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Finite-energy gate-based continuous-variable quantum computers can be efficiently simulated on qudit or qubit devices, with error that vanishes as the encoding dimension grows.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Finite-energy gate-based continuous-variable quantum circuits can be approximated on qudit or qubit computers with polynomial overhead, eliminating any superpolynomial advantage for this model.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A real, mostly rigorous CV-to-DV simulation theorem for the Gaussian+cubic gate model with explicit energy-dependent error bounds; honestly narrower in scope than the title claims, plus one fixable slip in the Theorem-1 constant — worth a serious referee. the 3 major comments →

arxiv 2510.08546 v2 pith:ZDZ74CLO submitted 2025-10-09 quant-ph

Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy

Alex Maltesson , Ludvig Rodung , Niklas Budinger , Giulia Ferrini , Cameron Calcluth This is my paper
classification quant-ph MSC 81P68
keywords continuous-variable quantum computingdiscrete-variable quantum computingqudit simulationstabilizer subsystem decompositionGKP encodingcubic phase gateenergy-constrained quantum computationcomputational equivalence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 paper aims to settle a long-standing question: whether continuous-variable (CV) quantum computers, which process information in infinite-dimensional bosonic modes, are computationally more powerful than discrete-variable (DV) qubit or qudit computers. It argues that under realistic constraints they are not, provided the CV circuit is built from Gaussian operations and cubic phase gates, operates under a bounded total energy, and uses finitely resolved, finitely ranged measurements. The central result, Theorem 1, bounds the error of approximating such a CV circuit by a d-dimensional qudit computer as 1207 E*^2 K n^2 / sqrt(d), where E* is the energy bound, K the number of cubic-phase rounds, and n the number of modes. Since choosing d polynomially in E*, K, n, and 1/epsilon makes this error arbitrarily small, the paper concludes that gate-based CV quantum computing offers no superpolynomial advantage over DV quantum computing in this realistic setting. The proof is constructive, giving an explicit recipe for translating CV states, gates, and measurements onto qudit hardware.

Core claim

Theorem 1 states that the outcomes of a realistic CVQC circuit—energy bounded by E* throughout, measurements at finite constant resolution and finite range—can be approximated by a qudit quantum computer with error at most 1207 E*^2 K n^2 / sqrt(d), where K is the number of interlaced rounds of Gaussian and cubic-phase gates and n is the number of modes. Because choosing d = (1207 K n^2 E*^2 / epsilon)^2 makes the error at most epsilon, the paper concludes that any polynomial-size, finite-energy CV circuit built from this gate set can be simulated to arbitrary precision on qudits, and via binary encoding on qubits, with polynomial overhead. The simulation is constructive: the stabilizer subs

What carries the argument

The load-bearing tool is the stabilizer subsystem decomposition (SSD), a map that takes a CV state and extracts a d-dimensional qudit state by projecting phase space onto the unit cells of a GKP lattice. The paper chains three CV models—realistic CVQC (RCVQC), cut-off CVQC (CCVQC), and modular CVQC (MCVQC)—and shows that CCVQC approximates RCVQC via a gentle-measurement and Markov argument, that MCVQC is exactly equivalent to CCVQC, and that MCVQC's measurement statistics coincide with performing SSD followed by computational-basis measurement. Gate-by-gate error bounds (Lemmas 2 through 9) then allow the triangle inequality to sum the total approximation error.

Load-bearing premise

The result depends on the claim that Gaussian operations together with cubic phase gates form a universal set for continuous-variable quantum computation, for which the paper notes no formal proof is given; if that universality fails, the theorem covers only this specific gate family.

