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Quantized Low-Rank Quantum State Tomography: Hyperbolic Quantization and Riemannian Least-Squares Recovery

HanQin Cai, Longxiu Huang, Juntao You

arXiv:2608.27503v1quant-phcs.IT

TL;DR

Low-rank quantum state tomography must recover states from Pauli responses whose storage and communication use finite bits, but quantization can bias those responses. The paper proposes mean-preserving HyperQuant and direct finite-bit least-squares recovery, together with QuantRGD for computation. It proves minimax distortion, nonasymptotic recovery, a bit–shot tradeoff, and linear convergence under explicit resource conditions.

  • Problem

    Finite-bit representation of Pauli batch responses introduces quantization distortion, raising whether quantized responses can retain unquantized statistical accuracy under resource budgets (M, N, B).

  • Method

    The paper combines a mean-preserving hyperbolic quantizer, rank-constrained least squares, and a Riemannian gradient method operating directly on quantized responses.

  • Results

    The paper establishes minimax distortion and nonasymptotic recovery guarantees, a bit–shot tradeoff retaining the unquantized error order, and linear convergence of QuantRGD to a statistical neighborhood.

  • Takeaways & Limitations

    Exact mean preservation keeps the finite-bit least-squares population target unchanged while supporting statistically and computationally provable recovery.

  • Takeaways & Limitations

    The guarantees operate under the rank-at-most-r density-matrix model and the stated bit–shot matching condition.

Abstract

from arXiv · show

We study low-rank quantum state tomography from finite-bit Pauli batch responses. To avoid bias introduced by generic quantization, we propose HyperQuant, a mean-preserving hyperbolic quantizer adapted to the second-moment scale of Pauli responses. We establish minimax distortion guarantees and show that exact mean preservation enables direct rank-constrained least-squares recovery without altering the population target. We derive nonasymptotic recovery guarantees and an explicit bit--shot tradeoff under which finite-bit responses retain the error order of unquantized batch averages using fewer response bits. For efficient computation, we develop QuantRGD, a Riemannian gradient method with provable linear convergence to the corresponding statistical neighborhood under explicit resource conditions. Numerical experiments validate the predicted quantization, recovery, and convergence behavior.

1 Introduction

This paper addresses low-rank quantum state tomography when Pauli batch responses are stored with finite precision, separating measurement, sampling, and communication resources. It introduces HyperQuant and QuantRGD to preserve statistical targets and enable efficient recovery from quantized responses.

  • Motivation: Finite-bit Pauli batch responses make precision and throughput explicit resources alongside Pauli settings and quantum copies.The model uses M settings, ℓ repeated measurements per setting, and b bits per encoded batch average.
  • Statistical recovery: Exact mean preservation permits direct rank-constrained least-squares recovery without altering the population target.The resulting guarantees account explicitly for finite-shot noise and quantization.
  • HyperQuant: HyperQuant is a mean-preserving hyperbolic quantizer adapted to the second-moment scale of Pauli responses.The construction targets minimax distortion under a weighted conditional variance criterion.
  • Statistical recovery: When ℓ≤d, O(log log(e + ℓ)) bits per Pauli batch suffice to retain the error order of unquantized batch averages under bit–shot matching.This establishes an explicit tradeoff between transmitted response bits and measurement shots.
  • Computational recovery: QuantRGD operates directly on finite-bit responses and converges linearly to a statistical neighborhood under explicit resource conditions.Each iteration costs O(Mdr + dr^2 + r^3) flops and uses O(dr) working memory.
  • Experiments: Numerical experiments validate HyperQuant's distortion advantage, finite-bit recovery under constrained response budgets, and QuantRGD's convergence performance.The comparisons include uniform quantization and representative unquantized methods.

2 Observation Model and Hyperbolic Quantization

The paper models low-rank quantum state tomography with finite-bit Pauli batch responses and introduces HyperQuant, a mean-preserving hyperbolic quantizer. Its design controls quantization distortion while preserving conditional unbiasedness and supports minimax guarantees.

