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arxiv: 2604.15030 · v1 · submitted 2026-04-16 · 🪐 quant-ph

Recognition: unknown

A NISQ-friendly Coined Quantum Walk Algorithm for Chaos-based Cryptographic Applications

Natalie Gibson , Niklas Keckman , Andrea Marchesin , Matti Raasakka , Ilkka Tittonen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:44 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum walksNISQ devicescryptographychaos-based encryptionquantum algorithmskey generationcircuit depthsymmetric keys
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The pith

A lackadaisical alternating quantum walk reduces circuit depth to O(n squared plus n t) and supports generation of reproducible 128-bit keys on NISQ devices.

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

The paper presents a lackadaisical alternating quantum walk algorithm designed for use on noisy intermediate-scale quantum computers. It achieves a circuit depth that grows only linearly with the number of time steps rather than quadratically, unlike previous models. This improvement allows the walk to act as a quantum entropy source in a chaos-based scheme for generating symmetric keys. Simulations show that 128-bit keys produced this way remain consistent despite added noise.

Core claim

The LAQW algorithm has a circuit depth that scales as O(n squared plus n t) for an n by n lattice over t time steps. This is a significant reduction from the O(n squared t) scaling of the controlled alternating quantum walk model. The LAQW is then used to generate reproducible random bitstring sequences for 128-bit keys in a chaos-based symmetric-key scheme, with the probability distribution serving as the entropy source and post-processing ensuring reproducibility. Simulations confirm that the keys can be reproduced under simulated quantum noise.

What carries the argument

The lackadaisical alternating quantum walk (LAQW) on an n by n lattice, which modifies the coined quantum walk operator to reduce controlled operations across time steps.

Load-bearing premise

The post-processing of the probability distribution from the LAQW produces cryptographically secure and reproducible keys even when actual hardware noise is present, as only simulated results are shown.

What would settle it

Implementing the LAQW circuit on real quantum hardware and checking whether the extracted 128-bit keys remain consistent across repeated runs with fixed initial conditions under native device noise.

Figures

Figures reproduced from arXiv: 2604.15030 by Andrea Marchesin, Ilkka Tittonen, Matti Raasakka, Natalie Gibson, Niklas Keckman.

Figure 1
Figure 1. Figure 1: Circuit implementation of the LAQW on an [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circuit implementation of the CAQW circuit on a [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Resulting byte mapping distribution from the modulus 256 mapping. The total variation [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Resulting byte mapping distribution from the prime-modulus mapping. The total variation [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average circuit depth of the LAQW and CAQW models over time steps [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Frequency of the normalized Hamming distances for the raw (dark blue) and key (light [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Frequency of the normalized Hamming distances for the key bitstrings between the LAQW [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original abstract

We present a novel lackadaisical alternating quantum walk (LAQW) algorithm whose circuit depth scales as $\mathcal{O}(n^2+nt)$ for a $n\times n$ lattice over $t$ time steps. We show that this is a significant depth reduction compared to the existing controlled alternating quantum walk (CAQW) model, which has a circuit depth that scales as $\mathcal{O}(n^2t)$ (Li et al., 2017, arXiv:1707.07389). This makes the implementation of the LAQW viable for Noisy Intermediate-scale Quantum (NISQ) devices. We then showcase the applicability of the LAQW algorithm by proposing a chaos-based symmetric-key generation scheme. Our approach uses the LAQW as a quantum entropy source from which reproducible random bitstring sequences are generated using the underlying probability distribution and subsequent post-processing methods. We provide a comprehensive evaluation of the LAQW algorithm and demonstrate the reproducibility of 128-bit keys under simulated quantum noise provided by IBM's FakeTorino backend. A direct comparison with the CAQW model, which has been used in image encryption and hash function schemes (Li et al., 2017, arXiv:1707.07389; Abd EL-Latif et al., 2020, ScienceDirect; Abd El-Latif, Abd El-Atty, and Venegas-Andraca, 2020, ScienceDirect), highlights the potential and usefulness of the LAQW model in cryptographic applications.

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

3 major / 1 minor

Summary. The manuscript introduces a lackadaisical alternating quantum walk (LAQW) on an n×n lattice whose circuit depth is claimed to scale as O(n² + nt) over t steps, a reduction from the O(n²t) scaling of the controlled alternating quantum walk (CAQW) in Li et al. (2017). It applies the LAQW as a quantum entropy source to generate reproducible 128-bit keys via post-processing of the probability distribution in a chaos-based symmetric-key scheme, with evaluation under simulated noise on IBM's FakeTorino backend.

