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Finding Gaussian Structure in Bosonic States
Alvan Arulandu, Sitan Chen, Ziyun Chen, Jerry Li, Eric Ma
TL;DR
The paper asks how to find the best pure Gaussian approximation to an arbitrary bosonic state. It combines Gaussian-measurement warm starts with refinement using non-Gaussian measurements and optimization, establishing protocols in both fidelity regimes and a tolerant Gaussianity tester for any threshold.
Problem
Agnostic tomography asks for a pure Gaussian state whose infidelity with an arbitrary bosonic state is at most ε above the best achievable, addressing states that may be non-Gaussian but close to Gaussian.
Source: §1 · p. 4Method
The protocols use Gaussian measurements with robust-statistics techniques to obtain a warm start, then refine it using non-Gaussian measurements and optimization.
Source: §1.2 · p. 7Results
The protocols cover high-fidelity and arbitrary low-fidelity inputs, and yield a tolerant tester distinguishing opt above c + ε from below c − ε for any c in (0,1).
Source: §1.1 · p. 5Takeaways & Limitations
The maximizing Gaussian basis can sanitize near-Gaussian states and organize the remaining non-Gaussian structure of more general states at low photon number.
Source: §1 · p. 5Unless NP ⊆ BQP, runtime poly(n,1/ε) is impossible for agnostic tomography; the low-fidelity guarantee scales exponentially in target precision.
Source: §1.1 · p. 6
Abstract
from arXiv · showhide
We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $\rho$, the goal is to output a pure Gaussian state whose infidelity with $\rho$ is at most $\mathrm{opt} + \epsilon$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/\epsilon$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/\epsilon)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + \epsilon$ or $\mathrm{opt} < c - \epsilon$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/\epsilon)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.
1 Introduction
The introduction motivates finding the Gaussian structure of arbitrary bosonic states and defines agnostic tomography as finding a pure Gaussian approximation within ε of the best achievable infidelity. It presents protocols for high- and low-fidelity settings, their measurement and optimization approach, and their place relative to prior work.
1 Introduction
- The maximizing Gaussian basis can sanitize near-Gaussian states and act as a skeleton in which the remaining non-Gaussian structure lies at low photon number.The introduction also connects maximum Gaussian fidelity to measuring non-Gaussianity and the utility of mean-field approximations.Source: §1 · pp. 4–5
1.1 Results
- The work gives agnostic tomography protocols for high- and low-fidelity regimes and a tolerant Gaussianity tester that distinguishes opt above c + ε from opt below c − ε for any c in (0,1).The high-fidelity guarantee applies when opt is below a universal constant; the low-fidelity protocol handles arbitrary states.Source: §1.1 · p. 5
- Unless NP ⊆ BQP, no agnostic-tomography algorithm can achieve runtime poly(n,1/ε); Gaussian measurements alone also cannot attain opt + ε in some settings without exponentially many copies.The stated lower bound for Gaussian-only protocols is exp(Ω(n^1/3)) copies in some settings.Source: §1.1 · p. 6
1.2 Our Techniques
- The shared approach uses Gaussian measurements and robust statistics for a warm start, then non-Gaussian measurements and optimization to refine the estimate.The high-fidelity refinement uses local convexity and gradient descent; the low-fidelity case uses global tensor optimization after projecting to a low-photon-number subspace.Source: §1.2 · pp. 6, 8
- For low fidelity, list-decodable Gaussian learning yields candidate bases; iterative unsqueezing and pruning leave a short list with a basis where the optimum has energy poly(1/α), independent of n and E.In this basis, the problem is projected to a bounded-photon-number subspace and reduced to tensor optimization.Source: §1.2 · pp. 7–8
1.3 Related work
- Earlier bosonic Gaussian-learning results treated exactly Gaussian inputs [BMEM25] [CMF+26], while this work studies agnostic inputs and estimates maximum fidelity over all Gaussian states.The paper describes its work as the first agnostic tomography study in the continuous-variable setting.Source: §1.3 · p. 9
2 Preliminaries
This section establishes matrix and bosonic-state notation, characterizes Gaussian states and measurements, and records robust-statistics tools used in the paper.
