Quantum Computing in Discrete- and Continuous-Variable Architectures

Shraddha Singh

REVIEW 3 major objections 4 minor 219 references

Quantum Computing in Discrete- and Continuous-Variable Architectures

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This thesis establishes a non-abelian composite pulse, the Gaussian-Controlled-Rotation, that cancels oscillator quantum-fluctuation errors in 4.5 times fewer gates than the standard BB1 pulse and makes deterministic, optimizer-free…

desk verdict Solid GCR pulse analysis; the state-preparation claims overstate how optimizer-free they are. read the letter →

arxiv 2507.01146 v1 pith:P7P4YJYQ submitted 2025-07-01 quant-ph

Shraddha Singh This is my paper

classification quant-ph MSC 81P6881P45 PACS 03.67.Lx03.67.Pp

keywords non-abelianquantumsignalprocessingGaussian-controlledrotationcompositepulsescontinuous-variablecomputinghybridCV-DVarchitectureGKPstatepreparationdeterministicprobabilisticerrorcorrection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The thesis argues that controlling a qubit rotation with non-commuting oscillator operators, rather than with a classical angle, makes quantum signal processing substantially more powerful. Its central object is the Gaussian-Controlled-Rotation (GCR), a two-pulse sequence that cancels rotation errors caused by the oscillator's quantum position and momentum fluctuations. The author shows GCR achieves the same error cancellation as the standard BB1 composite pulse while being at least 4.5 times shorter in circuit duration, and uses it to design analytical, deterministic preparations of squeezed, cat, and GKP states that match numerically optimized schemes. The same non-abelian framework is extended to readout, error-corrected gates, and a probabilistic analysis of photon-loss correction in finite-energy GKP codes. If correct, this gives hybrid oscillator-qubit architectures a practical route to high-fidelity control without costly classical optimization.

What carries the argument

The core object is the Gaussian-Controlled-Rotation composite pulse, GCR($\theta$), built from two conditional displacements whose control parameters are the non-commuting oscillator operators $\hat{x}$ and $\hat{p}$. The mechanism is a pre-correction: the momentum-controlled rotation $R_{\pi/2}(-\theta\Delta^2\hat{p}/|\alpha|)$ acts like a rotation about $\hat{x}$ with angle proportional to $\hat{x}-\alpha$, exactly canceling the first-order error of the position-controlled rotation, provided the ancilla qubit starts in $|g\rangle$. The error parameter is $\chi = \theta\Delta/2|\alpha|$, and the whole argument maps the composite pulse onto a flat-top square-wave response in phase space, giving the error scalings $P_e \sim 0.1\chi^6$ and $1-F_H \sim \chi^4/8$.

What would settle it

Run GCR($\theta$) on a squeezed coherent state with the ancilla prepared in $|e\rangle$ rather than $|g\rangle$; if the failure probability and target-rotation fidelity remain at the claimed $\chi^6$ and $\chi^4$ levels, the stated $|g\rangle$ precondition is not load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that Gaussian-Controlled-Rotation, defined as GCR($\theta$) = $R_0(-\theta \hat{x}/|\alpha|)\, R_{\pi/2}(-\theta\Delta^2 \hat{p}/|\alpha|)$, acts as a non-abelian composite pulse: applied to $|g\rangle |\alpha_\Delta\rangle$ it produces the target rotation $R_0(-\theta\alpha/|\alpha|)$ with failure probability $P_e \sim 0.1\chi^6$ and hybrid infidelity $1-F_H \sim \chi^4/8$, where $\chi = \theta\Delta/2|\alpha|$. The proof rests on the identity $\sigma_y|g\rangle = i\sigma_x|g\rangle$, which turns a momentum-controlled pre-rotation into a first-order correction of position-uncertainty errors. The thesis further claims this sequence enables deterministic, numerically-optimizer-free preparation of squeezed states, two- and four-legged cat states, and GKP states with fidelity comparable to the best numerical methods, and that the same non-abelian framework yields an analytical account of photon-loss correction in finite-energy GKP codes through a newly defined 'probabilistic distance.'

