Preparing approximate $N$-fold cat states with the phase space instruction set

Andrew Lucas; Shuyan Zhou

REVIEW 2 major objections 4 minor 24 references

Preparing approximate $N$-fold cat states with the phase space instruction set

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Preparing an N-fold rotationally symmetric Schrödinger cat state from a natural continuous-variable gate set requires circuit depth at least the Euler totient φ(N); for prime N, a depth-4N protocol matches the bound, settling the…

desk verdict Theorems 3.1 and 4.1 hold up and give a real depth separation for the phase space instruction set, but the compilation-inefficiency claim in Section 5 overreaches and should be cut or heavily qualified. read the letter →

arxiv 2608.07696 v1 pith:YL5FVHSY submitted 2026-08-07 quant-ph

Shuyan Zhou , Andrew Lucas This is my paper · ORCID

classification quant-ph MSC 81P6811R18 PACS 03.67.-a

keywords N-foldcatstatephasespaceinstructionsetcircuitdepthEulertotientfunctioncyclotomicpolynomialsbosonicpreparationqubit-dependentdisplacementDiophantinealignment

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

Preparing an N-fold Schrödinger cat state — a superposition of N coherent states arranged on a circle in phase space — is a basic task for bosonic quantum computing. This paper asks how many elementary instructions it takes when the only allowed gates are single-qubit rotations and displacements of the cavity field that depend on the qubit state, a natural phase space instruction set for circuit-QED hardware. The answer is a no-go theorem: for large coherent amplitude, any circuit that achieves small error must have depth at least φ(N), Euler's totient function, which grows like N/log log N. For prime N, the paper also gives an explicit depth-4N protocol that saturates this bound, with total runtime Θ(α), so the two bounds together pin down the exact asymptotic cost. The broader implication is that a universal gate set can be surprisingly inefficient at preparing even a simple family of states, and that converting bosonic circuits between different universal gate sets can carry a large overhead.

What carries the argument

Two mechanisms carry the argument. The lower bound rests on the cyclotomic identity dim_Q span_Q{$e^{{2πij/N}}$} = φ(N): the N-th roots of unity are the roots of the N-th cyclotomic polynomial, which is irreducible of degree φ(N) over Q, so at least φ(N) independent complex directions are needed just to place coherent-state centres at the N target points, and each phase-space displacement contributes only one such direction. The upper bound rests on the phase-alignment condition: a single real scaling λ must simultaneously bring every phase difference 2(θ* − θ_j) to within tolerance of the ideal angle θ_κ modulo 2π, and Proposition A.4 guarantees such a λ exists because the sine-difference frequencies are Q-linearly independent for almost every fixed angle φ, making the multiples of λ dense on the torus.

What would settle it

For the lower bound, fix N = 5 (so φ(5) = 4) and optimize the best possible depth-3 circuit from the phase space instruction set against the 5-fold cat state at α = 50; Theorem 3.1 predicts no such circuit can beat squared error (2N)^{-1} = 0.1, so finding one would refute it. For the upper bound, take N = 11 and a generic fixed angle φ, then search explicitly for the finite λ that satisfies the phase-alignment conditions (A.58) at the required tolerance; a systematic failure of this search across many generic φ would show the density-based existence guarantee is not realized in practice.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.1: if a circuit built from the phase space instruction set prepares the N-fold cat state |α_N⟩ with squared state-vector error below (2N)^{-1}, then for all sufficiently large |α| the circuit must contain at least φ(N) instructions. The reason is algebraic rather than kinematic: after M conditional displacements, all reachable coherent-state centres are signed sums Σ_j σ_j β_j of the M displacement parameters, so they lie in a Q-vector space of dimension at most M; but the N target centres α $e^{{2πij/N}}$ generate a Q-vector space of dimension φ(N), the degree of the cyclotomic field extension. For prime N, φ(N) = N − 1, and the companion Theorem 4.1 provides a matching depth-4N protocol whose four-gate V blocks each add one coherent component, with the required phase-alignment parameters supplied by a Diophantine density argument. Because successive large displacements merge into chords of the cat's polygon, the protocol's runtime is Θ(α), which Proposition 3.4 shows is optimal.