2 Preliminaries
- The Frechet derivative notation specifies directional derivatives and higher-order multilinear derivatives, including the Hessian quadratic form.Source: §2 · p. 10
2.1 Matrix decompositions
- A positive definite matrix admits a Williamson decomposition with a symplectic factor and positive diagonal symplectic eigenvalues.Source: §2.1 · p. 10
- Every complex symmetric matrix has a Takagi factorization whose diagonal entries are its singular values.Source: §2.1 · p. 10
2.2 Bosonic quantum systems
- Gaussian states are determined by their mean and covariance, and purity requires covariance of the form SS^T with S symplectic.Their covariance eigenvalues occur in reciprocal pairs.Source: §2.2 · p. 12
- Gaussian unitaries transform Gaussian states by affine changes to their means and covariance, while Gaussian measurements yield classical Gaussian samples on Gaussian inputs.Heterodyne measurement is the case with seed covariance I.Source: §2.2 · p. 12
- The Bargmann form uniquely parameterizes pure Gaussian states by a vector b and symmetric matrix A, with norm of A below one.The paper also records displacement and photon-number formulas in these coordinates.Source: §2.2 · pp. 13–14
- Near-vacuum parameter distance is related to trace distance, and these properties are used in the high-fidelity algorithm’s iterative refinement.Source: §2.2 · p. 15
2.3 Primitives from robust statistics
- Robust Gaussian estimation learns mean and covariance from distributions close in total variation, while list-decodable estimation returns candidates when the best Gaussian approximation may be poor.Source: §2.3 · p. 16
3 Agnostic learning in the high-fidelity regime
This section develops a high-fidelity approach that uses robust Gaussian tomography for a warm start, then non-Gaussian refinement to reach infidelity scaling opt + epsilon. It builds on near-vacuum geometry for gradient descent and establishes the costs and conditions for direct gradient measurements, which improve copy complexity over noisy finite-difference gradients.
3.1 Robust heterodyne tomography
- Robust heterodyne tomography replaces empirical mean and covariance with robust estimates to resist outliers, but its semi-agnostic error scales with the closest Gaussian state's energy.A two-sided covariance error can violate the symplectic condition, motivating a one-sided covariance estimator; the energy dependence is at least linear in n.Source: §3.1 · pp. 17–18
3.2 Robust iterative unsqueezing
- O(n^3 opt_tr^-2 log log ||V^-1||op log^5(1/delta)) copies suffice for robust iterative unsqueezing with Gaussian measurements, removing the inverse-covariance factor but retaining a linear mode factor.Each unsqueezing round uses O(n) copies for a constant-relative-error covariance estimate, so the initial energy dependence does not undermine robustness.Source: §3.2 · p. 19
3.3 Robust pure state projection
- ˜O(n^2 epsilon^-2 log log ||V^-1||op log^5(1/delta)) copies suffice for robust pure-state projection using exclusively Gaussian measurements, yielding a semi-agnostic guarantee without mode or energy factors.The resulting infidelity is bounded by opt_tr + O(zeta), which the passage expresses as ˜O(opt_tr); logarithmic factors in opt_tr remain.Source: §3.3 · pp. 19–20, §3.4 · p. 20
3.4 Robust iterative refinement via gradient descent
- Non-Gaussian iterative refinement achieves infidelity scaling opt + epsilon when the closest pure Gaussian state has infidelity below a universal threshold.The method starts from robust pure-state tomography and refines the estimate by gradient descent on the infidelity objective, using measurements and finite differences to estimate its gradient.Source: §3.4 · pp. 20–21
3.4.1 Near-vacuum geometry of the infidelity
- Near the vacuum, sufficiently small trace distance ensures controlled Hessian behavior and a smooth, strongly convex infidelity objective with a unique minimizer in a small parameter ball.The minimizer lies within one quarter of the ball's radius.Source: §3.4.1 · pp. 21–22