  • HyperQuant design: HyperQuant uses a mean-preserving stochastic quantizer whose alphabet is adapted to the second-moment scale of Pauli batch responses.Mean preservation prevents bias, while hyperbolic level placement controls additional quantization variance.
  • Observation model: Finite-bit tomography separates Pauli settings, quantum copies, and response bits as resources M, N = Mℓ, and B = Mb.Each sampled Pauli observable is measured over ℓ copies, averaged, and encoded using b bits.
  • HyperQuant design: A one-bit HyperQuant response is conditionally equivalent to a single Pauli outcome and cannot retain batch-averaging variance reduction, regardless of ℓ.For K = 2, the alphabet is {±1}, and mean preservation uniquely determines the response probabilities.
  • Minimax guarantees: HyperQuant is minimax optimal up to an absolute constant when K − 1 is a sufficiently large multiple of Aℓ,d.The guarantee is established under a weighted conditional variance criterion.
  • Minimax guarantees: In the high-shot pure-state regime, minimax scalar distortion has the correct dependence on alphabet size K and Pauli dimension d, up to absolute constants.The stated distortion order is log^2 d / K^2.

3 Rank-Constrained Least-Squares Recovery

Mean preservation lets quantized responses enter rank-constrained least squares without changing its population target. The resulting estimator and QuantRGD algorithm retain statistical accuracy under explicit bit–shot and resource conditions.

  • Population objective: Mean preservation makes the expected excess least-squares loss exactly one half the squared Frobenius distance from ρ⋆.Quantization changes observation fluctuations but not the population least-squares objective.
  • Statistical recovery: Every global minimizer of the rank-constrained least-squares problem is statistically accurate with sufficiently many random Pauli settings.Theorem 3 provides the guarantee with probability at least 1 − d^-10.
  • Bit–shot tradeoff: The variance proxy Vℓ,K,d combines finite-shot noise and quantization contributions, and bit–shot matching keeps the quantization contribution no larger than 1/ℓ.Under the matching condition, Vℓ,K,d ≤ 2/ℓ.
  • Bit–shot tradeoff: For ℓ ≤ d, only logarithmically many quantization levels are needed, while for ℓ > d the required alphabet size grows as √(ℓ/d) up to a logarithmic factor.These requirements arise from matching quantization variance to the shot-noise scale.
  • Bit–shot tradeoff: When ℓ = Θ(d), finite-bit responses reduce the response budget from O(M log d) for lossless counts to O(M log log d).The finite-bit budget retains the same order of least-squares accuracy under the stated condition.

4 Riemannian Optimization for Quantized QST

QuantRGD directly optimizes the quantized least-squares objective on fixed-rank density matrices, using Riemannian updates and projection to enforce the constraints. Under explicit resource conditions, it converges linearly to a statistical neighborhood determined by finite-shot noise and quantization.

  • QuantRGD: QuantRGD directly optimizes the least-squares objective formed from finite-bit responses while retaining the rank-constrained formulation enabled by mean preservation.
  • QuantRGD: Each iteration projects the Euclidean gradient onto the tangent space, takes a descent step, and projects back onto the density-matrix set.The update uses the Riemannian gradient and a positive stepsize.
  • Initialization: The spectral initializer backprojects quantized responses and applies rank-r spectral truncation before iterative optimization.The truncation retains the r eigencomponents with largest eigenvalues in magnitude.
  • Computational complexity: Each QuantRGD iteration costs O(Mdr + dr^2 + r^3) flops and requires O(dr) working memory.Under the convergence conditions, the projection can be reduced to an eigendecomposition of order at most 2r.
  • Resource conditions: The state-dependent setting requirement is at most e O(κ^2r^2d) when λ1(ρ⋆) ≥ 1/r, consistent with standard unquantized spectral-initialization scaling.
  • Convergence guarantees: Under explicit resource conditions and the bit–shot condition, QuantRGD enters a contraction region and converges linearly to the statistical neighborhood of ρ⋆.The iteration count to reach that neighborhood is logarithmic in the ratio between initialization error and statistical radius.

5 Numerical experiments

The experiments test HyperQuant's scalar distortion, finite-bit recovery, and QuantRGD convergence against unquantized and alternative quantized responses. HyperQuant closely follows predicted distortion and yields recovery and runtime behavior near unquantized baselines at suitable bit depths.

  • 5.1 Scalar quantization: The weighted criterion for HyperQuant follows the predicted K^-2 decay, with a value approximately 20 times smaller than uniform quantization at K = 16.
  • 5.1 Scalar quantization: The normalized Bayes distortion remains approximately between 0.5 and 0.75 across tested dimensions and bit depths.Its constant-order behavior agrees with the reported log2 d/[d(K−1)^2] scaling.
  • 5.2 Recovery performance: HyperQuant remains close to the unquantized batch average and improves when the response bit budget increases from four to five bits.Uniform quantization and raw-prefix transmission have larger recovery errors at the same bit budget.
  • 5.2 Recovery performance: Increasing b closes the recovery gap between HyperQuant and the unquantized response, while raw-prefix transmission remains less accurate because it averages only b physical outcomes.
  • 5.3 Convergence performance: QuantRGD with b = 5 closely tracks unquantized RGD in recovery error and runtime across four tested settings, whereas b = 4 reaches a higher finite-bit error floor.
  • 5.3 Convergence performance: Lossless transmission of the batch average requires approximately 2.2–2.6 times the response-bit budget of QuantRGD with b = 5 over the tested dimensions.