Load-bearing premise

The error cancellation works only when the ancilla qubit starts in the $|g\rangle$ eigenstate of $\sigma_z$, because the pre-correction relies on $\sigma_y|g\rangle = i\sigma_x|g\rangle$; any initialization error, leakage, or pre-rotation of the ancilla breaks the cancellation.

Editorial extensions

If this is right

Deterministic preparation of squeezed, cat, and GKP states becomes possible without numerical optimization, reducing classical processing and control hardware requirements.
GCR-based readout and error-corrected gate teleportation can achieve high fidelity for GKP qubits even with residual oscillator errors, provided the ancilla bias assumptions hold.
The probabilistic-distance analysis offers an analytical explanation of how recent beyond-break-even GKP experiments correct photon loss, and extends to square and hexagonal lattices and qudit encodings.
The non-abelian QSP framework generalizes toward non-abelian quantum singular value transformation, potentially unifying hybrid CV-DV algorithms in the same way QSVT unifies qubit algorithms.
Multi-mode extensions of GCR supply efficient entangling gates and error-corrected two-qubit rotations for GKP-based fault-tolerant architectures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit dependence on the initial $|g\rangle$ ancilla state suggests a direct experimental test: preparing the ancilla in $|e\rangle$ or a superposition should break the error cancellation, and verifying this would confirm the claimed mechanism rather than a generic pulse-sequencing effect.
The same pre-correction idea could be ported to other non-commuting pairs of control operators, such as number and phase, to design composite pulses for states that are not Gaussian superpositions.
The probabilistic-distance metric could be applied to other bosonic codes, such as binomial or cat codes, to compare their photon-loss correction performance on a common footing.
Because GCR preparation is deterministic yet allows a herald of ancilla errors, the protocols could be inserted into mid-circuit error-detection schedules without a numerical re-optimization of the whole circuit.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — you asked for a take on Singh's thesis. The real content is the GCR composite pulse: an analytic two-pulse sequence that converts oscillator position/momentum spread into a clean qubit rotation, with error scalings Pe ~ 0.1χ^6 and hybrid infidelity ~ χ^4/8, and a circuit duration at least 4.5× shorter than BB1(90). The derivation is self-contained, the scalings are checked with QuTiP, and the BB1 comparison is fair. That is the strongest part of the thesis, and it holds up on reading.

What is actually new here is narrower than the abstract implies. GCR itself is from the author's arXiv:2504.19992, so in the thesis it is a restatement. The genuinely new pieces are the Gaussian hierarchy (Sec. 2.3) and the probabilistic error-correction analysis for GKP photon loss (Ch. 5). The Gaussian hierarchy is a natural but useful frame — classifying CV operations by Hamiltonian polynomial degree and asking where GKP gates sit relative to the Clifford hierarchy. It raises good open questions, even if the classification is rough. The probabilistic distance idea could matter for comparing GKP lattices under photon loss, but the full derivation sits mostly in an unpublished manuscript, so I cannot fully vet it from this excerpt.

Soft spots, in proportion: the "without numerical optimization" claim is overstated. In Sec. 4.1 the squeezing schedule parameters α_k/Δ_k^2 are explicitly approximated numerically ("various approximations/numerical methods"), and the protocol's convergence plots depend on scanning a and c. Since GKP preparation in Sec. 4.3 builds on that squeezed state, the numerical dependence propagates to the headline GKP result. The claim should be qualified to "no numerical optimizer for the pulse structure," not no numerics anywhere. That is an honest presentation error, not a fatal one — the GCR gadget itself is analytic.

The other limitation is the ancilla |g> initialization: GCR's error cancellation uses σ_y|g> = iσ_x|g>, so any rotation or error before the gate breaks it. The thesis states this explicitly, but every state-prep protocol inherits it, so it is a hard operational condition for experimenters. No code or data is shipped either, which is a shame; the QuTiP plots could be reproduced with modest effort.

