Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?

It is straightforward to give a sum-over-paths expression for the transition amplitudes of a quantum circuit as long as the gates in the circuit are balanced, where to be balanced is to have all nonzero transition amplitudes of equal magnitude. Here we consider the question of whether, for such circuits, the relative phases of different discrete-time paths through the configuration space can be defined in terms of a classical action, as they are for continuous-time paths. We show how to do so for certain kinds of quantum circuits, namely, Clifford circuits where the elementary systems are continuous-variable systems or discrete systems of odd-prime dimension. These types of circuit are distinguished by having phase-space representations that serve to define their classical counterparts. For discrete systems, the phase-space coordinates are also discrete variables. We show that for each gate in the generating set, one can associate a symplectomorphism on the phase-space and to each of these one can associate a generating function, defined on two copies of the configuration space. For discrete systems, the latter association is achieved using tools from algebraic geometry. Finally, we show that if the action functional for a discrete-time path through a sequence of gates is defined using the sum of the corresponding generating functions, then it yields the correct relative phases for the path-sum expression. These results are likely to be relevant for quantizing physical theories where time is fundamentally discrete, characterizing the classical limit of discrete-time quantum dynamics, and proving complexity results for quantum circuits.


I. INTRODUCTION AND SUMMARY
The sum-over-paths methodology in quantum mechanics, pioneered by Richard Feynman, offers an alternative to the standard means of expressing quantum dynamics, just as the least-action formulation of classical dynamics offers an alternative to the standard Hamiltonian formulation [1]. In particular, it allows one to determine the probability amplitude of making a transition among states for any given (possibly time-dependent) Hamiltonian operator describing the quantum dynamics of the system. There is, however, a second type of problem to which it can be applied. Here, one is given a modular description of the quantum system's dynamics-for instance, a description of a quantum circuit with gates that are drawn from some fixed set of possibilities-and the goal is to compute the transition amplitudes of the overall circuit from a knowledge of the transition amplitudes of each gate.
The distinction between these two types of problems is best illustrated by an example. Suppose one is interested in the transverse position of an atom as it passes through an interferometer. It is then useful to treat different components in the interferometer as gates in a circuit. Determining the propagator associated to a particular gate given a knowledge of the Hamiltonian governing the dynamics of the atom as it passes through that gate is a problem of the first sort. Determining the propagator associated to the entire interferometric set-up given a knowledge of the propagators associated to each gate is a problem of the second sort. We shall refer to the two sorts of problems henceforth as the continuous-time scenario and the circuit scenario respectively. In either scenario, one can consider the system's degrees of freedom to be discrete or continuous. An interferometer is an example of a circuit acting on continuous degrees of freedom, while the circuits that are most commonly studied in the field of quantum computation involve discrete degrees of freedom.
A quantum circuit can be specified by a sequence of gates, where each gate is characterized by a unitary operator. The dynamics that occurs within each gate is generally not specified. This is because only the overall functionality of the gate is important for the functionality of the circuit as a whole, and there are many different choices of the dynamics within the gate that lead to the same functionality. For instance, a piece of polaroid and a birefringent crystal both allow one to achieve the overall functionality of a polarization filter, even though the evolution of the light within the two sorts of components is quite different. Given that the dynamics internal to each gate is irrelevant-and may in fact be unknown-the problem of computing the overall functionality of a circuit cannot be cast into the sum-over-paths methodology of the continuous-time scenario. Instead, one requires a sum-over-paths methodology that is explicitly catered to the circuit scenario, wherein each gate in the circuit is treated as a black box.
It is straightforward to express the transition amplitude of a circuit in terms of a sum or integral over discretetime paths. Suppose q is a label for the basis relative to which we compute amplitudes on a given system-called the configuration of that system. For a circuit acting on n systems, the configuration of the n systems is a vector q ≡ (q (1) , . . . , q (n) ), where q (i) is the configuration of the ith system. Suppose the circuit is a sequence of N unitaries, {Û k } N k=1 , so that the total unitary isÛ =Û NÛN −1 · · ·Û 2Û1 . It is then appropriate to discretize time into N steps. Denoting the configuration at time step k by q k ≡ (q (1) k , . . . , q (n) k ), a discrete-time path through the configuration space is a sequence of N + 1 configurations, γ = ( q 0 , q 1 , . . . , q N ), (I.1) Fig. 1 depicts a circuit acting on n systems with a gate depth of N and illustrates our labelling convention for the discrete-time paths. If the configuration is a continuous variable-for instance, if the Hilbert space of each system is L 2 (R) so that q (i) ∈ R-then we can insert resolutions of the identity between every pair of adjacent unitaries to obtain q N |Û | q 0 = N k=1 q k |Û k | q k−1 d q N −1 · · · d q 1 , (I. 2) where d q k ≡ dq (1) k . . . dq (n) k . Defining the amplitude associated with the path γ ∈ R n(N +1) as the amplitude q N |Û | q 0 can be expressed as the following integral over discrete-time paths where P 0 ( q 0 , q N ) denotes the space of discrete-time paths that begin at q 0 and end at q N and P0( q0, q N ) (·) dγ denotes (·) d q N −1 · · · d q 1 .
If, on the other hand, the configuration is a discrete variable-for instance, if the Hilbert space for each system is C d so that our label is discrete, q (i) ∈ Z d -then we have (I.5) Defining the amplitude for a discrete-time path γ ∈ (Z d ) n(N +1) by Eq. (I.3), we have q N |Û | q 0 = γ∈P0( q0, q N ) A(γ), (I. 6) where P 0 ( q 0 , q N ) denotes the space of discrete-time paths that begin at q 0 and end at q N . Under various circumstances, it is possible to restrict the set of paths appearing in the sum or integral. The paradigm example of this occurs in an interference experiment, where if the particle is known to pass through a screen containing slits, then one can restrict the path integral to those paths that pass through one of the slits. For instance, if the particle reaches the plane of the screen at the kth time step and the slit in the screen is at position x, then the transition amplitude for step k has the form q k |Û k |q k−1 ∝ δ(q k − q k−1 )δ(q k − x), where δ represents the Dirac-delta function. Integrating over q k , the delta function forces all paths to pass through the point q k = x, so that one can restrict the integral to these paths alone.
Another example of a restriction on the set of paths-the one that will be important here-is when a gate in the circuit maps the set of configurations to itself via some bijective map. In this case, we have q k |Û k | q k−1 ∝ δ( q k − f ( q k−1 )) for some bijective function f , and it is sufficient to restrict the sum over paths to those paths for which q k = f ( q k−1 ).
