Lie-algebraic classical simulations for quantum computing

TL;DR

The paper introduces $\,\mathfrak{g}\$-sim, a Lie-algebraic framework for classically simulating quantum dynamics that becomes efficient when the dynamical Lie algebra $\mathfrak{g}$ has polynomial size in the system. By exploiting invariant subspaces (irreps) and adjoint representations, it enables scalable evaluation of observables and gradients, including noiseless and Pauli-noise scenarios, and supports tasks such as VQA landscape analysis, pre-training, circuit synthesis, and supervised quantum ML. The authors demonstrate polynomial-resource simulations on algebras like $\mathfrak{g}_0$ for up to $n\approx 200$ qubits, and provide extensive demonstrations across variational training, pre-training strategies, and unitary compilation, revealing regimes where classical methods remain powerful and highlighting fundamental resource-theory distinctions from Wick-based approaches. This framework broadens the set of problems approachable by classical simulation, offering practical tools for benchmarking, circuit design, and QML on near-term quantum devices.

Abstract

The classical simulation of quantum dynamics plays an important role in our understanding of quantum complexity, and in the development of quantum technologies. Efficient techniques such as those based on the Gottesman-Knill theorem for Clifford circuits, tensor networks for low entanglement-generating circuits, or Wick's theorem for fermionic Gaussian states, have become central tools in quantum computing. In this work, we contribute to this body of knowledge by presenting a framework for classical simulations, dubbed "$\mathfrak{g}$-sim", which is based on the underlying Lie algebraic structure of the dynamical process. When the dimension of the algebra grows at most polynomially in the system size, there exists observables for which the simulation is efficient. Indeed, we show that $\mathfrak{g}$-sim enables new regimes for classical simulations, is able to deal with certain forms of noise in the evolution, as well as can be used to tackle several paradigmatic variational and non-variational quantum computing tasks. For the former, we perform Lie-algebraic simulations to train and optimize parametrized quantum circuits (thus effectively showing that some variational models can be dequantized), design enhanced parameter initialization strategies, solve tasks of quantum circuit synthesis, and train a quantum-phase classifier. For the latter, we report large-scale noiseless and noisy simulations on benchmark problems. By comparing the limitations of $\mathfrak{g}$-sim and certain Wick's theorem-based simulations, we find that the two methods become inefficient for different types of states or observables, hinting at the existence of distinct, non-equivalent, resources for classical simulation.

Figures (13)