Overall, the central GCR argument holds, but the thesis claims more than it delivers. For readers in bosonic QEC or hybrid control, Ch. 3 and the Gaussian hierarchy are worth the time. It deserves a serious referee — someone who knows the GKP literature and will push on the distinction between "analytic gadget" and "fully optimizer-free protocol." I would engage with it and cite the GCR result with the numerical fitting caveat.

Referee Report

3 major / 4 minor

Summary. This dissertation develops a framework for hybrid discrete-variable (DV) and continuous-variable (CV) quantum control, centered on 'non-abelian quantum signal processing' (NA-QSP). The central technical object is the Gaussian-Controlled-Rotation (GCR), a two-pulse non-abelian composite sequence that approximates a qubit rotation whose sign is set by the mean oscillator position, with failure probability Pe ~ 0.1 chi^6 and hybrid infidelity 1-FH ~ chi^4/8, and with a claimed circuit-duration advantage of at least 4.5x over the commuting-variable BB1(90) sequence (Secs. 3.2, 3.3, Eqs. 3.15, 3.29-3.33). The thesis applies GCR to deterministic preparation of squeezed, cat, and GKP states (Ch. 4), introduces a Gaussian hierarchy classification of CV operations (Sec. 2.3), develops a probabilistic error-correction framework for photon loss in finite-energy GKP codes (Ch. 5), and discusses error-corrected control of GKP qudits (Ch. 6).

Significance. If the central GCR analysis is correct, it is a genuinely useful analytical primitive: it gives explicit, checkable error scalings for a hybrid CV-DV gate, it is confirmed by QuTiP numerics in the present manuscript, and the analytical structure of the circuit is better suited to error tracking and generalization to qudits than purely numerical pulses. The proposed Gaussian hierarchy is a stimulating open-problem framework, even if much of it remains conjectural. The probabilistic-distance analysis of photon-loss correction also provides a concrete analytical lens on recent beyond-break-even GKP experiments. These strengths justify publication after revision, provided the load-bearing claim of 'preparation without numerical optimization' is made precise.

major comments (3)

[Sec. 4.1 and App. C.1] The abstract and Ch. 4 claim deterministic preparation of squeezed, cat, and GKP states 'without numerical optimization,' but Sec. 4.1 explicitly states that the correction parameter alpha_k/Delta_k^2 for each squeezing round must be obtained by approximating the state as Gaussian, and later states that the corrections to the linear slope 4 alpha/Delta^2 are 'approximated numerically (see App. C.1).' The convergence, fidelity, and circuit-duration results in Fig. 4.1(c)-(e) depend on this numerically fitted schedule, and the GKP protocol of Sec. 4.3 starts from the squeezed state prepared in Sec. 4.1, so the numerical dependence propagates to the headline GKP preparation result. This contradicts the unqualified claim in the abstract. Please either provide an analytic schedule for alpha_k as a function of Delta_k, or clearly qualify the claim (for example, 'no numerical state-by-state optimizer, though the squeezing schedule uses a Gaussian-fit approximation') and state that qualification in the abstract and protocol statements.
[Sec. 2.3, near Eqs. (2.90)-(2.96)] The text concludes that 'for GKP encoding, C_n subset of G_n for n >= 4,' but the argument given does not establish this containment. The example shows that a gate sqrt(H) in G4 lies outside the union of Clifford hierarchy levels, which proves G4 is not contained in C4 but does not prove that every Clifford-hierarchy-4 gate lies in G4. The recursive conjugacy-set argument also appears only to track a finite set of generated gates, not to characterize the full Clifford hierarchy. Since this section is explicitly framed as an open problem, the claim should be weakened to a conjecture unless a full proof is supplied.
[Sec. 3.2.1, Eq. (3.23)] The error-cancellation mechanism of GCR relies on the identity sigma_y|g> = i sigma_x|g>, which holds only when the ancilla qubit begins in the sigma_z eigenstate |g>. The text states this condition explicitly, which is good, but the protocols in Ch. 4 that use GCR as a pre-correction inherit this sensitivity without an error analysis. A small initial admixture of |e> will generically produce a first-order error in the correction parameter lambda, potentially spoiling the chi^4/chi^6 scalings. This does not invalidate the derivation, but the practical claim of deterministic, high-fidelity preparation should include an explicit sensitivity estimate or a stated assumption that ancilla initialization errors are negligible.