What would settle it

Compute, for a small system (n=1, K=1, moderate E*), the exact total-variation distance between the output of a cubic-phase CV circuit and its qudit approximation via SSD, and check it against the bound 1207 E*^2 / sqrt(d); a violation would disprove the theorem's constant. Alternatively, exhibit a finite-energy CV circuit outside the Gaussian-plus-cubic set (for instance one requiring a quartic phase gate) that cannot be approximated by SSD images of this gate set, which would show the universality premise fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If Theorem 1 is correct, a CV algorithm in the Gaussian-plus-cubic model with polynomial energy and gate count can be simulated on a DV device using only logarithmically many qubits per mode at fixed E*, with the total qubit count scaling polynomially in n, K, and 1/epsilon.
  • Existing classical simulation algorithms for DV quantum circuits become applicable to a broad class of CV circuits, extending beyond the all-Gaussian or highly symmetric cases that were previously tractable.
  • The result implies no exponential quantum speedup of gate-based CVQC over DVQC under the stated physical constraints, addressing the long-standing question of CV-versus-DV computational power.
  • The constructive mapping provides a practical pathway to port CV algorithms, including those using GKP-encoded states, onto qubit hardware with explicit resource bounds.
  • Because d = 2^k, the qubit error bound becomes 1207 E*^2 K n^2 / 2^{k/2}, so each additional qubit per mode reduces the error exponentially in k.
  • A concrete corollary is that space complexity for simulating fixed-energy CV circuits grows only logarithmically with the number of modes, while growing polynomially if the energy grows exponentially.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether the Gaussian-plus-cubic gate set is truly universal for CV quantum computation; if it later is proven universal, the theorem extends from a restricted family to all finite-energy polynomial-Hamiltonian CV circuits, but if not, the equivalence covers only this specific gate family.
  • The error constants in the theorem are likely far from tight; sharper per-gate bounds would reduce the qubit overhead, potentially making the translation practical for near-term bosonic hardware.
  • A natural testable extension is to replace the cubic phase gate with the Kerr gate, which the paper conjectures would obey similar energy-behaved bounds; verifying that would widen the equivalence to other experimentally relevant gate sets.
  • The framework suggests a route to classical simulation: instead of directly simulating Gaussian dynamics, one could run DV-style stabilizer simulations on the SSD image of the circuit, which may connect to resource-theoretic measures of non-Gaussianity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims that gate-based continuous-variable quantum computers, when subject to a finite energy bound, are efficiently simulable by discrete-variable (qudit/qubit) quantum computers. The proof proceeds through a chain of models: a realistic CV model (RCVQC) is shown to be approximated by a cut-off model (CCVQC), which is equivalent to a modular model (MCVQC), which is then approximated by a qudit model (DVQC) using the stabilizer subsystem decomposition. The main result, Theorem 1, gives an explicit error bound ϵ ≤ 1207 E*^2 K n^2 / sqrt(d), and Corollary 1 translates this to qubits. The paper also provides detailed energy-vs-parameter bounds and error analyses for the individual gates used in the construction.

Significance. If the proof chain is completed, the result is significant: it gives an explicit, energy-dependent simulation of a broad family of CV circuits on discrete-variable hardware, with polynomial overhead in the number of modes, the energy bound, and 1/ϵ. The paper's strengths include a fully explicit error budget, no fitted parameters, and a concrete construction of the DV gates (including the SSD mapping). The proof is modular and the constants are tracked, which is rare in this literature. However, the central claim is conditional on the unproven universality of the Gaussian-plus-cubic gate set, and one key lemma in the chain (Lemma 10) has a technical gap. These issues are fixable but are load-bearing for the paper's titular equivalence.

