Abstract We propose a formalism that captures the algebraic structure of many-body two-level quantum systems, and directly motivates an efficient numerical method. This formalism is based on the binary representation of the enumeration-indices of the elements of the corresponding Lie algebra. The action of arbitrarily large elements of that algebra reduces to a few bit-wise exclusive-or operations. This formalism naturally produces sparse representations of many-body density operators, the size of which we control through a dynamic truncation method. We demonstrate how this formalism applies to real-time evolution, dissipative Lindblad action, imaginary-time evolution, and projective measurement processes. We find that this approach to calculating quantum dynamics scales close to linearly with the number of non-zero components in the density operator. We refer to this exclusive-or represented quantum algebra as ORQA. As a proof of concept, we provide a numerical demonstration of this formalism by simulating quantum annealing processes for the maximum independent set problem for up to 22 two-level systems.
Abstract We propose a formalism that captures the algebraic structure of many-body two-level quantum systems, and directly motivates an efficient numerical method. This formalism is based on the binary representation of the enumeration-indices of the elements of the corresponding Lie algebra. The action of arbitrarily large elements of that algebra reduces to a few bit-wise exclusive-or operations. This formalism naturally produces sparse representations of many-body density operators, the size of which we control through a dynamic truncation method. We demonstrate how this formalism applies to real-time evolution, dissipative Lindblad action, imaginary-time evolution, and projective measurement processes. We find that this approach to calculating quantum dynamics scales close to linearly with the number of non-zero components in the density operator. We refer to this exclusive-or represented quantum algebra as ORQA. As a proof of concept, we provide a numerical demonstration of this formalism by simulating quantum annealing processes for the maximum independent set problem for up to 22 two-level systems.
High-performance numerical methods are essential not only for advancing quantum many-body physics but also for enabling integration with emerging quantum computing platforms. We present a scalable, general-purpose parallel algorithm for quantum simulations based on -represented quantum algebra (ORQA). This framework applies to arbitrary spin systems and naturally integrates with quantum circuit simulation in the Heisenberg picture, making it particularly relevant to recent large-scale experiments on superconducting qubit processors [Y. Kim ]. As a benchmark, we simulate the kicked Ising model on a 127-qubit heavy-hexagon lattice, tracking the time evolution of local magnetization using up to 1 × 10 12 Pauli strings. Executed on the supercomputer Fugaku, our simulations exhibit strong scaling up to 2 17 parallel processes with near-linear communication overhead. These results establish ORQA as a practical and high-performance tool for quantum many-body dynamics and highlight its potential for integration into hybrid quantum-classical computational frameworks, complementing recent advances in tensor network and surrogate simulation techniques.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
Efficient manipulation of Pauli-algebraic objects is a key bottleneck in the classical emulation and benchmarking of quantum algorithms for chemistry and many-body physics. This bottleneck appears in Hamiltonian construction, variational ansatz preparation, expectation-value and gradient evaluation, and real-time propagation, all of which require repeated Pauli-algebra operations. Here, we present a high-performance Pauli-algebra framework tailored to quantum many-body and quantum-chemical simulations. The framework combines compact binary symplectic encoding, canonical coefficient reduction, and grouped sparse operator representations that exploit shared bit-flip patterns among Pauli strings. The resulting Julia/C\texttt{++} implementation accelerates Pauli multiplication, Hamiltonian construction, and operator--state multiplication in sparse and symmetry-adapted many-electron spaces. Benchmarks demonstrate efficient Hamiltonian construction, large-active-space VQE and ADAPT-VQE calculations, and real-time variational dynamics on modern multicore CPU and GPU architectures. These results show that structure-aware Pauli-algebra engines provide a scalable classical backend for developing and benchmarking quantum algorithms in quantum chemistry and many-body simulation.
Abstract Simulating non-Gaussian quantum dynamics in open continuous-variable systems is notoriously challenging: exact methods scale exponentially, while approximations sacrifice non-classical features such as Wigner negativity. Here we introduce an exact finite-dimensional reduction that overcomes this curse of dimensionality. We prove that the algebra of extended Gaussian quasi-probability densities (EGQPDs) is closed under Lindblad evolution with linear jump operators, reducing the infinite-dimensional Wigner PDE to a closed system of ODEs that scales polynomially with the number of modes. A necessary and sufficient condition for closure is established: the algebra remains closed iff every jump operator is at most linear; otherwise the extended algebra with polynomial prefactors is required. We further show that the minimal number of Gaussian components defines a non-Gaussian rank—a discrete complexity measure monotonically non-increasing under the dynamics. The framework yields explicit ODEs for the component parameters, admits a rigorous density theorem and error propagation bound, and provides closed-form algebraic criteria for Gaussianity, purity, and entanglement. Seven numerical experiments validate the method across Gaussian and non-Gaussian states, entanglement decay, PT-symmetric exceptional-point dynamics, and complexity scaling, demonstrating machine-precision accuracy and exponential speedups over grid and Fock methods. The EGQPD framework provides a systematic, exact, and efficient toolbox for non-Gaussian open quantum dynamics.
In this paper, we study the non-Markovian quantum dynamics in quantum computations from the perspective of a Lie algebraic approach based on numerical analysis. By vectorizing the density matrix of quantum states, the non-Markovian evolutions can be represented with high-dimensional linear time-varying equations, where the time-varying parameters arise from the non-Markovian interactions between the quantum system and environment. We study the Magnus expansion of such linear time-varying quantum dynamics and clarify how the truncation errors for the first- and second-order Magnus expansions are influenced by the non-Markovian properties. Besides, when the quantum states are measured for filtering, the dynamics can be modeled as time-varying stochastic differential equations due to the existence of measurement noise. The Magnus expansions based on quantum stochastic filtering are different when the quantum measurement noises are modeled in an {Itô} or Stratonovich approach, rendering different truncation errors. Based on this, numerical simulations further demonstrate the efficiency of Magnus expansions in simulating non-Markovian quantum dynamics without or with stochasticity, and how the truncation errors are influenced by the Lie algebras in the Liouville space.
Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.