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Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

Ilan Tzitrin, J. Eli Bourassa, Nicolas C. Menicucci, Krishna Kumar Sabapathy

arXiv:1910.03673v2quant-phphysics.optics

TL;DR

Ideal GKP grid states encode qubits in oscillators, but their infinite energy motivates practical finite-energy approximations. The paper analyzes how these approximations affect encoded information and circuits, and studies photonic preparation; it reports fidelity-resource trade-offs and differing robustness to preparation uncertainties.

  • Problem

    The paper examines how realistic finite-energy approximations to ideal grid states affect encoded qubit quality and computational circuits.

    Source: §I · p. 2

  • Method

    The authors use modular subsystem decomposition to track logical errors and numerically optimize photonic GKP-state preparation using Gaussian states and photon-number-resolving measurements.

    Source: §I · p. 2, A. Preparation of non-Gaussian states · p. 16

  • Results

    The preparation study finds fidelity-probability trade-offs across circuit sizes and greater output-state robustness to squeezing uncertainty than to beamsplitter-angle uncertainty.

    Source: §IV · p. 24

  • Takeaways & Limitations

    The analysis provides figures of merit for assessing physical and logical information and benchmarks for using approximate GKP states in computational circuits.

    Source: §I · p. 2, §IV · p. 24

Abstract

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Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.

I. INTRODUCTION

GKP grid states encode qubits in continuous-variable systems and use translation symmetry to correct small displacement errors. The paper examines how finite-energy approximations affect encoded information and computational circuits, and reviews ideal square-lattice states.

I. INTRODUCTION

  • GKP codes encode logical qubits in continuous-variable systems and use translation symmetry to correct displacement errors in phase space.The paper reviews their role in fault-tolerant quantum computation and other applications.Source: §I · p. 2
  • The paper studies photonic preparation using squeezed displaced vacuum states, interferometers, and photon-number-resolving detectors, with postselection making the scheme probabilistic.It situates this approach among proposals and demonstrations for preparing GKP qubits across multiple platforms.Source: §I · p. 2
  • The paper analyzes how physical approximations to ideal grid states affect encoded qubits and computational circuits, using modular subsystem decomposition to track and mitigate logical errors.It also introduces figures of merit for physical and logical information and benchmarks for computational circuits.Source: §I · p. 2

A. Ideal GKP states

  • The paper focuses on square-lattice GKP states, whose ideal logical states are infinite combs of infinitely squeezed spikes spaced by 2√π in position.The logical 1 state is a position displacement of the logical 0 state by √π.Source: A. Ideal GKP states · p. 3
  • Ideal square-lattice GKP states correct position and momentum displacements up to √π/2 without confusing logical 0 and logical 1.Their phase-space Wigner functions are lattice delta functions, with lattice shape and spacing determined by the stabilizers.Source: A. Ideal GKP states · p. 3

B. Normalizable GKP states

Finite-energy GKP states can be formed by replacing ideal spikes with finite-width peaks and applying an overall envelope. The paper examines alternative normalization prescriptions and their effects on state properties and operations.

B. Normalizable GKP states

  • The canonical finite-energy approximation replaces delta spikes with Gaussians of width ∆ and applies an overall Gaussian envelope.In the common regime ∆ = κ, the states approach ideal GKP states as ∆ tends to zero.Source: B. Normalizable GKP states · pp. 3–4
  • Applying the non-unitary envelope E(ϵ) = e^-ϵn to an ideal state gives a compact normalizable-state prescription that approximates the Gaussian-peak form when ∆ = κ and both parameters are small.This operation can be viewed as interfering the ideal state and vacuum at a beamsplitter, then measuring and postselecting vacuum in one mode.Source: B. Normalizable GKP states · p. 4, Figure 2 · p. 4
  • Wigner logarithmic negativity decreases as ϵ increases, while the state becomes more Gaussian as its peaks become fewer and wider.Figure 3 shows the trend for the normalizable |0ϵ⟩ state.Source: Figure 3 · p. 4, B. Normalizable GKP states · p. 4
  • The paper also considers other normalization routes, including noise channels and weighted Fock-state superpositions, and examines which envelope functions best suit the preparation scheme.The operator used for the plotted normalizable states corresponds to g(n) = e^-ϵn.Source: B. Normalizable GKP states · pp. 4–5
  • The envelope operator and displacement generally do not commute, so displacing a normalizable logical 0 state need not produce the same state as enveloping an ideal logical 1 state.The paper adopts an explicit prescription for normalizable states to avoid this ambiguity.Source: B. Normalizable GKP states · p. 5

C. Modular subsystem decomposition

The modular subsystem decomposition separates a GKP-encoded logical qubit from an infinite-dimensional gauge space, clarifying how finite-energy envelopes affect logical information. The section then tracks how imperfect states and Gaussian operations alter the logical and physical state.

