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arxiv: 2507.20317 · v1 · submitted 2025-07-27 · 🪐 quant-ph

Efficient Gaussian State Preparation in Quantum Circuits

Yichen Xie , Nadav Ben-Ami This is my paper

Pith reviewed 2026-05-19 02:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Gaussian state preparationquantum Fourier transformstate preparationquantum circuitsgate complexityapproximate Gaussianquantum computing
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The pith

A quantum circuit prepares approximate Gaussian states by using single-qubit rotations followed by the quantum Fourier transform.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes preparing approximate Gaussian states on digital quantum computers by first using single-qubit rotations to create an exponential amplitude profile across the basis states. It then applies the quantum Fourier transform to convert this profile into one that approximates a Gaussian distribution in the computational basis. This achieves high fidelity with the desired Gaussian state. The approach also supports pruning small controlled-phase angles during the Fourier transform to lower the gate count to near linear in the number of qubits. A sympathetic reader would care because Gaussian states appear in quantum simulation, chemistry, and machine learning, where efficient preparation on gate-based hardware has been costly.

Core claim

The authors demonstrate that single-qubit rotations can form an exponential amplitude profile whose quantum Fourier transform yields an approximate discrete Gaussian distribution with high fidelity, and that pruning small controlled-phase angles in the transform reduces total gate complexity to near-linear scaling in the number of qubits while preserving the approximation quality.

What carries the argument

The composition of single-qubit rotations that generate an exponential amplitude profile followed by the quantum Fourier transform that maps the profile to a Gaussian-like distribution.

Load-bearing premise

The quantum Fourier transform of the exponential amplitude profile produces a close enough discrete Gaussian approximation and that pruning small controlled-phase gates does not add unacceptable error.

What would settle it

Run the circuit for four or five qubits on a simulator or device, measure the output probabilities, and verify whether the fidelity to the target discrete Gaussian stays above 0.95 and whether pruning changes that fidelity by more than a few percent.

Figures

Figures reproduced from arXiv: 2507.20317 by Nadav Ben-Ami, Yichen Xie.

Figure 1
Figure 1. Figure 1: The Gaussian preparation circuit we proposed with 5 qubits. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the Gaussian preparation circuit probability distribution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Number of gates to n for different QFT angle threshold δ, full QFT, and amplitude encoding. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 n 10 16 10 14 10 12 10 10 10 8 10 6 10 4 MSE MSE to n = 0.0982 = 0.0491 = 0.0245 = 0.0123 = 0.0061 = 0.0031 = 0.0015 Full QFT Amplitude Encoding [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MSE to n for different QFT angle threshold δ, full QFT, and amplitude encoding. The practical impact of achieving near-linear gate scaling cannot be overstated, because real devices remain limited by decoherence, crosstalk, and other noise sources that scale adversely with circuit depth [25]. A near-linear approach in the QFT block can thus allow for significantly larger n (as shown in [PITH_FULL_IMAGE:fi… view at source ↗
Figure 5
Figure 5. Figure 5: KL-Divergence to n for different QFT angle threshold δ, full QFT, and amplitude encoding. (y-axis is log scale) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 n 0.9972 0.9974 0.9976 0.9978 0.9980 0.9982 0.9984 0.9986 Fidelity Fidelity to n = 0.0982 = 0.0491 = 0.0245 = 0.0123 = 0.0061 = 0.0031 = 0.0015 Full QFT [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Fidelity to n for different QFT angle threshold δ and full QFT [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

Gaussian states hold a fundamental place in quantum mechanics, quantum information, and quantum computing. Many subfields, including quantum simulation of continuous-variable systems, quantum chemistry, and quantum machine learning, rely on the ability to accurately and efficiently prepare states that reflect a Gaussian profile in their probability amplitudes. Although Gaussian states are natural in continuous-variable systems, the practical interest in digital, gate-based quantum computers demands discrete approximations of Gaussian distributions over a computational basis of size \(2^n\). Because of the exponential scaling of naive amplitude-encoding approaches and the cost of certain block-encoding or Hamiltonian simulation techniques, a resource-efficient preparation of approximate Gaussian states is required. In this work, we propose and analyze a circuit-based approach that starts with single-qubit rotations to form an exponential amplitude profile and then applies the quantum Fourier transform to map those amplitudes into an approximate Gaussian distribution. We demonstrate that this procedure achieves high fidelity with the target Gaussian state while allowing optional pruning of small controlled-phase angles in the quantum Fourier transform, thus reducing gate complexity to near-linear in \(\mathcal{O}(n)\). We conclude that the proposed technique is a promising route to make Gaussian states accessible on noisy quantum hardware and to pave the way for scalable implementations on future devices. The implementation of this algorithm is available at the Classiq library: https://github.com/classiq/classiq-library.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide an efficient method for preparing approximate Gaussian states on quantum circuits by preparing an exponential amplitude profile with single-qubit rotations and then applying the quantum Fourier transform, with optional pruning of small controlled-phase gates to achieve O(n) complexity while maintaining high fidelity.

