PHYSICAL REVIEW A 96, 032331 (2017)

Semiclassical formulation of the Gottesman-Knill theorem and universal quantum computation

Lucas Kocia, Yifei Huang, and Peter Love* Department of Physics, Tufts University, Medford, Massachusetts 02155, USA (Received 17 February 2017; published 20 September 2017) We give a path-integral formulation of the time evolution of qudits of odd dimension. This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers ofh. ¯The largest power of h¯ required to describe the evolution is a traditional measure of classicality. We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order h¯ 0 and so are “classical,” just like propagation of Gaussians under harmonic Hamiltonians in the continuous case. Such operations have no dependence on phase or quantum interference. Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path-integral contributions truncated at order ¯ h1, a number that nevertheless scales exponentially with the number of qudits. The same sum in continuous systems has an infinite number of terms at orderh ¯ 1.

DOI: 10.1103/PhysRevA.96.032331

I. INTRODUCTION contain −I . The set of stabilizer states is preserved by elements of the Clifford group, that is the normalizer of the Pauli The study of contextuality in quantum information has group, and can be simulated efficiently. More precisely, by led to progress in our understanding of the Wigner function the Gottesman-Knill theorem, for n qubits, a quantum circuit for discrete systems. Using Wootters’ original derivation of of a Clifford gate can be simulated using O(n) operations on discrete Wigner functions [1], Mari et al. [2], Gross [3], and a classical computer. Measurements require O(n2) operations Howard et al. [4] have pushed forward a new perspective on with O(n2) bits of storage [8,9]. the quantum analysis of states and operators in finite Hilbert The reason that stabilizer evolution by Clifford gates can be spaces by considering their quasiprobability representation efficiently simulated classically has been explained in various on discrete phase space. Most notably, the positivity of ways. For instance, as already mentioned, stabilizer states have such representations has been shown to be equivalent to been shown to be noncontextual in qudit [4] and rebit [10] noncontextuality, a notion of classicality [4–7]. Quantum gates systems. One potential obstacle to proving this for qubits is that and states that exhibit these features are the stabilizer states qubit systems possess state-independent contextuality [11,12]. and Clifford operations used in quantum error correction and Of course, we know how to simulate qubit stabilizer states stabilizer codes. The noncontextuality of stabilizer states and and Clifford operations efficiently by the Gottesmann-Knill Clifford operations explains why they are amenable to efficient theorem [8,9]. For recent progress relating noncontextuality classical simulation [8,9]. to classical simulatability for qubits, we refer the reader to This progress raises the question of how these discrete techniques are connected to prior established methods for [13,14]. It has also been shown for dimensions greater than simulating quantum mechanics in phase space. A particularly two that a state of a discrete system is a stabilizer state if and relevant method is trajectory-based semiclassical propagation, only if its appropriately defined discrete Wigner function is which has been widely used in the continuous context. Perhaps, non-negative [3]. Therefore, when acted on by positive-definite when applied to the discrete case, semiclassical propagators operators, it can be considered as a proper positive-definite can lend their physical intuition to outstanding problems in (classical) distribution. quantum information. Conversely, concepts from quantum Here, we instead relate the concept of efficient classical h information may serve to illuminate the comparatively older simulation to the power of ¯that a path-integral treatment must field of continuous semiclassics. be expanded to in order to describe the quantum evolution Quantum information attempts to classify the “quantum- of interest. It is well known that Gaussian propagation in ness” of a system by the presence or absence of various continuous systems under harmonic Hamiltonians can be quantum resources. Semiclassical analysis proceeds by suc- described with a single contribution from the path integral h0 cessive approximation using ¯h as a small parameter, where truncated at order ¯ [15]. We show that the corresponding case the power of ¯h required is a measure of “quantumness.” Can in discrete systems exists where stabilizer states take the place these two views of quantum versus classical be related? In this of Gaussians and harmonic Hamiltonians that additionally paper, we build a bridge from the continuous semiclassical preserve the discrete phase space take the place of the general world to the discrete world and examine the classical-quantum continuous harmonic Hamiltonians. In the discrete case, we characteristics of discrete quantum gates found in circuit will only considerd-dimensional systems for oddd since their models and their stabilizer formalism. center representation (or Weyl formalism) is far simpler. Stabilizer states are eigenvalue one eigenvectors of a As a consequence, we will show that operations with commuting set of operators making up a group which does not Clifford gates on stabilizer states can be treated by a path integral independent of the magnitude of h¯ and are thus fundamentally classical. Such operations have no dependence *Corresponding author: [email protected] on phase or quantum interference. This can be viewed as a

