Embedding classical dynamics in a quantum computer

Consider a classical dynamical system on a compact metric space $X$, described by a dynamical flow map

$$\Phi^{t}:X\to X\quad\mbox{with}\quad t\in\mathbb{R},$$

(1)

as indicated by a horizontal arrow in the left-hand column of Fig. 2. The classical state space $X$ is embedded into the space of Borel probability measures $\mathcal{P}(X)$ (i.e., the classical statistical space) by means of the map $\delta$ sending $x\in X$ to the Dirac measure $\delta_{x}\in\mathcal{P}(X)$ supported at $x$. The dynamics acts naturally on the classical statistical space by the pushforward map on measures,

$$\Phi^{t}_{*}:\mathcal{P}(X)\to\mathcal{P}(X)\quad\mbox{with}\quad\Phi^{t}_{*}(\nu)=\nu\circ\Phi^{-t},$$

(2)

also known as the transfer or Perron-Frobenius operator [44, 45]. The map $\delta$ has the equivariance property $\Phi^{t}_{*}\circ\delta=\delta\circ\Phi^{t}$, represented by the top loop in the the left-hand column in Fig. 2.

Associated with the dynamical system are spaces of classical observables, which we take here to be spaces of complex-valued functions on $X$. A natural example is the space of continuous functions, denoted as $C(X)$, which also forms an (abelian) algebra with respect to the pointwise product of functions. The Koopman operator [64, 65], $U^{t}$, acts on observables in $C(X)$ by composition with the flow map, i.e.,

$$U^{t}:C(X)\to C(X)\quad\mbox{with}\quad U^{t}f=f\circ\Phi^{t};$$

(3)

see also Fig. 1. The horizontal arrow in the first line of the right-hand column in Fig. 2 represents the action of the Koopman operator on a subalgebra $\mathfrak{A}\subseteq C(X)$ that will be described in Sec. II.2 below.

In this context, a simulator of the system can be described as a procedure which takes as an input an observable $f\in C(X)$ and an initial condition $x\in X$, and produces as an output a function $\hat{f}^{(t)}(x)$ approximating the evolution $f(\Phi^{t}(x))$ of the observable under the dynamics. For instance, if $\Phi^{t}$ is the flow generated by a system of ODEs $\dot{x}=\vec{V}(x)$ on $\mathcal{X}=\mathbb{R}^{m}$, and $X\subset\mathcal{X}$ is an invariant subset of this flow (e.g., an attractor), a standard simulation approach is to construct a finite-difference approximation $\hat{\Phi}^{t}:\mathcal{X}\to\mathcal{X}$ of the dynamical flow based on a timestep $\Delta t$ (using interpolation to generate a continuous-time trajectory), and obtain $\hat{f}^{(t)}(x)=f(\hat{\Phi}^{t}(x))$ by evaluating the observable of interest $f$ on the approximate trajectory. The scheme then converges in a limit of $\Delta t\to 0$ by standard results in ODE theory and numerical analysis for observables $f$ of sufficient regularity.

From an observable-centric standpoint, a simulator of the system corresponds to a linear operator $\hat{U}^{t}$ approximating the Koopman operator $U^{t}$, giving $\hat{f}^{(t)}(x)=\hat{U}^{t}f(x)$. For instance, the ODE-based approximation just mentioned can be described in this way for $\hat{U}^{t}f=f\circ\hat{\Phi}^{t}$, but note that not every approximation of $U^{t}$ has to be of the form of a composition operator by a flow. Indeed, “lifting” the task of simulation from states to (classical) observables opens the possibility of using new approximation techniques, which in some cases can resolve computational bottlenecks, e.g., due to high dimensionality ($m$) of the ambient state space $\mathcal{X}$ [66]. Invariably, every practical simulator $\hat{U}^{t}$ is restricted to act on a space of observables of finite dimension, $N$ (e.g., a subspace of $C(X)$ or $L^{2}$). In general, the computation cost of acting with $\hat{U}^{t}$ on elements of this space scales as $N^{2}$, but can be reduced to $O(N)$ if $\hat{U}^{t}$ is efficiently represented by a diagonal matrix. The evaluation cost of observables, which corresponds to summation of an $N$-term basis expansion such as a Fourier series, is typically $O(N)$.

In what follows, rather than employing an approximation $\hat{U}^{t}$ acting on classical observables, our goal is to simulate the action of $U^{t}$ using a quantum mechanical system. As we will see, this can be achieved at a logarithmic cost of elementary quantum operations (gates); specifically, QECD allows simulation of spaces of classical observables of dimension $N=2^{n}$ using $O(n^{2})$ gates.

The QECD framework effecting the representation of the classical system by a quantum mechanical system employs the following key spaces:

1. 1.

   The classical state space $X$.

2. 2.

   A Banach ^∗-algebra $\mathfrak{A}\subseteq C(X)$ of classical observables.

3. 3.

   An infinite-dimensional RKHS $\mathcal{H}\subset\mathfrak{A}$.

4. 4.

   A finite-dimensional Hilbert space $\mathbb{B}_{n}$ associated with the quantum computer.