C. Modular subsystem decomposition

  • The decomposition splits the physical space into a two-dimensional logical space and an infinite-dimensional gauge space; ideal GKP states allow perfect logical recovery by tracing out the gauge mode.For normalizable states, the envelope generally entangles the logical and gauge modes, leaving a mixed logical qubit.Source: C. Modular subsystem decomposition · p. 5
  • As ϵ grows, logical-state trajectories converge on the vacuum point, while Bloch-sphere states near the equator are more distorted.The figure marks ϵ = 0.0625 ≈ ∆ = 12 dB and ϵ = 0.25 ≈ ∆ = 6 dB as reasonable values.Source: C. Modular subsystem decomposition · p. 5, Figure 4 · p. 6
  • The authors track envelope effects by transforming the Stokes vectors of ideal GKP qubits and renormalizing each vector to obtain the resulting Bloch vectors.The Stokes-vector components correspond to the identity and Pauli matrices; renormalization divides by the first coefficient.Source: C. Modular subsystem decomposition · p. 5

D. Operations on normalizable GKP states and error-tracking

  • For ideal GKP states, Gaussian operations implement logical Clifford gates, but a given logical gate can correspond to multiple physical operations.The physical operations may be non-unique because code stabilizers act trivially on the code space.Source: Table I · p. 6, D. Operations on normalizable GKP… · p. 7
  • On normalizable states, applying a Gaussian operation is equivalent to applying it to an ideal state and then applying a conjugated, modified envelope.The envelope is conjugated by the Gaussian unitary.Source: D. Operations on normalizable GKP… · p. 7
  • Imperfect GKP states and Gaussian operations damage both, with greater damage at larger ϵ and for operations involving higher powers of quadrature operators.Larger ϵ corresponds to fewer and broader peaks in the GKP state.Source: D. Operations on normalizable GKP… · p. 8

1. Prerecovery figures of merit

The section compares imperfect GKP states with ideal states using physical and logical fidelity, output-distribution distance, and energy cost. These measures show why physical-state overlap alone can miss preserved logical information.

1. Prerecovery figures of merit

  • Repeated logical X operations can reduce physical fidelity even when the ideal circuit is a logical identity.The finite-energy state lacks the ideal state's translational symmetry, and its physical fidelity can vanish under a rigid translation.Source: §1 · p. 8
  • Logical fidelity remains close to unity when the physical operation displaces only the gauge mode, illustrating the modular decomposition’s utility.Logical fidelity compares the reduced logical states rather than the full physical states.Source: §1 · pp. 8–9
  • Binned homodyne readout makes the distribution distance invariant to repeated X(2k√π) displacements in the toy circuit.The relevant comparison is between output probability distributions, not the pre-measurement states.Source: §1 · p. 9
  • Energy cost tracks physical resource requirements and increases with repeated displacements, while physical fidelity decreases.The energy measures include the mean and standard deviation of occupation number.Source: §1 · p. 9
  • The averaged GKP gate fidelity measures average logical fidelity against ideal implementations; increasing envelope strength worsens fidelity for identity and phase gates.The envelope itself creates a deviation from unit fidelity, and the deviation is greater for the phase gate.Source: §1 · pp. 9–10

2. Displacements in position

Position displacements implementing logical X can degrade physical fidelity while increasing energy cost. The section proposes circuit mappings and recalibration strategies to limit displacement while retaining logical and physical fidelity.

2. Displacements in position

  • Each successive X(√π) displacement increases mean energy cost, with the energy spread increasing more slowly than the mean.The plotted example uses |0ϵ⟩ with ϵ = 0.063 (Δ ≈ 12 dB).Source: §2 · p. 10
  • Higher energy can make normalizable GKP states more susceptible to loss, motivating approaches to manage displacement.Large displacements from the origin require more energy.Source: §2 · p. 10
  • For Gaussian circuits, the mean displacement can be classically computed and used to reoptimize a circuit to minimize its maximal displacement.The authors leave the details of this proposed compilation approach to future research.Source: §2 · p. 10
  • Recalibrating with X(−k) when the mean reaches a threshold is another proposed way to manage displacement.The authors propose combining this step with error correction.Source: §2 · p. 11
  • New physical mappings for logical X maintain logical and physical fidelities while minimizing energy cost.The authors report evidence that minimizing energy cost generally also improves other figures of merit.Source: §2 · p. 11

3. Displacements in momentum

Momentum displacements implementing logical Z have the same impact on normalizable GKP states as position displacements implementing logical X. The section recommends monitoring momentum changes and applying the mapping deterministically.

