Quantum circuit ynamics via path integrals: Is there a classical action for iscrete-time paths? Mark D. Penney, 1, Dax Enshan Koh, 2, an Robert W. Spekkens 3, 1 Mathematical Institute, University of Oxfor, Woostock Roa, Oxfor, UK OX2 4GG 2 Department of Mathematics, Massachusetts Institute of Technology, Cambrige, Massachusetts 02139, USA 3 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canaa N2L 2Y5 (Date: Monay 14 th August, 2017) It is straightforwar to compute the transition amplitues of a quantum circuit using the sum-overpaths methoology when the gates in the circuit are balance, where a balance gate is one for which all nonzero transition amplitues are of equal magnitue. Here we consier the question of whether, for such circuits, the relative phases of ifferent iscrete-time paths through the configuration space can be efine in terms of a classical action, as they are for continuous-time paths. We show how to o so for certain kins of quantum circuits, namely, Cliffor circuits where the elementary systems are continuous-variable systems or iscrete systems of o-prime imension. These types of circuit are istinguishe by having phase-space representations that serve to efine their classical counterparts. For iscrete systems, the phase-space coorinates are also iscrete variables. We show that for each gate in the generating set, one can associate a symplectomorphism on the phase-space an to each of these one can associate a generating function, efine on two copies of the configuration space. For iscrete systems, the latter association is achieve using tools from algebraic geometry. Finally, we show that if the action functional for a iscrete-time path through a sequence of gates is efine using the sum of the corresponing generating functions, then it yiels the correct relative phases for the path-sum expression. These results are likely to be relevant for quantizing physical theories where time is funamentally iscrete, characterizing the classical limit of iscrete-time quantum ynamics, an proving complexity results for quantum circuits. Contents I. Introuction an Summary 2 A. The question 2 B. Summary of results 5 C. Significance for unerstaning the quantum-classical istinction 8 II. Continuous variable Cliffor circuits 11 A. Sum-over-paths expression for CV Cliffor circuits 11 B. The iscrete-time analogue of the action functional for CV systems 15 C. Symplectomorphisms an generating functions for CV Cliffor gates 17 D. Main result 19 III. Quopit Cliffor circuits 19 A. Sum-over-paths expression for quopit Cliffor circuits 19 B. Symplectic representation of iscrete systems 21 C. The iscrete-time analogue of the action functional for iscrete systems 22 D. Symplectomorphisms an generating functions for quopit Cliffor gates 24 IV. Concluing Remarks 26 Acknowlegements 27 A. Proof of Corollary 1 28 Electronic aress: mark.penney@maths.ox.ac.uk Electronic aress: axkoh@mit.eu Electronic aress: rspekkens@perimeterinstitute.ca
B. Kähler ifferential forms 28 References 30 2 I. INTRODUCTION AND SUMMARY A. The question The sum-over-paths methoology in quantum mechanics, pioneere by Richar Feynman, offers an alternative to the stanar means of expressing quantum ynamics, just as the least-action formulation of classical ynamics offers an alternative to the stanar Hamiltonian formulation [1]. In particular, it allows one to etermine the probability amplitue of making a transition among states for any given (possibly time-epenent) Hamiltonian operator escribing the quantum ynamics of the system. There is, however, a secon type of problem to which it can be applie. Here, one is given a moular escription of the quantum system s ynamics for instance, a escription of a quantum circuit with gates that are rawn from some fixe set of possibilities an the goal is to compute the transition amplitues of the overall circuit from a knowlege of the transition amplitues of each gate. The istinction between these two types of problems is best illustrate by an example. Suppose one is intereste in the transverse position of an atom as it passes through an interferometer. It is then useful to treat ifferent components in the interferometer as gates in a circuit. Determining the propagator associate to a particular gate given a knowlege of the Hamiltonian governing the ynamics of the atom as it passes through that gate is a problem of the first sort. Determining the propagator associate to the entire interferometric set-up given a knowlege of the propagators associate to each gate is a problem of the secon sort. We shall refer to the two sorts of problems henceforth as the continuous-time scenario an the circuit scenario respectively. In either scenario, one can consier the system s egrees of freeom to be iscrete or continuous. An interferometer is an example of a circuit acting on continuous egrees of freeom, while the circuits that are most commonly stuie in the fiel of quantum computation involve iscrete egrees of freeom. A quantum circuit can be specifie by a sequence of gates, where each gate is characterize by a unitary operator. The ynamics that occurs within each gate is generally not specifie. This is because only the overall functionality of the gate is important for the functionality of the circuit as a whole, an there are many ifferent choices of the ynamics within the gate that lea to the same functionality. For instance, a piece of polaroi an 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 ifferent. Given that the ynamics internal to each gate is irrelevant an may in fact be unknown the problem of computing the overall functionality of a circuit cannot be cast into the sum-over-paths methoology of the continuous-time scenario. Instea, one requires a sum-over-paths methoology that is explicitly catere to the circuit scenario, wherein each gate in the circuit is treate as a black box. It is straightforwar to express the transition amplitue of a circuit in terms of a sum or integral over iscretetime paths. Suppose q is a label for the basis relative to which we compute amplitues on a given system calle 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, {Ûk} N k=1, so that the total unitary is Û = ÛNÛN 1 Û2Û1. It is then appropriate to iscretize time into N steps. Denoting the configuration at time step k by q k (q (1) k,..., q(n) k ), a iscrete-time path through the configuration space is a sequence of N + 1 configurations, γ = ( q 0, q 1,..., q N ). Fig. 1 epicts a circuit acting on n systems with a gate epth of N an illustrates our labelling convention for the iscrete-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 ientity between every pair of ajacent unitaries to obtain q N Û q 0 = N k=1 q k Ûk q k 1 q N 1 q 1, (I.1) (I.2) where q k q (1) k... q (n) k. Defining the amplitue associate with the path γ Rn(N+1) as A(γ) = N q k Ûk q k 1, k=1 (I.3)