The Hilbert spaces $\mathcal{H}$ and $\mathbb{B}_{n}$ have corresponding (non-abelian) algebras of bounded linear operators, $B(\mathcal{H})$ and $B(\mathbb{B}_{n})$, respectively, acting as quantum mechanical observables. Moreover, states on these algebras are represented by density operators, i.e., trace-class, positive operators of unit trace, acting on the respective Hilbert space. We denote the spaces of density operators on $\mathcal{H}$ and $\mathbb{B}_{n}$ by $Q(\mathcal{H})$ and $Q(\mathbb{B}_{n})$, respectively. Below, $n$ represents the number of qubits, thus the dimension of $\mathbb{B}_{n}$ is $2^{n}$.

The spaces of classical states and observables $X$ and $\mathfrak{A}$ are mapped into the spaces of quantum states and observables $Q(\mathbb{B}_{n})$ and $B(\mathbb{B}_{n})$, respectively; see Fig. 2. The following maps on states (left-hand column) and observables (right-hand column) transform the classical system into a quantum-mechanical one on $\mathbb{B}_{n}$:

• •

  We construct a map $\hat{\mathcal{F}}_{n}:X\to Q(\mathbb{B}_{n})$ from classical states (points) in $X$ to quantum states on $\mathbb{B}_{n}$. By analogy with the RKHS-valued feature maps in machine learning [67], $\hat{\mathcal{F}}_{n}$ will be referred to as a quantum feature map. To arrive at $\hat{\mathcal{F}}_{n}$, the classical statistical space $\mathcal{P}(X)$ is first embedded into the quantum mechanical state space $Q(\mathcal{H})$ associated with $\mathcal{H}$ through a map $P:\mathcal{P}(X)\to Q(\mathcal{H})$ (see (22) below). The composite map $\mathcal{F}:=P\circ\delta$ thus describes a one-to-one quantum feature map from $X$ into $Q(\mathcal{H})$. Next, the infinite-dimensional space $Q(\mathcal{H})$ is projected onto a finite-dimensional quantum state space $Q(\mathcal{H}_{n})$ associated with a $2^{n}$-dimensional subspace $\mathcal{H}_{n}\subset\mathcal{H}$ by means of a map $\bm{\Pi}^{\prime}_{n}:Q(\mathcal{H})\to Q(\mathcal{H}_{n})$. We refer to this level of description as matrix mechanical since all quantum states and observables are finite-rank operators, represented by $2^{n}\times 2^{n}$ matrices. To arrive at the quantum computational state space, we finally apply a unitary $\mathcal{W}_{n}:Q(\mathcal{H}_{n})\to Q(\mathbb{B}_{n})$, so that the full quantum feature map from $X$ to $Q(\mathbb{B}_{n})$ takes the form $\hat{\mathcal{F}}_{n}=\mathcal{W}_{n}\circ\bm{\Pi}^{\prime}_{n}\circ P\circ\delta$.

• •

  We construct a linear map $\hat{T}_{n}:\mathfrak{A}\to B(\mathbb{B}_{n})$ from classical observables in $\mathfrak{A}$ to quantum mechanical observables in $B(\mathbb{B}_{n})$. This map takes the form $\hat{T}_{n}=\mathcal{W}_{n}\circ\bm{\Pi}_{n}\circ T$, where $\bm{\Pi}_{n}:B(\mathcal{H})\to B(\mathcal{H}_{n})$ is a projection, so that $\hat{T}_{n}$ yields a quantum computational representation of classical observables passing through intermediate quantum mechanical and matrix mechanical representations. Here, $T:\mathfrak{A}\to B(\mathcal{H})$ is one-to-one on real-valued functions in $\mathfrak{A}$, and $Tf$ is self-adjoint whenever $f$ is real.

Next, we describe the maps governing the temporal evolution of states and observables, represented by horizontal arrows in Fig. 2:

• •

  At the quantum mechanical level, states in $Q(\mathcal{H})$ evolve under the operator $\Psi^{t}$ (horizontal arrow in the left-hand column) induced by a unitary Koopman operator $U^{t}=e^{tV}$ on $\mathcal{H}$. This evolution is generated by a skew-adjoint generator $V:D(V)\to\mathcal{H}$, defined on a dense subspace $D(V)\subset\mathcal{H}$ and possessing a countable spectrum of eigenfrequencies.

• •

  The generator $V$ is mapped to a self-adjoint Hamiltonian $H_{n}:\mathbb{B}_{n}\to\mathbb{B}_{n}$ given by $H_{n}=\mathcal{W}_{n}\bm{\Pi}_{n}V/i$. This Hamiltonian is decomposable as a sum $H_{n}=\sum_{j}G_{j}$ of mutually-commuting operators $G_{j}\in B(\mathbb{B}_{n})$, each of which is of pure tensor product form, $G_{j}=\bigotimes_{i=1}^{n}G_{ij}$. The latter property enables quantum parallelism in the unitary evolution $\hat{\Psi}^{t}_{n}:Q(\mathbb{B}_{n})\to Q(\mathbb{B}_{n})$ at the quantum computational level generated by $H_{n}$ (see horizontal arrow at the bottom of the left-hand column of Fig. 2). One of our main results is that $\hat{\Psi}^{t}$ can be implemented via a quantum circuit of size $O(n)$ and no interchannel communication (see Figs. 6 and 8).