Significance. If validated, this technique could significantly reduce the resources needed for Gaussian state preparation in applications such as quantum simulation of continuous-variable systems, quantum chemistry, and quantum machine learning on gate-based quantum computers. The open-source implementation in the Classiq library is a positive aspect for reproducibility.

major comments (2)
  1. [Initial amplitude profile preparation] The assertion that single-qubit rotations form an 'exponential amplitude profile' whose QFT yields a Gaussian is problematic. Independent single-qubit rotations prepare a product state whose amplitudes factorize according to the binary digits of the basis state index, typically resulting in amplitudes dependent on Hamming weight rather than a geometric sequence a_k ∝ r^k for integer k. This does not generally produce the claimed discrete Gaussian approximation after QFT. Please specify the exact circuit and amplitude formula used (likely in the Methods or Results section) and demonstrate how it achieves the geometric profile.
  2. [Pruning analysis] The section on gate pruning in the QFT lacks a rigorous error bound for the fidelity loss as a function of the pruning threshold, n, and the Gaussian parameter. Without this, the claim of controllable error with O(n) gates is not fully supported. Standard AQFT error analyses do not directly apply here.
minor comments (2)
  1. [Abstract] The abstract mentions 'high fidelity' and 'near-linear in O(n)' but does not include any specific numerical results, fidelity values, or comparisons to other methods. Adding these would improve clarity.
  2. [Implementation] While the GitHub link is provided, ensure that the code includes examples with explicit fidelity measurements for various n and pruning thresholds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. The feedback identifies key areas for clarification in the circuit construction and for strengthening the theoretical support of the pruning analysis. We address each point below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Initial amplitude profile preparation] The assertion that single-qubit rotations form an 'exponential amplitude profile' whose QFT yields a Gaussian is problematic. Independent single-qubit rotations prepare a product state whose amplitudes factorize according to the binary digits of the basis state index, typically resulting in amplitudes dependent on Hamming weight rather than a geometric sequence a_k ∝ r^k for integer k. This does not generally produce the claimed discrete Gaussian approximation after QFT. Please specify the exact circuit and amplitude formula used (likely in the Methods or Results section) and demonstrate how it achieves the geometric profile.

    Authors: We appreciate this clarification request. The single-qubit rotations use qubit-dependent angles chosen to produce exactly the geometric profile a_k ∝ r^k. For qubit j (bit weight 2^j), the rotation angle θ_j is selected so that the amplitude ratio for |1⟩ versus |0⟩ equals r^{2^j} (normalized by the overall factor). The resulting product-state amplitude for computational basis state |k⟩ with bits b_j is then ∏_j α_j(b_j) ∝ ∏_j (r^{2^j})^{b_j} = r^{∑ 2^j b_j} = r^k. This is a standard technique for preparing geometric states via product rotations. We will add the explicit angle formula, the short algebraic demonstration above, and a circuit diagram with the varying θ_j to the Methods section. revision: yes

  2. Referee: [Pruning analysis] The section on gate pruning in the QFT lacks a rigorous error bound for the fidelity loss as a function of the pruning threshold, n, and the Gaussian parameter. Without this, the claim of controllable error with O(n) gates is not fully supported. Standard AQFT error analyses do not directly apply here.

    Authors: We agree that an analytical error bound would strengthen the claims. While numerical experiments in the manuscript already demonstrate high fidelity (>0.99) for pruning thresholds that reduce gate count to O(n), we will derive a rigorous bound in the revised manuscript. The difference between the full and pruned QFT unitaries can be bounded in operator norm by the sum of the absolute values of the discarded controlled-phase angles. Propagating this through the initial state (whose norm is bounded independently of n for fixed Gaussian width) yields an explicit fidelity lower bound that depends on the pruning threshold, n, and the Gaussian parameter σ. This derivation will be added to the Pruning Analysis section, together with a brief comparison to why standard AQFT bounds require adjustment for the non-uniform initial state. revision: yes

Circularity Check

0 steps flagged

No circularity: method composes standard gates and compares to external Gaussian target

full rationale

The derivation begins with single-qubit rotations to prepare an exponential amplitude profile, applies the quantum Fourier transform, and evaluates fidelity against an independent target Gaussian distribution. No equation defines the output Gaussian in terms of the input profile or vice versa; the approximation is asserted by explicit construction and numerical demonstration rather than by renaming or self-referential fitting. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the provided description. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the standard postulates of quantum mechanics and the known circuit decomposition of the quantum Fourier transform; no new physical entities or ad-hoc constants are introduced beyond the usual choice of rotation angles and a pruning threshold.

free parameters (1)
  • pruning threshold for controlled-phase angles
    A cutoff value below which small angles are omitted; its specific numerical choice affects both gate count and fidelity but is not derived from first principles.
axioms (1)
  • domain assumption The quantum Fourier transform maps an exponential amplitude vector to a discrete Gaussian-like distribution with controllable error.
    Invoked when the authors state that the QFT step produces the target Gaussian profile.

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discussion (0)

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Forward citations

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