- Within B_R, the infidelity Hessian is within operator norm 1 of 2I, and the minimizer is unique and has ∥θ*∥_P ≤ R/4.The geometry argument also ensures every point in B_R describes a valid pure Gaussian state when R < 1/16.Source: §3.4.1 · pp. 24–25
3.4.2 Noisy objective estimate
- For θ ∈ B_R, objective estimation uses O(κ^-2 log(2/δ)) copies for additive error κ with probability at least 1−δ; gradient estimation uses O(n^6 κ^-4 log(n/δ)) copies for additive error κ in ∥·∥P with probability at least 1−δ.Gradient estimation assumes d_tr(ρ, |0⟩⟨0|) ≤ γ_0, θ ∈ B_(R/2), and κ ∈ (0, R/4]. Both estimates use a single Gaussian unitary followed by an on-off photon detector.Source: §3.4.2 · p. 25
3.4.3 Gradient descent
- Near-vacuum gradient descent uses ˜O(n^6 ε^-2 log(1/δ)) copies to output a pure Gaussian state with probability at least 1 − δ.The guarantee applies when d_tr(ρ, |0⟩⟨0|) ≤ γ_0; the proof shows a competitive minimizer lies within the descent region.Source: §3.4.3 · pp. 25–26
3.4.4 Direct gradient estimation
- Direct non-Gaussian gradient estimation reduces the near-vacuum algorithm’s copy complexity to ˜O(n^2 ε^-1 log(1/δ)).The protocol estimates the gradient at the vacuum, then conjugates the current iterate to vacuum, estimates the gradient, takes a step, and undoes the accumulated Gaussian unitaries.Source: §3.4.4 · pp. 27–28
4 Agnostic learning in the low-fidelity regime
This section develops low-fidelity agnostic learning by constructing candidate Gaussian bases and optimizing over Gaussian approximators using tensor, cutoff-block, and convex-net procedures. It combines these methods with preprocessing and validation to establish learning guarantees and runtime bounds.
4.1 List-decodable iterative unsqueezing
- Theorem 4.1 bounds the list size by ceil(8/(3α)) and gives a failure probability of at most η under its stated best-approximator conditions.With probability at least 1−η, the list contains a basis satisfying the theorem’s guarantee.Source: §4.1 · p. 29
4.1.2 Unsqueezing and the fit
- Along a branch following accurate candidates, each unsqueezing round roughly takes the approximator’s squeezing to its square root, and a final fit controls its photon number.The fit uses total estimation-error bounds across modes to place the estimated Gaussian state at the vacuum in the fitted basis.Source: §4.1 · p. 29, §4.1.2 · pp. 30–31
4.1.1 List estimates
- Heterodyne list decoding returns positive-definite-covariance candidates, with probability at least 9/10 one satisfying all three stated approximation bounds.Repeated runs on fresh copies reduce the probability that the combined list misses a good pair to at most 10^-b.Source: §4.1.1 · pp. 29–30
4.1.3 Pruning the list
- Pruning keeps bases with sufficient cutoff mass and removes near-duplicates; the basis distance is computable using polynomially many matrix operations.Small distance bounds photon number in the other basis, while large distance makes the low-photon subspaces nearly orthogonal, limiting how many separated bases survive.Source: §4.1.3 · pp. 31, 32, 33, 35
- If the best approximator has mean photon number at most h and fidelity at least α, a cutoff k ≥ 64h/α gives cutoff mass at least 3α/4.This ensures that useful bases survive the cutoff-mass test.Source: §4.1.3 · p. 35
- Pruning retains a candidate whose cutoff mass tr(ρP) is at least 3α/4.Source: §4.1.3 · p. 36
4.1.4 The algorithm and its analysis
- Each unsqueezing block keeps at most 8/(3α) bases while preserving a basis useful at the next squeezing level, provided its estimation lists and cutoff-mass estimates are good.Across R blocks, all unions and mass estimates succeed with probability at least 1 − η.Source: §4.1.4 · pp. 37, 39
4.2 Optimization at bounded photon number
- Theorem 4.9 constructs an optimizer competing with pure Gaussian states of photon number at most h, with output mean photon number at most 17(h + 1)^3.The input state σ is arbitrary.Source: §4.2 · p. 40
4.2.1 Setup
- The convex parameter domain D_h contains every pure Gaussian state with total photon number at most h, and its log-normalization is smooth and convex.Source: §4.2.1 · p. 40
4.2.2 Learning the low degree components