3. Displacements in momentum

  • Momentum displacements Z(√π) have the same physical-fidelity impact on normalizable GKP states as position displacements X(√π).The authors attribute this equivalence to the Fourier relation between the gates and the envelope’s commutation with the Fourier gate.Source: §3 · p. 11
  • Monitoring average momentum changes and applying the proposed mapping deterministically are recommended for logical Z.The alternative prescription using two quadratic phase gates is described as a worse solution.Source: §3 · p. 11

4. Shearing operations

Imperfect GKP states respond differently to physical gates: phase-gate shearing can damage physical and logical properties in distinct ways, while rotations and beamsplitters preserve the finite-energy envelope. SUM gates can entangle the logical subsystem, but the amount of entanglement depends on squeezing.

4. Shearing operations

  • Phase-gate shearing increases oscillations near wavefunction peaks and raises the energy spread faster than the mean energy.The plotted energy cost is for successive P applications to |0ϵ⟩ at ϵ = 0.063 (about 12 dB).Source: Figure 12 · p. 12
  • Higher-squeezing states better preserve logical information under the phase gate, even as their physical fidelity falls faster than that of lower-squeezing states.Lower-squeezing states have a wider central peak near q ≈ 0, where the phase gate adds little phase; the gate also warps the logical Bloch sphere, especially near the equator.Source: Figure 11 · p. 11, Figure 10 · p. 11, §4 · p. 12
  • Recompiling repeated logical phase gates can reduce costly shearing by replacing even powers with identity or logical Z, and three successive gates with P⁻¹.The authors leave the best optimization strategy for any particular circuit to future research.Source: §4 · p. 12

5. Rotations

  • Repeated perfect rotations, including the Fourier gate at φ = π/2, do not damage normalizable GKP states because they commute with the envelope operator.The Fourier gate implements the logical Hadamard operation.Source: §5 · p. 12

6. Squeezing and beamsplitters

  • Squeezers can damage normalizable GKP states comparably to the phase gate, whereas beamsplitters commute with the envelope and are not harmful.The SUM(g) decomposition includes squeezers; their parameter r varies monotonically with g, and positive g corresponds to negative r.Source: §6 · p. 12, §7 · p. 13

7. SUM gates

  • For modes initialized in |+ϵ⟩|0ϵ⟩ and acted on by the CNOT [SUM(1)] gate, approximately 4.5 dB of squeezing is required to produce entanglement in the logical subsystem.Source: §7 · p. 13

E. Error correction with normalizable GKP states

Normalizable GKP states can correct small displacement errors using Steane or Knill recovery, but finite-energy effects constrain correction and can produce logical errors. Their nonuniform peak probabilities can also provide information about the data state's displacement.

E. Error correction with normalizable GKP states

  • Steane recovery uses separable ancillae for syndrome measurements, whereas Knill recovery teleports the data state and corrects logical-Pauli byproducts; the Knill output is already on the GKP grid.With high-quality ancillae, Knill recovery can avoid applying two SUM gates to noisy data; this work focuses on Steane recovery.Source: Figure 14 · p. 13, E. Error correction with normalizable… · pp. 13–14, Figure 15 · p. 14
  • Finite-energy ancilla envelopes restrict the correctable error range; in the stated three-error setting, all shifts must be smaller than √π/6 for successful correction.A total position or momentum shift greater than √π/2 produces a logical bit-flip or phase-flip error, respectively.Source: E. Error correction with normalizable… · p. 14
  • The Glancy-Knill condition is necessary and sufficient for correcting displacement errors with the standard Steane and Knill schemes: successful correction requires a high probability of no logical error.A generic state with high no-error probability must have little modular spread in both position and momentum, approaching the ideal GKP form.Source: E. Error correction with normalizable… · p. 14
  • In normalizable states, unequal peak probabilities can sometimes identify a data peak and support Bayesian estimation of the data state's mean for recentering.For example, an outcome with n_a + n_d = 9 implies n_d = 3 with unity probability, but that outcome is vanishingly rare and does not necessarily locate the state center at 2√π.Source: E. Error correction with normalizable… · p. 15
  • GKP error correction cannot be expected to correct all errors; concatenating GKP qubits into codes such as surface codes can protect against more general errors.The authors suggest that the modular decomposition could help develop such codes.Source: E. Error correction with normalizable… · p. 15