3 q (1) 0 q (1) 1 q (1) 2 q (1) N 1 q (1) N q (2) 0 q (2) 1 q (2) 2 q (2) N 1 q (2) N q (3) 0 Û 1 q (3) 1 Û 2 q (3) 2 q (3) N 1 Û q (3) N N q (n) 0 q (n) 1 q (n) 2 q (n) N 1 q (n) N FIG. 1: A circuit consisting of a sequence of N unitaries, together with a parameterisation of the path through configuration space. the amplitue q N Û q 0 can be expresse as the following integral over iscrete-time paths q N Û q 0 = A(γ) γ, P 0( q 0, q N ) (I.4) where P 0 ( q 0, q N ) enotes the space of iscrete-time paths that begin at q 0 an en at q N an where P 0( q 0, q N ( ) γ ) enotes ( ) q N 1 q 1. If, on the other han, the configuration is a iscrete variable for instance, if the Hilbert space for each system is C so that our label is iscrete, q (i) Z then we have q N Û q 0 = q N 1 (Z ) n q 1 (Z ) n k=1 N q k Ûk q k 1. Defining the amplitue for a iscrete-time path γ (Z ) n(n+1) by Eq. (I.3), we have q N Û q 0 = A(γ), γ P 0( q 0, q N ) (I.5) (I.6) where P 0 ( q 0, q N ) enotes the space of iscrete-time paths that begin at q 0 an en at q N. Uner various circumstances, it is possible to restrict the set of paths appearing in the sum or integral. The paraigm 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 an the slit in the screen is at position x, then the transition amplitue for step k has the form q k Ûk q k 1 δ(q k q k 1 )δ(q k x), where δ represents the Dirac-elta function. Integrating over q k, the elta 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 Ûk q k 1 δ( q k f( q k 1 )) for some bijective function f, an 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 etermine the transition amplitues for a unitary that is generate by a Hamiltonian (possibly time-epenent) over some time interval. This is achieve by partitioning the time interval into a large number of small intervals an 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 A(γ) = N e is[γ], (I.7) where S[γ] is the classical action 1 of the path γ an N is a complex number that is inepenent of the path. Much of the success of the sum-over-paths methoology relies on the fact that only the phase an not the magnitue of the amplitue A(γ) is path-epenent, an this in turn is a consequence of taking the limit of small step size. 1 Here an elsewhere in the article, any action for a path on a continuous-variable configuration space will be assume to be imensionless when it appears in a path integral expression, that is, it will be expresse in units of in that context.
This fact generally fails to hol in the circuit scenario. If the gates of the circuit are black boxes, then the most fine-graine sequence into which the overall unitary associate with the circuit can be factorize is one wherein each element of the sequence correspons to a gate in the circuit. But for an arbitrary gate, the associate unitary U k has matrix elements q k U k q k 1 for which the phase an the magnitue may be epenent on q k an q k 1. Therefore, in general, both the phase an magnitue of the amplitue A(γ) may be epenent on the path γ 2 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 balance. The property of being balance is efine relative to the orthogonal basis use to efine the configuration space in the path sum. It hols when every transition among basis elements that has a nonzero probability of occurence uner the gate has the same probability of occurence. As an example, for a qubit wherein the configuration space in the path sum is the basis of eigenstates of the Z Pauli operator, the gate associate to the Z Pauli operator is balance, as is every gate associate to a linear combination of the X an Y Pauli operators. The Z Pauli leaves each basis element invariant, so that only the trivial transition has nonzero probability, while for the X an Y Paulis, each basis element is taken to an equal superposition of basis elements, so that the state transitions that have nonzero probability of occurence is the full set, an each occurs with equal probability. We now provie a precise efinition of the balance property. Although each unitary in the sequence {Ûk} N k=1 may in general act on all n systems, it is common to consier gate sets with gates that act on small subsets of the systems (as will be the case in the Cliffor circuits we stuy further on). We therefore efine the property of being balance for a gate Û where the number of inputs an outputs is m, which may iffer from n. Definition 1. Let q enote the basis elements for the inputs of the gate an let Q enote the basis elements for its outputs. The gate Û is sai to be balance if 4 q, Q R m (or (Z ) m ) : Q Û q = NeiS( q, Q) δ(g( q, Q)) (I.8) where N is a complex constant 3, S( q, Q) is a function of q an Q with values in the fiel R, g is a smooth map 4 R m R m R m (or (Z ) m (Z ) m (Z ) m ) an δ is a Dirac-elta function 5 on R m (or a Kronecker elta function on (Z ) m ). In other wors, for the subset of values of q an Q where the amplitue Q Û q is nonzero a subset that one can always specify through a conition of the form g( q, Q) = 0 this amplitue is equal in magnitue an iffers only in phase. For a circuit compose entirely of balance gates, the functional over iscrete-time paths appearing in the path sum