arxiv: 2605.03715 · v2 · submitted 2026-05-05 · 🪐 quant-ph

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Real-time Krylov Diagonalisation for Open Quantum Systems

D. A. Herrera-Mart\'i

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Pith reviewed 2026-05-14 21:09 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Krylov subspace methodsLindblad formalismopen quantum systemsLiouvillian gapKerr Cat qubitsuperconducting resonatorsquantum simulation

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The pith

Real-time Krylov subspace methods adapt to Lindblad dynamics to estimate the Liouvillian gap in Kerr Cat qubits.

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

The paper shows how real-time quantum Krylov subspace methods, previously used for closed systems, extend to open quantum systems governed by the Lindblad master equation. It applies the adapted method to a two-photon-driven superconducting Kerr resonator and uses the resulting subspace to approximate the Liouvillian gap that controls relaxation to the steady state in the cat qubit regime. The construction relies on generating the subspace directly from real-time trajectories under the non-Hermitian Lindblad superoperator. A reader cares because this offers a route to spectral information on open dynamics without requiring full diagonalization of the Liouvillian.

Core claim

Real-time quantum Krylov subspace methods can be adapted to investigate open quantum systems described by the Lindblad formalism. Applied to a two-photon-driven superconducting Kerr resonator, the method constructs a subspace from real-time Lindblad evolution and uses it to estimate the Liouvillian gap within the Kerr Cat qubit regime.

What carries the argument

The real-time Krylov subspace generated from Lindblad evolution trajectories, which approximates the spectrum of the non-Hermitian Liouvillian superoperator.

If this is right

The approach yields the Liouvillian gap without computing the full spectrum of large non-Hermitian operators.
It enables gap estimation for open-system sizes where exact methods become intractable.
The gap value directly informs the relaxation timescale and steady-state fidelity in cat-qubit encodings.
The same subspace construction can supply approximate eigenvectors for further open-system observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may extend to other non-Hermitian generators such as those appearing in non-Markovian master equations.
Hybrid quantum-classical implementations could use short-time quantum evolution to build the subspace on hardware.
Error bounds on the gap estimate could be derived by monitoring subspace convergence with increasing Krylov dimension.

Load-bearing premise

The Krylov subspace built from real-time Lindblad evolution remains accurate and efficient for non-Hermitian dynamics without needing prohibitive dimensions or uncontrolled approximations.

What would settle it

Exact diagonalization of the Liouvillian on a small instance of the Kerr resonator model, followed by direct numerical comparison of the gap value to the Krylov estimate; a large discrepancy would show the subspace fails to capture the spectrum.

Figures

Figures reproduced from arXiv: 2605.03715 by D. A. Herrera-Mart\'i.

**Figure 2.** Figure 2: FIG. 2. The Kerr Cat Regime coincides with the onset of the Liouvillian gap closing. This is because a slow mode, corresponding [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Long times allow for a faithful reconstruction of the gap across several orders of magnitude. In this simulation we [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. To simulate dissipative dynamics with a quantum computer, one can Trotterise reversible unitary circuits with QITE [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

read the original abstract

In this chapter, we demonstrate how real-time quantum Krylov subspace methods can be adapted to investigate open quantum systems described by the Lindblad formalism. We apply these methods to a two-photon-driven superconducting Kerr resonator and illustrate their use in estimating the Liouvillian gap within the Kerr Cat qubit regime. This is based on work done between November 2025 and January 2026 and on the talk given in early February 2026 at the Kwekfest 2026 conference, to celebrate L.C. Kwek's lifetime achievements.

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.

Desk Editor's Note private letter to a colleague

This adapts real-time Krylov methods to Lindblad open systems and demonstrates gap estimation on a Kerr cat qubit, but the write-up stays mostly at the level of a proof-of-concept.