6 Proofs of Main Results

The proofs establish pointwise variance bounds for mean-preserving quantizers and apply them to matching hyperbolic constructions, yielding the stated minimax result under the relevant second-moment constraint.

  • Quantization variance: Mean preservation implies Var(Q(x) | x) ≥ (x − q_j)(q_{j+1} − x) between adjacent alphabet levels.The bound follows from nonnegativity of (Z − q_j)(Z − q_{j+1}); adjacent stochastic rounding attains equality.
  • Quantization variance: The hyperbolic alphabet uses x = √ν sinh y with uniformly spaced transformed levels and adjacent stochastic rounding.The transformed interval width is τ = 2Aν/(K−1), enabling the upper-bound construction.
  • Minimax bounds: The upper and lower arguments combine, and specializing ν = νℓ,d yields the claimed minimax quantization result.The proof takes the infimum over admissible quantizers after setting the scale to νℓ,d.
  • Minimax bounds: A cell-length argument supplies a matching lower bound by finding a transformed cell of width at least 2Aν/(K−1).The argument centers on a point x within the widest cell and applies the pointwise variance bound.
  • Distributional distortion: The corollary extends the argument from pointwise distortion to distributions on [−1,1] with second moment at most νℓ,d.Point masses suffice when the target point is admissible; otherwise a two-point distribution enforces the second-moment constraint.

6.3 Proof of Theorem 2

Theorem 2’s proof derives a lower bound for arbitrary mean-preserving quantizers by constructing a multiscale pure-state prior whose Pauli coefficients span many scales.

  • Multiscale prior: A multiscale prior mixes Haar-random pure states over dyadic dimensions to prevent a K-level quantizer from concentrating resolution at one scale.The resulting coefficient scales range between d^-1/2 and a constant.
  • Coefficient distribution: For a fixed nonidentity Pauli on the active subsystem, Haar invariance reduces the coefficient distribution to a projection onto balanced eigenspaces.The projected squared norm follows a Beta distribution, leading to the stated density representation.
  • Lower-bound integration: The resulting mixture density is lower-bounded across scales, supporting integration of quantization error over intersected quantization cells.The proof uses dyadic choices of m and Stirling bounds to obtain a pointwise density lower bound.
  • Coefficient distribution: Ancilla structure restricts nonzero contributions to Pauli strings whose ancilla factors are identities or σz operators.Only a fraction 2^{n−k} of Pauli strings have a nonzero ancilla contribution.
  • Lower-bound integration: The proof concludes with E Var(Q(X) | X) ≥ c5 d^(n−2) and the corresponding logarithmic-width bound for all admissible quantizers.The lower bound is transferred from the constructed prior to the supremum over pure states.

6.4 Proof of Theorem 3

Theorem 3 combines Pauli restricted isometry with concentration of the score at the true state to control the rank-constrained least-squares estimator.

  • Proof strategy: The proof applies the Pauli RIP at rank level s = min{2r,d} with failure probability controlled by a high-probability event.The required sample condition is M ≥ C2 r d log^7 d.
  • Score concentration: Mean preservation makes the quantization residual conditionally mean-zero, while bounded responses permit a matrix Bernstein argument.The residual satisfies |ξ_i| ≤ 2, and the resulting matrices are independent, mean-zero, and Hermitian.
  • Proof strategy: The score at the true state is controlled in the rank-2r restricted norm with probability at least 1 − 2d^-12.The bound scales with d, √r, and the maximum response-noise scale.
  • Least-squares control: For the estimator difference Δ, rank(Δ) ≤ 2r, so the restricted isometry bound applies to the least-squares optimality inequality.The proof compares the estimator objective with the feasible true state.
  • Least-squares control: Von Neumann’s trace inequality and Cauchy–Schwarz convert the score inner product into a Frobenius-norm error bound.After division by ||Δ||_F, the estimator error is bounded by four times the restricted score norm.
  • Probability guarantee: A union bound gives total failure probability at most d^-10 under the stated resource condition.The two failure probabilities are d^-11 and 2d^-12.

