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

Physical Review Letters

Modeling an Adiabatic Quantum Computer via an Exact Map to a Gas of Particles

Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

Published Mar 23, 2007• 27 citations
Publisher

Abstract

We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}{|}^{\ensuremath{\gamma}}$, where $|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}|$ is the adiabaticity parameter. The scaling exponent for the escape probability is $\ensuremath{\gamma}=\frac{1}{2}$ for all levels, except the edge (bottom and top) ones, where $\ensuremath{\gamma}\ensuremath{\lesssim}\frac{1}{3}$. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.

DOI
10.1103/physrevlett.98.120503

Related papers

Modeling an adiabatic quantum computer

Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than |\dotλ|^γ, where |\dotλ| is the adiabaticity parameter. The scaling exponent for the escape probability is γ= 1/2 for all levels, except the edge (bottom and top) ones, where γ

arXiv:quant-ph/0701161•Jan 22, 2007•1 citation

Alternative adiabatic quantum dynamics with algorithmic applications

Joseph Cunningham, Jérémie Roland

In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.

quant-pharXiv:2605.30110•May 29, 2026•0 citations

Pechukas-Yukawa approach to the evolution of the quantum state of a parametrically perturbed system

Mumnuna A. Qureshi, Johnny Q. Zhong, Zihad Qureshi +3 more

We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.

arXiv:1706.05243•Physical Review A•Mar 16, 2018•3 citations

How the High-energy Part of the Spectrum Affects the Adiabatic Computation Gap

Yosi Atia, Dorit Aharonov

Towards better understanding of how to design efficient adiabatic quantum algorithms, we study how the adiabatic gap depends on the spectra of the initial and final Hamiltonians in a natural family of test-bed examples. We show that perhaps counter-intuitively, changing the energy in the initial and final Hamiltonians of only highly excited states (we do this by assigning all eigenstates above a certain cutoff the same value), can turn the adiabatic algorithm from being successful to failing. Interestingly, our system exhibits a phase transition; when the cutoff in the spectrum becomes smaller than roughly $n/2$, $n$ being the number of qubits, the behavior transitions from a successful adiabatic process to a failed one. To analyze this behavior, and provide an upper bound on both the minimal gap as well as the success of the adiabatic algorithm, we introduce the notion of "escape rate", which quantifies the rate by which the system escapes the initial ground state (a related notion was also used by Ilin and Lychkovskiy in arXiv:1805.04083). Our results indicate a phenomenon that is interesting on its own right: an adiabatic evolution may be robust to bounded-rank perturbations, even when the latter closes the gap or makes it exponentially small.

arXiv:1906.02581•Jun 6, 2019•0 citations

Direct mapping of quantum circuits to adiabatic algorithms

M. Stewart Siu

We give a method of mapping any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. It is a direct mapping as we use the same initial, intermediate and final states as in the circuit. Hence we show constructively the existence of efficient adiabatic evolution paths for a class of problems. Since the physical implementation of unitary gates in the circuit model requires precise timing control, the equivalent adiabatic construction may also provide a more realizable alternative.

•Sep 3, 2004•0 citations

Adiabatic quantum computation with a one-dimensional projector Hamiltonian

Avatar Tulsi

Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyze a particular class of quantum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum-energy gap, which governs the time required for a successful evolution, is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.

arXiv:0806.0385•Physical Review A•Nov 23, 2009•27 citations
We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than ∣\ensuremathλ\ifmmode˙\else\textperiodcentered\fi∣\ensuremathγ|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}{|}^{\ensuremath{\gamma}}∣\ensuremathλ\ifmmode˙\else\textperiodcentered\fi​∣\ensuremathγ, where ∣\ensuremathλ\ifmmode˙\else\textperiodcentered\fi∣|\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\lambda}}|∣\ensuremathλ\ifmmode˙\else\textperiodcentered\fi​∣ is the adiabaticity parameter. The scaling exponent for the escape probability is \ensuremathγ=12\ensuremath{\gamma}=\frac{1}{2}\ensuremathγ=21​ for all levels, except the edge (bottom and top) ones, where \ensuremathγ\ensuremath≲13\ensuremath{\gamma}\ensuremath{\lesssim}\frac{1}{3}\ensuremathγ\ensuremath≲31​. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.
We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than |\dotλ|^γ, where |\dotλ| is the adiabaticity parameter. The scaling exponent for the escape probability is γ= 1/2 for all levels, except the edge (bottom and top) ones, where γ
In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes. As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022]. We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.
We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.
Towards better understanding of how to design efficient adiabatic quantum algorithms, we study how the adiabatic gap depends on the spectra of the initial and final Hamiltonians in a natural family of test-bed examples. We show that perhaps counter-intuitively, changing the energy in the initial and final Hamiltonians of only highly excited states (we do this by assigning all eigenstates above a certain cutoff the same value), can turn the adiabatic algorithm from being successful to failing. Interestingly, our system exhibits a phase transition; when the cutoff in the spectrum becomes smaller than roughly n/2n/2n/2, nnn being the number of qubits, the behavior transitions from a successful adiabatic process to a failed one. To analyze this behavior, and provide an upper bound on both the minimal gap as well as the success of the adiabatic algorithm, we introduce the notion of "escape rate", which quantifies the rate by which the system escapes the initial ground state (a related notion was also used by Ilin and Lychkovskiy in arXiv:1805.04083). Our results indicate a phenomenon that is interesting on its own right: an adiabatic evolution may be robust to bounded-rank perturbations, even when the latter closes the gap or makes it exponentially small.
We give a method of mapping any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. It is a direct mapping as we use the same initial, intermediate and final states as in the circuit. Hence we show constructively the existence of efficient adiabatic evolution paths for a class of problems. Since the physical implementation of unitary gates in the circuit model requires precise timing control, the equivalent adiabatic construction may also provide a more realizable alternative.
Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyze a particular class of quantum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum-energy gap, which governs the time required for a successful evolution, is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.