• •

  The horizontal arrow at the quantum mechanical level represents the action of the Heisenberg evolution operator $\mathcal{U}^{t}:B(\mathcal{H})\to B(\mathcal{H})$. Under the assumption that the RKHS $\mathcal{H}$ is invariant under the Koopman operator, $\mathcal{U}^{t}$ acts on $B(\mathcal{H})$ by conjugation with $U^{t}$, i.e., $\mathcal{U}^{t}A=U^{t}AU^{t*}$.

• •

  The corresponding Heisenberg evolution operator at the quantum computational level, $\hat{\mathcal{U}}^{t}_{n}:B(\mathbb{B}_{n})\to B(\mathbb{B}_{n})$, acting on quantum mechanical observables on the Hilbert space $\mathbb{B}_{n}$, is represented by the horizontal arrow at the bottom of the right-hand column. This operator is obtained by projection of $\mathcal{U}^{t}$, viz. $\hat{\mathcal{U}}_{n}^{t}=\mathcal{W}_{n}\bm{\Pi}_{n}\mathcal{U}^{t}$.

Given a classical initial condition $x\in X$, the quantum computational system constructed by QECD makes probabilistic predictions $\hat{f}^{(t)}_{n}(x)$ of $f(\Phi^{t}(x))$ through quantum mechanical measurement of the projection-valued measure (PVM) [4, 68] associated with the quantum register on the quantum state $\hat{\rho}^{(t)}_{x,n}:=\hat{\Psi}^{t}_{n}(\hat{\rho}_{x,n})$, where $\hat{\rho}_{x,n}=\hat{\mathcal{F}}_{n}(x)$. The state $\hat{\rho}_{x,n}$ is prepared by means of a circuit of size $O(n)$, which is applied to the standard initial state vector of the quantum computer. Furthermore, the measurement step is effected by performing a rotation $\hat{\rho}^{(t)}_{x,n}\mapsto\tilde{\rho}^{(t)}_{x,n}$ by a QFT, which is implementable via a circuit of size $O(n^{2})$. An ensemble of such measurements then approximates the quantum mechanical expectation value

$$\langle\hat{T}_{n}f\rangle_{\hat{\rho}^{(t)}_{x,n}}:=f^{(t)}_{n}(x).$$

(4)

The function $x\mapsto f^{(t)}_{n}(x)$ converges in turn uniformly to the true classical evolution, i.e., $U^{t}f(x)$, in the large-qubit limit, $n\to\infty$. We will return to these points in a more detailed discussion in Secs. V–VIII.

In summary, the key distinguishing aspects of QECD are as follows:

1. wide

   Dynamical consistency. The predictions made by the quantum quantum computational system via (4) converge to the true classical evolution as the number of qubits $n$ increases. In particular, since $\dim\mathbb{B}_{n}=2^{n}$, the convergence is exponentially fast in $n$.

2. wiide

   Quantum efficiency. The full circuit implementation of the scheme, including state preparation, dynamical evolution, and measurement, requires a circuit of size $O(n^{2})$ and depth $O(n)$. Since, as just mentioned, the dimension of $\mathbb{B}_{n}$ increases exponentially with $n$, the quantum computational system constructed by QECD has an exponential advantage over classical simulators of the underlying $2^{n}$-dimensional subspace of classical observables.

3. wiiide

   State preparation. The quantum computational state $\hat{\rho}_{x,n}$ corresponding to classical state $x$ is prepared by passing the standard initial state vector of the quantum computer through a circuit of size $O(n)$ and depth $O(1)$. This overcomes the expensive (potentially exponential) state preparation problem affecting many quantum computational algorithms.

4. wivde

   Measurement process. The process of querying the system to obtain predictions is a standard projective measurement of the quantum register. Importantly, no quantum state tomography or auxiliary classical computation is needed to retrieve the relevant information.

In the ensuing sections, we lay out the properties of the classical system under study (Sec. III), and describe the conversion to the quantum computational system using QECD (Secs. IV–VIII).

We focus on the class of continuous, measure-preserving, ergodic flows with a pure point spectrum generated by finitely many eigenfrequencies and continuous corresponding eigenfunctions. Every such system is topologically conjugate (for our purposes, equivalent) to an ergodic rotation on a $d$-dimensional torus, so we will set $X=\mathbb{T}^{d}$ without loss of generality. Using the notation $x=(\theta^{1},\ldots,\theta^{d})$ to represent a point $x\in\mathbb{T}^{d}$, where $\theta^{j}\in[0,2\pi)$ are canonical angle coordinates, the dynamics is described by the flow map

$$\Phi^{t}(x)=(\theta^{1}+\alpha_{1}t,\ldots,\theta^{d}+\alpha_{d}t)\mod 2\pi,$$

(5)

where $\alpha_{1},\ldots,\alpha_{d}$ are positive, rationally independent (incommensurate) frequency parameters. This dynamical system is also known as a linear flow on the $d$-torus, but note that $\mathbb{T}^{d}$ is not a linear space. In dimension $d>1$, the orbits $\Phi^{t}(x)$ of the dynamics do not close by incommensurability of the $\alpha_{j}$, each forming a dense subset of the torus (i.e., a given orbit passes by any point in $\mathbb{T}^{d}$ at an arbitrarily small distance). The case $d=2$ is shown for two choices of $\alpha_{j}$ in Fig. 3, illustrating the difference between ergodic and non-ergodic dynamics. In dimension $d=1$, the flow map corresponds to a harmonic oscillator on the circle, $\mathbb{T}^{1}=S^{1}$, where each orbit is periodic and samples the whole space.