has the form γ A(γ) = Ne is(γ) δ(g(γ)) (I.9) where N is a complex number that is path-inepenent, S(γ) is a real-value function of γ, an δ(g(γ)) is a Dirac elta (or Kronecker elta) function that specifies the paths of nonzero amplitue. Specifically, if the unitary at time step k is mae up entirely of balance gates, so that q k Ûk q k 1 = N k e is k( q k 1, q k ) δ(g k ( q k 1, q k )), then we have N N N k, k=1 S(γ) N S k ( q k 1, q k ), k=1 δ(g(γ)) N δ(g k ( q k 1, q k )). Denoting by P( q 0, q N ) the space of paths of nonzero amplitue that start at q 0 an en at q N, one then has for continuous variables, q N Û q 0 = N e is(γ) γ, (I.11) P( q 0, q N ) k=1 (I.10) 2 Functionals A(γ) wherein both the phase an the magnitue are path-epenent (corresponing to a complex action functional) have seen applications in the sum-over-paths methoology for continuous-time ynamics (e.g.,[2, 3]). Whether similar generalizations of the stanar sum-over-paths methoology can be of use in the circuit scenario is an interesting question which we o not pursue here. 3 Because N is complex, this ecomposition is not unique. In what follows, we will aopt the convention of excluing any constant terms from S( Q, q) an putting the resulting factor into N. 4 Due to technical aspects of the theory of istributions, there are restrictions that one shoul place on this function so that the resulting istribution is well-efine. As these restrictions are satisfie by all of the functions that we explicitly consier in this paper, we shall ignore these conitions an irect the intereste reaer to Ref. [4] (Chapter 6). 5 Here we make the convention that when g( q, Q) = 0 for all ( q, Q), δ(g( q, Q)) is the constant istribution with value 1. Inee, one can think of the δ function as a map which takes functions g( q, Q) to istributions on ( q, Q) an so we are free to aopt this convention.
5 where P( q 0, q N ) ( ) γ is integration over P( q 0, q N ) with respect to the measure inuce by δ(g(γ)) 6. variables, q N Û q 0 = N e is(γ). P( q 0, q N ) For iscrete (I.12) The notion of balance gates was introuce in the context of iscrete systems by Dawson et al. [5], who were also the first to consier the sum-over-paths methoology in the circuit scenario 7. They note that certain gate sets that are universal for quantum computation such as the gate set consisting of only the Haamar an Toffoli gates are comprise entirely of balance gates. As such, circuits built from this gate set can be analyze by a sum-overpaths methoology, an this was use to provie simple proofs of some known complexity results, for example that BQP PP. Bacon et al. [6] extene their work by consiering algebraic circuits efine by a gate set consisting of three phase-changing gates an a Fourier transform gate. Because the elements of this gate set are also all balance, it is possible to apply the sum-over-paths methoology to algebraic circuits as well. Certain well-stuie families of circuits, known as Cliffor circuits, also have gate sets comprise entirely of balance gates 8. Cliffor circuits were first introuce in the context of qubits [8], but were subsequently generalize to continuous variable systems [9] an -level systems for > 2 (quits) [10, 11]. Dawson et al. note that the balance property hel for qubit Cliffor circuits. It is not ifficult to see that it hols for continuous variable (CV) an quit Cliffor circuits as well. In all such circuits inee, any circuit consisting entirely of balance gates the sum-over-paths methoology provies an alternative way of computing transition amplitues for the whole circuit from a knowlege of the transition amplitues of each gate. In this article, we are not, however, intereste in the sum-over-paths approach for its use as an alternative technique of solving quantum ynamics for circuits, but rather for the novel perspective that it offers on the ifference between quantum an classical theories of that circuit. In the case of continuous-time ynamics, the brige between the classical an the quantum theory is mae through the phase factor S[γ] that is assigne to a path γ in the path integral expression for the ynamics; it is simply the classical action of the path γ. Specifically, in the case of n systems escribe by continuous variables unergoing continuous-time ynamics over a time interval [0, T ], a path γ is specifie, in the limit of small step-size, as a function q : [0, T ] R n, an the classical action of this path is the integral of the Lagrangian of the system along the path, S[γ] T 0 L( q(t), q(t)) t. (I.13) We here aress the following question: for iscrete-time ynamics, can the phase S[γ] appearing in the path integral expression be unerstoo as the action functional for a iscrete-time path γ in a classical counterpart of the quantum circuit? The main result of our article is a emonstration that it can be so unerstoo for certain kins of circuits. B. Summary of results Our emonstration that the phase of a iscrete-time path can be unerstoo in terms of an action functional necessitates solving two problems about iscrete-time classical ynamics: 1. Defining an action functional for iscrete-time classical ynamics, 2. Determining the iscrete-time classical ynamics associate to a given type of iscrete-time quantum ynamics. 6 This measure is not simply the inuce measure on P( q 0, q N ) as a subspace of P 0 ( q 0, q N ) = R n(n+1). It also takes into account the graients of the functions g k at those points. For more etails, see Ref. [4] (Chapter 6). 7 The efinition of a balance gate use in [5], that the non-zero matrix elements Q Û q all have the same absolute value, is equivalent to the one we have provie. 8 It turns out that for the quopit Cliffor circuits an continuous-variable circuits that we consier, balanceness is a property not just of the Cliffor gates, but also of any unitary that is implemente by a circuit compose of those gates. For a proof of the balanceness of unitary operations implemente by quopit Cliffor gates, see [7]. The proof for CV Cliffor gates is similar.
