistant from each other, and can be a new source of classical fields or serve as multiple phase references. To implement independently tunable homodyne detection on multiple spectral modes, the relative phase of each tooth must be variable. This can be implemented using a waveshaper [91].
Though we are primarily interested in describing quantum computing with the 3D cluster state, we first briefly summarize what is known about the other cases. The 0D case is not sufficiently connected for use in oneway quantum computing. The 1D case is a resource for single-mode one-way quantum computing, as described in Ref. [37]. The 2D case is a universal resource, and can implement multimode gates via the one-way quantumcomputing protocol described in Ref. [34]. The quantum circuits that can be implemented on this resource are local in (1+1) dimensions. Below, we provide a one-way quantum-computing protocol for the 3D resource state. Our protocol is capable of implementing local quantum circuits in (2+1) dimensions, which should improve quantum circuit compilation relative to 2D resources.
Recall also that the states described in the previous sections were technically not CV cluster states. This does not cause any issues because the generation circuit only consists of two-mode squeezing and 50:50 beamsplitters that do not mix the position part and momentum part of each quadrature. For this reason, the states generated are equivalent to CV cluster states by application of a π/4-phase delay on all modes. This change can be incorporated into the measurement device. See Appendix. A and Ref. [29] for more details. For the procedure described below, we assume that these phase delays have already been implemented, and thus, the states described will be CV cluster states.

A. Preliminaries
Before constructing a model for one-way quantum computing using the 3D cluster state, we requires some additional definitions.
The first step is to define the relevant modes that the CV cluster state is made from in terms of the infinitesimal spectral modes described in the previous section. These can be written as an integral over the squeezing spectrum weighted by a normalized Kernel function, K(ω), within the frequency range [−∆ω/2, ∆ω/2], with time index t n , spectral index l and a spatial mode index s.
We collect all the field operators δâ s, l, t n (ω), into a vector a:â The subscript ofâ j denotes the modes on a particular graph representation of CV cluster state. The entanglement was previously characterized by spatial, spectral, and temporal modes, but now is only characterized by a single subscript j. The length ofâ is set by the number of entangled modes in Eq. (21).
Relative to these operators, recall that the quadrature operators can be defined viaâ j = (q j + ip j )/ √ 2. We denote the s th basis state of the operatorr j as |s r l , wherê r will usually be a position or momentum operator, or a linear combination of the two. Now we define some useful gates to construct our measurement-based protocol. The 50:50 beamsplitter gate (BSG) between modes ( j , k) can be written aŝ Note that this gate is not invariant under a swapping of the inputs. Graphically, this is represented by a red arrow from mode j to mode k.
A specific combination of these results in a balanced mixing of four modes, which we call the foursplitter gatê It will become convenient to use the matrix representation of the Heisenberg-picture evolution for these gates, i.e.,B † jk where The phase delay is written aŝ The single mode squeezer is given the following nonstandard definition: where s is known as the squeezing factor, which is the ordinary squeezing gate with squeezing paramater r = ln |s| followed by a π phase delay if s < 0.
The displacement operator is defined aŝ Finally, it will be convenient to define the following single-mode Gaussian unitarŷ 1. Square cluster state The following square cluster state plays a key role in the analysis of our one-way quantum-computing protocol. It can be generated by sending one mode from each of a pair of two-mode CV cluster states through a 50:50 BSG (33) where C = 1/ √ 2 on the left and C = 1 on the center and right.