It is important to note that if the dynamical system is not presented in the form of a torus rotation, then standard constructions from ergodic theory may be used to transform it into the form in (5). These constructions are based entirely on spectral objects (i.e., eigenfunctions and eigenfrequencies) associated with the Koopman operator of the system. See Sec. I D in the SM for further details [63]. The same constructions allow one to treat the case where $X$ is a periodic or quasiperiodic attractor of a dynamical flow $\Phi^{t}:\mathcal{X}\to\mathcal{X}$ on a higher-dimensional space $\mathcal{X}\supseteq X$. By virtue of these facts, the quantum mechanical framework described in this paper can readily handle simulations of observables of general measure-preserving, ergodic flows with pure point spectrum. Relevant examples include ODE models on $\mathcal{X}=\mathbb{R}^{m}$ with quasiperiodic attractors [69], as well as PDE models where $\mathcal{X}$ is an infinite-dimensional function space. The latter class includes many pattern-forming physical systems such as thermal convection flows [70], plasmas [71], and reaction-diffusion systems [72] in moderate-forcing regimes.

At any dimension $d$, the flow in (5) is measure-preserving and ergodic for a probability measure $\mu$ given by the normalized Haar measure. The dynamics of classical observables $f:X\to\mathbb{C}$ is governed by the Koopman operator $U^{t}$, which is a linear operator, acting by composition with the dynamical flow in accordance with (3) [44, 73, 45]. The Koopman operator acts as an isometry on the Banach space of continuous functions on $X$, i.e., $\lVert U^{t}f\rVert_{C(X)}=\lVert f\rVert_{C(X)}$, where $\lVert f\rVert_{C(X)}=\max_{x\in X}\lvert f(x)\rvert$ is the uniform norm. In addition, $U^{t}$ lifts to a unitary operator on the Hilbert space $L^{2}(\mu)$ associated with the invariant measure. That is, using $\langle f,g\rangle_{L^{2}(\mu)}=\int_{X}f^{*}g\,d\mu$ to denote the $L^{2}(\mu)$ inner product, we have $\langle U^{t}f,U^{t}g\rangle_{L^{2}(\mu)}=\langle f,g\rangle_{L^{2}(\mu)}$ for all $f,g\in L^{2}(\mu)$, which implies, in conjunction with the invertibility of $\Phi^{t}$, that

$$U^{t*}={U^{t}}^{-1}.$$

Here, $U^{t*}$ denotes the operator adjoint, which is also frequently denoted as $(U^{t})^{\dagger}$. The collection $\{U^{t}:L^{2}(\mu)\to L^{2}(\mu)\}_{t\in\mathbb{R}}$ then forms a strongly continuous unitary group under composition of operators ¹¹1This implies (i) $U^{s}\circ U^{t}=U^{s+t}$, (ii) $(U^{t})^{-1}=U^{-t}$, and (iii) $\lim_{t\to 0}U^{t}f=f$ for all $s,t\in\mathbb{R}$ and $f\in L^{2}(\mu)$.. See again Fig. 1.

By Stone’s theorem on one-parameter unitary evolution groups [75], the Koopman group on $L^{2}(\mu)$ has a skew-adjoint infinitesimal generator, i.e., an operator $V:D(V)\to L^{2}(\mu)$ defined on a dense subspace $D(V)\subset L^{2}(\mu)$ satisfying

$$V^{*}=-V\quad\mbox{and}\quad Vf=\lim_{t\to 0}\frac{U^{t}f-f}{t},$$

(6)

for all $f\in D(V)$. The generator gives the Koopman operator at any time $t$ by exponentiation,

$$U^{t}=e^{tV}.$$

(7)

Modulo multiplication by $1/i$ to render it self-adjoint, it plays an analogous role to a quantum mechanical Hamiltonian generating the unitary Heisenberg evolution operators.

As already noted, the torus rotation in (5) is a canonical representative of a class of continuous-time continuous dynamical systems on topological spaces with quasiperiodic dynamics generated by finitely many basic frequencies. This means that every such system can be transformed into an ergodic torus rotation of a suitable dimension by a homeomorphism (continuous, invertible map with continuous inverse). By specializing to this class of systems (as opposed to a more general measure-preserving, ergodic flow), we gain two important properties:

1. wide

   The dynamics has no mixing (chaotic) component. This implies that the spectrum of the Koopman operator for this system acting on $L^{2}(\mu)$, or a suitable RKHS as in what follows, is of “pure point” type, obviating complications arising from the presence of continuous spectrum as would be the case under mixing dynamics.