- For every Gaussian approximator in D_h, the cutoff K scales as (h + 1)^3 + (h + 1) log(1/ε), and individual-copy measurements estimate the input cutoff block within ε/16 in operator norm.The input state may have arbitrary photon number; the cutoff applies to the candidate Gaussian approximators.Source: §4.2.2 · pp. 42–43
4.2.3 Global optimization of the normalized overlap
- A deterministic normalized-overlap optimizer uses at most C_h,K,δ(n + 1)^cK real-arithmetic operations, reducing the numerator search to a bounded-dimensional subspace and optimizing the denominator convexly.The subspace dimension is independent of n; the algorithm searches a net over its coordinates.Source: §4.2.3 · pp. 44–45
- Algorithm 3 approximately maximizes Gaussian overlap over D_h by optimizing convex J_z objectives for points in a net of projected coordinates.Each objective has a bounded-norm subgradient oracle and computable projection; the resulting arithmetic cost is C_{h,K,δ}(n + 1)^{cK} operations for universal c.Source: §4.2.3 · pp. 46–48
4.3 Main result
- With probability at least 1 − δ, the low-fidelity learner outputs a pure Gaussian state whose infidelity is at most opt + ε, given an upper bound E0 on the optimum state's photon number.The guarantee applies to an arbitrary n-mode bosonic state and ε ∈ (0, 1].Source: §4.3 · pp. 48–49
- The algorithm preprocesses the state, optimizes candidate Gaussian states in transformed bases, and validates them on fresh copies, returning the candidate with the largest estimated fidelity.Vacuum testing after undoing each candidate's Gaussian unitary estimates its fidelity; Hoeffding's inequality gives O(ε^-2 log(6W/δ)) validation copies per candidate.Source: §4.3 · p. 49
- The learner's arithmetic cost is Cε(n + 1)^kε times a universal polynomial in 1 + log(1/δ) and 1 + log log(e + E0), with copy bounds absorbing exp(poly(1/ε)) into (n + 1)^poly(1/ε).The net search contributes arithmetic operations; the complexity bounds do not depend on opt.Source: §4.3 · p. 49
5 Agnostic learning with Gaussian measurements
The section shows that Gaussian measurements can improve a warm start to O(opt) + ε, but non-Gaussian measurements are necessary for opt + ε guarantees with efficient copy complexity. It develops robust-statistics tools for this refinement and proves an exponential copy lower bound for Gaussian-only protocols in some settings.
5.1 Refining to O(opt) + ε with Gaussian measurements
- Gaussian measurements achieve O(opt) + ε infidelity using Õ(n^2 ε^-1 log log ||V^-1||op log^5(1/δ)) copies, but opt + ε is not efficiently achievable with Gaussian measurements in some settings.The Gaussian-only guarantee assumes the state is within the stated high-fidelity threshold of a pure Gaussian state.Source: §5 · p. 50, §5.1 · p. 50, §5.2 · p. 54
- Hellinger closeness to a target Gaussian yields a decomposition into a bounded-covariance distribution with a nearby mean and a contaminating remainder.This conversion supports robust estimation even when the retained distribution is not itself Gaussian.Source: §5.1 · p. 51
- For bounded-covariance distributions under Hellinger misspecification, a polynomial-time mean estimator uses Õ(ε^-2 d log(1/δ)) samples.The result is assembled from the robust estimation lemma and proposition.Source: §5.1 · p. 52
- For a Gaussian target, robust estimation also learns its mean and covariance using Õ(ε^-2 d^2 log(1/δ)) samples.The section uses this covariance estimate alongside the mean estimate to refine the Gaussian-state warm start.Source: §5.1 · pp. 52–53
5.2 Hardness for Gaussian measurements
- Some high-fidelity instances require at least η^-1 exp(Ω(n^1/3)) copies for Gaussian-only protocols to achieve opt + cη^2 infidelity.The lower bound allows arbitrary adaptivity and classical processing in the Gaussian-measurement protocol.Source: §5.2 · pp. 53, 54, 56
6 Hardness in the low-fidelity regime
This section reduces maximum clique to agnostic learning of pure Gaussian states, establishing that a poly(n, 1/ε)-time learner would imply NP ⊆ BQP. The reduction uses a one-photon projector to constrain Gaussian states to a family encoding the graph optimization.