Our solutions to these two problems will be outline in the rest of the introuction an elaborate upon in the main text. Once these two founations are lai, it becomes straightforwar to verify that for certain types of quantum circuits, the action functional for their classical counterparts is what etermines the relative phases of paths in the sum-over-paths expression for the quantum ynamics. Defining an action functional for iscrete-time ynamics. To unerstan what counts as a goo efinition of an action functional in iscrete-time ynamics, we review the role that the action functional plays in continuous-time ynamics. In fact, it plays two relate roles: It etermines the classical trajectories via a least-action principle; It generates the symplectomorphism associate with evolution over a time interval via its evaluation on the classical trajectories over that interval; The first role is well-known, while the secon role is worth reviewing. Suppose that one is consiering continuoustime ynamics efine by a Hamiltonian H( q, p) over a time interval [0, T ] an let ( q cl (t), p cl (t)) be the solution to Hamilton s equations with initial value ( q, p). Then the ynamics over the given time interval is escribe by the symplectomorphism φ : ( q, p) ( Q, P ) by setting Q = q cl (T ) an P = p cl (T ). Note that q cl (t) can also be characterize as the solution to the Euler-Lagrange equations with bounary conitions q cl (0) = q an q cl (T ) = Q for the Lagrangian L( q, q) associate to H( q, p). Consier the function of q an Q that one obtains by evaluating the action functional on the classical trajectory through configuration space that begins at q an ens at Q, enote γ cl, 6 G φ ( q, Q) S[γ cl ] = T 0 L( q cl (t), q cl (t)) t. It is well-known ([12], Chapter 9) that this function generates the symplectomorphism φ in the sense that (I.14) p (i) = G φ( q, Q) q (i), P (i) = G φ( q, Q) Q (i). (I.15) In other wors, the generating function of a symplectomorphism inuce by Hamiltonian ynamics is exactly the action functional evaluate on the classical trajectories. Now consier iscrete-time classical ynamics for a continuous-variable system, such as the classical interferometer escribe earlier. Suppose that the nature of the ynamics is specifie moularly, as a sequence of symplectomorphisms on the system s phase space. For both the case of a physical theory wherein time is funamentally iscrete an the case of circuit ynamics wherein the internal ynamics of each gate is unknown, there is no Lagrangian escribing a continuous-time ynamics within a given time-step. Nonetheless, the association between generating functions an actions escribe above provies a means of efining an action without reference to such a Lagrangian. Specifically, if the symplectomorphism for the kth time-step is enote φ k an has a generating function G φk ( q k 1, q k ), one can simply interpret the latter as the total action of the transition from q k 1 to q k uner φ k. One can then efine the action functional for the iscrete-time path γ = ( q 0,..., q N ) as the sum over time-steps of the actions associate to each transition, N S(γ) = S( q 0,..., q N ) = G φk ( q k 1, q k ). k=1 (I.16) The reason it is appropriate to ientify this as the iscrete-time action functional is that, as we will prove in Proposition 1, it plays one of the roles that the continuous-time action functional plays, namely, that if one evaluates it on the classical iscrete-time trajectories, one obtains the generating function for the symplectomorphism associate to the overall transition between the initial time an the final time. We will show that this result about iscrete-time ynamics goes through not just for continuous variables but for iscrete variables as well. In the latter case, we imagine that a single system has a phase space with coorinates taking values in the ring Z, so that the phase space of n such systems is (Z ) 2n, an the iscrete-time ynamics is specifie as a sequence of symplectomorphisms on this phase space. At first glance, it is not obvious that the connection between generating functions an actions that was escribe above can be leverage to efine a iscrete-time action functional in this case. The ifficulty is that, as is evient from Eq. (I.15), generating functions are typically characterise in a way that makes explicit use of ifferential calculus, an for iscrete variables, one oes not have a notion of ifferentiation in the usual sense. To resolve this issue, we call upon the algebra-geometry corresponence. While this has many incarnations in various areas of mathematics, the unerlying iea is that one can frequently establish a ictionary which translates geometric
structures of a space into structures on its algebra of functions, an vice-versa. The particular instantiation we nee is the uality between the geometry of so-calle affine schemes an the algebra of commutative rings ([13], Chapter II.2). Uner this uality, the algebraic counterpart of an affine space over Z is an algebra of polynomials with coefficients in Z, where the number of variables equals the imension of the affine space. Explicitly, for the classical phase space of n systems with coorinates q (i), p (i), each taking values in Z, one has the corresponence (Z ) 2n Z [q (1),..., q (n), p (1),..., p (n) ] =: Z [ q, p]. (I.17) We exploit this corresponence by ientifying what structure on the algebra of polynomials is the ual of a ifferential structure on the iscrete space. Fortunately this question has long been answere by algebraic geometers. The structure goes by the name of the Kähler ifferential forms on the algebra Z [ q, p] ([13], Chapter II.8). The algebra-geometry corresponence forces us to efine generating functions for the ynamics of iscrete systems not as polynomial functions over the reals, but rather as polynomial functions over Z, 7 G( g, Q) Z [ q, Q]. (I.18) For example, for any symplectomorphism φ associate to a single system, that is, φ : Z Z Z Z an (Q, P ) = φ[(q, p)], the generating function is a polynomial in two variables, q an Q, with coefficients in Z, G φ (q, Q) : Z Z Z. (I.19) Similarly for symplectomorphisms on pairs of systems, φ : (Z ) 2 (Z ) 2 (Z ) 2 (Z ) 2, the associate generating function is a polynomial with coefficients in Z, G φ ( q, Q) : (Z ) 2 (Z ) 2 Z. (I.20) It follows that if we efine the total action of a given transition uner a symplectomorphism by evaluating the associate generating function on the given initial an final configurations, then the action will be Z -value. We can then efine a Z -value action functional for a iscrete-time path as the sum over time-steps of the action of the transition at each time-step, so that Eq. (I.16) applies, but is Z -value. Just as in the continuous-variable case, if one evaluates this iscrete-time action functional on the classical iscrete-time trajectories, one obtains the generating function for the symplectomorphism associate to the overall ynamics of the whole circuit, as we show in Lemma 3. It is in this sense that it is appropriate to unerstan the sum of the generating functions as an action functional, even for iscrete variables. 