Load-bearing premise

The matching depth-4N protocol rests on the assumption that a single scalar parameter can be chosen to align all the required phase angles at once; the proof shows this parameter exists for almost every geometric configuration, but not for every one, and it does not say how to find it in practice.

Editorial extensions

If this is right

For any fixed N, the minimum circuit depth needed to prepare an N-fold cat state with the phase space instruction set grows at least as φ(N) ≳ N/log log N, so the cost is nearly linear in the number of cat legs rather than logarithmic.
When N is prime, the depth-4N protocol saturates the lower bound, so the asymptotic cost of preparing prime-fold cats is settled: Θ(N) depth and Θ(α) runtime.
The runtime Θ(α) is optimal — no protocol can do better, because separating the peaks of a cat with finite fidelity already forces a total displacement of order α (Proposition 3.4).
There is no efficient runtime-preserving compilation between the phase space instruction set and a quantum-signal-processing gate set: the same cat state that a QSP circuit prepares in O(N) time provably needs Θ(α) runtime and near-linear depth in the phase space instruction set (Theorem 5.1).
The construction extends to generalized cat states with arbitrary complex coefficients and phases at the same depth 4N (Appendix A.9).

Reading between the lines

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

The cyclotomic argument is a template for other target states whose peaks occupy points of high algebraic degree over Q: grids or lattices of coherent states would presumably lower-bound circuit depth by the dimension of the field generated by their coordinates, not merely by the number of peaks.
The matching protocol is an existence result: the phase-alignment parameter λ is guaranteed by torus density, but the proof supplies no algorithm to find it, so at experimentally relevant α the protocol likely needs to be wrapped in numerical optimization, as the paper's own pipeline does.
The small-α numerics suggest a practical, testable claim: for α ≳ 7 the number-theoretic protocol is a better seed for pulse optimization than random initialization, and benchmarking it against SNAP- or QSP-based preparation at the same α would show whether this structural cost translates into wall-clock control time.
The perimeter-merging trick that keeps runtime Θ(α) — adjacent conditional displacements combine into short chords of the cat's polygon — could be reused in other multi-step bosonic protocols where successive displacements are nearly collinear.

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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

Dear colleague,

The thing to know: the paper's core is sound and worth engaging. Theorem 3.1 gives a depth lower bound Ω(φ(N)) for preparing N-fold cat states with the phase space instruction set, and Theorem 4.1 gives a matching depth-4N protocol for prime N with runtime Θ(α). The abstract's further claim about cross-gate-set compilation inefficiency does not follow from the theorems and should be walked back.

What is new: the cyclotomic dimension argument (Proposition 3.2) used as a control obstruction is a nice idea, and the Diophantine phase-alignment construction in the appendix is a genuine existence proof for the matching protocol. The runtime merging trick—combining adjacent conditional displacements so the total displacement cost wraps the polygon perimeter rather than traversing radius N times—is clean and gives the Θ(α) runtime. The error analysis via telescoping sums is coherent. Credit where due: these are real results, not numerical speculation; the numerics in Section 4.2 are a demonstration and are not load-bearing.

Soft spots, in order of importance. Section 5's Theorem 5.1 is not proved. The argument compares the QSP state-preparation circuit's O(N) cost with the S-gate state-preparation lower bound; that says the two gate sets prepare the same family of states at different cost, not that converting an existing Urot circuit into S-gates is inefficient. The Trotter estimate (5.3)–(5.5) is a heuristic upper bound on one compilation strategy, and the authors admit a cleverer cross-compilation might exist. The abstract's "further imply" overreaches. This is a load-bearing gap only for the compilation narrative; Theorems 3.1 and 4.1 stand independently. Second, the Diophantine phase-alignment step is an existence result via Kronecker density; it gives no explicit λ and the thresholds α* are unquantified and pointwise in N (Remark 4.2 acknowledges this). That is acceptable for an asymptotic theorem, but it means the protocol is not yet a hands-on recipe. Third, no code or data accompanies the GRAPE claims, so Section 4.2 is not reproducible as written; minor. There is also a small conjugation slip in the definition of θ_j in Lemma A.1, which the reader flagged; it doesn't affect the bounds.