major comments (3)
  1. [Sec. I and Conclusion] The paper explicitly states that the Gaussian-plus-cubic gate set was 'presented as universal' in Ref. [6] but that 'a formal proof is not given' (Sec. I), and the Conclusion limits the analysis to this set, leaving arbitrary polynomial gates (e.g., Kerr) to future work. Since Theorem 1 and Corollary 1 are proved only for this restricted circuit family, the title and abstract overclaim an equivalence of 'gate-based CV quantum computers' in general. This is not an internal inconsistency, but it makes the main claim conditional on an unproven external assumption. The authors should either supply or cite a formal proof of universality for this gate set, or explicitly reframe the paper's claims as applying to the Gaussian-plus-cubic family.
  2. [Lemma 10 proof, Eqs. (50)-(56)] The application of the Gentle measurement Lemma in the proof of Lemma 10 does not appear valid as written. In Eq. (51), ρ_j is defined with projectors Λ_d already applied on both sides, so Λ_d ρ_j Λ_d = ρ_j. The Gentle measurement bound is then applied to a quantity ∥ρ_j − (1/N_j) Λ_d ρ_j Λ_d∥_1, which for j>0 is not the comparison needed: ρ_j is not the unprojected state before the j-th projector. A correct proof should compare, at each step, the unprojected state before the projection with the renormalized projected state, and then relate the resulting sequence to the RCVQC and CCVQC evolutions. As it stands, the proof does not establish Lemma 10, and Lemma 10 is an essential step in Theorem 1. This gap is likely fixable, but the argument must be rewritten.
  3. [Lemma 10 statement vs. proof] The statement of Lemma 10 gives the bound ϵ_RC ≤ 2L sqrt(E*/dπ), but the proof concludes with ϵ_RC ≤ 2(L+1) sqrt(E*/dπ). The factor L+1 is later used in Theorem 1. The statement and proof should be reconciled, and the bound in the theorem should use the correct factor.
minor comments (4)
  1. [Theorem 1 proof, Eq. (75)] The simplification of the two error contributions uses 22 sqrt(π/8) E*^2, but the preceding inequality 22Kn^2 sqrt(E*/dπ) ≤ 22 sqrt(π/8) E*^2 K n^2 / sqrt(d) is not valid for all E* ≥ 1/2 (e.g., E*=1/2, d=2). The final constant 1207 still appears to be sufficient, but the proof should either restrict the line to E* ≥ 1 or replace it with a direct worst-case bound, e.g., using 22 sqrt(E*/π d) ≤ 36 E*^2 / sqrt(d) for E* ≥ 1/2, d ≥ 2, so that 1171 + 36 = 1207.
  2. [Corollary 1] The exact implementation of arbitrary diagonal unitaries on k qubits requires O(2^k) gates, as noted. This is polynomial in E* and 1/ϵ after substituting k, but the practical overhead should be stated explicitly when summarizing the polynomial-time claim: the gate count is exponential in the number of qubits per mode, though that number is logarithmic in E*/ϵ.
  3. [Sec. III B] The sentence 'In Ref. [7], this set was proven to be equivalent to those specified by operations of the form Eq. (23)' appears to conflict with the earlier statement in Sec. I that no formal proof is given (citing Refs. [7,8]). Please clarify whether Ref. [7] or Ref. [53] is intended, and avoid the ambiguity.
  4. [Eq. (44)] The overflow operator K~_- is called a 'Kraus operator representing the failure to measure an outcome.' It is a POVM element, not a Kraus operator; the terminology should be corrected.

Circularity Check

0 steps flagged

No circularity: the error analysis is self-contained; the only imported premise is the externally disclosed universality of the Gaussian+cubic gate set.

full rationale

I walked the derivation chain RCVQC->CCVQC->MCVQC->DVQC. The only imported premise is the definition of gate-based CVQC as Gaussian operations interlaced with cubic phase gates; the paper explicitly states that the formal universality proof is not given and cites independent works (Refs. [6,7,8]) rather than its own prior results. The error budget is self-contained: Lemma 10 bounds the cutoff error using the gentle measurement lemma and Markov's inequality; Lemma 11 and Lemma 12 are exact identities between modular measurements and SSD computational-basis measurements; Lemma 13 sums per-gate SSD approximation errors derived in Appendix C from commutator bounds; Theorem 1 is a triangle inequality over these lemmas. The parameter d appears both as the number of RCVQC measurement bins and as the qudit dimension, but this is a natural encoding rather than a fitted input; the theorem is a genuine statement about how the approximation error scales with d. Self-citations (Refs. [26,32-34,46]) are background or application references and are not load-bearing for the main proof. The remaining universality caveat is an external, disclosed assumption, which is a scope/correctness concern rather than circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claim rests on no fitted numerical parameters. The model parameters E*, n, K, and d are inputs, not fitted values. The main load-bearing assumptions are the universality of the Gaussian+cubic gate set, the formal validity of the SSD/GKP construction, and the realism of the bounded-energy/finite-resolution measurement model. No new physical entities are introduced.