9 We have shown that our iscrete-time action functional plays the secon role that its continuous-time counterpart plays, but what about the first role? Does it also efine classical iscrete-time trajectories by a least-action principle? A bit of reflection shows that the question is not well-pose without substantial revision to the statement of the principle. For iscrete egrees of freeom in particular, the normal manner of thinking about it in terms of an extremization of the action functional oes not work. As we note above, the algebra-geometry corresponence forces the action to be Z -value in this case, an Z is not an orere set, so that there is no possibility of efining a principle of extremality in terms of it. It may be possible, however, to recast the principle in terms of stationary points in the space of paths rather than in terms of extremization. To o so, one must translate to the iscrete-variable context the normal story about how to express the variation of the action inuce by a small variation in the path. But it is not even clear what is the appropriate sort of variation of the path to consier. Presumably, if this question can be answere, then stationary points can be ientifie using a scheme wherein Kahler ifferential forms play the role of normal erivatives. In the case of continuous egrees of freeom, where the action functional is real-value, the question is well-pose, but we o not have the answer. Whether our proposal for a iscrete-time action functional can recover classical paths by some principle of stationary action (for either iscrete or continuous egrees of freeom) remains an open question. Because it remains unclear whether it is possible to unerstan iscrete-time classical ynamics in terms of a principle of stationary action, in this work we focus on emonstrating that our propose functional over iscrete-time paths plays the secon role that the action functional over continous-time paths plays in classical ynamics, that of efining a generating function. It is in this sense that our propose functional oes inee merit the title of an action functional. 9 Note that our proposal agrees with that of Baez an Gilliam [14] concerning what sorts of mathematical object shoul represent a iscrete-time an iscrete-variable analogue of the action functional.
Determining the classical counterpart of quantum iscrete-time ynamics. For certain kins of quantum circuits involving only balance gates, we can ientify the natural classical counterpart an its escription in terms of a phase space an symplectomorphisms. In particular, we o so for CV Cliffor circuits an for quit Cliffor circuits where the imension of the elementary systems is an o prime. The CV case is the easiest to consier because there it is obvious which classical phase space to associate to a given quantum state space an (at least for the CV Cliffor gates) which symplectomorphisms to associate to given quantum unitaries. This is achieve using the Wigner representation, which efines a classical moel of the quantum ynamics (specifically, a noncontextual hien variable moel) [15]. The Wigner representation associates a symplectomorphism to each of the elementary gates of a CV Cliffor circuit, an from the latter we obtain a generating function. In this way, we can ientify the action functional for the overall circuit. We then procee to show (in Theorem 2) that this action functional yiels the correct phases in the path sum expression for the quantum ynamics, that is, we show that it yiels the functional S(γ) of Eq. (I.9). To achieve an analogous interpretation of S(γ) in Eq. (I.9) for iscrete rather than continuous-variable systems in a circuit scenario, one must first of all etermine what symplectic space shoul be associate with a given iscrete quantum system. This is not evient a priori because for iscrete systems, such as the intrinsic spin egree of freeom, the quantum ynamics was not obtaine historically by quantizing a classical theory of iscrete variables. However, it turns out that for certain quit Cliffor circuits, there is clarity about what is the natural classical counterpart. These are the so-calle quopit Cliffor circuits, where a quopit is a quit where the imension is an o prime [16]. What is special about quopit Cliffor circuits is that they are known to amit a noncontextual hien variable moel, an this moel provies the classical counterpart of the quantum ynamics 10. For quopit Cliffor circuits, the moel is provie by a iscrete analogue of the Wigner representation propose by Gross, where the Cliffor operations are represente as transformations of an affine space over the finite fiel Z [17]. 11 The iscrete Wigner representation associates to every elementary gate of a quopit Cliffor circuit a symplectomorphism on the iscrete phase space. 12 Using the techniques escribe earlier, one can efine generating functions in terms of these an consequently also an action functional. We show (in Theorem 4) that this choice oes inee yiel the correct expressions for the quantum ynamics, that is, we show that the functional S(γ) of Eq. (I.9) can inee be written as a sum of the generating functions of the symplectomorphisms associate to the gates. The iea of looking at sums of generating functions as generalisations of the action functional is in part inspire by a little-known paper of Dirac [18] wherein he explores the possibility of a Lagrangian approach to quantum mechanics. While Dirac was not successful at reformulating quantum mechanics, he i notice certain formal similarities between the generating functions of symplectomorphisms an the infinitesimal generators of unitary operations that implement a change of basis, an it was this work that ultimately inspire Feynman s formulation of the path integral (as Feynman notes in his Nobel lecture). By exploring this connection in the iscrete-time scenario, our results serve to clarify the precise role of generating functions in the sum-over-paths formulation of quantum theory. 8 C. Significance for unerstaning the quantum-classical istinction The physical relevance of whether or not the exponent in the path integral expression can be intereprete as a classical action functional is this: if it can be so interprete, then one has built a brige between a classical theory an its corresponing quantum theory for iscrete-time ynamics. In particular, one obtains insights into two aspects of this brige: 1. Schemes for quantization, that is, how to efine the quantum counterpart of a given classical evolution, 10 This is not at os with the fact that the full quantum theory fails to amit of such a moel because Cliffor circuits o not realize arbitrary unitaries. 11 In fact, Gross s Wigner representation is efine for all o imensions, not just o prime. In that case Z is no longer a fiel, which significantly increases the ifficulty of working with this representation. Nonetheless, the elementary gate set for composite imensions has been worke out by Hostens [11], but it is much more complicate an we o not consier it here. We believe that similar results as ours shoul hol for arbitrary o imensions. 12 The symplectomorphisms arising from both the iscrete an continuous variable Wigner representations are all affine, that is, a composition of a linear symplectic map an a phase-space translation. We nonetheless propose, in Sections II B an III C, the iscrete-time analogues of action functionals for ynamics given by arbitrary (i.e., not necessarily affine) symplectomorphisms. As the formalism employe naturally accommoates this broaer class of ynamics it as no further mathematical technicalities while greatly expaning the range of applicability of our proposal.