minor comments (4)

[Fig. 4.2 caption] The caption contains the typo 'cat ctates'; this should read 'cat states.'
[Fig. 7.3 caption] The caption contains the typo 'succcess probability'; this should be corrected to 'success probability.'
[Fig. 5.6 caption] The caption contains the typo 'V oronoi cells'; the spacing should be corrected to 'Voronoi cells.'
[Sec. 4.1, Fig. 4.1(d)-(e)] The figure and text report a specific Fisher information (F=53.5) and infidelity (0.008) for the faster protocol, but it is not immediately clear from the figure whether these values correspond to post-selected or non-post-selected results; adding a note in the caption would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the GCR derivation is self-contained and benchmarked externally, but the 'without numerical optimization' wording is unsupported by the numerically approximated squeezing schedule.

full rationale

The central GCR result is derived rather than assumed. In Sec. 3.2.1, the pre-rotation is explicitly constructed with lambda = theta*Delta^2/|alpha| so that the momentum-controlled rotation converts to a first-order position correction on |g> (Eq. 3.24), and Sec. 3.2.2 computes P_e ~ 0.1 chi^6 and 1 - F_H ~ chi^4/8 from Hermite expansions (Eqs. 3.29-3.32), with numerical verification in Fig. 3.2. The target rotation R_0(-theta*alpha/|alpha|) is not inserted as an input; it is the ideal limit of the constructed gate. The circuit-depth comparison with BB1(90) (Eq. 3.33) is arithmetic on conditional-displacement amplitudes, not a fitted result. State preparation benchmarks are made against external numerical schemes (Refs. [5,116]) and against the classical BB1 pulse, so the central claims have independent content. Self-citations to Refs. [1,2,31] appear frequently, but the thesis itself contains the relevant derivations, so these citations are not load-bearing circularity. The |g>-initialization condition is a stated precondition, not a hidden circular input. The main caveat is that Sec. 4.1 states that the squeezing schedule alpha_k/Delta_k^2 'can be computed using various approximations/numerical methods' and later says 'we approximate corrections to the linear slope 4*alpha/Delta^2 numerically,' which conflicts with the abstract's claim of preparation 'without numerical optimization' and propagates to the GKP protocol that starts from that squeezed state. This is an overclaim about analyticity of the method, not a circular reduction: the Chapter 3 error-cancellation proof does not assume the squeezing results or the target fidelities. Score 2 reflects the minor, non-load-bearing self-citations and the unsupported optimizer-free wording, with no genuine circular step identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the paper contributes frameworks (NA-QSP, Gaussian hierarchy, probabilistic distance) which are mathematical constructs, not entities with independent empirical handles. The main hand-selected inputs are the squeezing schedule coefficients and the GKP envelope width.

free parameters (2)

Squeezing schedule exponents a and c = a in {0.06, 0.13, 0.27}, c in [-3, 0] or c = 2
The deterministic squeezing protocol in Sec. 4.1 requires choosing |α|_k = a Δ_k^c; fidelity and circuit duration trade-off depend on these hand-picked values, and the text says linear-slope corrections are approximated numerically (App. C.1).
Finite-energy GKP envelope parameter Δ = 0.34 for the GKP preparation benchmarks
Δ is chosen to match recent experiments [3,5] and is used as a code-design input rather than fitted to the target claim, but it determines the circuit parameters in Sec. 4.3.

