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:2601.14400 · New Journal of Physics

Pauli Propagation for Imaginary-Time Evolution

Rafael Gómez-Lurbe, Armando Pérez

quant-phSubmitted Jan 20, 2026• Announced Jan 22, 2026• 0 citations• Revised Aug 23, 2026 (v3)
v1 · Jan 20, 2026v2 · Jan 30, 2026v3 · Aug 23, 2026
arXivHTMLPDF

Abstract

We extend the Pauli Propagation framework to imaginary-time evolution and introduce imaginary-time Pauli Propagation (ITPP), an operator-based algorithm for approximating thermal and ground-state properties directly in the Pauli basis. We derive explicit imaginary-time propagation rules for Pauli strings and analyze the main approximation errors arising from Trotterization and Pauli-space truncation, including bounds that account for the non-unitary nature of the evolution. Benchmarking ITPP on the one-dimensional transverse field Ising model, we find that the method performs best in the high-temperature regime, where thermal states admit a compressed Pauli-basis approximation, while the required number of Pauli terms grows rapidly as the system approaches the ground-state limit. We further compare with tensor network simulations and study finite-temperature scalability through the number of retained Pauli terms required to reach a fixed target accuracy. These results establish ITPP as a complementary framework for imaginary-time simulation and suggest that combining imaginary-time and real-time Pauli Propagation could provide a pathway toward simulating more general non-unitary and open quantum system dynamics within a unified Pauli-based framework.

DOI
10.1088/1367-2630/ae9c65

Related papers

Thermal State Simulation with Pauli and Majorana Propagation

Manuel S. Rudolph, Armando Angrisani, Andrew Wright +3 more

We introduce a propagation-based approach to thermal state simulation by adapting Pauli and Majorana propagation to imaginary-time evolution in the Schrödinger picture. Our key observation is that high-temperature states can be sparse in the Pauli or Majorana bases, approaching the identity at infinite temperature. By formulating imaginary-time evolution directly in these operator bases and evolving from the maximally mixed state, we access a continuum of temperatures where the state remains efficiently representable. We provide analytic guarantees for small-coefficient truncation and Pauli-weight (Majorana-length) truncation strategies by quantifying the error growth and the impact of backflow. Large-scale numerics on the 1D J1-J2 model (energies) and the triangular-lattice Hubbard model (static correlations) validate efficiency at high temperatures.

quant-pharXiv:2602.04878•Feb 5, 2026•0 citations

Non-unitary Trotter circuits for imaginary time evolution

Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo +1 more

We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic imaginary time evolution (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator - that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, the suggested block encoding in this paper results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the transverse Ising model and the fermionic Hubbard model and is demonstrated to converge to the ground state of the system.

arXiv:2304.07917•Apr 16, 2023•0 citations

Finite Imaginary-Time Evolution for Polynomial Unconstrained Binary Optimization

Jaehee Kim, Juhyeon Kim, Gwonhak Lee +4 more

Imaginary-time evolution is a standard primitive for ground-state preparation but is nonunitary, precluding direct quantum implementation. We develop Finite Imaginary-Time Evolution (FinITE), a finite-beta construction for diagonal Pauli-Z cost Hamiltonians arising from polynomial unconstrained binary optimization (PUBO) instances, including QUBO and HUBO cases. FinITE uses the linear-combination-of-unitaries (LCU) framework to implement a scaled imaginary-time propagator. The commuting Pauli-Z structure makes termwise block-encodings compose without product-formula error, and higher-order Pauli-Z terms are handled directly without quadratization. The structure yields an exact finite-beta identity between the LCU success probability and the ground-subspace fidelity. Combined with a gap-based fidelity lower bound, the identity yields a closed-form sufficient imaginary-time threshold beta-star for a chosen target fidelity. The threshold depends on estimates of the spectral gap and the initial ground-subspace overlap. Because the LCU success event is flagged by a known ancilla outcome, we integrate fixed-point amplitude amplification with an explicit query-complexity bound. Statevector simulations verify the identity on a five-vertex MaxCut (QUBO) and an eight-qubit cubic HUBO instance, and shot-based simulations on the MaxCut instance illustrate the predicted finite-beta threshold and amplification procedure.

quant-pharXiv:2604.27482•May 1, 2026•0 citations

Non-unitary Trotter circuits for imaginary time evolution

Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo +1 more

Abstract We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state (GS) of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic ITE (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator—that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, the suggested block encoding in this paper results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the transverse Ising model and the fermionic Hubbard model and is demonstrated to converge to the GS of the system.

Quantum Science and Technology•Jun 4, 2024•13 citations

Efficient computation of real-time correlators using Pauli Propagation

Alexander F. Kemper, Raghav G. Jha, Arnab Bachhar +3 more

Pauli propagation has shown promise for classically simulating quantum dynamics by evolving observables directly in the Heisenberg picture. In this work, we investigate its use for computing real-time two-point time-ordered correlators in one- and two-dimensional quantum systems. One major limitation of Pauli propagation is the rapid growth in the number of Pauli strings beyond short times. We overcome this limitation by combining accurate short-time Pauli-propagation data with time extension methods based on a positivity condition and the observation that the dynamics are often dominated by a small number of characteristic frequencies. This combined approach extends correlation functions far beyond the directly accessible time window while avoiding the exponential proliferation of Pauli operators. We demonstrate that the resulting correlators retain the relevant dynamical and spectral information. Our results broaden the regime in which classical methods can reliably probe the real-time dynamics of interacting quantum many-body systems.

quant-pharXiv:2607.24924•Jul 29, 2026•0 citations

Non-Unitary Trotter Circuits for Imaginary Time Evolution

Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo +1 more

We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state (GS) of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic ITE (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator-that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, our suggested block encoding results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the fermionic Hubbard model and is demonstrated to converge to the GS of the system [1].

IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•2 citations
We extend the Pauli Propagation framework to imaginary-time evolution and introduce imaginary-time Pauli Propagation (ITPP), an operator-based algorithm for approximating thermal and ground-state properties directly in the Pauli basis. We derive explicit imaginary-time propagation rules for Pauli strings and analyze the main approximation errors arising from Trotterization and Pauli-space truncation, including bounds that account for the non-unitary nature of the evolution. Benchmarking ITPP on the one-dimensional transverse field Ising model, we find that the method performs best in the high-temperature regime, where thermal states admit a compressed Pauli-basis approximation, while the required number of Pauli terms grows rapidly as the system approaches the ground-state limit. We further compare with tensor network simulations and study finite-temperature scalability through the number of retained Pauli terms required to reach a fixed target accuracy. These results establish ITPP as a complementary framework for imaginary-time simulation and suggest that combining imaginary-time and real-time Pauli Propagation could provide a pathway toward simulating more general non-unitary and open quantum system dynamics within a unified Pauli-based framework.
We introduce a propagation-based approach to thermal state simulation by adapting Pauli and Majorana propagation to imaginary-time evolution in the Schrödinger picture. Our key observation is that high-temperature states can be sparse in the Pauli or Majorana bases, approaching the identity at infinite temperature. By formulating imaginary-time evolution directly in these operator bases and evolving from the maximally mixed state, we access a continuum of temperatures where the state remains efficiently representable. We provide analytic guarantees for small-coefficient truncation and Pauli-weight (Majorana-length) truncation strategies by quantifying the error growth and the impact of backflow. Large-scale numerics on the 1D J1-J2 model (energies) and the triangular-lattice Hubbard model (static correlations) validate efficiency at high temperatures.
We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic imaginary time evolution (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator - that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, the suggested block encoding in this paper results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the transverse Ising model and the fermionic Hubbard model and is demonstrated to converge to the ground state of the system.
Imaginary-time evolution is a standard primitive for ground-state preparation but is nonunitary, precluding direct quantum implementation. We develop Finite Imaginary-Time Evolution (FinITE), a finite-beta construction for diagonal Pauli-Z cost Hamiltonians arising from polynomial unconstrained binary optimization (PUBO) instances, including QUBO and HUBO cases. FinITE uses the linear-combination-of-unitaries (LCU) framework to implement a scaled imaginary-time propagator. The commuting Pauli-Z structure makes termwise block-encodings compose without product-formula error, and higher-order Pauli-Z terms are handled directly without quadratization. The structure yields an exact finite-beta identity between the LCU success probability and the ground-subspace fidelity. Combined with a gap-based fidelity lower bound, the identity yields a closed-form sufficient imaginary-time threshold beta-star for a chosen target fidelity. The threshold depends on estimates of the spectral gap and the initial ground-subspace overlap. Because the LCU success event is flagged by a known ancilla outcome, we integrate fixed-point amplitude amplification with an explicit query-complexity bound. Statevector simulations verify the identity on a five-vertex MaxCut (QUBO) and an eight-qubit cubic HUBO instance, and shot-based simulations on the MaxCut instance illustrate the predicted finite-beta threshold and amplification procedure.
Abstract We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state (GS) of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic ITE (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator—that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, the suggested block encoding in this paper results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the transverse Ising model and the fermionic Hubbard model and is demonstrated to converge to the GS of the system.
Pauli propagation has shown promise for classically simulating quantum dynamics by evolving observables directly in the Heisenberg picture. In this work, we investigate its use for computing real-time two-point time-ordered correlators in one- and two-dimensional quantum systems. One major limitation of Pauli propagation is the rapid growth in the number of Pauli strings beyond short times. We overcome this limitation by combining accurate short-time Pauli-propagation data with time extension methods based on a positivity condition and the observation that the dynamics are often dominated by a small number of characteristic frequencies. This combined approach extends correlation functions far beyond the directly accessible time window while avoiding the exponential proliferation of Pauli operators. We demonstrate that the resulting correlators retain the relevant dynamical and spectral information. Our results broaden the regime in which classical methods can reliably probe the real-time dynamics of interacting quantum many-body systems.
We propose an imaginary time equivalent of the well-established Pauli gadget primitive for Trotter-decomposed real time evolution, using mid-circuit measurements on a single ancilla qubit. Imaginary time evolution (ITE) is widely used for obtaining the ground state (GS) of a system on classical hardware, computing thermal averages, and as a component of quantum algorithms that perform non-unitary evolution. Near-term implementations on quantum hardware rely on heuristics, compromising their accuracy. As a result, there is growing interest in the development of more natively quantum algorithms. Since it is not possible to implement a non-unitary gate deterministically, we resort to the implementation of probabilistic ITE (PITE) algorithms, which rely on a unitary quantum circuit to simulate a block encoding of the ITE operator-that is, they rely on successful ancillary measurements to evolve the system non-unitarily. Compared with previous PITE proposals, our suggested block encoding results in shorter circuits and is simpler to implement, requiring only a slight modification of the Pauli gadget primitive. This scheme was tested on the fermionic Hubbard model and is demonstrated to converge to the GS of the system [1].