Verdict: the core lower bound and matching construction deserve serious refereeing. I would send it out. The authors should be asked to fix Theorem 5.1 (either prove a real compilation lower bound or reframe it as a conjecture/observation) and to state the nonconstructive nature of λ more prominently.

For reading group: yes, the cyclotomic obstruction is worth discussing. I'd cite Theorems 3.1 and 4.1 in future work.

Referee Report

2 major / 4 minor

Summary. The paper studies the circuit depth and runtime needed to prepare N-fold rotationally invariant Schrödinger cat states using the 'phase space instruction set' (unconditional and qubit-dependent displacements plus single-qubit rotations on a single boson). The main results are Theorem 3.1, a lower bound of Ω(φ(N)) on circuit depth for sufficiently large coherent-state amplitude α, and Theorem 4.1, a matching depth-4N construction for prime N whose runtime is Θ(α) for fixed N and target error. The construction relies on a Diophantine phase-alignment condition whose solvability is proved by a Kronecker-density argument. The paper also presents numerical demonstrations for N=3 and N=5 at moderate α and, in Section 5, claims that the results imply a strong inefficiency in compiling between the phase-space instruction set and a gate set containing Urot(θ).

Significance. If the two main theorems stand, the paper gives a rare asymptotic separation in continuous-variable state preparation: a simple family of states that is surprisingly expensive to prepare with a natural universal gate set, with the lower bound coming from cyclotomic polynomial structure rather than from quantum speed limits. The lower-bound proof is parameter-free and the upper-bound protocol is structurally explicit, with closed-form protocols supplied for N=3 and N=5 and numerical evidence at finite α. The order-of-limits caveat in Remark 4.2 is honestly stated. The compilation-inefficiency claim in Section 5, however, is not established with the same rigor and needs to be either proved properly or removed from the abstract.

major comments (2)

[Section 5, Theorem 5.1] The proof as written does not establish the theorem, and the abstract's final claim overreaches. The comparison between the O(N)-runtime QSP protocol of [17] and the Θ(N)-depth, Θ(α)-runtime bounds for the phase-space instruction set is a comparison of state-preparation costs in two different gate sets; by itself it does not bound the cost of compiling one circuit into the other. The Trotter estimates in Eqs. (5.3)-(5.5) only describe a particular first-order synthesis and cannot rule out a smarter compiler; the text even concedes 'there may be a more efficient cross-compilation'. The theorem can be repaired, but only by supplying the missing lower-bound argument: if a compiler mapped the QSP circuit of (5.2) to an S-circuit with runtime o(α), that compiled circuit would prepare the same N-fold cat state with fidelity above 2/3 and runtime o(α), contradicting Proposition 3.4. The proof should be rewritten around that contradiction, with a precise definition of 'efficient compilation' and the allowed error scaling, and the abstract should be conditional on that formalized statement.
[Section 4.1 and Appendix A.4] The text calls the proof of Theorem 4.1 'constructive', but the crucial parameter λ in the phase-alignment condition is shown to exist only through Kronecker density (Proposition A.4), which is nonconstructive and gives no algorithm to find λ for general prime N. This does not invalidate the existential content of Theorem 4.1, but it does mean the paper provides an existence proof plus explicit instances for N=3 and N=5, not a fully constructive recipe for every prime N. The wording in Section 4.1 should be softened accordingly, or an explicit Diophantine search procedure should be supplied.

minor comments (4)