6.5 Proof of Lemma 1

The proof of Lemma 1 characterizes projection onto the rank-constrained density-matrix set through eigenvalue thresholding and establishes stability and rank properties.

  • Projection formula: For X in the rank-constrained set, the projection is obtained by projecting the leading eigenvalues of W onto the probability simplex.The ordering constraint can be omitted because simplex projection preserves the ordering of the eigenvalues.
  • Projection stability: The metric projection satisfies ||Π_Dr(W) − ρ||_F ≤ 2||W − ρ||_F for every feasible ρ.This follows from projection optimality and the triangle inequality.
  • Rank stability: When ||W − ρ||_op ≤ λ_r(ρ)/4, Weyl’s inequality separates the leading r-dimensional eigenspace from the remainder.The proof obtains λ_r(W) > λ_{r+1}(W) under this spectral-gap condition.
  • Rank stability: A threshold τ0 is chosen so the shifted leading eigenvalues form an interior point of the probability simplex.The construction defines μ_j = λ_j(W) − τ0 and ensures all μ_j are positive and sum to one.
  • Projection formula: Strict convexity gives a unique simplex minimizer with exactly r positive eigenvalues, completing the rank characterization.This establishes the stated projection representation and rank property.

6.6 Proof of Theorem 4

The proof establishes initialization and deterministic contraction for the rank-constrained Riemannian procedure, using concentration, spectral truncation, restricted-isometry control, and gradient bounds.

  • Initialization: Theorem 4’s backprojection analysis combines concentration bounds, Pauli isotropy and orthogonality, matrix variance control, and matrix Bernstein’s inequality.These estimates are combined with a union bound to obtain the stated high-probability initialization bound.
  • Initialization: Spectral truncation preserves rank-r positive semidefiniteness when the rth eigenvalue remains separated from the remaining spectrum.Weyl’s inequality establishes the eigenvalue separation, while Eckart–Young–Mirsky controls the truncation error.
  • Contraction: The deterministic contraction proof tracks the Frobenius error and tangent-space projection, then uses rank bounds and restricted-isometry polarization to control the projected gradient.The argument applies when the iterate remains within the specified basin and the rank-4r RIP condition holds.
  • Contraction: Exact line search and the tangent-space gradient bound yield a recursion showing basin invariance and rank-r projected iterates.Iterating the recursion gives the contraction toward a statistical neighborhood governed by the gradient at the target.

6.8 Proof of Theorem 5

The proof of Theorem 5 verifies restricted isometry, initialization, and score conditions under the stated sample-size assumptions, then combines them to obtain the theorem’s guarantee.

  • Restricted isometry: The rank-4r restricted-isometry condition holds with probability at least 1 − d^-12 when the theorem’s sample-size condition is satisfied.This follows by applying Lemma 2 with s = min{4r, d}, δ = 1/64, and η = d^-12.
  • Initialization: The initialization condition is verified by substituting the bounds on V_ℓ,K,d and N = Mℓ into the established initialization estimate.The proof then applies the theorem’s sample-size conditions and chooses C5 sufficiently large.
  • Score condition: The score condition follows from concentration together with the bound ∥∇L_M(ρ⋆)∥_(2r) ≤ 1/64 λ_r(ρ⋆).Substitution into the contraction result yields (22) whenever C5 ≥ 52.
  • Conclusion: The union bound over restricted isometry, initialization, and score events gives failure probability at most 2d^-10.This combines the three verified resource-dependent conditions used in the theorem.

7 Conclusion

The paper studies low-rank quantum state tomography with finite-bit Pauli batch responses and develops quantization, recovery, and optimization guarantees validated numerically.

  • Conclusion: HyperQuant is a mean-preserving hyperbolic quantizer that keeps finite-bit responses conditionally unbiased while controlling quantization variance.The paper establishes minimax distortion guarantees for this quantization scheme.
  • Conclusion: Exact mean preservation supports direct rank-constrained least-squares recovery with nonasymptotic guarantees and a favorable bit–shot tradeoff relative to unquantized batch transmission.The finite-bit estimator can retain the error order of unquantized batch averages under the derived tradeoff.
  • Conclusion: QuantRGD is a Riemannian gradient method with provable linear convergence to the statistical neighborhood under explicit resource conditions.Numerical experiments validate the predicted quantization, recovery, and convergence performance.
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