axioms (5)
  • domain assumption The set of Gaussian operations plus cubic phase gates is universal for gate-based CVQC (Eqs. 23 and 34).
    Invoked in Sec. I and Sec. IV.A; the paper states that a formal proof is not given (Refs. [7,8]) yet uses this set as the model for all CV algorithms.
  • domain assumption The stabilizer subsystem decomposition (SSD) and GKP projectors are well-defined as bounded operations on arbitrary finite-energy CV states.
    Used throughout Sec. IV and Appendix C; GKP states are non-normalizable Dirac combs, and the operator-norm bound in Eqs. (C20)-(C24) is informal.
  • domain assumption Energy boundedness and finite-resolution/finite-range measurements are realistic constraints for CV hardware.
    Definition 2 and the motivation in Sec. II; if energy can grow superpolynomially or if measurements are ideal, the claimed equivalence may fail.
  • domain assumption DVQC may use arbitrary DV operations; in the qubit corollary, arbitrary diagonal qudit gates can be implemented exactly on qubits.
    Definition 1 and Corollary 1; no finite gate-set compilation overhead is analyzed.
  • standard math Standard inequalities and lemmas (Gentle measurement Lemma, Schatten norm inequalities, Markov's inequality, triangle inequality) hold.
    Used in Lemma 10 and Appendix C; these are standard background results and are cited.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy." pith.science (2026). https://pith.science/paper/ZDZ74CLO

@misc{pith2026251008546,
  author       = {Pith},
  title        = {Pith review of: Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDZ74CLO}},
  note         = {Machine review of arXiv:2510.08546}
}
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read the original abstract

Continuous systems are studied in many branches of modern physics, such as high-energy physics, cosmology, condensed matter physics, quantum chemistry, and field theories. Such systems are expected to benefit from the substantial advantages in computational power of quantum computers. The continuous-variable paradigm of quantum computation provides the most natural computational formalism for these tasks. However, most existing quantum hardware is based on discrete-variable systems. We address this fundamental discrepancy by providing a rigorous framework for translating native continuous-variable algorithms onto qubit-based quantum processors. This mapping is constructed from a gate-based model of continuous-variable quantum computers, consisting of states and operations built from a polynomial sequence of elementary gates in a finite set, with total energy polynomial in the number of modes. We prove that, under realistic constraints, a gate-based model of continuous-variable quantum computers can be efficiently simulated using discrete-variable devices, thereby establishing a computational equivalence between these paradigms.

Figures

Figures reproduced from arXiv: 2510.08546 by Alex Maltesson, Cameron Calcluth, Giulia Ferrini, Ludvig Rodung, Niklas Budinger.

Figure 1
Figure 1. Figure 1: FIG. 1. This figure summarizes the succession of proofs we perform [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. In this Figure, we show the circuit diagram consisting of a sequence of operations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. In this figure, we show the decomposition of an arbitrary colored gate with index [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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    Quadrature non-Gaussianity and nonzero stellar rank are shown to be witnesses of particle entanglement rather than photon addition, with a generalized basis-dependent stellar rank proposed.

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    Proof of maximum energy increase of squeezing Lemma 14.The energyE ˆS(r) ˆρˆS†(r) of a stateˆρafter applying squeezing ˆS(r)with parameterris bounded in terms of the energy of the initial state ase −2|r|E ˆρ≤E ˆS(r) ˆρˆS†(r) ≤e 2|r|E ˆρ. Proof.First note that ⟨ˆn⟩ˆS(r) ˆρˆS†(r) = Tr h ˆS(r)ˆρˆS†(r)ˆn i = Tr ˆρ cosh(r)ˆa† + sinh(r)ˆa cosh(r)ˆa+ sinh(r)ˆa† ...