In the continuous-time scenario, one seeks to determine the transition amplitudes for a unitary that is generated by a Hamiltonian (possibly time-dependent) over some time interval. This is achieved by partitioning the time interval into a large number of small intervals and factorizing the unitary into a sequence of unitaries, one for each time step. In the limit of small step size, it is well known that the functional over paths appearing in the path sum has the form where S[γ] is the classical action of the path γ and N is a complex number that is independent of the path. The sum-over-paths methodology relies critically on the fact that only the phase and not the magnitude of the amplitude A(γ) is path-dependent, and this in turn is a consequence of taking the limit of small step size. This fact generally fails to hold in the circuit scenario. If the gates of the circuit are black boxes, then the most fine-grained sequence into which the overall unitary associated with the circuit can be factorized is one wherein each element of the sequence corresponds to a gate in the circuit. But for an arbitrary gate, the associated unitary U k has matrix elements q k |U k | q k−1 for which the phase and the magnitude may be dependent on q k and q k−1 . Therefore, in general, both the phase and magnitude of the amplitude A(γ) may be dependent on the path γ.
Nonetheless, the functional that appears in the path sum can have a form analogous to Eq. (I.7) for specific types of quantum circuits. This occurs when all of the gates appearing in the circuit have the property of being balanced. Although each unitary in the sequence {Û k } N k=1 may in general act on all n systems, it is common to consider gate sets with gates that act on small subsets of the systems (as will be the case in the Clifford circuits we study further on). We therefore define the property of being balanced for a gateÛ where the number of inputs and outputs is m, which may differ from n. Definition 1. Let | q denote the basis elements for the inputs of the gate and let | Q denote the basis elements for its outputs. The gateÛ is said to be balanced if where N is a complex constant 1 , S( q, Q) is a function of q and Q with values in the field R, g is a smooth map 2 and δ is a Dirac-delta function on R m (or a Kronecker delta function on (Z d ) m ). In other words, for the subset of values of q and Q where the amplitude Q|Û | q is nonzero-a subset that one can always specify through a condition of the form g( q, Q) = 0-this amplitude is equal in magnitude and differs only in phase.
For a circuit composed entirely of balanced gates, the functional over discrete-time paths appearing in the path sum has the form where N is a complex number that is path-independent, S(γ) is a real-valued function of γ, and δ(g(γ)) is a Dirac delta (or Kronecker delta) function that specifies the paths of nonzero amplitude. Specifically, if the unitary at time step k is made up entirely of balanced gates, so that q k |Û k | q k−1 = N k e iS k ( q k−1 , q k ) δ(g k ( q k−1 , q k )), then we have Denoting by P( q 0 , q N ) the space of paths of nonzero amplitude that start at q 0 and end at q N , one then has for continuous variables, where P( q0, q N ) (·) dγ is integration over P( q 0 , q N ) with respect to the measure induced by δ(g(γ)) 3 . For discrete variables, (I.12) The notion of balanced gates was introduced in the context of discrete systems by Dawson et al. [3], who were also the first to consider the sum-over-paths methodology in the circuit scenario 4 . They noted that certain gate sets that are universal for quantum computation-such as the gate set consisting of only the Hadamard and Toffoli gates-are comprised entirely of balanced gates. As such, circuits built from this gate set can be analyzed by a sum-overpaths methodology, and this was used to provide simple proofs of some known complexity results, for example that BQP ⊆ PP. Bacon et al. [4] extended their work by considering algebraic circuits defined by a gate set consisting of three phase-changing gates and a Fourier transform gate. Because the elements of this gate set are also all balanced, it is possible to apply the sum-over-paths methodology to algebraic circuits as well.
Certain well-studied families of circuits, known as Clifford circuits, also have gate sets comprised entirely of balanced gates. Clifford circuits were first introduced in the context of qubits [5], but were subsequently generalized to continuous variable systems [6] and d-level systems for d > 2 (qudits) [7,8]. Dawson et al. noted that the balanced property held for qubit Clifford circuits. It is not difficult to see that it holds for continuous variable (CV) and qudit Clifford circuits as well.
In all such circuits-indeed, any circuit consisting entirely of balanced gates-the sum-over-paths methodology provides an alternative way of computing transition amplitudes for the whole circuit from a knowledge of the transition amplitudes of each gate.
In this article, we are not, however, interested in the sum-over-paths approach for its use as an alternative technique of solving quantum dynamics for circuits, but rather for the novel perspective that it offers on the difference between quantum and classical theories of that circuit.
In the case of continuous-time dynamics, the bridge between the classical and the quantum theory is made through the phase factor S[γ] that is assigned to a path γ in the path integral expression for the dynamics; it is simply the classical action of the path γ. Specifically, in the case of n systems described by continuous variables undergoing continuous-time dynamics over a time interval [0, T ], a path γ is specified, in the limit of small step-size, as a function q : [0, T ] → R n , and the classical action of this path is the integral of the Lagrangian of the system along the path, Can the phase factor S[γ] appearing in the path integral expression for discrete-time dynamics be identified as a natural quantity in an associated classical theory? In particular, can the functional S[γ] appearing in Eq. (I.9) be understood as the analogue of the action functional for a discrete-time path γ in a classical counterpart of the quantum circuit?
The main result of our article is a demonstration that it can be so understood for certain kinds of circuits. The CV case is the easiest to consider because there it is obvious which classical state space to associate to a given quantum state space and (at least for the CV Clifford gates) which classical dynamics to associate to a given quantum dynamics. So we start with the CV case.
As noted earlier, in the circuit scenario, one generally does not have a description of the dynamics internal to a gate. Nonetheless, we will begin by considering the case wherein we do have such a fine-grained description. Suppose that for the dynamics internal to a given gate, the associated classical Hamiltonian is H( q, p) and let ( q cl (t), p cl (t)) be the solution to Hamilton's equations with initial value ( q, p). Then the effective dynamics over the time interval [T i , T f ] is described by the symplectomorphism φ : ( q, p) → ( Q, P ) by setting Q = q cl (T f ) and P = p cl (T f ). Note that q cl (t) can also be characterized as the solution to the Euler-Lagrange equations with boundary conditions q cl (T i ) = q and q cl (T f ) = Q for the Lagrangian L( q,˙ q) associated to H( q, p). It is well-known ( [9], Chapter 9) that the function (I.14) generates the symplectomorphism φ in the sense that In other words, the generating function of a symplectomorphism coming from Hamiltonian dynamics is exactly the action functional evaluated on the classical trajectories. It follows that if, for the quantum dynamics internal to a gateÛ , the integral over paths from q at t = T i to Q at t = T f is equivalent to taking just the path from q to Q that satisfies the classical equations of motion, denoted γ cl , then the overall propagator for the gate reduces to the form of Eq. (I.8) where S( q, Q) = T f Ti L( q cl (t),˙ q cl (t)) dt which, by Eq. (I.14), implies that the phase factor is given by the generating function of the symplectomorphism φ associated to the gate, Suppose this is the case for all the gates in the circuit. If the symplectomorphism associated with the unitary at time step k is φ k , then the functional over discrete-time paths through the whole circuit, γ = ( q 0 , . . . , q N ) → A(γ), is of the form of Eq. (I.9) where This analysis suggests an approach to the case where one has only a black-box description of a gate's functionality rather than a description of the internal dynamics. Specifically, if one can identify the effective symplectomorphism φ associated to each gate in the classical counterpart of the quantum circuit, then one can take the generating function for φ, G φ ( q, Q), as the phase factor S( q, Q) in the expression for the amplitude, Eq. (I.8). The action functional for the discrete-time path is then taken to be the sum of the generating functions associated to the sequence of unitaries, as in Eq. (I.17).