[Section 4.2] There is a typo: 'asume' should be 'assume'.
[Theorem 4.1 and Eq. (2.9)] The theorem says a protocol prepares |αN⟩ 'with squared state-vector 1−ε, i.e. achieves (2.9)', but Eq. (2.9) is written in terms of the infidelity h, and the relation between h and the squared state-vector error ε is nonlinear. The notation should be aligned.
[Section 4.2, gate-level refinement] The text describes the gate-level optimization as 'cutoff-free' because it uses a finite coherent-branch representation; this is accurate for the branch representation, but the representation is still finite (512 branches), so the phrase 'cutoff-free' should be qualified to avoid implying an exact infinite-dimensional optimization.
[Appendix A.9] There is a stray sentence fragment 'Calculating it explicitly, we get Finally' before Eq. (A.127); the exposition there should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: The main lower and upper bounds are derived from independent number theory and explicit coherent-state estimates; Section 5's compilation claim is an unsupported inference rather than a circular reduction.

full rationale

The central lower bound (Theorem 3.1) is proved from the cyclotomic fact dim_Q span_Q Ω_N = φ(N), combined with coherent-state overlap estimates; it does not use any fitted parameter or any result imported from the authors' prior work. The upper bound (Theorem 4.1) is an explicit protocol whose phase-alignment step is established via Kronecker density and Q-linear independence (Propositions A.4, A.7, A.9); the quantities ε_ph and t_k are error budgets and existence choices from number theory, not parameters fitted to the target state. The numerical GRAPE experiments in Section 4.2 are demonstrations and are not inputs to any theorem statement. Section 5's claim of compilation inefficiency is indeed not established by the supplied proof: comparing state-preparation costs between gate sets does not by itself lower-bound the cost of cross-compiling a given circuit, and the authors themselves hedge that 'there may be a more efficient cross-compilation.' However, that is a logical gap in an auxiliary claim, not circularity: the conclusion is not identical to its premises by construction, and no fitted quantity is renamed as a prediction. The paper contains no load-bearing self-citation chain, and the derivation chain is self-contained against external benchmarks. Therefore the circularity score is 0; the Section 5 concern belongs to correctness risk, not to circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the V gate is a composite circuit primitive, not a new particle, force, or degree of freedom. The central claim depends only on standard CV quantum mechanics plus classical number-theoretic facts.

free parameters (3)

epsilon_ph (Diophantine phase tolerance) = 0.029 and 0.065 in the N=5 numerics; rho h / (2 pi (N-1) sqrt(N)) in the proof
Hand-chosen tolerance controlling phase-alignment error versus availability of lambda; affects numerical fidelity but not the asymptotic theorem.
rho (error-budget split) = arbitrary in (0,1), e.g. 1/2
Arbitrary division of total error h between phase-alignment and implementation error in A.7; no data fitting.
Diophantine scaling parameters t_k = existential, not explicit for general prime N; explicit for N=3 and N=5
The proof only guarantees existence via Kronecker density; the protocol depends on their finite values. These are construction parameters, not fitted to external data.

assumptions (5)

standard math Cyclotomic polynomial Phi_N is irreducible over Q and has degree phi(N); the N-th roots of unity span a phi(N)-dimensional Q-vector space.
Used in Proposition 3.2 and Corollary 3.3 to establish the lower bound.
standard math phi(N) > N / (2 log log N) for all sufficiently large N (Hardy-Wright bound).
Used after Theorem 3.1 to translate the depth bound into N / log log N scaling.
standard math Kronecker density: a one-parameter subgroup {lambda gamma} is dense in the torus when the coordinates gamma_j are Q-linearly independent.
Proposition A.4 relies on this to assert existence of lambda satisfying all phase-alignment inequalities.
domain assumption Coherent states, displacement algebra, and error bounds from Section 2.1, e.g. ||D(z)|0> - D(z)D(epsilon)|0>|| <= |epsilon|.
Standard CV quantum mechanics; underlies the trigonometric approximation and all fidelity estimates.
domain assumption Large-alpha separation: for |alpha| above thresholds, coherent-state peaks of an N-fold cat are nearly orthogonal and normalization is within 1% of N.
Used in Theorem 3.1 and Proposition 3.4; explicitly conditioned on sufficiently large |alpha|.