assumptions (5)

domain assumption Ancilla qubit starts in |g>, an eigenstate of σ_z, for the GCR pre-correction to convert a σ_y rotation into a σ_x rotation.
Used in Sec. 3.2.1 around Eq. (3.22); the text states the scheme only works for initial qubit state |g>.
domain assumption Oscillator states are Gaussian wave functions with known position uncertainty Δ, so rotation errors from quantum fluctuations are linear in (x-α) to first order.
Sec. 3.1.1 Eq. (3.1) and Sec. 3.2.1; the error cancellation is derived from the derivative action on a Gaussian envelope.
domain assumption The phase-space instruction set (conditional displacements CD and qubit rotations R) is the available universal gate set.
Sec. 2.4.1 Eq. (2.97); all GCR constructions are built from these gates.
ad hoc to paper The Gaussian hierarchy definition G_n = {U | [H(D), H(U)] = H(U') ⇒ U' in G_{n-1}} is a valid classification analog of the Clifford hierarchy.
Sec. 2.3 Eq. (2.90) proposes this definition; the paper presents it as an open-problem framing, not as a theorem.
domain assumption Photon loss can be analyzed through finite-energy GKP error spaces and a probabilistic distance.
Chapter 5 (abstract and Sec. 5.3); the full derivation is not visible in this excerpt, but the framework is introduced to explain break-even experiments.

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Cite this review

Pith. "Pith review of Quantum Computing in Discrete- and Continuous-Variable Architectures." pith.science (2026). https://pith.science/paper/P7P4YJYQ

@misc{pith2026250701146,
  author       = {Pith},
  title        = {Pith review of: Quantum Computing in Discrete- and Continuous-Variable Architectures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7P4YJYQ}},
  note         = {Machine review of arXiv:2507.01146}
}

read the original abstract

This thesis develops a theoretical framework for hybrid continuous-variable (CV) and discrete-variable (DV) quantum systems, with emphasis on quantum control, state preparation, and error correction. A central contribution is non-abelian quantum signal processing (NA-QSP), a generalization of quantum signal processing to settings where control parameters are non-commuting operators. Within this framework, we introduce the Gaussian-Controlled-Rotation (GCR) protocol, which enables high-fidelity control of CV states using DV ancillae. This approach allows for deterministic preparation of squeezed, cat, and Gottesman-Kitaev-Preskill (GKP) states without numerical optimization. Two previously unpublished contributions are included: (i) Chapter 2.3 introduces the Gaussian hierarchy, a classification of CV operations analogous to the Clifford hierarchy, offering a new lens for understanding CV gate sets; (ii) Chapter 5 presents an analytical framework for correcting photon loss in finite-energy GKP codes, introducing the notion of probabilistic error correction, providing insight into recent GKP experiments surpassing break-even thresholds. Overall, this work lays the groundwork for scalable, fault-tolerant quantum computation in hybrid CV-DV architectures, with applications to logical gate synthesis, readout, and hybrid algorithm design using ancilla oscillators.

Figures

Figures reproduced from arXiv: 2507.01146 by the authors.

**Figure 1.1.** Thesis flowchart. The blue bubbles in the flowchart represent the key topics covered in this thesis and their interconnections. The gray bubbles attached to each chapter highlight intriguing discussion points related to the corresponding topic. Chapter 2 introduces the building blocks of hybrid systems, covering DV and CV state spaces, quantum channels, and operations. It highlights the absence of a structured hier… view at source ↗

**Figure 2.1.** The hybrid CV-DV space. (a) Qubit Bloch sphere (see Sec. 2.1). The Bloch sphere represents the state space of qubits. Pure quantum states lie on the surface of the 3D object (i.e. on the unit 2-sphere) while the mixed states lie inside it (i.e., the solid ball). The three axes denote the eigenstates (or ‘basis states’) of the Pauli operators which lie on the anti-nodal points. For example, the eigenstates of the σz … view at source ↗

**Figure 2.2.** Visualization of CV states. Wigner function plots for (a) Gaussian and (b) non-Gaussian states. • Wave function (marginal probability): The wave function squared |ψ(x)| 2 = | ⟨x|ψ⟩ |2 gives the marginal probability distribution of the state along the position (x) of the oscillator. A similar distribution can be obtained along the momentum (p) of the oscillator by taking the Fourier transform of the wave function, … view at source ↗

