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Bulletin of the American Physical Society

Quantum compiler for classical dynamics

Dimitrios Giannakis, A. Ourmazd, Joanna Sławińska, Jöerg Schumacher

Published Dec 11, 2020• 0 citations
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Abstract

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting \emph{quantum compiler} enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The compiler is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H$. Furthermore, an embedding of classical observables into self-adjoint operators on $\mathcal H$ is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of the Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit with an $O(n)$ gates and no interchannel communication. Furthermore, the circuit features a quantum Fourier transform stage with $O(n^2)$ gates, which generates predictions by measurement in the standard computational basis. We implement the framework in Qiskit, and demonstrate its consistency for prototypical periodic and quasiperiodic systems.

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We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting \emph{quantum compiler} enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The compiler is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, H\mathcal HH. Furthermore, an embedding of classical observables into self-adjoint operators on H\mathcal HH is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of the Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of H\mathcal HH, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a 2n2^n2n-dimensional tensor product Hilbert space associated with nnn qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an nnn-channel quantum circuit with an O(n)O(n)O(n) gates and no interchannel communication. Furthermore, the circuit features a quantum Fourier transform stage with O(n2)O(n^2)O(n2) gates, which generates predictions by measurement in the standard computational basis. We implement the framework in Qiskit, and demonstrate its consistency for prototypical periodic and quasiperiodic systems.
We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, H\mathcal{H}H. Furthermore, an embedding of classical observables into self-adjoint operators on H\mathcal{H}H is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of H\mathcal{H}H, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a 2n{2}^{n}2n-dimensional tensor product Hilbert space associated with nnn qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an nnn-channel quantum circuit of size O(n)O(n)O(n) and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size O(n)O(n)O(n), and a quantum Fourier transform stage of size O(n2)O({n}^{2})O(n2), which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension 2n{2}^{n}2n using circuits of size O(n2)O({n}^{2})O(n2). We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.
The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many authors, often independently, into a framework now broadly known as Koopman-von Neumann representation of classical dynamics. This article surveys aspects of this framework for measure-preserving ergodic dynamical systems and connects it with recent approximation techniques for Koopman and transfer operators that are amenable to data-driven numerical implementation. In broad terms, these methods are based on representations of (i) classical observables as elements of an algebra of operators acting on a Hilbert space; and (ii) classical probability measures as elements of the state space of that algebra, with lifted versions of the Koopman and transfer operators inducing dynamical evolution of observables and states, respectively. A common theme underlying the techniques surveyed here is the use of reproducing kernel Hilbert spaces with coalgebra structure (so-called "reproducing kernel Hilbert algebras'') that aids the quantum representation of classical objects, as well as the use of Fock spaces to build approximation schemes with high expressivity and structure preservation properties (notably, preservation of positivity and multiplicativity of composition operators). Applications to quantum algorithms for approximating the Koopman evolution of observables in systems with pure point spectra are also discussed.
We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. Each qubit (a two-level quantum system) is initially mapped to a vector in an eight dimensional probability space (equivalently, to a classical random variable with eight probabilistic outcomes). The key idea in this mapping is to store both the amplitude and phase information of the complex coefficients that describe the qubit state in the probabilities. Due to the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces, nnn qubits are then mapped to a tensor product of nnn 8-dimensional probabilistic vectors (i.e., the Hilbert space of dimension 2n2^n2n is mapped to a probability space of dimension 8n8^n8n). After this initial mapping, we show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations on these classical random variables. The key defining feature of both the mapping to the probability space and the transformations in this space (i.e., operations on the random variables) is that they are not linear, but instead affine. Using this architecture, the evolution of the 2n2^n2n complex coefficients of the quantum system can be tracked in the joint fully-correlated probabilities of the polynomial number of random variables. We then give specific procedures for implementing (1) the Deutsch-Jozsa algorithm, and (2) the Quantum Fourier Transform in the probability space. Identical to the Quantum case, simulating the Quantum Fourier Transform in the probability space requires O(n)O(n)O(n) probabilistic bits and O(n2)O(n^2)O(n2) (i.e., quadratic in the number of quantum bits) operations.
We propose a gate-based quantum algorithm for the prediction step of Bayesian state estimation based on the Fokker-Planck equation on a discretized position-velocity state space. The probability density is encoded in the amplitudes of a quantum state, enabling a compact representation of high-dimensional distributions. Exploiting the circulant structure of finite-difference operators, the evolution is realized in the spectral domain using quantum Fourier transforms and phase rotations. A key result is that the drift component can be implemented exactly in amplitude space, leading to an accurate reproduction of the classical transport dynamics. In contrast, the diffusion term does not admit a linear representation in amplitude space due to the nonlinear relation between probability density and wave function. To enable a quantum implementation, we introduce a unitary surrogate based on a Wick rotation, transforming diffusion into a dispersive phase evolution. This yields a fully unitary propagation that can be implemented efficiently on a gate-based quantum computer. The proposed method is evaluated numerically for different scenarios and shows strong agreement with the exact solution of the Fokker-Planck equation. The approach demonstrates the potential of quantum computing for Bayesian state estimation, as the representable state space grows exponentially with the number of qubits. This allows the efficient representation and propagation of probability densities that would otherwise require complex tensor decompositions on classical hardware, making the method a promising candidate for high-dimensional filtering problems.
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.
We introduce Quantum Index Algebra (QIA) as a finite, index-based algebraic framework for representing and manipulating quantum operators on Hilbert spaces of dimension 2m2^m2m. In QIA, operators are expressed as structured combinations of basis elements indexed by Boolean codes, allowing products, commutators, and conjugations to be computed through finite rules on discrete indices rather than through dense matrix arithmetic. This representation unifies combinatorial index structure, explicit matrix realization, and transformation properties under Walsh-Hadamard-type transforms within a single formalism. Using QIA and its associated block-matrix realization, we reformulate the Bernstein-Vazirani hidden-string problem in its phase-oracle form entirely within a real, finite-dimensional algebraic setting. We show that, under structured oracle access, the QIA procedure reproduces the Bernstein-Vazirani algorithm exactly and achieves the same asymptotic query complexity and circuit depth as the standard quantum algorithm. In particular, the hidden string is recovered by symbolic manipulation of a sparse algebraic representation of the oracle rather than by numerical simulation of quantum amplitudes. Our results demonstrate that the apparent quantum speed-up in this setting is a consequence of operator structure rather than Hilbert-space dimensionality alone. QIA thus provides a precise language for separating genuinely quantum resources from those arising from algebraic and combinatorial structures and offers a new perspective on the classical simulability of structured quantum circuits.