2. The efinition of intrinsically quantum behaviour, that is, etermining when a given experiment fails to amit of a classical explanation. Although the first aspect concerns a map from classical to quantum, while the secon concerns a map from quantum to classical (or the lack thereof), there remains an important istinction between them. For the problem of quantization, one has a particular classical phase-space an Hamiltonian in min. For the problem of eterming whether an experiment amits of a classical explanation, on the other han, one woul like to be permissive about the nature of the classical phase-space an Hamiltonian that can appear in the explanation. We iscuss each aspect in turn. Quantization. For continuous-time ynamics, the sum-over-paths approach provies a means of making inferences from the Lagrangian escription of the classical ynamics to its quantum ynamics. Inee, it is this sort of problem that has riven the evelopment of the vast technical machinery an wie-ranging applications of the path integral. The sum-over-paths methoology arguably provies the most fruitful approach to the problem of quantization. In particular, recall that, unlike canonical quantization, the path integral formulation of quantum theory can be applie to classical theories that have a Lagrangian formulation but no Hamiltonian formulation. Consequently, the path integral approach has a broaer scope. In terms of applications, unerstaning the sum-over-paths methoology for the circuit scenario may provie a means to irectly quantize certain classical circuits. Stuying the quantum generalization of various types of circuits, coes, an other tools from classical computer science has been a source of many innovations in the fiel of quantum computation. The present work eviates from the traitional approach to such generalizations insofar as it requires a phase-space escription of the classical circuit. We hope that the novelty of this perspective may offer new insights. A sum-over-paths methoology for the circuit scenario can also aress the problem of quantizing theories of physics wherein time is funamentally iscrete. Many have espouse the iea that iscrete-time ynamics might be the correct basis for physics while the stanar continuous-time ynamics might be merely a useful approximation thereto. In the classical context, the iea has been pursue through the stuy of cellular automata. The fact that cellular automata have a ynamical law that is similar to laws of physics in being both time-inepenent an spatially local, an the fact that many choices of this law yiel ynamics having features that are strongly reminiscent of physics, incluing a funamental limit to the spee of propagation of influences, the possibility of evolving stable structures an complex structures, an computational universality, has motivate many to pursue a reconstruction of physical theories in terms of them [19 21]. In the quantum context, the fact that it is in principle impossible to resolve spatial istances an times arbitrarily finely [22, 23] an the iea that lengths an times, like other observables in quantum theory, ought to take values in a iscrete spectrum [24, 25] also motivate researchers to pursue formulations of physics wherein space an time are funamentally iscrete, implying iscrete-time rather than continuous-time ynamics. Some have also consiere quantum cellular automata (see, e.g., [26, 27]) as a basis for physics [28]. Generally, researchers pursuing iscrete-time ynamics as the basis of physics favour the assumption that the internal egrees of freeom of systems shoul be taken to be iscrete. Nonetheless, a iscrete-time ynamics for continuous egrees of freeom is another consistent option. For instance, in the quantum context, one can consier scalar fiels as the systems which evolve over iscrete time, an in the classical context, one can consier cellular automata where the internal state of each cell is a continuous variable [21]. It is our hope that the results in this article might provie some insight into how to achieve quantization in such iscrete-time classical theories. For instance, if one constructs a cellular automaton wherein the state-space of a cell can be unerstoo as a phase space an the upate rule (the iscrete-time ynamics) can be unerstoo as a symplectomorphism, then our results provie a way of etermining the quantize version of that cellular automaton. In this sense, our work connects most with an approach to classical iscrete-time ynamics terme iscrete mechanical systems, where, unlike stanar cellular automata, one aims for a iscrete generalization of Hamiltonian an Lagrangian escriptions of the ynamics [14]. Note that if one starts from classical iscrete-time theories wherein the ynamics is generate by a set of symplectomorphisms, an one uses the sum-over-paths methoology outline here to etermine the quantum counterpart of this theory, then the iscrete-time quantum theories that one obtains can always be formulate as a kin of quantum circuit. Importantly, however, not every sort of quantum circuit that the Hilbert-space formalism of quantum theory permits us to efine can arise in this fashion. This metho of quantization can only yiel quantum circuits wherein each of the funamental gates has the property of being balance. Our work therefore provies some reason for thinking that this restricte class of quantum circuits may be a better starting point for any research program that aims to buil a quantum formulation of physics wherein space an time are funamentally iscrete. Intrinsically quantum behaviour. Discussions of the istinction between quantum an classical ynamics often appeal to the path integral methoology: a given quantum ynamics is thought to amit of a classical moel (hence to not be intrinsically quantum) if the typical action scale is large compare to Planck s constant, such that the 9