We here show that for CV Clifford circuits, each gate in the generating set can indeed be associated to a symplectomorphism. This is achieved using the Wigner representation, which defines a noncontextual hidden variable model of the quantum dynamics [10]. We proceed to show that by using the prescription described above one obtains the correct expressions for the quantum dynamics, that is, we show (in Theorem 2) that the functional S(γ) of Eq. (I.9) is indeed a sum of the generating functions of these symplectomorphisms.
To achieve an analogous interpretation of S(γ) in Eq. (I.9) for discrete rather than continuous-variable systems in a circuit scenario, one must first of all determine what symplectic space should be associated with a given discrete quantum system. This is not evident a priori because for discrete systems, such as the intrinsic spin degree of freedom, the quantum dynamics was not obtained historically by quantizing a classical theory of discrete variables and as such the classical counterpart of this quantum dynamics is less clear.
However, it turns out that for certain qudit Clifford circuits, there is clarity about what is the natural classical counterpart. These are the so-called "quopit" Clifford circuits, where a quopit is a qudit where the dimension d is an odd prime [11]. What is special about quopit Clifford circuits is that they are known to admit a noncontextual hidden variable model, and this model provides the classical counterpart of the quantum dynamics 5 . For quopit Clifford circuits, the model is provided by a discrete analogue of the Wigner representation proposed by Gross, where the Clifford operations are represented as transformations of an affine space over the finite field Z d [12]. 6 Whereas it is a simple matter to go from the symplectomorphism associated to each gate to its generating function in the case of CV Clifford circuits, the quopit case is not so straightforward. The difficulty is that, by Eq. (I. 15), generating functions are characterised in a way which makes explicit use of differential calculus. The classical counterpart of a quopit Clifford circuit is clearly discrete and so it does not make sense to consider derivatives of functions in the usual sense.
To resolve this issue we call upon the algebra-geometry correspondence. While this has many incarnations in various areas of mathematics, the underlying idea is that one can frequently establish a dictionary which translates geometric structures of a space into structures on its algebra of functions, and vice versa. The particular instantiation we need is the duality between the geometry of so-called affine schemes and the algebra of commutative rings ( [13], Chapter II.2).
Under this duality the algebraic counterpart of an affine space over Z d is an algebra of polynomials with coefficients in Z d , where the number of variables equals the dimension of the affine space. Explicitly, for the classical phase space for n quopits one has the correspondence We exploit this correspondence by identifying what structure on the algebra of polynomials is the dual of a differential structure on the discrete space. Fortunately this question has long been answered by algebraic geometers. The structure goes by the name of the Kähler differential forms on the algebra Z d [ q, p] ( [13], Chapter II.8).
We show that this theory is sufficiently robust to define generating functions for the elementary gates of quopit Clifford circuits. There is, however, an important difference between the generating functions for CV and quopit Clifford circuits. Since the classical counterpart of a quopit Clifford circuit is discrete, the algebra-geometry correspondence forces us to define generating functions not as real-valued functions G( q, Q) of the configuration variables, but rather as polynomial functions For example, for any symplectomorphism φ associated to a single-system gate in the generating set, that is, φ : , the generating function is a polynomial in two variables, q and Q, with coefficients in Z d and hence defines a function Similarly for gates in the generating set that act on pairs of systems, φ : 2 , the associated generating function defines a function, Regardless of the number of systems on which a gate acts, the associated generating function G( q, Q) will take values in Z d . On the other hand, the function S( q, Q) appearing in the expression for the amplitude, Eq. (I.8), takes values in R. Therefore, if the amplitude is to be determined by the generating function, we require that For a quopit Clifford circuit, if the symplectomorphism associated with the gate at time step k is denoted by φ k , then the functional over discrete-time paths that appears in the path sum, γ = ( q 0 , . . . , q N ) → A(γ), can be taken to be of the form of Eq. (I.9) where We show (in Theorem 4) that this choice does indeed yield the correct expressions for the quantum dynamics, that is, we show that the functional S(γ) of Eq. (I.9) can indeed be written as a sum of the generating functions of the symplectomorphisms associated to the gates. Note that our proposal agrees with that of Baez and Gilliam [14] concerning what sorts of mathematical object should represent a discrete-time and discrete-variable analogue of the action functional.
The idea of looking at sums of generating functions as generalisations of the action functional is in part inspired by a little-known paper of Dirac [15] wherein he explores the possibility of a Lagrangian approach to quantum mechanics. While Dirac was not successful at reformulating quantum mechanics, he did notice certain formal similarities between the generating functions of symplectomorphisms and the infinitesimal generators of unitary operations that implement a change of basis, and it was this work that ultimately inspired Feynman's formulation of the path integral (as he notes in his Nobel lecture). Our results serve to clarify the precise role of generating functions in the sum-over-paths formulation of quantum theory.
We end our introduction with some comments on the potential applications of our results. For continuous-time dynamics, the sum-over-paths approach provides a means of making inferences from the Lagrangian description of the classical dynamics to its quantum dynamics. (Indeed, it is this sort of problem that has driven the development of the vast technical machinery and wide-ranging applications of the path integral.) In other words, the sum-over-paths methodology provides an approach to the problem of quantization. Recall that, unlike canonical quantization, the path integral formulation of quantum theory can be applied to classical theories that have a Lagrangian formulation but no Hamiltonian formulation. Consequently, the path integral approach has a broader scope.
In the field of quantum computation, determining which classical circuit is the analogue of a given quantum circuit is important for understanding whether the behaviour of the quantum circuit is surprising or not [16]. As such, another motivation for understanding the classical-quantum contrast in the circuit scenario is to address questions about whether the predictions of the quantum circuit are really nonclassical. Insights into this question could, for instance, shed light on whether a given computational architecture, such as the one implemented by D-wave, has inherently nonclassical features [17]. Furthermore, it has recently been shown that certain types of nonclassicality-Bell-inequality violations for instance-can constitute resources for cryptographic and computational tasks [18][19][20][21]. A broader perspective on nonclassicality in the circuit scenario, therefore, can help to identify the resources for quantum computational speed-up. A sum-over-paths methodology in the circuit scenario may even provide a means to directly "quantize" certain classical circuits.
More importantly, a sum-over-paths methodology for the circuit scenario might have relevance for theories of physics wherein time is fundamentally discrete. Many have espoused the notion that space-time might be fundamentally discrete. In the quantum context, this idea has been pursued in the field of quantum gravity [22]. In the classical context, it has been pursued through the study of cellular automata [23,24] and discrete mechanical systems [14]. In physical theories where time is discrete rather than continuous, the internal degree of freedom of systems are often also taken to be discrete, but they can just as easily be considered to be continuous: in the quantum context, one can consider scalar fields as the systems which evolve over discrete time; in the classical context, one can consider cellular automata where the internal state of each cell is a continuous variable [24].