6 Hardness in the low-fidelity regime
- Unless NP ⊆ BQP, no quantum algorithm runs in poly(n, 1/ε) time and achieves opt + ε infidelity for arbitrary n-mode states.The theorem gives the learner success probability at least 0.99.Source: §6 · p. 57
- The one-photon projector favors a structured family of pure Gaussian states, while a graph-dependent perturbation selects the member associated with the graph's maximum-clique objective.The projector's near-maximizers are close to states of the form |ψ*(x)⟩; within that family, the graph projector encodes the clique quantity.Source: §6 · pp. 58–59
- A hypothetical agnostic learner would estimate the encoded graph optimum accurately enough to recover the integer maximum-clique size in polynomial time.The reduction specifies λ = O(n^-2) and ε = O(n^-5) to obtain the resulting efficient algorithm.Source: §6 · p. 60
AI Usage Disclosure
- The authors disclose that GPT tools contributed to the framework and technical ideas, while the authors worked out details with the tools and claim responsibility for errors and omissions.The manuscript preparation also used Claude Opus 5.5 and Fable 5.1.Source: AI Usage Disclosure · p. 60
A.1.1 Proof of Lemma 2.5
The appendix derives Gaussian-state parameter and photon-number formulas, then establishes spectral and normalization properties of a matrix used in the representation.
A.1.1 Proof of Lemma 2.5
- The appendix obtains the relation b = β − Āβ and solves it for β using the positive-definite inverse I − ĀA.Triangle inequalities applied to the relation also establish the stated bound on β.Source: §A.1.1 · p. 68
- Centering the annihilation operators at the state mean and applying a Takagi factorization provides the steps for evaluating the total-number-operator expectation.The unitary change of coordinates preserves the relevant vector norm in the derivation.Source: §A.1.1 · pp. 68–69
A.1.2 Proof of Lemma 2.6
- The eigenvalues of M(A) are 1 − s_j and 1 + s_j, which are positive because s_j ≤ ||A||op < 1.Source: §A.1.2 · p. 69
- The normalization formula is derived by rewriting the complex integral over real coordinates, translating its center, and applying the Gaussian integral.Taking logarithms yields the stated expression.Source: §A.1.2 · pp. 69–70
A.2.1 Proof of Theorem 3.2
The proof estimates Gaussian-state parameters from heterodyne samples and converts parameter accuracy into a trace-distance guarantee.
A.2.1 Proof of Theorem 3.2
- Heterodyne outcomes are within opt_tr in total variation of a classical Gaussian with mean μ_h and covariance Σ_h, so robust Gaussian learning estimates both parameters.The covariance estimate is converted to a state covariance by V̂ = 2Σ̂_h − I.Source: §A.2.1 · p. 70
- The estimated covariance defines a valid Gaussian state because the relative covariance bound ensures V̂ + iΩ ⪰ 0.Source: §A.2.1 · p. 70
- The resulting trace-distance error is bounded by Õ(opt_tr) times the closest state's energy.The proof bounds it by (n + tr V^-1) · Õ(opt_tr), then uses the energy bound on tr V^-1.Source: §A.2.1 · p. 70
A.2.2 Proof of Theorem 3.4
The proof verifies that the robust estimators meet the conditions needed for the unsqueezing property underlying Theorem 3.3. Unitary invariance preserves the trace-distance bound across iterations, and each robust heterodyne tomography call determines the copy cost.
A.2.2 Proof of Theorem 3.4
- The unsqueezing property used in Theorem 3.3 is substantiated by Lemmas 12 and 13 of [BMEM25].The proof says the required estimator conditions follow from (87) and Proposition 2.9, with trace distance preserved across iterations by unitary invariance.Source: §A.2.2 · p. 71
- Each robust heterodyne tomography call uses O(n^2 opttr^-2 log^5(1/delta)) copies.The stated copy complexity is for each usage of the tomography procedure.Source: §A.2.2 · p. 71
A.2.3 Proof of Lemma 3.7
The proof converts absolute heterodyne-outcome error guarantees into relative parameter estimates, then uses those estimates to establish closeness of the corresponding Gaussian states. The covariance estimate is controlled after removing its diagonal factor, and phase-space closeness yields trace-distance closeness under a small-error condition.