• Figure 1: Theoretical framework and applications of $\mathfrak{g}$-sim.(a) Essential components of the Lie-algebraic representation of quantum dynamics. First, we need a description of the Lie algebra $\mathfrak{g}$ associated with a unitary evolution of interest \ref{['eq:lie_alg']}, as well as an irreducible representation (irrep) $\mathcal{L}_\lambda$: a subspace of the operator space that is invariant under the action of $\mathcal{G}=e^{\mathfrak{g}}$ (Definition. \ref{['def:invariant']}). Here we depict the elements of $\mathcal{L}_\lambda$ as a collection of black geometric shapes. Next, we need to know how these elements inter-connect to each other via the dynamics. This is determined entirely by the so-called representation elements \ref{['eqn:structure_factors']}. The final ingredient of $\mathfrak{g}{\text{-} }{\rm sim}$ is a classical description of the input state, i.e., the expectation values with respect to the elements of $\mathcal{L}_\lambda$\ref{['eq:descr_inout']}. (b) Comparison of naïve and Lie-algebraic approaches to quantum dynamics in the Heisenberg picture. In the naïve approach, an observable $A$ is viewed as a linear operator acting on an $n$-qubit Hilbert space $\mathcal{H}$, with $\operatorname{dim}(\mathcal{H})=2^n$. The observable is represented as a $2^n\times 2^n$ matrix whose coefficients evolve according to the von Neumann equation of motion, and thus the approach always scales as $\Theta(4^n)$ and is not classically scalable. In the Lie-algebraic approach, the observable is instead viewed as a linear combination of distinct basis terms in $\mathfrak{g}$, whose coefficients couple to each other over time via the representation elements. Whenever $\operatorname{dim}(\mathfrak{g})\in\mathcal{O}(\operatorname{poly}(n))$, $\mathfrak{g}{\text{-} }{\rm sim}$ yields a scalable classical simulation framework for irreps with $\dim(\mathcal{L}_\lambda)\in\mathcal{O}(\operatorname{poly}(n))$. (c) Summary of the potential applications of $\mathfrak{g}{\text{-} }{\rm sim}$. These efficient classical simulations have utility in simulation and optimization of quantum systems, including pre-training of VQA or QML problems, compiling unitary processes to compact quantum circuits, and characterizing QML optimization landscapes.
• Figure 2: Noiseless and noisy dynamics of a 200-qubit magic state using $\mathfrak{g}{\text{-} }{\rm sim}$. We simulate the Trotterized dynamics of an initial magic state \ref{['eqn:initial_magic_state']} under a TFXY spin-chain Hamiltonian \ref{['eqn:hamiltonian_tfxy_randomfields']}, for $n=200$ qubits and $\tau=2.81$ in $\mathfrak{g}{\text{-} }{\rm sim}$. We report values for correlators of the form $\langle \sigma^y_j \sigma^x_{j+1} \rangle$ along the dynamics. (a) In the absence of noise, we see that the correlations propagate across the whole chain. This simulation ran in 16 minutes on a single CPU core. Inset: Small oscillations in the correlators reach the far end of the chain, confirming that the light cone of the dynamics reaches from the first to last qubit. (b) Including noise in the dynamics, in the form of random 2-qubit Pauli channels \ref{['eq:pnc']} acting after each entangling gate with a fault probability of $p=3\times10^{-4}$, we see that the dissipation weakens the correlations by $6$ orders of magnitude.
• Figure 3: Performance benchmarks of $\mathfrak{g}$-sim. (a) Circuit used for the benchmark at $n=3$ qubits. The set of generators are presented in Eq. \ref{['eqn:g_tfxy']}. Here, $R_Z$ denotes a rotation about the $Z$ axis, and $R_{\mu\nu}$ a rotation generated by the Pauli operator $\sigma^\mu\sigma^\nu$. We compare the memory (b) and compute time (c) requirements for state vector simulations (blue) and $\mathfrak{g}$-sim (orange) for the unitary shown in panel (a).
• Figure 4: Overparametrization in large system sizes.(a) Convergence traces at $n=50$ qubits of the approximate training error $\epsilon_{\operatorname{TFXY}}(\bm{\theta})$. (b) The probability of converging to $\epsilon_{\operatorname{TFXY}}<10^{-4}$ for uniform random initialization of $\bm{\theta}$ as measured by 50 samples, at varied circuit depths, with corresponding number of parameters $N_p$ reported as a fraction of $\operatorname{dim}(\mathfrak{g}_0)$, and for $n$ ranging from 20 to 50 qubits.
• Figure 5: Pre-training VQE for LTFIM using $\mathfrak{g}$-sim.(a, b) Mean training error $\epsilon_{\operatorname{LTFIM}}(\bm{\theta},\bm{\phi})$ (Eq. \ref{['eqn:error_ltfim']}) for VQE with random initialization (blue) and pre-trained with $\mathfrak{g}$-sim (orange), (a) at $n=12$ qubits and varying longitudinal field strengths $h_x$ and (b) at field strength $h_x=-1.0$ and varying system sizes $n$. Dashed lines represent initial ansatz configurations, while solid lines represent trained ansätze. (c, d) Comparison of gradient variances at varying system size $n$ and longitudinal field strengths $h_x$ for (c) uniform random parameter initialization and (d) $\mathfrak{g}$-sim pre-training. Shaded bars represent bootstrapped 95% confidence intervals.

Theorems & Definitions (18)

• Definition 1: Dynamical Lie algebra
• Definition 2: Dynamical Lie group
• Definition 3: Invariant subspace
• Definition 4: adjoint representation of $\mathfrak{g}$ in the $\lambda$-th irrep
• Definition 5: Adjoint representation of $\mathcal{G}$ in the $\lambda$-th irrep
• Definition 6: Center of a group
• Theorem 1
• Corollary 1
• Corollary 2
• Definition 7: Normalizer of $\mathcal{L}_\lambda$