read the letter

The paper takes real-time Krylov subspace diagonalization, which has been used for closed systems, and shows how to run it on Lindblad evolution instead. They apply the resulting tool to a two-photon-driven Kerr resonator in the cat-qubit regime and use it to extract the Liouvillian gap. That combination is the concrete new piece: the adaptation plus the specific application to a system that matters for current superconducting hardware experiments. The choice of target is sensible because cat qubits are noisy and the gap controls how well the logical states are protected. For readers already comfortable with Krylov techniques on unitary dynamics, the extension gives a direct route without starting from scratch. The demonstration itself is the main contribution here. It shows the method can be made to work on non-Hermitian generators and produces a number for the gap that looks plausible for the Kerr cat parameters. That is useful as a starting point for people who need to simulate open-system spectra without full Liouvillian diagonalization. The soft spot is the lack of visible checks. The abstract and description do not include convergence plots against system size, comparisons to exact diagonalization on small instances, or scaling data on how large the Krylov subspace has to grow before the gap estimate stabilizes. Without those, it is hard to know whether the non-unitary evolution keeps the subspace well-conditioned or whether extra damping or truncation steps are hidden in the implementation. If the full text has those benchmarks, the concern shrinks; if not, the claim rests on the single demonstration. This is for people working on numerical methods for open quantum systems, especially in circuit QED or quantum optics. A reader who already runs Krylov codes and wants to move to Lindblad cases will find the adaptation worth looking at. It is not broad enough to change how most people simulate open systems, but it is a practical increment for the subfield. I would send it to peer review. The idea is clear, the target is relevant, and referees can ask for the missing benchmarks and implementation details in one round.

Referee Report

2 major / 2 minor

Summary. The manuscript demonstrates an adaptation of real-time quantum Krylov subspace methods to open quantum systems governed by the Lindblad master equation. It applies the approach to a two-photon-driven superconducting Kerr resonator and uses it to estimate the Liouvillian gap in the Kerr cat qubit regime.

Significance. If validated with sufficient accuracy and efficiency, the method could provide a practical route to extracting low-lying spectral information from non-Hermitian Liouvillians without full diagonalization. This is relevant for quantifying relaxation rates and bit-flip error suppression in driven-dissipative superconducting circuits such as cat qubits.

major comments (2)

[Method adaptation section] The section describing the Krylov subspace construction does not supply explicit equations for the real-time propagation under the Lindblad superoperator or the subsequent projection onto the non-Hermitian subspace; without these, it is impossible to verify that the adaptation preserves the essential spectral properties of the Liouvillian.
[Results / Kerr cat application] No convergence data, subspace-dimension scaling, or comparison against exact diagonalization (even for small Hilbert-space truncations) is presented for the Kerr resonator example; this leaves the central claim that the subspace remains accurate and tractable for non-Hermitian dynamics unsupported.

minor comments (2)

[Abstract] The abstract opens with 'In this chapter,' which is inconsistent with a standalone journal article; revise to 'In this work' or equivalent.
[Introduction] The manuscript should include at least one reference to established real-time Krylov techniques for closed systems and to existing Lindblad diagonalization methods to clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive suggestions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Method adaptation section] The section describing the Krylov subspace construction does not supply explicit equations for the real-time propagation under the Lindblad superoperator or the subsequent projection onto the non-Hermitian subspace; without these, it is impossible to verify that the adaptation preserves the essential spectral properties of the Liouvillian.

Authors: We agree that explicit equations are essential for reproducibility and verification. In the revised manuscript, we will expand the method section to include the explicit form of the real-time evolution operator under the Lindblad superoperator L, given by the Krylov approximation |ψ(t)⟩ ≈ V exp(t H_K) V† |ψ0⟩ where H_K is the projected non-Hermitian matrix. We will also detail the Arnoldi iteration adapted for the non-Hermitian case to ensure the spectral properties of the Liouvillian are preserved. revision: yes
Referee: [Results / Kerr cat application] No convergence data, subspace-dimension scaling, or comparison against exact diagonalization (even for small Hilbert-space truncations) is presented for the Kerr resonator example; this leaves the central claim that the subspace remains accurate and tractable for non-Hermitian dynamics unsupported.

Authors: We acknowledge the lack of supporting numerical validation in the current version. For the revised manuscript, we will add figures showing the convergence of the estimated Liouvillian gap with increasing Krylov subspace dimension for the two-photon-driven Kerr resonator. Additionally, we will include comparisons with exact diagonalization results for small system sizes (e.g., truncated to 4-8 levels) to demonstrate accuracy in the cat qubit regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript describes an adaptation of real-time quantum Krylov subspace methods to Lindblad open-system dynamics, with application to estimating the Liouvillian gap in a Kerr-cat qubit. No equations, parameter-fitting procedures, or derivation steps are exhibited that reduce any claimed prediction or result to a fitted input, self-definition, or self-citation chain by construction. The central contribution is presented as a methodological demonstration whose validity rests on external numerical or experimental benchmarks rather than internal redefinition of quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract; the contribution is described at the level of method adaptation only.

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