It is our hope that the results in this article might provide some insight into how to make the correct association between discrete-time classical theories and discrete-time quantum theories.
The outline of the paper is as follows. The continuous variable and quopit Clifford circuits are considered separately in Sections II and III, respectively. This is because, although the end results are quite similar, the mathematics involved is quite different. After introducing these circuits, in Sections II A and III A, we explicitly describe the resulting sumover-paths expressions for transition amplitudes in Theorems 1 and 3. The remainder of each Section is devoted to showing that the functionals S(γ) (of Eq. (I.9)) are the generating functionals of the corresponding symplectic representations. More specifically, for the continuous variable case, we introduce generating functions in Section II B and then, in Section II C, we introduce the symplectic representation and prove Theorem 2. For the quopit case, we introduce the symplectic representation via the discrete Wigner transform in Section III B. Then, in Section III C, we introduce Kähler differentials and use them to define the corresponding generating functional. Finally, in Section III D, we prove Theorem 4. As some of the mathematics used in Section III may not be familiar to some readers, we have included an Appendix with some additional background information.
Hats (x) indicate operators. Subscripts (x k ) will be used to index time steps. Z d denotes the ring of integers modulo d. Complex conjugation will be represented by an overbar (z).

A. Sum-over-paths expression for CV Clifford circuits
We turn our attention to applying the sum-over-paths methodology to a particular example of a family of quantum circuits for which every gate in the generating set is balanced: the subset of qantum circuits known as continuous variable (CV) Clifford circuits. These have previously been studied as the appropriate generalization of qubit Clifford circuits for continuous variables [6]. In fact, it has been shown that such circuits can be efficiently simulated on a classical computer, an extension of the Gottesman-Knill Theorem from qubits to CV systems [25]. Our interest in CV Clifford circuits comes from a more foundational perspective, namely, that they can be described by a noncontextual hidden variable model [10], which provides the means by which we identify an action functional over the paths, as we shall see in Sec. II C.
The goal of this section is to determine, for an arbitrary CV Clifford circuit, a sum-over-paths expression for its transition amplitudes as in Eq. I.11. We then show that the exponent of the phase factor associated to each allowed path can be understood as a discrete-time generalisation of the action functional. We introduce this notion in Section II B and then, in Section II C, we prove that it agrees with our calculation of the aforementioned phase factor. An n-system CV Clifford circuit consists of preparations and measurements in the configuration basis of L 2 (R n ) and an elementary gate set consisting of the following 1-system and 2-system gates: ).F is called the Fourier gate and corresponds to evolution for unit duration under the Hamiltonian for a harmonic oscillator with mass 2 π and frequency π 2 (it is the analogue of the Hadamard gate in a qubit Clifford circuit). It is intuitively understood as a rotation in phase space by π/2.P (η) is called a phase gate (by analogy to the phase gate in a qubit Clifford circuit), and corresponds to a phase-space squeezing operation (via a position-dependent boost). TheX(τ ) gate (a generalization of the Pauli-X gate in a qubit Clifford circuit) implements a translation of the configuration by τ . Finally,Σ is called the sum gate (it is the analogue of the CNOT gate in a qubit Clifford circuit) and can be understood as a translation of the second system by an amount equal to the coordinate of the first. The intuitive phase-space accounts that we have just provided for these quantum gates will be shown, in Sec. II C, to be an accurate description of the associated symplectomorphisms.
Bartlett et al. [25] have shown that CV Clifford circuits can (up to a global phase factor) implement all and only those unitaries lying in the so-called n-system CV Clifford group, C n . To define C n , one must first introduce the n-system CV Pauli group G n . Denote the group of unitaries on L 2 (R n ) by U L 2 (R n ) . G n is the subgroup of U L 2 (R n ) that is generated by {X i (τ ),Ẑ i (σ) : τ, σ ∈ R, i ∈ {1, . . . , n}} whereX i (τ ) is the operator that translates system i by τ andẐ i (σ) := e iσqi is the operator that boosts system i by σ.
Definition 2. The n-system CV Clifford group, C n , is defined to be the normaliser of the n-system CV Pauli group inside U L 2 (R n ) , that is, Consider a given n-system CV Clifford circuit C implementing a unitaryÛ ∈ C n . To calculate the amplitudes for each path through the configuration space, we first need the matrix elements for the elementary gates. Lemma 1. The matrix elements for the elementary CV Clifford gates are: It follows that all of these gates are balanced. Proof.
• We use the fact thatF corresponds to evolution for unit duration under the Hamiltonian for a harmonic oscillator with mass 2 π and frequency π 2 , the matrix elements of which are well-known (e.g. problem 3-8 in [1]), to infer that It follows that • The matrix elements ofP (η) are trivial to compute: • Finally, forΣ one has It follows that For any system at any time-step where the circuit has no gate acting, we shall describe the gate as identity and denote it by 1. The identity gate is a special case ofX(τ ) where τ = 0 and a special case ofP (η) where η = 0, so that we can infer from Lemma 1 that its contribution to the amplitude is simply δ (Q − q). (Note that although we could have simply represented the identity gate byX(0) orP (0) in a description of the circuit, it is more straightforward to treat it distinctly).
For a CV Clifford circuit C, the amplitude of any path γ through configuration space is given, according to Eq. (I.9), by where N C , S C (γ) and δ(g C (γ)) can be decomposed into contributions from each time-step in the manner described by Eq. (I.10). It should be noted that gates acting on different systems during the same time-step contribute to the overall amplitude exactly as they would if they acted at consecutive time-steps, and hence their contributions also combine in the fashion described by Eq. (I.10). Lemma 1 then allows us to express the contribution of each gate to the amplitude explicitly. We see that eachF gate introduces a path-independent complex factor of 1−i 2 √ π to the normalization and eachF † gate a factor of 1+i 2 √ π . Let q(gate) denote the configuration at the input of a gate for path γ, while Q(gate) denotes the configuration at its output for path γ. In terms of this notation, eachF gate introduces a phase factor of e −iQ(gate)q(gate) , eachF † gate a phase factor of e iQ(gate)q(gate) , and eachP (η) gate a phase factor of e −i η 2 q(gate) 2 . Finally, we get nontrivial constraints on the allowed paths from the 1,P (η),X(τ ),Σ andΣ † gates. These results are summarized in the following theorem. Here, F gates denote a sum over allF gates in C, and similarly for any other sort of gate, and #(F ) (#(F † )) denotes the number ofF gates (F † gates).