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    Proof.For this proof, we will use the decomposition from Ref

    Proof of maximum energy increase of shear gate Lemma 15.The energy of a stateˆρafter applying the shear operator with parametersis bound byE ˆP(s) ˆρˆP †(s) ≤(1 +s) 2E ˆρ. Proof.For this proof, we will use the decomposition from Ref. [58] ˆP(s) =e i s 2 ˆq2 = ˆR(θ) ˆS(r) ˆR(θ′)(A10) where r=arcsinh s 2 = ln " s 2 + r s2 4 + 1 # ,tan(θ) =e r, θ ′ =θ− π 2 ....

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    Proof.For this proof, we will again use the decomposition provided in Ref

    Proof of maximum energy increase of controlled-Zgate Lemma 16.The energy of a stateˆρafter applying the controlled-Zoperator with parametersis bound byE CZb kl(s) ˆρCZb † kl(s) ≤ (s+ 1) 4E ˆρ. Proof.For this proof, we will again use the decomposition provided in Ref. [58] CZb kl(s) =e isˆqk ˆql = ˆFkBSb kl(θ) ˆF † k ˆSk(r) ˆSl(r) ˆFkBSb kl(θ′) ˆF † k ,(A1...

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    Maximum value of squeezing With the maximum possible energy of a state beingE ∗, we can consider what happens to the energy of the lowest energy state (i.e., the vacuum state) when squeezed byr. We can bound the squeezing parameter as the maximum value of squeezing that can be applied before reaching a level of energy that is too high for the system to co...

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    ˆρ ˆq2 + ˆp+ 3γˆq2 2 −1 2 !# + 1 2 = Tr

    Proof of maximum cubicity of cubic phase gates Lemma 17.The maximum value of the cubic phase gate parameterγis bound in terms of the maximum energy of the system as γ≤8E ∗3/2. Proof. E ˆC(γ) ˆρˆC(γ) † =⟨ˆn⟩ˆC(γ) ˆρˆC(γ) † + 1 2 = Tr h ˆC(γ)ˆρˆC(γ) † ˆn i + 1 2 = Tr ˆρˆC(γ) † ˆq2 + ˆp2 −1 2 ˆC(γ) + 1 2 = Tr " ˆρ ˆq2 + ˆp+ 3γˆq2 2 −1 2 !# + 1 2 = Tr " ˆρ ˆq...

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    Proof.For the CV Fourier transform ˆF, the corresponding DV Fourier transform ˆFd is equivalent, such that the Fourier transform is preserved under the action of the SSD

    Proof of error of Fourier transform In this Section, we will prove Lemma 2. Proof.For the CV Fourier transform ˆF, the corresponding DV Fourier transform ˆFd is equivalent, such that the Fourier transform is preserved under the action of the SSD. Expanding the second term in Eq. (C2), gives ˆF † d TrS( ˆFˆρˆF †) ˆFd = 1 ℓ ˆF † d Z T2 ℓ dt ˆΠ ˆD(−t) ˆFˆρˆF...

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    ˆρ, mY k=1 ˆqik # 1 = ˆΛd

    Error of generalized phase gate To obtain the error for the simulation of displacement, the shear gate, the controlled-Zgate, and the cubic phase gate, we introduce what we call the generalized phase gate. In CV , this gate introduces a phase depending on a polynomial of the position operators. In DV , the phase depends on a polynomial of the computationa...

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    We begin by providing a proof of Lemma 3

    Proof of error of displacement Using the results of the previous Section, we now look at error bounds for specific gates. We begin by providing a proof of Lemma 3. Proof.The single-modep-displacement withf(ˆq) =sˆqwhich turns out to be exact as ∥[ˆρ, sˆq−s(ˆq+tq)]∥1 = 0.(C37) Since the CV displacement, ˆD(r), defined in Eq. 12, can be decomposed as ˆZ(q) ...