Figures from the paper (26 more)

**Figure 2.3.** Figure 2.3: Bloch-Messiah decomposition of Gaussian operations. Decomposition of (a) TMS r, π 2 , and (b) SUM(λ) gate using photon-number preserving beam-splitter gates and single-mode squeezing gates. The transformation from the initial mode quadrature operators on the left …

**Figure 2.4.** Figure 2.4: The structure of this thesis in light of a bottom-up architecture for hybrid CV-DV quantum processors. See text for details. ous layers of an architecture stack include the physical layer which constitutes the building block or the hardware layer. For each layer in t…

**Figure 3.1.** Figure 3.1: Framework of composite pulses in phase space and its applications. The blue curves show the Gaussian probability distribution | ⟨x|ψ⟩ |2 of the oscillator position, and the green arrows indicate the spin orientation of the ancilla qubit for the state |g⟩⊗|α∆⟩ (see Eq…

**Figure 3.2.** Figure 3.2: Performance of non-abelian composite pulse sequence GCR(θ) in quantum phase space. (a) Comparison against BB1(θ) for the case of θ = π/2 and ∆ = 1 so that χ = π 4|α| . The colored lines denote the various merits of correctness (failure probability: solid, infidelit…

**Figure 3.3.** Figure 3.3: Readout binning using Gaussian-controlled-BB1 pulse sequence, BB1(GCR). All plots follow the same legend. (a) Readout binning for the case of |α| = √ π/2. The plot gives the probability to measure a +1 outcome upon σy measurement, post the BB1(GCR) and BB1 in red an…

**Figure 4.1.** Figure 4.1: Deterministic preparation of squeezed states. (a) Deterministic squeezing protocol with incremental GCR. (b) Narration of GCR as a squeezing gadget S(∆, ∆′ ). The plots show how this sequence introduces a small amount of squeezing while unentangling the qubit from t…

**Figure 4.2.** Figure 4.2: Deterministic preparation of two-legged cat states. (a) Deterministic cat state preparation requires a unentangling sequence given by U. (b) Entanglingunentangling gadgets using GCR. (c) We show numerical results with options of no correction (U = Ry(θ ′x/ˆ |α|) in…

**Figure 4.3.** Figure 4.3: Deterministic GKP logical |+Z⟩ state preparation. (a) Circuit components for GKP preparation with ∆ = 0.34 (as used in recent experiments [3, 5]). S denotes the squeezing circuit from Fig.4.1, Ck = Uke −i √ 2πpσˆ z represents the gate sequence given by Eq. (4.25) in …

**Figure 4.4.** Figure 4.4 [PITH_FULL_IMAGE:figures/full_fig_p124_4_4.png]

**Figure 5.1.** Figure 5.1 [PITH_FULL_IMAGE:figures/full_fig_p136_5_1.png]

**Figure 5.2.** Figure 5.2: Logical GKP codeword |0⟩GKP and corresponding error word are shown in the first two plots. The maximally mixed state in the logical codespace ρGKP = |0⟩GKP⟨0|GKP+|1⟩GKP⟨1|GKP 2 and the error space aρˆ GKPaˆ † are shown in the next two plots. All axes are labeled in u…

**Figure 5.3.** Figure 5.3: Norms of GKP states and error words in relation to probabilistic distance. The squared ratio of the norm of codeword |0⟩ (N0) and the norm of error word aˆ |0⟩ (N1). The analytical curve shown in blue is obtained using Eqs. 5.64-5.67. The numerical curve is obtained …

**Figure 5.4.** Figure 5.4: Probabilistic error correction of photon loss on a square GKP state using SBS. (Left) We plot the orthogonality of the codeword and the error word (|+H⟩GKP , aˆ |+H⟩GKP) (purple), and the logically orthogonal codewords (|+H⟩GKP , |−H⟩GKP) (red). Here, 1 on the y-axi…

