Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.16260 · Quantum

Multi-product Zeno effect achieving higher order convergence rates

Tim Möbus

Published Jun 30, 2026• 0 citations
arXivHTMLPDF

Abstract

The quantum Zeno effect is a fundamental mechanism for implementing the effective dynamics of projected Hamiltonian and Lindbladian systems. It approximates the target projected evolution by interleaving Hamiltonian or Lindblad dynamics with quantum operations associated with the desired subspace. In contrast to the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is typically limited to order 1 / n . In this work, we improve this convergence rate by employing a multi-product formula, thereby achieving arbitrarily high-order convergence of the form 1 / n K + 1 . This yields an improved approximation scheme for Zeno-like expectation values via an efficient post-processing method. The approach combines a modified Chernoff lemma, an adapted Dunford-Segal approximation, holomorphic functional calculus, and Chebyshev interpolation. We illustrate the method with the bosonic cat code and also consider the broader class of systems governed by the Bang-Bang decoupling method.

DOI
10.22331/q-2026-06-30-2148

Related papers

Higher-order Zeno sequences

Kasra Rajabzadeh Dizaji, Leeseok Kim, Milad Marvian +1 more

The quantum Zeno effect typically refers to freezing the dynamics of a quantum system through frequent observations. In general, quantum Zeno dynamics is obtained with an error of order $\mathcal{O}(1/N)$, where $N$ is the number of projective measurements performed within a fixed evolution time. In this work, we develop higher-order Zeno sequences that achieve faster convergence to Zeno dynamics, yielding an improved error scaling of $\mathcal{O}(1/N^{2k})$, where $k$ describes the order of the Zeno sequence. This is achieved by relating higher-order Zeno sequences to higher-order Trotter formulas that achieve similar convergence behavior. We leverage this relation to develop higher-order Zeno sequences for different manifestations of the quantum Zeno effect, including frequent projective measurements and unitary kicks. We go on to discuss achieving quantum Zeno dynamics through periodic control fields of high frequency. We explicitly develop control fields that yield a second-order type improvement in the Zeno error scaling and present shorter Zeno sequences. Finally, we discuss the connection to randomized and Uhrig dynamical decoupling to develop more efficient implementations in the weak coupling regime.

quant-pharXiv:2511.20792•Physical Review Research•Nov 27, 2025•0 citations

Quantum Zeno effect and dynamics

Paolo Facchi A, Marilena Ligabò

If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.

arXiv:0911.2201•Journal of Mathematical Physics•Feb 1, 2010•42 citations

Approximating exponentials of commutators by optimized product formulas

Fernando Casas, Alejandro Escorihuela-Tomàs, Pablo A. M. Casares

Abstract Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area.In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previouslypublished alternatives. They can also be used as basic methods in recursive procedures to increasethe order of approximation.We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.

arXiv:2407.10533•Quantum Information Processing•Jan 31, 2025•3 citations

Dual channel multi-product formulas

Seung Park, Sangjin Lee, Kyunghyun Baek

Product-formula (PF) based quantum simulation is a promising approach for simulating quantum systems on near-term quantum computers. Achieving a desired simulation precision typically requires a polynomially increasing number of Trotter steps, which remains challenging due to the limited performance of current quantum hardware. To alleviate this issue, post-processing techniques such as the multi-product formula (MPF) have been introduced to suppress algorithmic errors within restricted hardware resources. In this work, we propose a dual-channel multi-product formula that achieves a two-fold improvement in Trotter error scaling. As a result, our method enables the target simulation precision to be reached with approximately half the circuit depth compared to conventional MPF schemes. Importantly, the reduced circuit depth directly translates into lower physical error mitigation overhead when implemented on real quantum hardware. We demonstrate that, for a fixed CNOT count as a measure of quantum circuit, our proposal yields significantly smaller algorithmic errors, while the sampling error remains essentially unchanged.

quant-pharXiv:2602.01713•Feb 3, 2026•0 citations

Optimal Convergence Rate in the Quantum Zeno Effect for Open Quantum Systems in Infinite Dimensions

Tim Möbus, Cambyse Rouzé

Abstract In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g., a measurement, used to restrict the time evolution (due, for example, to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a $$C_0$$ C 0 -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate $$\mathcal {O}(\tfrac{1}{n})$$ O ( 1 n ) of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type $$\sqrt{n}$$ n -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator, and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.

arXiv:2111.13911•Annales Henri Poincaré•Oct 26, 2022•10 citations

Constant-Depth Multi-Product Formula for Trotter Error Mitigation in Near-Term Digital Quantum Simulation