Theorem 1. For an n-system CV Clifford circuit C implementing the overall unitaryÛ ∈ C n , the transition amplitudes can be computed by the sum-over-paths expression , and and where P C ( q0, q N ) (·) dγ denotes integration over q 1 , . . . , q N −1 subject to the following constraints  What is important for us is the functional form of S C (γ) because we seek to show that it can be interpreted as a generalised action functional through the theory of generating functions. Before doing so, however, we pause to present a scheme for implementing the constraint on the allowed paths, inspired by the one presented in Dawson et al. [3], and which proceeds by providing an explicit parameterization of the space of allowed paths.
For the gates 1,P (η),X(τ ),Σ andΣ † , one sees that for each configuration of the input(s), there is a unique configuration of the output(s) having non-zero amplitude. For the gatesF andF † , on the other hand, for any configuration of the input, every configuration of the output has a non-zero amplitude. For the latter gates, therefore, we must introduce a free parameter for the configuration at their output.
It follows that the number L of parameters sufficient to describe the space of allowed paths is just the sum of the number ofF gates and the number ofF † gates, L ≡ #(F )+#(F † ). We will call these the free configuration parameters and denote them by x 1 , . . . , x L , with the collection represented by the vector x ≡ (x 1 , . . . , x L ). (Note that, unlike elsewhere in this article, the subscript in this notation does not indicate the time-step to which the configuration pertains; it is merely an index for the free parameters.) It follows that every allowed path can be expressed as a function of these parameters, γ( x). However, the path γ( x) may not be an allowed path for all choices of values for the L parameters, and so the parameters are constrained to live in some subspace of R L .
To visualize the free configuration parameters and the constraints they satisfy, it is useful to annotate the circuit. The general prescription, which we illustrate with a concrete example in Fig. 2, is as follows: 1. Label the configurations of the input systems by {q , and the configurations of the output systems by {q 2. For every system immediately following anF gate or anF † gate, label the configuration of that system by x l , where l ranges from 1 to L, the number of such gates in the circuit.
3. For every system immediately following aP (η) orX(τ ) gate, and every pair of systems immediately following â Σ orΣ † gate, do not introduce a new label for the configurations of those systems, but rather specify, for each output of the gate, its functional dependence on the inputs of the gate (according to the functional relations determined in Lemma 1).
FIG. 2: An example of a 3-system CV Clifford circuit consisting of the following sequence of elementary gates. First, anF gate is implemented on the first system and aP (η) gate is implemented on the second system. Next, one has aΣ gate controlled on the second system and acting on the third system and then aΣ † gate controlled on the first system and acting on the second system. Finally, one has anX(τ ) gate acting on the first system and anF † gate acting on the third system. Also indicated is the labelling of the configurations of the systems described in the text.
Finally, constraints on the free configuarion parameters arise from the final boundary condition at the output of the circuit. For every i ∈ {1, . . . , n}, define B (i) ( x) to be the configuration of the ith system at the output of the circuit as a function of the free configuration parameters, x. The form of this function can depend on the configurations of the input systems, q 0 , which are given as initial conditions, as well as the τ and η parameters of the X(τ ) and P (η) gates which are given by the specification of the circuit. In our example of Fig. 2, for instance, (II.11) For a general circuit C, the vector of free configuration parameters, x, is constrained to the set F C ( q 0 , q f ), where In general, each constraint equation on x defines an affine hyperplane in R L . As such, F C ( q 0 , q f ) describes the (possibly empty) intersection of these affine hyperplanes.
For the example of Fig. 2, for instance, the set is Note that in our example, the free configuration parameter x 2 is fixed directly by the final boundary condition, so that one need not have introduced it. Indeed, one can restrict the free configuration parameters to the systems that are at the output of nonterminalF andF † gates (where nonterminal means not the last gate acting on a given system). This does not, however, change the complexity of solving the constraints. Given this parameterization of the allowed paths, we can rewrite Eq. (II.8) as where F C ( q0, q f ) (·) d L x denotes integration over the subspace given as the intersection of affine hyperplanes within R L that are picked out by the constraints on x 1 , . . . , x L in the definition of F C ( q 0 , q f ).
Within this integral, the phase of an allowed path is specified as a function of the free parameters x by adapting the functional form of Eq. (II.9) to the labelling scheme described above. For instance, in our example, the phase of the path detemined by free parameters x is B. The discrete-time analogue of the action functional for CV systems Let Ω j (R 2n ) denote the vector space of all j-forms on the phase space R 2n , and let Ω(R 2n ) = ⊕ 2n j=0 Ω j (R 2n ) denote the algebra of all differential forms on R 2n . Introducing canonical coordinates ( q, p), the 2-form ω ∈ Ω 2 (R 2n ) defined by is a symplectic form because it is non-degenerate and dω = 0. A smooth function φ : R 2n → R 2n , ( q, p) → ( Q, P ) is said to be a symplectomorphism if There is a canonical 1-form, which satisfies ω = − dθ. One can restate the condition for φ to be a symplectomorphism, Eq. (II. 16), in terms of this canonical 1-form θ: φ is a symplectomorphism if and only if there is aG( q, p) ∈ C ∞ (R 2n ) such that We call such aG( q, p) a generating function associated to the symplectomorphism φ ([9], Chapter 9). We note that the existence of a generating function for every symplectomorphism depends on the fact that every closed 1-form on R 2n is exact. Note that generating functions are only unique up to addition of scalars. If the q and Q variables can be taken to be independent, then one can expressG( q, p) purely in terms of q and Q, i.e., G( q, Q) :=G( q, p( q, Q)). It then follows from Eq. (II.18) that which is the sense in which G( q, Q) generates the symplectomorphism φ. Because q and Q can indeed be taken to be independent in all of the cases we will consider, whenever we refer to the generating function, we mean G( q, Q). Note that for a symplectomorphism φ : ( q, p) → ( Q, P ) that results from a continuous-time Hamiltonian dynamics acting over a finite time interval, the generating function of that symplectomorphism, G(q, Q), is simply the action of the classical trajectory over that time interval which has q as the initial configuration and Q as the final configuration, as in Eq. (I.14).
We now apply the proposal of Eq. (I.17) from the introduction, namely, that the analogue of the action functional for discrete-time dynamics is the sum of the generating functions associated to the symplectomorphisms that make up the discrete-time dynamics.
Consider an n-system CV Clifford circuit with N time-steps, so that the space of paths is R n(N +1) . Let Φ = {φ k } N k=1 be the sequence of symplectomorphisms of R 2n associated to the circuit, and denote the generating function associated to φ k by 7 The definition of an action functional for discrete-time paths of CV systems, proposed in Eq. (I.17) of the introduction, specifies that the action functional for this circuit should be as follows.
Definition 3. The action functional over paths in R n(N +1) that is associated to the sequence of symplectomorphisms We now turn to determining the explicit form of the generating functions associated to each of the gates in the generating set of the CV Clifford group.
These will be called the elementary CV Clifford symplectomorphisms.