III. PHOTONIC STATE PREPARATION AND CHARACTERIZATION

The paper develops a photonic preparation approach using Gaussian inputs, interferometers, and photon-number-resolving postselection to produce non-Gaussian GKP approximations. It represents target states as Gaussian operations on truncated Fock-state cores and optimizes those cores and their preparation circuits.

A. Preparation of non-Gaussian states

  • Photon-number-resolving measurements on modes of an interfered Gaussian input prepare a non-Gaussian state in the unmeasured modes, with specific detection outcomes selected by postselection.The framework is analogous to Gaussian BosonSampling and encompasses photon subtraction and addition; this work provides a thorough analysis of GKP preparation with these circuits.Source: A. Preparation of non-Gaussian states · p. 16
  • For a desired Fock superposition, the detected photon total can restrict candidate postselection patterns, while the proposed coefficient bound gives an upper bound on circuit modes to search.For N modes, the conjectured maximum number of independent coefficients after measuring N − 1 modes is (N + 2)(N − 1)/2.Source: A. Preparation of non-Gaussian states · p. 16

B. Core states for GKP

  • Gaussian operations on the core can be absorbed into and redecomposed within the preparation circuit, avoiding comparatively harder inline squeezing.When physical resources constrain nmax, targeting a Gaussian-transformed core can also attain higher fidelity to the desired state than directly truncating its Fock expansion.Source: B. Core states for GKP · pp. 16–17
  • Approximate GKP states are constructed by applying Gaussian operations to truncated Fock-state cores; for the real, symmetric targets considered, optimization needs only squeezing and even Fock coefficients.The method maximizes fidelity to the chosen normalizable target; the authors found that step-function cutoffs had more support in regions where the wavefunctions should vanish than the target states at equal cutoffs.Source: B. Core states for GKP · p. 17
  • Approximation fidelity improves with larger core cutoff nmax at fixed target squeezing, but worsens as target squeezing increases at fixed nmax.The comparison covers even nmax values from 2 to 12 photons and target squeezing from 3 to 11 dB for |0⟩.Source: B. Core states for GKP · p. 17

C. Characterization of approximate states

The paper evaluates approximate GKP states using energy, orthogonality, error-correction, and logical-encoding measures, then describes circuit optimization for their preparation. These measures can differ: high fidelity to a target does not guarantee comparable error-correction performance or logical encoding.

1. Average energy

  • Average photon number rises with target squeezing and core cutoff but stays below five photons; different approximate states across the GKP Bloch sphere have comparable resource requirements.For circuit GKP states, average energy translates into the initial squeezing required on each mode; similar resources also create a trade-off between preparing more state types and using more gate elements.Source: Figure 20 · p. 18, §1 · p. 18

2. Orthogonality

  • As target squeezing increases, approximate logical states become less orthogonal because truncated cores introduce oscillations between peaks, increasing overlap.Those oscillations occur in regions where the target wavefunctions should be close to zero.Source: §2 · p. 19, Figure 21 · p. 19

3. Glancy-Knill property

  • Despite fidelity above 93%, the nmax = 12 approximation reduces P_no err from the target's 74% to 58%.Direct optimization of P_no err raises it from 57% to 61% for nmax = 12 and r ≈ 1.87 dB, but the resulting state is less well confined to logical bins than the fidelity-optimized state.Source: Figure 22 · p. 19, §3 · p. 19, Figure 23 · p. 20

4. Logical subsystem Bloch sphere

  • For nmax = 2, |0A⟩ and |+A⟩ can lie near their target Bloch-sphere positions despite low fidelity, but their nearly Gaussian form is not expected to support universal computation.Mapping all four considered states well requires higher nmax.Source: §4 · p. 20, Figure 24 · p. 21

1. Algorithm for optimal circuits

  • The circuit search trains Gaussian input parameters, interferometer settings, and photon-number postselection to produce a truncated core, then redecomposes the circuit to move squeezing into the source.The procedure uses global and local optimization with Strawberry Fields and Walrus simulators.Source: §1 · p. 20

2. Circuit optimization results

The circuit study finds only marginal fidelity gains beyond three modes, while testing parameter stability and optical loss. Its conclusions also place the photonic preparation results alongside a separate analysis of state-preparation errors, assuming ideal gates and measurements.