amplitues of paths which eviate from the action-extremizing path ten to cancel. It is not at all clear, however, how this notion of quantum-classical corresponence might be applie to a iscrete-time evolution, such as arises in a quantum circuit. Meanwhile, the notions of local causality [29] an of noncontextuality [30, 31] are naturally suite to the question of whether a given quantum circuit amits of a classical explanation. In this approach, one consiers the set of possible circuits that can be built up from the elementary gate set, an one inquires about the possibility of explaining the experimental ata for each of these in terms of a locally causal moel or in terms of a noncontextual moel. The notion of a noncontextual moel of an experiment was generalize in Ref. [32] an the generalization was shown to be equivalent to the existence of a nonnegative quasiprobability representation [33, 34], another popular notion of classicality. Note that the notion of classicality that emerges in such work oes not involve the ratio of typical actions to Planck s constant. It is our view that we will not have unerstoo the quantum-classical istinction until such time as we have a notion of intrinsically quantum behaviour that is inepenent of how one formulates quantum theory (whether it be with path integrals or Schröinger ynamics, for instance) an of whether one is consiering continuous-time or iscrete-time ynamics. Insofar as our work contributes to unerstaning the quantum-classical istinction in the path integral formulation of iscrete-time ynamics, it is a step on the path towars such a unifie notion of intrinsic quantumness. The iea that a quantum ynamics shoul be eeme effectively classical only if the associate sum over paths is ominate by a single path (the action-extremizing one) has recently been challenge by Kent [35]. Our work provies further reason to be sceptical of this notion of classicality, inepenently of the reasons provie by Kent. If one unerstans a quantum ynamics to amit of a classical explanation when it amits of a noncontextual moel (or, equivalently, a nonnegative Wigner representation), then Cliffor circuit quantum ynamics amit of a classical explanation. An yet, as we will show here, the path-sum expression for such ynamics cannot be reuce to the contribution of a single path. Our work therefore provies a starting point for a re-assessment of what is the correct notion of the classical limit in the path integral formulation of quantum theory. Finally, the question of which families of quantum circuits amit of a classical moel has practical significance: it can help to ientify the resources that are responsible for quantum-over-classical avantages for information processing. It has recently been shown that certain types of nonclassicality can constitute resources for cryptographic an computational tasks. For instance, Bell-inequality violations are a resource for evice-inepenent cryptography[36, 37]. Furthermore, failing to amit of a noncontextual moel (equivalently, failing to amit of a positive quasiprobability representation) has recently been implicate in quantum computational spee-up [38 40]. A broaer perspective on nonclassicality in the circuit scenario promises more such insights. It may also help to etermine whether a given computational architecture, such as the one implemente by D-wave, has intrinsically quantum features or not[41]. Outline. The outline of the paper is as follows. The continuous variable an quopit Cliffor circuits are consiere separately in Sections II an III, respectively. This is because, although the en results are quite similar, the mathematics involve is quite ifferent. After introucing these circuits, in Sections II A an III A, we explicitly escribe the resulting sum-over-paths expressions for transition amplitues in Theorems 1 an 3. The remainer of each Section is evote to showing that the functionals S(γ) (of Eq. (I.9)) are the generating functionals of the corresponing symplectic representations. More specifically, for the continuous variable case, we introuce generating functions in Section II B an then, in Section II C, we introuce the symplectic representation an prove Theorem 2. For the quopit case, we introuce the symplectic representation via the iscrete Wigner transform in Section III B. Then, in Section III C, we introuce Kähler ifferentials an use them to efine the corresponing generating functional. Finally, in Section III D, we prove Theorem 4. Section IV offers some concluing remarks. As some of the mathematics use in Section III may not be familiar to some reaers, we have inclue an Appenix with some aitional backgroun information. Notational Conventions. Arrows ( x) inicate vector quantities an brackete superscripts (x (i) ) their components. Hats (ˆx) inicate operators. Subscripts (x k ) will be use to inex time steps. Z enotes the ring of integers moulo. Complex conjugation will be represente by an overbar (z). 10