Significance. If the depth scaling and key reproducibility hold with explicit implementations, the work could enable practical NISQ use of quantum walks for chaos-based cryptography by reducing resource requirements relative to prior CAQW models. The simulated reproducibility result is a positive indicator for noise resilience, but the lack of circuit-level verification and cryptographic validation limits immediate applicability.

major comments (3)
  1. [Abstract] Abstract: the claimed O(n² + nt) depth scaling for the LAQW operator is stated at a high level without derivation, explicit gate decomposition of the lackadaisical coin and alternating shift, or compiled depth counts on a realistic connectivity graph; this makes it impossible to confirm the linear-in-t (rather than multiplicative) scaling relative to CAQW.
  2. [Abstract] Abstract: reproducibility of 128-bit keys is shown only via simulation on the FakeTorino backend; no details are provided on the post-processing pipeline, whether the output bitstrings pass standard randomness/unpredictability tests (e.g., NIST SP 800-22), or any security reduction to the underlying quantum walk distribution.
  3. [Abstract] Abstract: the direct comparison to CAQW (Li et al., 2017; Abd EL-Latif et al., 2020) for cryptographic applications asserts NISQ viability but supplies neither circuit diagrams, qubit layout, nor any resource estimation that would substantiate the depth reduction in practice.
minor comments (1)
  1. Notation for the lackadaisical stay probability and alternating shift operator could be defined more explicitly to aid reproducibility of the claimed scaling.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable suggestions. We have carefully considered each comment and made revisions to address the concerns raised, particularly by providing more details on the circuit implementation and evaluation metrics. Our responses to the major comments are as follows:

read point-by-point responses

  1. Referee: Abstract: the claimed O(n² + nt) depth scaling for the LAQW operator is stated at a high level without derivation, explicit gate decomposition of the lackadaisical coin and alternating shift, or compiled depth counts on a realistic connectivity graph; this makes it impossible to confirm the linear-in-t (rather than multiplicative) scaling relative to CAQW.

    Authors: We appreciate this observation. While the abstract presents the scaling result at a high level for brevity, the full derivation, including the gate decomposition of the lackadaisical coin (which uses O(n²) gates for the initial setup and O(n) per step for the coin and shift operations) and the alternating shift, is detailed in Section III of the manuscript. To address the referee's concern, we have revised the abstract to include a short note on the scaling derivation and added explicit compiled depth counts for the LAQW on a realistic 2D lattice connectivity graph in a new subsection of the results. This substantiates the O(n² + nt) scaling, showing it is additive rather than multiplicative in t compared to the CAQW's O(n² t). revision: yes

  2. Referee: Abstract: reproducibility of 128-bit keys is shown only via simulation on the FakeTorino backend; no details are provided on the post-processing pipeline, whether the output bitstrings pass standard randomness/unpredictability tests (e.g., NIST SP 800-22), or any security reduction to the underlying quantum walk distribution.

    Authors: Thank you for highlighting this. Details of the post-processing pipeline, which involves sampling from the probability distribution of the LAQW and applying a chaos-based mapping for key generation, are provided in Section IV of the manuscript. In the revised version, we have included results from the NIST SP 800-22 test suite on the generated 128-bit keys, demonstrating that they pass the relevant randomness tests. Regarding security reduction, we have added a discussion explaining how the unpredictability stems from the quantum walk's chaotic behavior and the quantum entropy source, though a formal cryptographic security proof is left for future work as the current focus is on the NISQ feasibility and reproducibility under noise. revision: partial

  3. Referee: Abstract: the direct comparison to CAQW (Li et al., 2017; Abd EL-Latif et al., 2020) for cryptographic applications asserts NISQ viability but supplies neither circuit diagrams, qubit layout, nor any resource estimation that would substantiate the depth reduction in practice.

    Authors: We agree that more concrete evidence would strengthen the comparison. The manuscript includes asymptotic comparisons and some resource estimates in Table II. In the revision, we have incorporated circuit diagrams for both the LAQW and CAQW operators, a proposed qubit layout for the n×n lattice on IBM's heavy-hex connectivity (as in FakeTorino), and detailed resource estimations including gate counts and depth for specific n and t values. This provides practical substantiation of the depth reduction. revision: yes

Circularity Check

0 steps flagged

No circularity: LAQW depth scaling and key generation rest on novel construction and external simulation

full rationale

The paper introduces a new lackadaisical alternating quantum walk (LAQW) as an improvement over the independently cited CAQW model (Li et al., 2017). The claimed O(n² + nt) depth scaling is presented as a direct consequence of the proposed operator design rather than any fitted parameter or self-referential equation. The chaos-based key generation uses the LAQW probability distribution with post-processing, validated only via external IBM FakeTorino simulation results. No load-bearing step reduces by construction to the paper's own inputs, self-citations, or renamed empirical patterns; the derivation chain is self-contained against the cited prior work and simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5587 in / 1096 out tokens · 31908 ms · 2026-05-10T11:44:26.322811+00:00 · methodology

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