2. wiide

   The state space $X$ is a smooth, closed manifold with the structure of a connected, abelian Lie group. The abelian group structure, in particular, renders this system amenable to analysis with Fourier analytic tools.

Below, we use a $d$-dimensional vector $j=(j_{1},\ldots,j_{d})\in\mathbb{Z}^{d}$ to represent a generic multi-index, and

$$\phi_{j}(x)=\prod_{m=1}^{d}\varphi_{j_{m}}(\theta^{m})\quad\mbox{with}\quad\varphi_{l}(\theta)=e^{il\theta},$$

(8)

to represent the Fourier functions on $\mathbb{T}^{d}$. In Sec. XI, we will discuss possible avenues for extending the framework presented here to other classes of dynamical systems, such as mixing dynamical systems with continuous spectra of the Koopman operators.

According to the scheme described in Sec. II.2, we perform quantum conversion of an (abelian) algebra $\mathfrak{A}$ of classical observables, i.e., a space of complex-valued functions on $X$ which is closed under the pointwise product of functions. We construct $\mathfrak{A}$ such that it is a subalgebra of $C(X)$ with additional (here, $C^{\infty}$) regularity and RKHS structure. This structure is induced by a smooth, positive-definite kernel function $\tilde{k}:X\times X\to\mathbb{R}$, which has the following properties for every point $x\in X$ and function $f\in\mathfrak{A}$:

1. 1.

   The kernel section $\tilde{k}_{x}:=\tilde{k}(x,\cdot)$ lies in $\mathfrak{A}$.

2. 2.

   Pointwise evaluation, $x\mapsto f(x)$, is continuous, and satisfies

   $$f(x)=\langle\tilde{k}_{x},f\rangle_{\mathfrak{A}},$$

   (9)

   where $\langle\cdot,\cdot\rangle_{\mathfrak{A}}$ is the inner product of $\mathfrak{A}$.

Equation (9) is known as the reproducing property, and underlies the many useful properties of RKHSs for tasks such as function approximation and learning. Note, in particular, that $L^{2}$ spaces, which are more commonly employed in Koopman operator theory and numerical techniques (see Sec. III.1), do not have a property analogous to (9). In fact, pointwise evaluation is not even defined for the $L^{2}(\mu)$ Hilbert space on $\mathbb{T}^{d}$. See Refs. [76, 46, 47] for detailed expositions on RKHS theory.

Our construction of $\mathfrak{A}$ follows Ref. [50]. We begin by setting parameters $p\in(0,1)$ and $\tau>0$, and defining the map $\lvert\cdot\rvert_{p}:\mathbb{Z}^{d}\to\mathbb{R}_{+}$,

$$\lvert j\rvert_{p}:=\lvert j_{1}\rvert^{p}+\ldots+\lvert j_{d}\rvert^{p},$$

and the functions $\psi_{j}\in C(X)$,

$$\psi_{j}:=e^{-\tau\lvert j\rvert_{p}/2}\phi_{j}\quad\text{with}\quad j\in\mathbb{Z}^{d}.$$

We then define a kernel $\tilde{k}:X\times X\to\mathbb{R}_{+}$ via the series

$$\tilde{k}(x,x^{\prime})=\sum_{j\in\mathbb{Z}^{d}}\psi^{*}_{j}(x)\psi_{j}(x^{\prime}),$$

(10)

where the sum over $j$ converges uniformly on $X\times X$ to a smooth function. Intuitively, $\tau$ can be thought of as a locality parameter for the kernel, meaning that as $\tau$ decreases $\tilde{k}(x,x^{\prime})$ becomes increasingly concentrated near $x=x^{\prime}$, approaching a $\delta$-function as $\tau\to 0$.

An important property of the kernel that holds for any $\tau>0$ is that it is translation-invariant on the abelian group $X=\mathbb{T}^{d}$. That is, using additive notation to represent the binary group operation on $X$, we have

$$\tilde{k}(x+y,x^{\prime}+y)=\tilde{k}(x,x^{\prime}),\quad\forall x,x^{\prime},y\in X.$$

(11)

In particular, setting $y=\Phi^{t}(e)$, where $e$ is the identity element of $X$, and noticing that the dynamical flow from (5) satisfies $\Phi^{t}(x)=x+\Phi^{t}(e)$, we deduce the dynamical invariance property

$$\tilde{k}(\Phi^{t}(x),\Phi^{t}(x^{\prime}))=\tilde{k}(x,x^{\prime}),\quad\forall x,x^{\prime}\in X,\quad\forall t\in\mathbb{R}.$$

In Ref. [50] it was shown that for every $p>0$ and $\tau>0$, the kernel $\tilde{k}$ in (10) is a strictly positive-definite kernel on $X$, so it induces an RKHS, $\mathfrak{A}$, which is a dense subspace of $C(X)$. One can verify that the collection $\{\psi_{j}:j\in\mathbb{Z}^{d}\}$ forms an orthonormal basis of $\mathfrak{A}$, consisting of scaled Fourier functions, so every observable $f\in\mathfrak{A}$ admits the expansion

$$f=\sum_{j\in\mathbb{Z}^{d}}\tilde{f}_{j}\psi_{j}=\sum_{j\in\mathbb{Z}^{d}}\tilde{f}_{j}e^{-\tau\lvert j\rvert_{p}/2}\phi_{j},$$

where the sum over $j$ converges in $\mathfrak{A}$ norm. The above manifests the fact that $\mathfrak{A}$ contains continuous functions with Fourier coefficients decaying faster than any polynomial, implying in turn that every element of $\mathfrak{A}$ is a smooth function in $C^{\infty}(X)$.