Proof. These follow by direct computation using the definition in Eq. (II.18) • For theF gate, • For theF † gate, • For theP (η) gate, (II.34) • For theX(τ ) gate, • For theΣ gate, • For theΣ † gate, Note that instances of the identity gate correspond to the identity symplectomorphism (q, p) → (q, p) and have generating function equal to 0. Note also that if gates act in parallel on different systems, the symplectomorphism for the overall gate is simply the composition of the symplectomorphisms of the component gates. and the generating function for the overall gate is simply the sum of the generating functions of the component gates.
Let C be an n-system CV Clifford circuit consisting of N time-steps and wherein all the gates are CV Clifford gates. To such a circuit, there is an associated sequence of symplectomorphisms of R 2n , denoted Φ = {φ k } N k=1 , where each of the φ k is composed of the elementary CV Clifford symplectomorphisms described in Lemma 2. Then, according to our Definition 3, the action functional over discrete-time paths associated to Φ, denoted γ → S Φ (γ), is the sum of the generating functions associated to these symplectomorphisms. Specifically, Lemma 3 implies that where we have adopted the notational convention introduced above Theorem 1. Comparison with Eq. (II.9) establishes our main result for CV Clifford circuits.
Theorem 2. Consider an n-system CV Clifford circuit C, associated in quantum theory with a unitaryÛ ∈ C n and associated, in its classical counterpart, to a symplectomorphism Φ. The functional S Φ (γ) that specifies, via Definition 3, the action of the discrete-time path γ through the classical counterpart of the circuit is precisely equal to the functional S C (γ) that defines the phase assigned to γ in the sum-over-paths expression for the transition amplitude of the quantum circuit, Eq. (I.11).

A. Sum-over-paths expression for quopit Clifford circuits
We turn now to quopit Clifford circuits. Clifford circuits for collections of discrete systems of arbitrary dimension d were first introduced by Gottesman in [7] as a higher dimensional version of the qubit stabiliser codes for fault-tolerant quantum computation (and it was shown that the Gottesman-Knill theorem extends to these higher dimensions). A qudit of dimension d equal to an odd prime has been termed a "quopit" [11]. Hence, quopit Clifford circuits are Clifford circuits wherein the elementary systems are dimension d for d an odd prime. This Section will follow a structure similar to Section II. We begin by determining a sum-over-paths expression for transitions amplitudes of quopit Clifford circuits, as in Eq. (I.11), thus identifying the functional over paths appearing in the exponent of the phase factor. We then address the question of whether this functional admits of an interpretation in terms of a generalised action functional, just as was done in Sections II B and II C. A discrete phase space representation of quopit Clifford circuits is described in Section III B, In Section III C, it is shown how to define generating functions for symplectomorphisms on a discrete phase space using tools from algebraic geometry. In Section III D, the symplectomorphisms associated to the gates in the elementary gate set are identified, and, using the tools of Section III C, we find the associated generating functions. Finally, we show that an action functional defined via the sum of these generating functions coincides with the functional appearing in the exponent of the phase factor for the sum-over-paths expression of the circuit dynamics.
An n-quopit Clifford circuit consists of preparations and measurements in the computational basis of (C d ) ⊗n , where d is an odd prime, and has elementary gate set and arithmetic operations on elements of Z d are done modulo d. We callF the Fourier gate,R the Phase gate and Σ the Sum gate. It has been shown by Clark [26] that this gate set can (up to a global phase factor) implement any unitary lying in the n-quopit Clifford group, which we denote C d,n . To define this group, we must introduce the n-quopit Pauli group, denoted G d,n , the d-dimensional generalization of the qubit Pauli group. This is the subgroup of U (C d ) ⊗n generated by {χ(q i ), χ(p i ) : i ∈ {1, . . . , n}} and e iπ d 1, where i labels the quopits and where for a given quopit, Definition 4. The n-quopit Clifford group, C d,n , is defined to be the normaliser of the n-quopit Pauli group G d,n inside U (C d ) ⊗n , that is, C d,n := N (G d,n ).
It is evident, therefore, that these gates are balanced.
These identities are straightforward to verify. If, at some time-step, a quopit has no gate acting on it, we shall describe the gate as identity and denote it by 1. The identity gate can be obtained by acting the Fourier gate twice in succession, so that one can infer from lemma 1 and a short calculation that its contibution to the amplitude is what one expects, namely, δ Q,q Following argumentation parallel to that provided in Section II, except where the variables take values in Z d as opposed to R, we obtain the following result.
Theorem 3. Given an n-quopit Clifford circuit C implementing a unitaryÛ ∈ C d,n the transition amplitudes can be computed by the sum-over-paths expression, and where the set of allowed paths is given by P C ( q 0 , q f ), defined as the set of paths satisfying the following constraints ∀ 1,R gates : Q(gate) = q(gate) ∀Σ gates : Q (1) (gate) = q (1) (gate), Q (2) (gate) = q (1) (gate) + q (2) (gate). (III.7) For the remainder of this Section, we will show that S C (γ) can be understood as a generalised action functional. As before, however, we pause here to describe a method of implementing the constraint to the allowed paths in terms of a parameterization. We denote the initial configurations by q 0 and the final configurations by q f and for eachF gate in the circuit we introduce a free configuration parameter at its output. We denote the free configuration parameters as x ≡ (x 1 , . . . , x L ), where L = #(F ). Letting B (i) ( x) denote the configuration of the i-th system at the output of the circuit, one has that the space of allowed values of x is (III.8) Just as was found for the continuous case, each of the above equations defines an affine hypersurface in Z L d , and so F C ( q 0 , q f ) is a (possibly empty) subset of Z L d given by the intersection of n affine hypersurfaces. Given this parameterization, we have (III.9)

B. Symplectic representation of discrete systems
Consider the vector space (Z d ) 2n with basis (q (1) , . . . , q (n) , p (1) , . . . , p (n) ). One can introduce a symplectic inner product on this space in the usual fashion: Letting It is readily verified that [·, ·] is skew-symmetric and non-degenerate. As such, [ u, v] defines a symplectic inner product and (Z d ) 2n can be understood as a discrete phase space. An element S ∈ End(Z 2n d ) is said to be symplectic if it preserves the symplectic inner product, i.e., if for each The collection of all such elements forms the symplectic group, denoted by Sp(2n, Z d ). Furthermore, elements D a ∈ End(Z 2n d ) such that ∀ u ∈ Z 2n d , D a u = u + a where a ∈ Z 2n d are said to be phase-space displacements, and the collection of all such elements forms the group Z 2n d . Combinations of the latter two sorts of elements form the group Sp(2n, Z d ) Z 2n d , which we term the symplectic affine group.