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Cite this review

Pith. "Pith review of Preparing approximate $N$-fold cat states with the phase space instruction set." pith.science (2026). https://pith.science/paper/YL5FVHSY

@misc{pith2026260807696,
  author       = {Pith},
  title        = {Pith review of: Preparing approximate $N$-fold cat states with the phase space instruction set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YL5FVHSY}},
  note         = {Machine review of arXiv:2608.07696}
}

abstract

The phase space instruction set is a continuous-variable universal gate set involving single-qubit rotations and qubit-dependent displacements on a single boson. Using these gates, we prove that a circuit depth $\mathrm{\Omega}(\varphi(N))$ is necessary to approximately prepare a large $N$-fold rotationally invariant Schr\"odinger cat state; here $\varphi(N) \gtrsim N/\log\log N$ is the Euler totient function. A protocol saturating this asymptotic bound on circuit depth is obtained for every prime number $N$. This protocol has an asymptotically optimal runtime, when the gates are generated by Hamiltonian evolution. Our results provide a sharp example where a universal gate set is surprisingly inefficient at preparing a simple family of states, and further imply that converting bosonic circuits between different universal gate sets can be extremely inefficient.

Figures

Figures reproduced from arXiv: 2608.07696 by the authors.

**Figure 1.** One step in the N = 5 cat-state protocol, illustrated in phase space. From left to right: (i) the conditional displacement D(−a2Z/2), which shifts the already-prepared peaks on the |0⟩ branch by −a2/2 and the lower-branch vacuum component by +a2/2; (ii) the Y-controlled displacement D(b2Y ), which generates the phase responses θ0, θ1, and θ∗; (iii) the qubit rotation e iθ2Y , which rotates the common ancilla directi… view at source ↗

**Figure 2.** Fidelity obtained directly from the protocol as a function of the cat radius [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Initial pulse obtained by translating the optimized coherent-branch gate sequence, shown [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Reduced oscillator Wigner functions for the preparation of the five-component cat state [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: Comparison between direct GRAPE optimization from 20 random pulses vs. the [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: View of the phase-alignment condition for one addition gate. The black arrow is the ideal [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [17]

Engineering Non-Gaussian Bosonic Gates through Quantum Signal Processing,

Pak-Tik Fong and Hoi-Kwan Lau, “Engineering Non-Gaussian Bosonic Gates through Quantum Signal Processing,” (2025), arXiv:2508.20261 [quant-ph]

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[2] [1]

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work page Pith review arXiv 2017

[3] [2]

Deterministically Encoding Quantum Information Using 100-Photon Schr¨ odinger Cat States,

Brian Vlastakis, Gerhard Kirchmair, Zaki Leghtas, Simon E. Nigg, Luigi Frunzio, S. M. Girvin, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically Encoding Quantum Information Using 100-Photon Schr¨ odinger Cat States,” Science342, 607–610 (2013)

work page 2013

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Cavity State Manipulation Using Photon-Number Selective Phase Gates,

R. W. Heeres, B. Vlastakis, E. Holland, S. Krastanov, V. V. Albert, L. Frunzio, L. Jiang, and R. J. Schoelkopf, “Cavity State Manipulation Using Photon-Number Selective Phase Gates,” Physical Review Letters115, 137002 (2015)

work page 2015

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Universal Control of an Oscillator with Dispersive Coupling to a Qubit,

S. Krastanov, V. V. Albert, C. Shen, C.-L. Zou, R. W. Heeres, B. Vlastakis, R. J. Schoelkopf, and L. Jiang, “Universal Control of an Oscillator with Dispersive Coupling to a Qubit,” Physical Review A 92, 040303 (2015)

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work page 1963

[7] [6]

Generating Quantum Mechanical Superpositions of Macroscopically Distin- guishable States via Amplitude Dispersion,

B. Yurke and D. Stoler, “Generating Quantum Mechanical Superpositions of Macroscopically Distin- guishable States via Amplitude Dispersion,” Physical Review Letters57, 13–16 (1986)

work page 1986

[8] [7]

Dynamically Protected Cat-Qubits: A New Paradigm for Universal Quantum Computation,

M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, “Dynamically Protected Cat-Qubits: A New Paradigm for Universal Quantum Computation,” New Journal of Physics16, 045014 (2014)

work page 2014