It can also be shown that the RKHS induced by $\tilde{k}$ acquires an important special property which is not shared by generic RKHSs—namely, it becomes an abelian, unital, Banach ^∗-algebra under pointwise multiplication of functions. We list the defining properties for completeness in Appendix A.1. In Ref. [50], the space $\mathfrak{A}$ was referred to as a reproducing kernel Hilbert algebra (RKHA) as it enjoys the properties of both RKHSs and Banach algebras. In particular, a distinguishing aspect of $\mathfrak{A}$ is that it simultaneously has Hilbert space structure (as $L^{2}(\mu)$) and Banach ^∗-algebra structure (as $C(X)$), while also allowing pointwise evaluation by continuous functionals, (i.e., the reproducing property in (9)). The RKHAs associated with the family of kernels in (10) are examples of harmonic Hilbert spaces on locally compact abelian groups [48], and are also closely related (by Fourier transforms) to weighted convolution algebras [49] on the dual group $\mathbb{Z}^{d}$ of $X=\mathbb{T}^{d}$.

Table 1 summarizes the properties of $\mathfrak{A}$ and other function spaces on $X$ employed in this work. In what follows, we shall let $\mathfrak{A}_{\text{sa}}$ denote the set of self-adjoint elements of $\mathfrak{A}$, i.e., the elements $f\in\mathfrak{A}$ satisfying $f^{*}=f$. Since the ^∗ operation of $\mathfrak{A}$ corresponds to complex conjugation of functions, it follows that $\mathfrak{A}_{\text{sa}}$ contains the real-valued functions in $\mathfrak{A}$. Note that if $f=\sum_{j\in\mathbb{Z}^{d}}\tilde{f}_{j}\psi_{j}$ is an element of $\mathfrak{A}_{\text{sa}}$, then its expansion coefficients in the $\psi_{j}$ basis satisfy $\tilde{f}^{*}_{j}=\tilde{f}_{-j}$.

Next, we state a product formula for the orthonormal basis functions $\psi_{j}$, which follows directly from their definition, viz.

	$\displaystyle\psi_{j}\psi_{l}$	$\displaystyle=c_{jl}\psi_{j+l}\quad\mbox{with}$
	$\displaystyle c_{jl}$	$\displaystyle=\exp\left(-\tau\frac{\lvert j\rvert_{p}+\lvert l\rvert_{p}-\lvert j+l\rvert_{p}}{2}\right).$		(12)

In the above, we interpret the coefficients $c_{jl}$ as “structure constants” of the RKHA $\mathfrak{A}$. Figure 4 displays representative matrices formed by the $c_{jl}$ in dimension $d=1$ and 2 for $p=1/4$ and $\tau=1/4$.

In the special case $d=1$, we will let $\mathfrak{A}^{(1)}$ be the RKHA on the circle $S^{1}\equiv\mathbb{T}^{1}$ constructed as above. We denote the reproducing kernel of $\mathfrak{A}^{(1)}$ by $\tilde{k}^{(1)}$, and let $\psi_{j}^{(1)}$, $j\in\mathbb{Z}$, be the corresponding orthonormal basis functions with $\psi_{j}^{(1)}(\theta)=e^{-\lvert j\rvert^{p}\tau/2}\varphi_{j}(\theta)$. It then follows that $\mathfrak{A}$ admits the tensor product factorization

$$\mathfrak{A}=\bigotimes_{i=1}^{d}\mathfrak{A}^{(1)},$$

(13)

and the reproducing kernel and orthonormal basis functions of $\mathfrak{A}$ similarly factorize as

	$\displaystyle\tilde{k}(x,x^{\prime})$	$\displaystyle=\prod_{i=1}^{d}\tilde{k}^{(1)}(\theta^{i},\theta^{i\prime}),$
	$\displaystyle\psi_{j}(x)$	$\displaystyle=\prod_{i=1}^{d}\psi^{(1)}_{j_{i}}(\theta^{i}),$

where $j=(j_{1},\ldots,j_{d})$, and $\theta^{i}$, $\theta^{i\prime}$ are canonical angle coordinates of the points $x=(\theta^{1},\ldots,\theta^{d})$, $x^{\prime}=(\theta^{1\prime},\ldots,\theta^{d\prime})$, respectively (see also (8)).