**Figure 5.5.** Figure 5.5: Logical GKP codeword |0⟩GKP for the hexagonal code and corresponding error word are shown in the first two plots. The maximally mixed state in the logical codespace ρGKP = |0⟩GKP⟨0|GKP+|1⟩GKP⟨1|GKP 2 and the error space aρˆ GKPaˆ † are shown in the next two plots. Al…

**Figure 5.6.** Figure 5.6: Voronoi cells of square and hexagonal GKP codes. (a) Regions marked for errors that are safe for Pauli P eigenstates for hexagonal codes (left) and square codes (right) using the SBS stabilization scheme for displacement errors. The dark blue region marks I, that is,…

**Figure 5.7.** Figure 5.7: Probabilistic correction of hexagonal GKP lattices and comparison with square GKP codes. (a) Plots corresponding to Figs. 5.4-5.3 for the hexagonal GKP encoding. We can see agreement in numerical and analytical schemes. The performance, in this case, matches the per…

**Figure 6.1.** Figure 6.1: Finite-energy GKP readout [117] and stabilization [126] protocols in nonabelian QSP framework. Interpretation of SBS circuit along the position quadrature as logical identity on the GKP codewords. The circuit is divided into the entangling and unentangling gadgets. …

**Figure 6.2.** Figure 6.2: Logical readout of GKP states with correctable errors. The GKP readout procedure maps the logical state onto the ancilla qubit states g and e, which are then measured. (a) Solid curves show finite-energy GKP codewords; dotted curves show the corresponding displaced …

**Figure 6.3.** Figure 6.3: Error-suppressed GKP gate teleportation. (a) Error-corrected gate teleportation of logical Z(θ) (X(θ)) gate by an entangling-un-entangling sequence obtained from stabilizer of the logical {0, 1} ({+, −}) basis. (b) Toy model of a pieceable circuit to mitigate effect…

**Figure 6.4.** Figure 6.4: Pieceable GKP entangling operations. Circuit for logical Pauli operations using two ancillae. Here, |Φ⟩ = EC[ZZGKP(π + θ)|ψ, ψ⟩GKP]. For θ = π/4, we can prepare a Bell state if |ψ, ψ⟩GKP = |++⟩GKP. The fidelity of this Bell state preparation is 0.9997 (as per Sec. 6.…

**Figure 7.1.** Figure 7.1: Illustration of the protocol for magic state preparation in an XZZX code. (a) Rectangular XZZX code with data qubits on the vertices of a rotated grid. The stabilizers are the product of two Pauli X and two Pauli Z operators on qubits arranged on the vertices around…

**Figure 7.2.** Figure 7.2: Logical error rate and success probability for magic state injection using cat codes. Logical error rate (ε raw L ) and success rate after dm rounds of error correction in stage II with noise model A (CX slower than CZ) so that pCX = 20p + 120p/η. The bias is η = 104…

**Figure 7.3.** Figure 7.3: (a,b) Logical error rate (ε raw L ) and success rate after dm rounds of error correction in stage II with noise model B (CX as fast as CZ) so that pCX = 2p+ 12p/η. The bias is η = 104 and the code size in stage I is dx,1 × dz,1 = 1 × 3. Stage II code sizes dx,2 × dz…

**Figure 7.4.** Figure 7.4: XL and ZL error rate in the magic state for η = 104 (a,b) and η = 103 (c,d) for noise model (A). The black dashed lines in (b,d) is found by fitting ZL error rate in the magic state prepared using our scheme, at low p and large distances, to Ap2 . In (b) we use the s…

**Figure 7.5.** Figure 7.5: (a) Qubit arrangement in stage I of the standard scheme used for comparison [PITH_FULL_IMAGE:figures/full_fig_p199_7_5.png]

**Figure 7.6.** Figure 7.6: Quantum phase estimation in hybrid oscillator-qubit architecture. (a) Phase estimation can be performed using the control sequence described in Sec. 7.2.2 as confirmed by this figure, for α = 1 in Eqs. (7.3-7.6). (b) The error bar of this operation relies on the sque…

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