Physical modes and distributed modes
As with other multilayered CV cluster states [34,37,92], the description of one-way quantum computing can be simplified by expressing it in terms of so-called distributed modes, which defines a nonlocal tensor product structure for each macronode. Each of the physical modes {a, b, c, d} within a given macronode is mapped to a distinct distributed mode in {α, β, γ, δ} via Expressing the 3D cluster state in terms of the distributed modes simplifies the graph substantially. It becomes a disjoint collection of square cluster states as shown in Fig. 12.
In our protocol we encode input states into half of the macronodes of a given time step-specifically the red ones shown in Fig. 6. The gray macronodes serve as "routers" that control the application of entangling gates as the inputs distribute through the cluster state in the time direction. Besides simplifying the graph, the distributed modes play a special role in defining how each input is encoded within a given red macronode on a given time slice. More concretely, we will choose to encode each input into either the 'α' or the 'γ' distributed mode. In fact, it will be convenient to change whether the input resides in either the 'α' or 'γ' mode from time step to time step.
Our procotol for one-way quantum computing involves local homodyne measurements with respect to the physical modes. We denote the measured bases aŝ where the angle θ is controlled by the relative phase between physical mode j and its corresponding local oscillator. Local measurements on the physical modes (as shown in Fig. 13 (a)) translate to quantum gates followed by local measurements on the distributed modes (as shown in Fig. 13 (b)). However, for special choices of the measurement angles, this measurement can appear as partially separable (see Fig. 13 (c)) or completely separable (see Fig. 13 (d)). This follows from the following identity in Ref. [34]: It will also be useful to note that introducing an arbitrary permutation on the four modes before measurement shown in Fig. 13 (b) is equivalent to swapping some of the measurement bases after the four splitter and changing the sign of some outcomes [92]. LetP be an operator that permutes modes {α, β, γ, δ}. Then where the left-hand side is the Heisenberg picture evolution of the vector of observables measured in the circuit from Fig. 13 (b) through the permutation gateP. The right-hand side shows that this is equivalent to multiplying the vector of observables on the right-hand side by a 4 × 4 matrix M, which is the product of a permutation matrix and a diagonal matrix with entries in {−1, 1} [92]. Therefore, introducing a permutation gate before such a measurement is equivalent to swapping the measurement bases at the homodyne detectors in Fig. 13 (a), and flipping the sign of some outcomes.

Quantum computing via teleportation
Given an input, a two-mode CV cluster state, and a measurement that implements a 50:50 BSG followed by local homodyne detection, i.e.,  (38) we can implement the gatê via teleportation, where we have assumed infinite squeezing [29,37]. Finite squeezing effects for this gate can be included via the analysis described in Ref. [37]. The teleportation-induced gateV is a crucial factor for the one-way quantum-computing protocol described below. For convenience, we assume all measurement outcomes equal to zero. The true evolution will only differ from this case by a final displacement since all gates described are Gaussian. Now, we consider performing measurements on the top red and four grey macronodes in Fig. 12 (b). Each  15. (a) The two square graphs from Fig. 12 have been redrawn side by side. Note that the time arrow is from left to right, rather than from top to bottom. The green modes on the left are the distributed modes that can contain an input. The red arrows denote the application of a 50:50 BSG, and the black arrows indicate the local basis measurement after all such BSGs. (b) Alternative representation of (a). The square graph cluster state is equivalent to two pairs stitched together by a 50:50 BSG (see Eq. (33)). This picture more clearly shows that the measurement pattern implements sequential teleportation (see Eq. (38)). macronode is measured as shown in Fig. 13 (a) and (b). For the red macronode, we set θ a = θ c , and θ b = θ d , and the measurement can be modelled as shown in Fig. 13 (c). Subsequently, we choose to measure all grey macronodes in either thep(±π/4) basis, where the sign of the angle is determined with respect to Fig. 14 (a). Since each mode within a given grey macronode is measured in the same basis, measurement of the physical modes can be modelled as shown in Fig. 13 (d). The overall measurement pattern can now be summarized as shown in Fig. 15 (a).
Equivalently, we could use the description in Fig. 15  (b), which shows more clearly that this measurement pattern implements two rounds of teleportation (see Eq. (38)). Thus, we can write the gate implemented aŝ (40) where the ± sign depends on whether the red macronode is within a green shaded region or an unshaded region in Fig. 14 (b), respectively. Note that after teleportation, the input resides in the bottom red macronode of Fig. 12 (b). Evolving under Eq. (40) is equivalent to using two decoupled copies of the dual rail wire [37]. Two successive rounds of this measurement pattern are sufficient to implement arbitrary single-mode Gaussian unitary gates for inputs in either the 'α' and 'γ' distributed modes [37,93].
Another important measurement-based operation for our one-way quantum computing protocol is a swap between modes 'α' and 'γ', thus changing the distributed mode in which the logical information resides. As described in Eq. (37), a swap before a macronode measurement is equivalent to permuting the homodyne angles and post-processing. By swapping between the 'α' and 'γ' distributed modes, one can use the BMZI in Fig. 5 to insert an input state, such as a GKP ancilla state, into either distributed mode of any macronode in the cluster state.