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    Proof of error of shear gate Next, we provide a proof of Lemma 4. Proof.The trace distance between a state with the single-mode phase gate applied before and after the SSD— where the single- mode phase gate is defined asf(ˆq) = s 2 ˆq2—is bounded by 1 2 ˆP f d TrS(ˆρ)ˆP f† d −Tr S( ˆP f ˆρˆP f† ) 1 ≤ 1 2ℓ Z T2 ℓ dt∥[ˆρ, stq ˆq]∥1 ≤ |s| p 2E ˆρ ℓ Z T2 ℓ dt...

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    Proof of error of cubic phase gate We now provide a proof of Lemma 5. Proof.The error of the single-mode cubic phase gate given byf(ˆq) =γˆq 3 can be bounded by 1 2 ˆP f d TrS(ˆρ)ˆP f† d −Tr S( ˆP f ˆρˆP f† ) 1 ≤ 3 ℓ Z T2 ℓ dt [ˆρ, γt2 q ˆq] 1 + [ˆρ, γtq ˆq2] 1 ≤ 3|γ| p 2E ˆρ ℓ Z T2 ℓ dt|t2 q|+dℓ|t q| ≤3|γ| p 2E ˆρ Z Tℓ dtq|t2 q|+dℓ|t q| =3|γ| p 2E ˆρ ℓ3 ...

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    Proof of error of controlled-Zgate Here we provide a proof of Lemma 6. Proof.A simple two-mode interaction is provided by the controlled-Zgate withf(ˆq k,ˆql) =sˆqk ˆql bounded by 1 2 ˆP f d TrS(ˆρ)ˆP f† d −Tr S( ˆP f ˆρˆP f† ) 1 ≤ 1 2ℓ2 Z T4 ℓ dt [ˆρ, stqj ˆqi] 1 +∥[ˆρ, stqk ˆql]∥1 = 1 2ℓ2 |s| Z T4 ℓ dttqk ∥[ˆρ,ˆqk]∥1 +t ql ∥[ˆρ,ˆql]∥1 = 1 2ℓ2 |s|2 √ 2E ...

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    25 Proof.We consider a rotation of an arbitrary angleθ

    Proof of error of rotation Here we provide a proof of Lemma 7. 25 Proof.We consider a rotation of an arbitrary angleθ. Note that without loss of generality, we can restrict to the case that θ∈[− π 4 , π 4 )because we can generate rotations that are multiples ofπ/2using the Fourier transform. Neglecting the global phase in the definition of rotation from E...

  73. [73]

    Proof.We consider a beamsplitter of arbitrary angleθ

    Proof of error of Mach-Zehnder interferometer Here we provide the proof of Lemma 8. Proof.We consider a beamsplitter of arbitrary angleθ. Note that without loss of generality, we can restrict to the case that θ∈[− π 4 , π 4 )because we can cover all other angles using swap gates, which contribute no error in the trace distance. We will use the following d...

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    With the trace distance for a controlled-Zgate given by Lemma 6 2|s| √ E(π/d) 3/2,(C50) and given that, as specified by Lemma 16, the maximum energy of the new state is upper bounded by E ˆρ′ ≤(1 +s) 4 E ˆρ,(C51) we can follow the same steps as for the proof of rotation. We see that |s1|+ (1 +s 1)2|s2|+ (1 +s 1)2(1 +s 2)2|s1|<5(C52) and we find 1 2 TrS(BS...

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    Proof of error of squeezing Here we provide a proof of Lemma 9. Proof.We will use the following decomposition of squeezing into Fourier transforms and shear gates [60] ˆS(r) = ˆF ˆP(s 3) ˆF ˆP(s 2) ˆF ˆP(s 1)(C55) wherebys 1 =s 3 =e r ands 2 =e −r. Note that this is the same decomposition as the one given in Eq. (C42) for rotation, except with different p...

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.