Following Gross's work on the discrete Wigner representation, one can represent elements of the computational basis as probability distributations on the discrete phase space Z 2n d and elements of the Clifford group as elements of Sp(2n, Z d ) Z 2n d acting thereon [12]. This allows one to define a symplectic representation of quopit Clifford circuits. The discrete Wigner transformation associates to each density operatorρ ∈ B((C d ) ⊗n ) the quasi-probability distribution on Z 2n d defined by For computational basis elements | q 0 q 0 |, a simple calculation shows that W | q0 q0| is the uniform distribution supported on the phase space line {( q 0 , p) : p ∈ Z n d }. Similarly, for an element | p 0 P p 0 | of the momentum basis (the eigenbasis of χ(p)), W | p0 P p0| is uniformly supported on the phase space line {( q, p 0 ) : q ∈ Z n d }. For eachÛ ∈ C d,n there is an (S, a) ∈ Sp(2n, Z d ) Z 2n d such that µ(S, a) = e iθÛ for some θ. 3. For any density operatorρ and any v ∈ Z 2n d , W µ(S, a)ρµ(S, a) † (S v + a) = Wρ( v) (covariance property) Note that property 2 in Proposition 1 guarantees the existence of a symplectomorphism (S, a) for every element of the Clifford group, but not necessarily its uniqueness. Nonetheless, such uniqueness does in fact hold.

Corollary 1.
To eachÛ ∈ C d,n , there is a unique (S, a) ∈ Sp(2n, Z d ) Z 2n d such that µ(S, a) = e iθÛ for some θ. The proof of this corollary is included in Appendix A. Therefore, given a quopit Clifford circuit C, there is a sequence of symplectomorphisms of Z 2n d , Φ C = {φ k } N k=1 , where each φ k is the elementary symplectomorphism associated to one of the elementary gates composing C. We will now see that it is possible to define generating functions for these symplectomorphisms using a theory of differential forms on the affine space Z 2n d .
C. The discrete-time analogue of the action functional for discrete systems On first thought, one might think that one can only define symplectic structures on smooth manifolds. However, a careful examination of the material presented in Section II B shows that it was not the manifold structure itself which was important but rather the existence of an algebra of differential forms. To generalise to the symplectic vector space Z 2n d it therefore suffices to construct an analogue of differential forms in this context. Fortunately, the well-known Kähler differentials in algebraic geometry were invented for exactly this purpose ( [13] Chapter II.8). Rather then delve headfirst into the theory of Kähler differentials, in this Section we will instead give a concrete description which more than suffices for our purposes. For interested readers, more details about the underlying mathematical structure are provided in Appendix B.
One begins by considering the algebra of polynomials in 2n variables over Z d , Elements of Z d [ q, p] can be formally differentiated using the usual formulas for differentiating polynomial functions, except that one must remember to do all arithmetic operations modulo d. One can then define the algebra of Kähler differential forms on Z 2n d , denoted Ω(Z 2n d ), as follows. They are Z d -linear combinations of terms of the form f i1,...,i k ,j1,...j l dq (i1) ∧ · · · ∧ dq (i k ) ∧ dp (j1) ∧ · · · ∧ dp (j l ) , (III.14) subject to the same relations as the usual differential forms on R 2n . One can similarly decompose the Kähler differentials as where Ω j (Z 2n d ) is the vector space of Kähler j-forms. Finally, there is a differential d : 16) which is defined just as it is for differential forms on R 2n , except that the usual derivative is replaced with the formal derivative explained above. Just as one does for R 2n , we can extend the symplectic inner product on Z 2n d , defined in Eq. (III.11), by introducing a symplectic form ω ∈ Ω 2 (Z 2n d ), This form satisfies dω = 0 and is nondegenerate in the sense outlined in Appendix B. While we will not need the latter property for this paper, we feel that the fundamental role it plays in the formulation of classical dynamics on symplectic manifolds warrants its proof for the affine symplectic spaces we are considering.
In this context, a morphism Z 2n d → Z 2n d is a function ( q, p) → ( Q, P ) such that the components of Q and P can be written as polynomials in the components of q and p with coefficients in Z d . With this definition, we say that a morphism ( q, p) → ( Q, P ) is symplectic if and only if n i=1 dq (i) ∧ dp (i) = n i=1 dQ (i) ( q, p) ∧ dP (i) ( q, p).
This only defines the generating function up to addition by a constant. To remove this ambiguity we will choose the generating function to have no degree 0 components.
Notice that there exist forms which are closed but not exact and so one cannot necessarily associate a generating function to each symplectomorphism φ. We will see in Section III D the elementary quopit Clifford symplectomorphisms do indeed have associated generating functions.
Just as in the continuous case, it may be possible to rewrite the generating functionG( q, p) in terms of q and Q. This can be done exactly when the polynomial expressions for Q = Q( q, p) can be inverted to express p in terms of q and Q. As we will see, for the affine symplectomorphism associated to elements of the elementary quopit Clifford gates, it is always possible to do this inversion. As such, from now on when we refer to the generating function, we mean G( q, Q) :=G( q, p( q, Q)).
Finally, consider an n-quopit Clifford circuit consisting of a sequence of N gates, so that the space of paths through configuration space is Z be the sequence of symplectomorphisms of Z 2n d associated to each gate, and denote the generating function associated to φ k by (III.20) Based on the proposed definition of action functional for discrete-time paths of discrete systems, presented in Eq. (I.23) of the introduction, the action functional for a quopit Clifford circuit is as follows.
Definition 6. The action functional over paths in Z It remains to determine the precise form of the generating functions for the gates in the generating set of a quopit Clifford circuit.

D. Symplectomorphisms and generating functions for quopit Clifford gates
In Section III B, we saw how the discrete Wigner transform provides a representation of a given n-quopit N timestep Clifford circuit C on the discrete phase space Z 2n d in terms of a sequence of symplectomorphisms Φ = {φ k } N k=1 . Further, in Section III C we defined the action functional S Φ of such a sequence in terms of generating functions of the individual elements φ k . To show that the phase functional S C (γ) appearing in the path sum agrees with the action functional S Φ (γ), it remains only to compute the symplectomorphisms and generating functions of the elementary quopit Clifford gates.
Proof. These are determined by direct calculation using property 3 of Proposition 1, making frequent use of the identity, (III.22) • For theF gate, one has, where in the second last line, we used the change of variables x = b − s.
Proof. Much like Lemma 3, this follows by direct computation from the definition in Eq. (III. 19).
As before, instances of the identity gate correspond to the identity symplectomorphism (q, p) → (q, p) and have generating function equal to 0, and if gates act in parallel on different systems, the generating function for the overall gate is simply the sum of the generating functions of the component gates.
Let C be an n-system quopit Clifford circuit constituting a sequence of N time-steps, and let Φ = {φ k } N k=1 denote the sequence of symplectomorphisms of Z 2n d associated to the circuit. Using the quopit Clifford symplectomorphisms described in Lemma 5, and Definition 6, the action functional over discrete-time paths associated to Φ, denoted γ → S Φ (γ), is the sum of the generating functions associated to these symplectomorphisms. Specifically, Lemma 6 implies that which clearly corresponds to the phase factor in Eq. (III.6).