A.2.3 Proof of Lemma 3.7
- For a pure Gaussian covariance, reciprocal-pair eigenvalues and the bound on ∥V^-1∥op imply I/2 ⪯ V ⪯ 2I and 0 ⪯ Σh ⪯ 3V/2.The proof then obtains Vraw ⪰ (1/2 - 2zeta)I and ∥Vraw - V∥F ≤ 2zeta for zeta < 1/4.Source: §A.2.3 · p. 71
- Removing the diagonal factor from Vraw = Shat Dhat Shat^T changes the covariance estimate by O(zeta) in Frobenius norm.The proof introduces A(W) = -ΩWΩW and compares its values on Vraw and V to bound this perturbation.Source: §A.2.3 · pp. 71–72
- The resulting relative covariance bound, combined with the relative mean bound, gives total-variation distance O(zeta) between the associated Gaussian phase-space distributions.The covariance comparison uses ∥V -1/2(Vhat - V)V -1/2∥F ≲ zeta.Source: §A.2.3 · p. 72
- For the two pure states, their trace distance is at most 150 times the total-variation distance between N(m,V/2) and N(ˆm,ˆV/2), provided that distance is less than 1/600.Source: §A.2.3 · p. 72
A.2.4 Proof of Lemma 3.11
These appendix proofs establish derivative and estimator bounds used in the paper's arguments, along with covariance and Gaussian-transformation properties. The four lemmas cover normalized-state derivatives, a zeroth-order estimator, projection-based precision estimation, and a transformation of a centered quadratic Hamiltonian.
A.2.4 Proof of Lemma 3.11
- For Lemma 3.11, the derivative bound holds with CD = 176 or better.The proof works with the unnormalized vector e^(Ktheta)|0⟩ and bounds products by applying Lemma 3.10 to their operators.Source: §A.2.4 · pp. 72–74
A.2.5 Proof of Lemma 3.13
- For Lemma 3.13, the proof estimates fρ at theta and coordinate shifts in the parameter space to construct a zeroth-order estimator.It uses step size h = kappa/(3d), target accuracy kappa0 = kappa^2/(12d), and failure probability delta/(d+1).Source: §A.2.5 · pp. 74–75
A.3.1 Proof of Lemma 4.3
- For Lemma 4.3, projection optimality and the triangle inequality bound the Frobenius error between K0 and K by ≲ A t^3.The proof identifies the true precision K = Σ^-1 as belonging to K_A.Source: §A.3.1 · p. 75
A.3.2 Proof of Lemma 4.4
- For Lemma 4.4, a displacement removes u without changing the spectrum, and a passive mode change diagonalizes C into reciprocal pairs (e^(2rj), e^(-2rj)).The Williamson decomposition of M then supplies a Gaussian transformation of the centered quadratic Hamiltonian.Source: §A.3.2 · p. 76
A.3.3 Proof of Lemma 4.10
The proof characterizes the domain D_h through three constraints and establishes its convexity. Its operator-norm constraint also ensures the condition required for the Bargmann state.
A.3.3 Proof of Lemma 4.10
- The domain is an intersection of three norm balls in the real vector space P_n, so it is convex.Its operator-norm constraint ensures ∥A∥op < 1, as required for the Bargmann state.Source: §A.3.3 · p. 76
- If ν(πb,A) ≤ h, nonnegativity of each term in (9) implies the stated bounds, which are precisely the three constraints defining D_h.The proof concludes the characterization using these constraints.Source: §A.3.3 · p. 76
A.3.4 Proof of Lemma 4.13
The proof estimates the cutoff-state matrix entry by entry, then projects the estimate onto the set of subnormalized states. Hoeffding bounds give the stated copy complexity and accuracy guarantee.