2. Circuit optimization results

  • Beyond three modes, fidelity gains are marginal while success probability vanishes, despite added tunable parameters.Table II reports best fidelities and corresponding success probabilities for circuits targeting |0Δ⟩ with Δ = 10 dB.Source: §2 · p. 21, Table II · p. 22
  • As nmax increases, the circuit output’s Wigner peaks better approach |0Δ⟩, with the smallest difference near the phase-space origin.Differences remain apparent farther from the origin.Source: §2 · p. 22, Figure 25 · p. 23

1. Stability analysis of optical elements

  • The optimal circuit tolerates larger relative squeezing-parameter instability than beamsplitter-parameter instability.The stability analysis varies relative errors around the optimal squeezing and beamsplitter parameters.Source: §1 · p. 22, Figure 26 · p. 23

2. Effects of photon loss

  • With loss, Euler decomposition no longer applies exactly, and considering fewer detected photons expands the postselection search, which the authors leave for future study.The paper describes its proposed lossy-circuit strategy as suboptimal.Source: §2 · p. 23
  • At η = 0.02 dB, reoptimization changes fidelity from 82% to 81% after redecomposing the circuit and reinserting loss.The reoptimization cost also includes Wigner log negativity to preserve non-Gaussian properties.Source: §2 · pp. 22–23

IV. CONCLUSIONS

  • The computational analysis isolates state-preparation errors, assumes error-free gates and measurements, and uses modular subsystem decomposition to track the encoded logical qubit.The authors discuss physical and logical figures of merit, compilation, error mitigation, and Steane-type error correction with gauge-mode recalibration.Source: §IV · pp. 23–24

2. Effects of photon loss

  • As nmax increases, loss becomes more detrimental to fidelity; success probability stays relatively stable, while Wigner log negativity decreases with loss.The analysis applies a single loss parameter to optical components, outcoupling, and PNR detectors.Source: §2 · p. 22, Figure 27 · p. 24

IV. CONCLUSIONS

  • The photonic analysis estimates resources and quality for approximate grid states, including robustness to squeezing uncertainty and degradation under loss.The authors state that their analytical and numerical tools inform practical error-correction protocols, while fault-tolerant computation is a longer-term prospect.Source: §IV · p. 24

Appendix A: Notation, nomenclature, convention, and units

The appendix fixes notation for logical and physical operations and normalizable GKP states, then gives operational constructions for the envelope, logical readout, and its conjugation with Gaussian operations.

Appendix A: Notation, nomenclature, convention, and units

  • A bar marks a logical gate, while an unbarred gate acts physically; for |ψΔ,κ⟩, κ^-1 is the envelope width, and low Δ and κ correspond to large positive dB values.The appendix also specifies conventions for squeezing parameters and peak-width conversions.Source: Appendix A · p. 25

1. Dilation for the envelope operator E(ϵ)

  • The envelope operator e^-ϵn can be implemented by mixing an ideal GKP state with vacuum on a beamsplitter, post-selecting vacuum in the second mode, and associating e^-ϵ with transmissivity t.Source: §1 · pp. 25–26

2. Readout from the logical subsystem

  • Binned homodyne measurement of q provides computational-basis readout of the logical subsystem, with square-lattice bins grouped into binary outcomes.The paper identifies the POVMs with displacements of GKP states and describes the resulting operation as a logical-subsystem measurement.Source: §2 · p. 26

3. Conjugation and commutation with E(ϵ)

  • The envelope conjugates across exponentials of polynomials in creation and annihilation operators by changing U to E(-ϵ)UE(ϵ), but its unbounded-operator form requires mathematical care.Table V lists physical operators and their conjugated envelope operators.Source: §3 · pp. 26–27, Table V · p. 26

4. Measures of non-Gaussianity

The section describes Wigner negativity and its logarithmic measure as indicators of non-Gaussianity, then presents a probability construction for assessing correctable displacement regions with normalizable GKP states.