11 II. CONTINUOUS VARIABLE CLIFFORD CIRCUITS A. Sum-over-paths expression for CV Cliffor circuits We turn our attention to applying the sum-over-paths methoology to a particular example of a family of quantum circuits for which every gate in the generating set is balance: the subset of quantum circuits known as continuous variable (CV) Cliffor circuits. These have previously been stuie as the appropriate generalization of qubit Cliffor circuits for continuous variables [9]. In fact, it has been shown that such circuits can be efficiently simulate on a classical computer, an extension of the Gottesman-Knill Theorem from qubits to CV systems [42]. Our interest in CV Cliffor circuits comes from a more founational perspective, namely, that they can be escribe by a noncontextual hien variable moel [15], which provies the means by which we ientify an action functional over the paths, as we shall see in Sec. II C. The goal of this section is to etermine, for an arbitrary CV Cliffor circuit, a sum-over-paths expression for its transition amplitues as in Eq. I.11. We then show that the exponent of the phase factor associate to each allowe path can be unerstoo as a iscrete-time generalisation of the action functional. We introuce this notion in Section II B an then, in Section II C, we prove that it agrees with our calculation of the aforementione phase factor. An n-system CV Cliffor circuit consists of preparations an measurements in the configuration basis of L 2 (R n ) an an elementary gate set consisting of the following 1-system an 2-system gates: ˆF = e i π 4 (ˆq2 +ˆp 2 ) ˆP (η) = e i η 2 ˆq2, η R ˆX(τ) = e iτ ˆp, τ R ˆΣ = e iˆq(1) ˆp (2), (II.1) along with ˆF an ˆΣ ( ˆP (η) an ˆX (τ) are alreay inclue as ˆP ( η) an ˆX( τ)). ˆF is calle the Fourier gate an correspons to evolution for unit uration uner the Hamiltonian for a harmonic oscillator with mass 2 π an frequency π 2 (it is the analogue of the Haamar gate in a qubit Cliffor circuit). It is intuitively unerstoo as a rotation in phase space by π/2. ˆP (η) is calle a phase gate (by analogy to the phase gate in a qubit Cliffor circuit), an correspons to a phase-space squeezing operation (via a position-epenent boost). The ˆX(τ) gate (a generalization of the Pauli-X gate in a qubit Cliffor circuit) implements a translation of the configuration by τ. Finally, ˆΣ is calle the sum gate (it is the analogue of the CNOT gate in a qubit Cliffor circuit) an can be unerstoo as a translation of the secon system by an amount equal to the coorinate of the first. The intuitive phase-space accounts that we have just provie for these quantum gates will be shown, in Sec. II C, to be an accurate escription of the associate symplectomorphisms. Bartlett et al. [42] have shown that CV Cliffor circuits can (up to a global phase factor) implement all an only those unitaries lying in the so-calle n-system CV Cliffor group, C n. To efine C n, one must first introuce 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 generate by { ˆX i (τ), Ẑi(σ) : τ, σ R, i {1,..., n}} where ˆX i (τ) is the operator that translates system i by τ an Ẑi(σ) := e iσˆqi is the operator that boosts system i by σ. Definition 2. The n-system CV Cliffor group, C n, is efine to be the normaliser of the n-system CV Pauli group insie U ( L 2 (R n ) ), that is, C n := N (G n ). Note that C n has U(1) as a subgroup given by the operators e iφ 1. Then Bartlett et al. [42] prove that the set { ˆFi, ˆP (η) i, ˆX(τ) i, Σ i,j : η, τ R, i, j {1,..., n}} (II.2) are a generating set for the group C n /U(1). Consier a given n-system CV Cliffor circuit C implementing a unitary Û C n. To calculate the amplitues for each path through the configuration space, we first nee the matrix elements for the elementary gates. Lemma 1. The matrix elements for the elementary CV Cliffor gates are: Q ˆF q = 1 i 2 π e iqq ; Q ˆF q = 1+i 2 π eiqq ;
12 Q ˆP (η) q = e i η 2 q2 δ (Q q); Q ˆX(τ) q = δ (Q (q + τ)); Q (1), Q (2) ˆΣ q (1), q (2) = δ ( Q (1) q (1)) δ ( Q (2) (q (2) + q (1) ) ) ; Q (1), Q (2) ˆΣ q (1), q (2) = δ ( Q (1) q (1)) δ ( Q (2) (q (2) q (1) ) ). It follows that all of these gates are balance. Proof. We use the fact that ˆF correspons to evolution for unit uration uner the Hamiltonian for a harmonic oscillator with mass 2 π an 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 of ˆP (η) are trivial to compute: Q ˆF q = 1 i 2 π e iqq. (II.3) Q ˆF q = q ˆF Q = 1 + i 2 π eiqq. (II.4) Q ˆP (η) q = e i η 2 q2 δ (Q q). (II.5) For ˆX(τ), one has Q ˆX(τ) q = p Q e iτ ˆp p p q = 1 p e ip(q q τ) 2π = δ (Q (q + τ)). (II.6) Finally, for ˆΣ one has Q (1), Q (2) ˆΣ q (1), q (2) = p Q (1), Q (2) ˆΣ q (1), p p q (2) = 1 p e ip( q(2) q (1) ) Q (1), Q (2) q (1), p 2π It follows that = 1 ( 2π δ Q (1) q (1)) p e ip( q(2) q (1) +Q (2) ) ( = δ Q (1) q (1)) ( ) δ Q (2) (q (2) + q (1) ). (II.7) Q (1), Q (2) ˆΣ q (1), q (2) = q (1), q (2) ˆΣ Q (1), Q (2) ( = δ Q (1) q (1)) ( ) δ Q (2) (q (2) q (1) ). (II.8) For any system at any time-step where the circuit has no gate acting, we shall escribe the gate as ientity an enote it by 1. The ientity gate is a special case of ˆX(τ) where τ = 0 an a special case of ˆP (η) where η = 0, so that we can infer from Lemma 1 that its contribution to the amplitue is simply δ (Q q). (Note that although we coul have simply represente the ientity gate by ˆX(0) or ˆP (0) in a escription of the circuit, it is more straightforwar to treat it istinctly).