From an operator-theoretic perspective, simulating the dynamical evolution of a continuous classical observable $f\in C(X)$ can be understood as approximating the Koopman operator $U^{t}$ on $C(X)$; for, if $U^{t}$ were known one could use it to compute $U^{t}f(x)=f(\Phi^{t}(x))$ for every observable $f\in C(X)$, time $t\in\mathbb{R}$, and initial condition $x\in X$ (cf. Sec. II). Yet, despite its theoretical appeal, consistently approximating the Koopman operator on $C(X)$ is challenging in practice, as this space lacks the Hilbert space structure underpinning commonly employed operations used in numerical techniques, such as orthogonal projections (see Table 1). For a measure-preserving, ergodic dynamical system such as the torus rotation in (5), a natural alternative is to consider the unitary Koopman operator on the $L^{2}(\mu)$ Hilbert space associated with the invariant measure $\mu$. While this choice addresses the absence of orthogonal projections on $C(X)$, $L^{2}(\mu)$ lacks the notion of pointwise evaluation of functions, so one must correspondingly abandon the notion of pointwise forecasting in this space.

In light of the above considerations, RKHSs emerge as attractive candidates of spaces of classical observables in which to perform simulation, as they allow pointwise evaluation through the reproducing property in (9) while having a Hilbert space structure. Unfortunately, an obstruction to using RKHSs in dynamical systems forecasting is that a general RKHS $\mathcal{H}$ on $X$ need not be preserved under the dynamics, even if the reproducing kernel $k$ is continuous. That is, in general, if $f:X\to\mathbb{C}$ lies in an RKHS, the composition $f\circ\Phi^{t}$ need not lie in the same space, and thus the Koopman operator is not well-defined as an operator mapping the RKHS into itself [27]. Intuitively, this is because membership of a function $f$ in an RKHS generally imposes stringent requirements in its regularity, as we discussed for example in Sec. IV.1 with the rapid decay of Fourier coefficients, which need not be preserved by the dynamical flow.

An exception to this obstruction occurs when the reproducing kernel is translation-invariant, which holds true for the class of kernels introduced in Sec. III.2 (see (11)). In fact, it can be shown [55] that the RKHA $\mathfrak{A}$ associated with the kernel $\tilde{k}$ in (10),is invariant under the Koopman operator $U^{t}$ for all $t\in\mathbb{R}$, and $U^{t}:\mathfrak{A}\to\mathfrak{A}$ is unitary and strongly continuous. Analogously to the $L^{2}(\mu)$ case, the evolution group $\{U^{t}:\mathfrak{A}\to\mathfrak{A}\}_{t\in\mathbb{R}}$ is uniquely characterized through its skew-adjoint generator $V:D(V)\to\mathfrak{A}$, defined on a dense subspace $D(V)\subset\mathfrak{A}$, and acting on observables as displayed in (6).

For the torus rotation in (5), $V$ is diagonalizable in the $\{\psi_{j}\}$ basis of $\mathfrak{A}$. That is, for $j=(j_{1},\ldots,j_{d})\in\mathbb{Z}^{d}$, we have

$$V\psi_{j}=i\omega_{j}\psi_{j},$$

where $\omega_{j}$ is a real eigenfrequency given by

$$\omega_{j}=j_{1}\alpha_{1}+\ldots+j_{d}\alpha_{d}.$$

(14)

Moreover, $V$ admits a decomposition into mutually commuting, skew-adjoint generators $V_{1},\ldots,V_{d}$ satisfying

$$V_{l}\psi_{j}=ij_{l}\alpha_{l}\psi_{j}\quad\mbox{with}\quad l=1,\dots,d.$$

(15)

In particular, since $\{\psi_{j}\}$ is an orthonormal basis, (15) completely characterizes $V_{l}$, and we have

$$\begin{gathered}V=V_{1}+\ldots+V_{d},\\ [V_{j},V_{l}]=0,\quad[V_{j},V]=0.\end{gathered}$$

(16)

It should be noted that the Koopman generator on $L^{2}(\mu)$ admits a similar decomposition to (16); see e.g., Ref. [25] for further details. Analogously to the $L^{2}(\mu)$ case, the Koopman operator on $\mathfrak{A}$ can be recovered at any $t\in\mathbb{R}$ from the generator by exponentiation as given in (7).

We choose $\mathcal{H}$ as an infinite-dimensional subspace of the RKHA $\mathfrak{A}$ containing zero-mean functions. For that, we introduce the (infinite) index set

$$J=\{(j_{1},\ldots,j_{d})\in\mathbb{Z}^{d}:j_{i}\neq 0\},$$

(17)

and define $\mathcal{H}$ as the corresponding infinite-dimensional closed subspace

$$\mathcal{H}=\overline{\operatorname{span}\{\psi_{j}:j\in J\}}.$$

The space $\mathcal{H}$ is then an RKHS with the reproducing kernel

$$k(x,x^{\prime})=\sum_{j\in J}\psi_{j}^{*}(x)\psi_{j}(x^{\prime}).$$

(18)

In particular, for every $f\in\mathcal{H}$, which is necessarily an element of $\mathfrak{A}$, the reproducing property in (9) reads

$$f(x)=\langle k_{x},f\rangle_{\mathcal{H}}=\langle\tilde{k}_{x},f\rangle_{\mathfrak{A}},$$

where $k_{x}:=k(x,\cdot)$ is the section of the kernel $k$ at $x\in X$, and $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ denotes the inner product of $\mathcal{H}$.