Theorem 4. Consider an n-quopit Clifford circuit C, associated in quantum theory with a unitaryÛ ∈ C d,n and associated, in its classical counterpart, to a symplectomorphism Φ. The functional S Φ (γ) that specifies, via Definition 6, the action of the discrete-time path γ through the classical counterpart of the circuit is precisely equal to the functional S C (γ) that defines the phase assigned to γ in the sum-over-paths expression for the transition amplitude of the quantum circuit.

IV. CONCLUDING REMARKS
The sum-over-paths methodology has demonstrated its utility for proving relationships between quantum complexity classes. Just as it was applied by Dawson et al. to prove upper bounds on the power of arbitrary quantum circuits, so it may be applied, using the results of this paper, to prove upper bounds on the power of Clifford circuits. In particular, it can be used to show that the outcome probabilities of Clifford circuits can be computed efficiently on a classical computer. Following the terminology introduced in Ref. [27], this means that the problem of simulating such circuits is in PSTR(n), thereby providing an alternative proof of case (iv) in Table 1 of Ref. [27], if the results there are extended to quopit systems.
One can also consider other restricted models of quantum circuits, like matchgate circuits, which are circuits consisting of a certain class of two-qubit gates. Such circuits have been shown to be classically simulable under particular conditions [28][29][30]. An open question raised in [30] is whether one can understand the classical simulability of such circuits via a classical hidden-variable model. In light of our work, a further question that can be asked is whether such circuits can be described in terms of a classical action, if one describes such circuits using the sum-overpaths formulation.
In the continuous-time scenario the action functional S(γ) plays a number of separate but related roles. The most important of these roles are: 1. it determines the classical trajectories via a least-action principle; 2. it generates the symplectomorphism associated to a given time interval via its evaluation on the classical trajectories over that time interval; 3. it defines the amplitude for a given path in the path integral expression for the system's quantum dynamics.
Our proposal for the action functional in the circuit scenario successfully fulfills roles 2 and 3. It is then natural to ask in which sense it can fulfill the first role. A Lagrangian formulation of discrete-time dynamics was introduced by Baez and Gilliam [14]. Despite focusing on systems having discrete degrees of freedom the formalism presented therein can easily be extended to include continuous degrees of freedom. Their approach to discrete systems bears many similarities to ours, in that it has its foundations in algebraic geometry and the theory of Kähler differentials. Indeed, the paper of Baez and Gilliam was a significant source of inspiration for the current work. The action functionals considered by Baez and Gilliam take the same mathematical form as ours, namely a sum of polynomials in variables describing adjacent time steps. However, it is unclear how, given a Clifford circuit, to relate the dynamics of its Wigner representation to the dynamics which are the solutions to the discrete Euler-Lagrange equations for the associated action functional. The reason for this is that the former is a Hamiltonian description while the latter is a Lagrangian one. To relate the two, one would need to develop a generalisation of the Legendre transform for discrete time dynamics. This is an interesting question that merits further study.
Our work may also shed light on the question of the classical limit of quantum dynamics in the sum-over-paths methdology. It is generally thought to be the case that the correct notion of the classical limit in this context is that only a single path should appear in the sum. However, this notion has recently been challenged by Kent [31]. Another reason to be sceptical of the standard notion, independently of those provided by Kent, is that the phasespace representation of Clifford circuit dynamics can be understood to provide a classical account of such dynamics, and yet the path-sum expression still assigns non-zero amplitude to more than one path, as we've seen here. Our work, therefore, provides a starting point for a reevaluation of the classical limit in the sum-over-paths approach to quantum dynamics.
Our results also suggest a sum-other-paths approach to the quantization of cellular automata. Specifically, if the state-space of a cell in the cellular automaton can be understood as a phase space and the update rule (the discrete-time dynamics) can be understood as a symplectomorphism, then our results provide a way of determining the quantized version of that cellular automaton.
When considering discrete variables, we have focused on Clifford circuits for systems having dimensions which are primes larger than 2. As qubit Clifford circuits are the most familiar for those in the quantum computing community, it is natural to ask to what extent our results extend to this case. The elementary gates for qubit Clifford circuits arê |q, q + q q, q |, (IV.1) which are known as the Hadamard, Phase and CNOT gates, respectively. Because these gates are balanced, one can write a sum-over-paths expression for the transition amplitudes as in Eq. (I.11). One can readily see that the phase factor of a path through such a circuit is of the form is a polynomial with coefficients in the ring Z 4 . The surprise here is that this is a polynomial over Z 4 rather than Z 2 . It is surprising because the classical configuration variable associated to the computation basis of a qubit, which defines the space of paths in the path-sum, takes its values in Z 2 rather than Z 4 . Related to this fact, there is a significant obstruction to understanding S(γ) as an action functional for a classical counterpart to qubit Clifford circuits. For quopit Clifford circuits, we obtained the classical counterpart by looking at Gross's discrete Wigner representation [12]. The property of this representation that we exploited is its so-called Clifford covariance, proven by Gross and recalled in our Proposition 1. It is due to this covariance property that one can represent the elementary gates by symplectomorphisms of the discrete phase space. However, the Wigner representation introduced by Gross is not defined for qubit systems. And while there are alternate approaches to Wigner representations of qubit systems, such as those introduced by Gibbons et al. [32], these representations are only required to be covariant under phase-space displacements, that is, unitaries generated by the Weyl operators {e ipq : p ∈ R} and {e iqp : q ∈ R}, rather than the full Clifford group. Covariance under the full Clifford group is a very strong requirement to place on a Wigner representation. For n quopit systems, where d denotes the dimension, there are (d n ) d n +1 distinct Wigner representations that are covariant under phase-space displacements, but Gross's Wigner representation is the unique representation which is Clifford covariant. For qubits, it is an even stronger requirement; indeed, it has recently been shown by Zhu [33] that it is impossible to define a Clifford-covariant Wigner representation for qubits.
This rules out using a Wigner representation of qubit Clifford circuits to determine their classical counterpart. While we have no alternate proposal at present, the algebra-geometry correspondence gives some hint as to what the structure of the discrete phase space must be. In particular, since S(γ) is a polynomial with coefficients in Z 4 , the discrete phase space should be some space over Z 4 . Further, since in the sum over paths one only sums over the amplitude on paths with configuration variables that take their values in Z 2 , the space will not simply be (Z 4 ) 2n , but rather something more complicated such as a subspace or quotient thereof. Indeed, phase spaces of this sort have been previously considered in the literature [34], . Wallman and Bartlett [35] define a positive quasi-probability representation of single qubit Clifford circuits having underlying phase space Unfortunately the permutation underlying the phase gate cannot be written as a polynomial map of (q, p) and hence it is not possible to apply the techniques presented in this paper. Nonetheless, exploring spaces over Z 4 as phase spaces for qubit Clifford circuits does suggest a new line of inquiry, namely, for each n ≥ 0 one could look for a symplectic space over Z 4 carrying an action of the n-qubit Clifford group such that the associated action functional is the one appearing in the expression for the amplitude of a path.