Entangling gates
Next we describe how to implement multimode (a.k.a. entangling) gates between inputs encoded within adjacent red macronodes. We consider the four red macronodes adjacent to a particular grey macronode. In order to perform an entangling operation between any subset of the inputs on these four red macronodes, the only change relative to the single-mode gates described in the previous section is that modes that make up the central grey macronode are measured in different bases. For concreteness, we consider a particular subgraph of the 3D cluster state, as shown in Fig. 16.
We will assume that if a grey macronode is used to implement entangling gates, then none of the adjacent grey macronodes are used to do so as well. With respect to Fig. 16 (c), this means that modes in macronodes 'E', 'F', 'G', 'K', 'L', and 'M' are all measured in thep(±π/4) basis, and modes in macronodes 'H' and 'J' are measured in thep(∓π/4) basis, as described in the previous section. In order for inputs in macronodes 'A' and 'D' ('B' and 'C') to participate in the entangling gate, they will be assumed to reside in the 'γ' ('α') distributed mode. Note that if the input happened to be in the other of the two distributed modes, a swap can be employed as described in the previous section.
Excepting special cases such as those mentioned in Fig. 13 (b) and (c), generic angles θ Ia , θ Ib , θ Ic , and θ Id will result in measurements as shown in Fig. 13 (b). The 50:50 BSGs acting before the measurement device in Fig. 13 (b) are represented graphically in Fig. 16 (e). By employing a series of identities for 50:50 BSGs acting on entangled pairs, we can move the BSGs so that they act on other modes, thereby reducing the measurement on macronode 'I' to one that is local with respect to the distributed mode tensor product structure. This technique is known as beamsplitter gymnastics [34]. Fig. 17 shows how to do this forB Iα,Iβ andB Iα,Iγ , and similarly, Fig. 18 shows how to do this forB Iγ,Iδ and B Iβ,Iδ . The beamsplitter positioning in Fig. 17(e) and Fig. 18(b) are such thet all beamsplitters lie between teleportation steps, and thus, the evolution is merely a sequence of logical BSGs and teleportation. The total evolution can be written aŝ Any inputs present in the alternative distributed modes j ∈ {Aα, Bγ, Cγ, Dα} evolve according to the single-mode protocol described in the previous section Therefore, by the applying the swap degree of freedom between 'α' and 'γ' distributed modes, we can control participation in the entangling gate.
Restricting θ Ib = θ Id and θ Ia = θ Ic simplifiesŴ tô R Aγ (π)R Dγ (π)Ĝ ± Dγ,Aγ (θ Da , θ Db , θ Aa , θ Ab , θ Ia , θ Ib ) ×Ĝ ± Bα,Cα (θ Ba , θ Bb , θ Ca , θ Cb , θ Ia , θ Ib ), which implements a pair of two-mode gates between pairs of modes (Dγ, Aγ) and (Bα, Cα).  The goal is to move the BSG between the grey modes in the purple oval (6 and 7), replacing it with an operation that only on other modes. (b) By following a similar sequence of steps as in Fig. 17 (or equivalently, using the proof shown in Ref. [34]) we arrive at this alternative description. (c) The leftmost dotted beamsplitter can be moved to the left and we can substitute the square graphs back in by four applications of Eq. (33). At this stage, the 50:50 BSGB 6,7 had been replaced with the two dotted 50:50 BSGs that act on either sides of the square graphs, thereby achieving our goal.
By applying further restrictions, the gatesĜ can be reduced into a more familiar and tunableĈ Z (g) = e igq⊗q gate, ∀g ∈ R, which is a standard entangling gate in continuous-variable quantum computing. In particular where we have used the protocol from Ref. [34]. Thus, the four mode gateŴ can be reduced to pairs ofĈ Z gates with respect to any pairings of modes {Aα, Bγ, Cγ, Dα}. By leveraging the swap degrees of freedom at red macronodes as well as the above measurement restrictions at grey macronodes, we have shown that it is possible to implement either two (Ĝ) or four (Ŵ) mode entangling gates between nearest neighbor input states.