By excluding zero indices from the index set $J$, every element $f$ of $\mathcal{H}$ has zero mean, $\int_{X}f\,d\mu=0$, as noted above. The reason for adopting this particular definition for $\mathcal{H}$, instead of, e.g., working with the entire space $\mathfrak{A}$, is that later on it will facilitate construction of $2^{n}$-dimensional subspaces $\mathcal{H}_{n}\subset\mathcal{H}$ suitable for quantum computation (see Sec. V). In what follows, $\Pi:\mathfrak{A}\to\mathfrak{A}$ will denote the orthogonal projection with $\operatorname{ran}\Pi=\mathcal{H}$. Moreover, we set

	$\displaystyle\kappa$	$\displaystyle=k(x,x)=\sum_{j\in J}e^{-\tau\lvert j\rvert_{p}},$
	$\displaystyle\tilde{\kappa}$	$\displaystyle=\tilde{k}(x,x)=\sum_{j\in\mathbb{Z}^{d}}e^{-\tau\lvert j\rvert_{p}},$

where these definitions are independent of the point $x\in X$ by (11). We also note that, by construction, $\mathcal{H}$ is a Koopman-invariant subspace of $\mathfrak{A}$, so we may define unitary Koopman operators $U^{t}:\mathcal{H}\to\mathcal{H}$ by restriction of $U^{t}:\mathfrak{A}\to\mathfrak{A}$ from Sec. III.3.

For our purposes, a key property that the RKHS structure of $\mathcal{H}$ endows is the feature map, which is the continuous map $F:X\to\mathcal{H}$ mapping classical state $x\in X$ to the RKHS function

$$F(x)=k_{x}.$$

(19)

It can be shown that for the choice of kernel in (18), $F$ is an injective map, and the functions $\{F(x)\in\mathcal{H}:x\in X\}$ are linearly independent. It is then natural to think of the normalized feature vectors

$$\xi_{x}:=\frac{k_{x}}{\lVert k_{x}\rVert_{\mathcal{H}}}=\frac{k_{x}}{\kappa}$$

(20)

as “wavefunctions” corresponding to classical states $x\in X$.

We can generalize this idea by associating every such wavefunction $\xi_{x}$ with the pure quantum state $\rho_{x}=\langle\xi_{x},\cdot\rangle_{\mathcal{H}}\xi_{x}$. The mapping $\mathcal{F}:X\to Q(\mathcal{H})$ with

$$\mathcal{F}(x)=\rho_{x}$$

(21)

then describes an embedding of the classical state space $X$ into quantum mechanical states in $Q(\mathcal{H})$, which we refer to as a quantum feature map. Note that there is no loss of information in representing $x\in X$ by $\rho_{x}\in Q(\mathcal{H})$. Moreover, $\mathcal{F}$ can be understood as a composition $\mathcal{F}=P\circ\delta$, where $\delta:X\to\mathcal{P}(X)$ maps classical state $x\in X$ to the Dirac probability measure $\delta_{x}\in\mathcal{P}(X)$, and $P:\mathcal{P}(X)\to Q(\mathcal{H})$ is a map from classical probability measures on $X$ to quantum states on $\mathcal{H}$, such that

$$P(p)=\int_{X}\rho_{x}\,dp(x).$$

(22)

The map $P$ describes an embedding of the state space $X$ into the space of probability measures $\mathcal{P}(X)$, i.e., the classical statistical level in the left-hand column of Fig. 2. See Ref. [50] for further details on the properties of this map.

By virtue of it being an RKHS, we can also define classical and quantum feature maps for the RKHA $\mathfrak{A}$. Specifically, we set $\tilde{F}:X\to\mathfrak{A}$ and $\tilde{\mathcal{F}}:X\to Q(\mathfrak{A})$, where

$$\tilde{F}(x)=\langle\tilde{k}_{x},\cdot\rangle_{\mathfrak{A}},\quad\tilde{\mathcal{F}}(x)=\langle\tilde{\xi}_{x},\cdot\rangle_{\mathfrak{A}}\tilde{\xi}_{x},$$

(23)

and $\tilde{\xi}_{x}=\tilde{k}_{x}/\lVert\tilde{k}_{x}\rVert_{\mathfrak{A}}$. The feature maps $\tilde{F}$ and $\tilde{\mathcal{F}}$ have analogous properties to $F$ and $\mathcal{F}$, respectively, which we do not discuss here in the interest of brevity.

The quantum mechanical representation of classical observables in $\mathfrak{A}$ is considerably facilitated by the Banach algebra structure of that space. In Sec. IV.3.1, we leverage that structure to build representation maps from functions in $\mathfrak{A}$ to bounded linear operators in $B(\mathfrak{A})$. Then, in Sec. IV.3.2, we consider associated representations mapping into bounded linear operators on the RKHS $\mathcal{H}$ (which is a strict subspace of $\mathfrak{A}$), arriving at the map $T:\mathfrak{A}\to B(\mathcal{H})$ depicted in the right-hand column of Fig. 2. Additional details on the construction are provided in Appendix A.