4. Measures of non-Gaussianity

  • Wigner negativity is the area of the negative part of the Wigner function, and its logarithmic form is used as a non-Gaussianity measure.Source: §4 · p. 27
  • A pure state with zero Wigner negativity must be Gaussian, but a mixed state can be non-Gaussian while having a positive Wigner function.Source: §4 · p. 27

5. Glancy-Knill probability of no error

  • The Glancy-Knill construction assesses probability within correctable displacement regions of width √π/6, separating position and momentum regions without overcounting.The paper describes this as the probability of successful error correction with normalizable GKP states.Source: §5 · p. 27

Appendix C: Numerical techniques

The numerical appendix describes discretized wavefunctions and optimization procedures for approximate GKP states and photonic GBS circuits. It also notes that optimizer parameters can jump between solutions even when fidelity changes smoothly.

1. Algorithm 1 details

  • The target GKP wavefunction is built on a discretized position range spanning seven standard deviations, with 100 grid points per peak and Gaussian peaks weighted by the logical state.The wavefunction is normalized by numerical integration.Source: §1 · p. 28
  • Algorithm 1 optimizes fidelity between the target and a squeezed finite Fock-state superposition using local minimization and stochastic basinhopping.The core coefficients are normalized and restricted to even components up to nmax; the local solution is a candidate, not a guaranteed global optimum.Source: §1 · p. 28
  • Optimal squeezing and truncated-core coefficients can jump between parameter regions as Δ changes, even while fidelity and other global state properties vary smoothly.The authors report that basinhopping helps search across these regions.Source: §1 · p. 28

2. Algorithm 2 details

  • Algorithm 2 uses core-state coefficients, circuit modes, and a PNR postselection pattern to optimize a GBS device, returning circuit parameters, fidelity, and success probability.Its redecomposition moves the approximate-state squeezing to the beginning of the circuit rather than applying it inline after the interferometer.Source: §2 · pp. 28–29
  • The global search adds postselection probability with weight 0.1 to fidelity, then local search raises that weight to 1 to favor higher-probability solutions nearby.The global search uses basinhopping; the local search explores the neighborhood of its candidate solution.Source: §2 · p. 29
  • For each postselection pattern, the procedure evaluates the Gaussian circuit state, computes the pattern probability from the reduced measured-mode state, and uses the output-state overlap for fidelity.Source: §2 · p. 29

Appendix D: Approximate GKP Z, X, and H eigenstates

Approximate GKP-state fidelity generally improves with larger core-state cutoffs, while the corresponding energy rises; the numerical results indicate comparable preparation resources across the examined Bloch-sphere states. The analysis also relates Fock-state structure to fidelity trends and examines how approximate states affect projectors and displacement-error correction.

1. Further comments on |0A⟩

  • For |0A⟩ up to about 8 dB, nmax = 4 and 6 give nearly identical results because n = 6 is unnecessary and increasing n = 8 with nmax = 6 also increases n = 6.The n = 2 and 4 states are most probable after vacuum below about 8 dB; beyond that, n = 8 becomes more probable, and its contribution creates the outer peaks while n = 2 and 4 narrow the central peak.Source: §1 · pp. 29–30, Figure 28 · p. 30, Figure 29 · p. 30
  • Above 8 dB, the nmax = 6 solution jumps between similar-fidelity parameter-space basins, producing only slight changes in fidelity and average photon number.The reported fidelity is already about 90% in this range, where all core-state coefficients are nonzero.Source: §1 · p. 30

1. Projectors

  • For the |0A⟩ and displaced-|0A⟩ projectors targeting 10 dB states with nmax = 12, displacement breaks the target projector's symmetry, but this does not affect logical encoding.Preparing both approximate logical states preserves the target projector symmetry; the approximate states have about 95% fidelity, with peaks blurrier farther from the origin.Source: §1 · p. 31, Figure 31 · p. 32

2. Results for |1A⟩, |+A⟩, and |H+A⟩

  • Across the examined states, increasing nmax improves fidelity but raises energy, while the |0A⟩, |+A⟩, and |H+A⟩ preparations require comparable resources.The |+A⟩ state is effectively as resource-intensive as |0A⟩, and |H+A⟩ has squeezing and average photon numbers of the same order.Source: §2 · pp. 30–31, Figure 30 · p. 32

2. Quantum error correction matrix

  • In the QEC-matrix comparison, approximate-state error blobs grow as nmax decreases, and blobs farther from the origin smear enough that only the first multiples of √π/2 remain correctable.The figure compares normalizable 10 dB states with approximate states using nmax = 12, 8, and 4; for normalizable states, small displacements are correctable, while displacements by √π along q or p correspond to bit or phase flips.Source: §2 · p. 31, Figure 32 · p. 33
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