Chan Bin Bark, Sangjin Lee, Kwang Jun Ahn +3 more

Digital quantum simulation of many-body dynamics faces a tension between algorithmic Trotter error and physical noise that accumulates with circuit depth. Typical higher-order product formulas mitigate the algorithmic error at the cost of deeper circuits. Our benchmark shows that, within the limited physical error budget, the feasible advantage is confined to absolute errors well below the $10^{-2}$ scale, which vanishes under noise levels of current quantum hardware. In sharp contrast to the previous Trotter error mitigation methods, we introduce a constant-depth multi-product formula (cd-MPF) that suppresses the Trotter error by combining multiple circuits at fixed circuit depth. We identify an auxiliary parameter $α$, which reshapes the Trotter error terms while leaving the target evolution invariant. The classical linear combination of the measured expectation values cancels the leading $(Δt)^{2}$ algorithmic contribution and steepens the Trotter-error scaling with the two-qubit circuit depth $d$ from $d^{-2}$ to $d^{-4}$. Combined with physical-noise mitigation, our method can serve as a key ingredient for realizing long-time quantum dynamics simulation on near-term hardware.

quant-pharXiv:2607.12225•Jul 15, 2026•0 citations
The quantum Zeno effect is a fundamental mechanism for implementing the effective dynamics of projected Hamiltonian and Lindbladian systems. It approximates the target projected evolution by interleaving Hamiltonian or Lindblad dynamics with quantum operations associated with the desired subspace. In contrast to the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is typically limited to order 1 / n . In this work, we improve this convergence rate by employing a multi-product formula, thereby achieving arbitrarily high-order convergence of the form 1 / n K + 1 . This yields an improved approximation scheme for Zeno-like expectation values via an efficient post-processing method. The approach combines a modified Chernoff lemma, an adapted Dunford-Segal approximation, holomorphic functional calculus, and Chebyshev interpolation. We illustrate the method with the bosonic cat code and also consider the broader class of systems governed by the Bang-Bang decoupling method.
The quantum Zeno effect typically refers to freezing the dynamics of a quantum system through frequent observations. In general, quantum Zeno dynamics is obtained with an error of order O(1/N)\mathcal{O}(1/N)O(1/N), where NNN is the number of projective measurements performed within a fixed evolution time. In this work, we develop higher-order Zeno sequences that achieve faster convergence to Zeno dynamics, yielding an improved error scaling of O(1/N2k)\mathcal{O}(1/N^{2k})O(1/N2k), where kkk describes the order of the Zeno sequence. This is achieved by relating higher-order Zeno sequences to higher-order Trotter formulas that achieve similar convergence behavior. We leverage this relation to develop higher-order Zeno sequences for different manifestations of the quantum Zeno effect, including frequent projective measurements and unitary kicks. We go on to discuss achieving quantum Zeno dynamics through periodic control fields of high frequency. We explicitly develop control fields that yield a second-order type improvement in the Zeno error scaling and present shorter Zeno sequences. Finally, we discuss the connection to randomized and Uhrig dynamical decoupling to develop more efficient implementations in the weak coupling regime.
If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.
Abstract Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area.In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previouslypublished alternatives. They can also be used as basic methods in recursive procedures to increasethe order of approximation.We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.
Product-formula (PF) based quantum simulation is a promising approach for simulating quantum systems on near-term quantum computers. Achieving a desired simulation precision typically requires a polynomially increasing number of Trotter steps, which remains challenging due to the limited performance of current quantum hardware. To alleviate this issue, post-processing techniques such as the multi-product formula (MPF) have been introduced to suppress algorithmic errors within restricted hardware resources. In this work, we propose a dual-channel multi-product formula that achieves a two-fold improvement in Trotter error scaling. As a result, our method enables the target simulation precision to be reached with approximately half the circuit depth compared to conventional MPF schemes. Importantly, the reduced circuit depth directly translates into lower physical error mitigation overhead when implemented on real quantum hardware. We demonstrate that, for a fixed CNOT count as a measure of quantum circuit, our proposal yields significantly smaller algorithmic errors, while the sampling error remains essentially unchanged.
Abstract In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g., a measurement, used to restrict the time evolution (due, for example, to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a C0C_0C0​ C 0 -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate O(1n)\mathcal {O}(\tfrac{1}{n})O(n1​) O ( 1 n ) of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type n\sqrt{n}n​ n -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator, and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.
Digital quantum simulation of many-body dynamics faces a tension between algorithmic Trotter error and physical noise that accumulates with circuit depth. Typical higher-order product formulas mitigate the algorithmic error at the cost of deeper circuits. Our benchmark shows that, within the limited physical error budget, the feasible advantage is confined to absolute errors well below the 10−210^{-2}10−2 scale, which vanishes under noise levels of current quantum hardware. In sharp contrast to the previous Trotter error mitigation methods, we introduce a constant-depth multi-product formula (cd-MPF) that suppresses the Trotter error by combining multiple circuits at fixed circuit depth. We identify an auxiliary parameter ααα, which reshapes the Trotter error terms while leaving the target evolution invariant. The classical linear combination of the measured expectation values cancels the leading (Δt)2(Δt)^{2}(Δt)2 algorithmic contribution and steepens the Trotter-error scaling with the two-qubit circuit depth ddd from d−2d^{-2}d−2 to d−4d^{-4}d−4. Combined with physical-noise mitigation, our method can serve as a key ingredient for realizing long-time quantum dynamics simulation on near-term hardware.