B. Universal quantum computing and error correction against finite squeezing effects
Using finitely squeezed continuous-variable cluster states will result in Gaussian noise [37], the strength of which is set by the available amount of squeezing in the cluster state. This effect can be combated using non-Gaussian quantum error correction, such as a supply of GKP qubits, provided that the squeezing in the CV cluster state is sufficiently high. A 20.5 dB upper bound on the amount of squeezing required was given in Ref. [38], however, it is likely that this bound can be improved by incorporating quantum error correction strategies compatible with 3D entangled resource states [40][41][42][43].
A non-Gaussian resource is also required to extend the above one-way quantum-computing protocolwhich can only implement Gaussian unitary gates, and hence, is classically simulable-to a universal model. With the access to a supply of GKP qubits, we can generate all of the necessary ingredients for universal quantum computing with Gaussian operations [94]. In principle, all required Gaussian operations can be implemented using the 3D cluster state with homodyne detection via the gate set described above. Though implementing these gates on the 3D cluster state will additionally introduce Gaussian noise that arises due to having only finite squeezing, this can be corrected by using additional GKP ancilla states injected into the state at regular intervals [38].

C. Full Architecture
All the ingredients introduced above, in conjunction with classical control, yields a universal architecture for quantum computing, as sketched in Fig. 19 and further described below.
The quantum MRs and linear optical components are configured to generate a 3D CV cluster state described in Sec. II B 4. To implement Gaussian quantum gates, homodyne measurements (with fully tunable and independent local oscillator phases) are performed simultaneously on all spectral modes and sequentially on all temporal modes. To do so, a classical Kerr-Soliton frequency comb is shaped by classical processing circuits so that each frequency tooth carries a designated phase to address its corresponding quantum spectral mode. The processed classical frequency comb interferes with the 3D cluster state at a 50:50 beamsplitter, whose outputs are frequency demultiplexed by wavelength-division multiplexers (WDMs). An array of detectors perform balanced measurements. The measurement outcome is processed by a classical algorithm that determines the basis settings for homodyne measurements on the next batch of temporal modes.

V. CONCLUSION
We have proposed and analyzed a scalable and experimentally viable architecture for generating timefrequency-multiplexed 3D CV cluster states and utilizing them for large-scale quantum computing in integrated photonic circuits. Our architecture inherits the compactness of previous bulk-optical approaches, and thus, only requires a handful of squeezed light sources, multimode interference couplers, delay lines, and Mach-Zehnder interferometers. The squeezed light is produced via a χ (3) nonlinearity enhanced by MRs. This platform offers two key advantages: 1) only constant length de-lay lines are required to grow the resource state in the time direction; 2) and we can employ frequency multiplexing to extend the state in the frequency domain, and it is possible to address a large number (>2000) spectral modes via a frequency-comb soliton local oscillator. This claim is backed up by our numerical analysis, which found compatible physical device parameters to generate such a 3D cluster state. The extension of previous schemes [27,29,33,34,53] to the 3D case is significant as it opens the door topological error correction for Bosonic qubits [41,95].
Our architecture has the added benefit that it can be reprogrammed to generate cluster states of dimension less than three. This makes it compatible with previously studied protocols for lower dimensional cluster states [92,96,97].
This graphical description can be made even simpler when describing states whose non-zero graph edges all have same magnitude and differ only by a sign. We will not show self loops, which will all take value i cosh2r, where r is a parameter that describes the overall squeezing used to produce the state. Blue and yellow edges are all weight ∓iC sinh2r, respectively, where C is a real rescaling parameter specified in the relevant figure captions throughout this Article. The states represented by these graphs are not CV cluster states. However, all states described in this paper have the property that they are made from two-mode squeezed states and beamsplitters that do not mix the position and momentum quadratures. As a result, they can be converted to CV cluster states with self-loop weights i sech2r and edge weights ±C tanh2r, respectively, by applying π/4-phase delays prior to each measurement, which can be incorporated into the local oscillator(for a proof, see Theorem 1 in Ref. [29]). Due to this fact, we will treat each Gaussian state as if it were such a CV cluster state. These conventions are the same as those used and described in Refs. [29,34,92].
Some examples of the simplified graphical notation are given in Fig. 20.
The evolution of Gaussian pure states under Gaussian unitaries can be incorporated into this formalism as a graphical update rule [98]. One transformation that is particularly useful for understanding the construction of CV cluster states is the beamsplitter rule: , where the arrow points from mode j to mode k, indicating the application of a 50:50 beamsplitterB jk . Let C by the rescaling parameter for the left hand side. Then the rescaling parameter for the right hand side is C/ √ 2 [27]. Note that the beamsplitter "copies" each link, up to a change in sign.