A.3.4 Proof of Lemma 4.13
- The entry estimates use diagonal projectors and observables for the real and imaginary parts of off-diagonal entries, with expectations equal to the corresponding matrix entries.Each observable has outcomes in [-1, 1].Source: §A.3.4 · p. 77
- Hoeffding's inequality and a union bound ensure all empirical means have error at most delta with probability at least 1 - eta.The proof sets delta = epsilon/(64D).Source: §A.3.4 · p. 77
- O(D^4 epsilon^-2 log(2D/eta)) copies suffice to estimate the cutoff-state matrix entries.Here D is the cutoff-space dimension.Source: §A.3.4 · p. 77
- Projecting the Hermitian estimate onto the closed convex set of subnormalized states yields an estimate at most epsilon/16 from sigma_K.The projection can be computed by diagonalizing the estimate and projecting its eigenvalues onto the nonnegative vectors with sum at most one.Source: §A.3.4 · p. 77
A.4.1 Proof of Lemma 5.7
The proof invokes a moment-matching construction for unit-mean random variables, then specializes its support parameters for the polynomial bound.
A.4.1 Proof of Lemma 5.7
- Lemma 7 of and Section 2.11.1 of provide two unit-mean random variables matching the first k moments.Their support is contained in {0} union [1 + nu, lambda].Source: §A.4.1 · p. 78
- The moment-matching variables are supported on zero and an interval whose endpoints are determined by nu and lambda.With the proof's choice nu = 0, the interval begins at 1.Source: §A.4.1 · p. 78
- The proof sets nu = 0 and lambda = Ck^2, with P_L denoting real polynomials of degree at most L.Source: §A.4.1 · p. 78
A.5.1 Proof of Lemma 6.3
The proof establishes stability of the one-mode Gaussian-state maximizer and extends the bound to arbitrary modes by a passive-unitary reduction to one mode.
A.5.1 Proof of Lemma 6.3
- The one-mode parameter pair (beta*, alpha*) uniquely attains equality in the bound, with Re alpha* = -|alpha*| = -1/2 and beta*^2 = 3/2.Source: §A.5.1 · p. 78
- Near the maximizer, positive Hessian curvature gives parameter stability, which the proof converts into an infidelity bound proportional to p*1 - p1.The infidelity bound applies when the parameter distance is less than xi.Source: §A.5.1 · pp. 78–79
- Outside the local ball, compactness and uniqueness of the maximizer complete the one-mode stability argument.The proof restricts this argument to parameters where p1 is at least p*1/2.Source: §A.5.1 · p. 79
- For arbitrary modes, a passive unitary aligns b with the first mode, and projection onto that mode transfers the single-mode stability bound to x = b/||b||.Source: §A.5.1 · pp. 79–80
B Tensor representation of the truncated numerator
The appendix represents the truncated numerator by tensors and reduces its effective dimension; it also gives subgradient and projection oracles for the optimization procedure.
B Tensor representation of the truncated numerator
- The truncated numerator is represented by a tensor whose Frobenius norm is bounded independently of n.The proof notes that bounding the numerator values alone would not give this coefficient bound.Source: B Tensor representation of the… · p. 81
- Each (r, s) summand uses r copies of A and s copies of b and lies in the (2r + s)-photon sector.The constant coordinate of w(theta) supplies the remaining K - r - s input factors.Source: B Tensor representation of the… · p. 81
- A computable subspace W of dimension at most 8(K + 1)^6(Gamma/xi)^2 approximates g(v) by g(P_W v) within xi for ||v||2 <= 1.The reduction uses strong tensor directions obtained through flattenings and singular vectors.Source: B Tensor representation of the… · p. 82
- The tensor and matrix construction costs are stated as (n + 1)^O(K), including the SVD computations.Source: B Tensor representation of the… · p. 82
C Subgradient and projection oracles
- The penalty subgradient is zero inside the radius-t region and otherwise has norm at most lambda/Rh by the stated Lipschitz bound.Source: C Subgradient and projection oracles · p. 83
- The matrix projection preserves the input's Takagi factor and reduces to projecting its singular-value vector onto the intersection of an l2 ball and an l-infinity ball.The vector projection takes polynomially many real-arithmetic operations.Source: C Subgradient and projection oracles · p. 83