2469-9926/2017/96(3)/032331(13) 032331-1 ©2017 American Physical Society LUCAS KOCIA, YIFEI HUANG, AND PETER LOVE PHYSICAL REVIEW A 96, 032331 (2017) restatement of the Gottesman-Knill theorem in terms of powers of h¯ . We also consider more general propagation for discrete quantum systems. Quantum propagation in continuous sys- tems can be treated by a sum consisting of an infinite number of contributions from the path integral truncated at orderh 1¯. In discrete systems, we show that the corresponding sum consists of a finite number of terms, albeit one that scales exponentially with the number of qudits. FIG. 1. Translation of (a) a position state and (b) a momentum state along the chord (ξp ,ξq) in phase space. This work also answers a question posed by the recent work of Penney et al. that explored a “sum-over-paths” expression for Clifford circuits in terms of symplectomorphisms on phase Using the canonical commutation relation and eAˆ+ Bˆ = space and associated generating actions. Penney et al. raised Aˆ Bˆ − 1 [A,ˆ Bˆ ] e e e 2 if [A,ˆ Bˆ ] is a constant, it follows that the question of how to relate the dynamics of the Wigner i representation of (stabilizer) states to the dynamics which Zˆ Xˆ = e h¯ Xˆ Z.ˆ (4) are the solutions of the discrete Euler-Lagrange equations This is known as the Weyl relation and shows that the product for an associated functional [16]. By relying on the well- of a shift and a boost (a generalized translation) in phase space established center-chord (or Wigner-Weyl-Moyal) formalism is only unique up to a phase governed by h¯. in continuous [17] and discrete systems [18], we show how We proceed to introduce the chord representation of oper- the dynamics of Wigner representations are governed by ators and states [17]. The generalized phase space translation such solutions related to a “center generating” function and operator (often called the Weyl operator) is defined as a product that these solutions are harmonic and classical in nature. of the shift and boost: Subsequent to the work presented here, the same group i − ξp ·ξq ξp ξq attributed the ability to efficiently simulate Clifford gates to Tˆ (ξp ,ξ q) = e 2 h¯ Zˆ Xˆ , (5) their associated quadratic equations [19]. 2n where ξ (ξp ,ξ q) define the chord phase space. We begin by giving an overview of the center-chord ≡ ∈ R Tˆ ξ ,ξ ξ representation in continuous systems in Sec. II. Then, Sec. III ( p q) is a translation by the chord in phase space. This introduces the expansion of the path integral in powers of ¯h. can be seen by examining its effect on position and momentum This leads us to show what “classical” simulability of states states: i ξq in the continuous case corresponds to and to what higher h (q+ )·ξp Tˆ (ξp ,ξ q)|q = e ¯ 2 |q + ξ q (6) order of h¯ an expansion is necessary to treat any quantum operator. Section IV then introduces the discrete variable and i ξp case and defines its corresponding conjugate position and − ( p+ )·ξq Tˆ (ξp ,ξ q)| p = e h¯ 2 | p + ξ p , (7) momentum operators. The path integral in discrete systems is then introduced in Sec. V and, in Sec. VI, we define the which are shown in Fig. 1. Changing the order of shiftsXˆ and Clifford group and stabilizer states. We prove that stabilizer boosts Zˆ changes the phase of the translation in phase space state propagation within the Clifford group is captured fully up by ξ, as given by the Weyl relation above [Eq. (4)]. to order ¯h0 and so is efficiently simulable classically. Section An operator Aˆ can be expressed as a linear combination of VII shows that extending the Clifford group to a universal gate these translations: set necessitates an expansion of the semiclassical propagator ∞ ∞ 1 to a finite sum at order h¯ . Finally, we close the paper with a Aˆ = dξ p dξ q Aξ (ξp ,ξ q)Tˆ (ξp ,ξ q), (8) discussion and directions for future work in Sec. VIII. −∞ −∞ where the weights are † II. CENTER-CHORD REPRESENTATION Aξ (ξp ,ξ q) = Tr (Tˆ (ξp ,ξ q) Aˆ). (9) IN CONTINUOUS SYSTEMS These weights give the chord representation of Aˆ. If Aˆ is a We define position operatorsqˆ, qˆ|q = q |q , and momen- state, the function Aξ is also called the characteristic function p p Fˆ qFˆ † tum operators ˆ as their Fourier transform, ˆ = ˆ , where of the state. The Weyl function, or center representation, is dual to the ∞ ∞ n 2π i chord representation. It is defined in terms of reflections instead ˆ 2 d d . F = h p q exp − p · q | p q| (1) Rˆ −∞ −∞ h¯ of translations. We can define the reflection operator as the symplectic Fourier transform of the translation operator: Since [q,ˆ pˆ] = i h¯, these operators produce a particularly ∞ i T simple Lie algebra and are the generators of a Lie group. −n ξ J x Rˆ(x p ,xq) = (2πh¯) dξ eh¯ Tˆ (ξ), (10) In this Lie group we can define the “boost” operator −∞ δpp i qδppˆ i q δpp , 2n Zˆ |q = e h¯ |q = e h¯ |q , (2) where x ≡ (x p xq) ∈ R are a continuous set of Weyl phase- space points or centers and J is the symplectic matrix and the “shift” operator 0 −I n δpq − i pδpqˆ J = (11) Xˆ |q = e h¯ |q = |q + δpq. (3) I n 0

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Wigner functions with the center representation, and the center representation as dual to the chord representation motivates the development of both center (Wigner) and chord representations for discrete systems in Sec. IV.

III. PATH-INTEGRAL PROPAGATION IN CONTINUOUS SYSTEMS FIG. 2. Reflection of (a) a position state and (b) a momentum Propagation from one quantum state to another can be x ,x state across the center ( p q) in phase space. expressed in terms of the path-integral formalism of the quantum propagator. For one degree of freedom, with an for I n the n-dimensional identity. The association of this initial position q and final position q , evolving under the operator with reflection can be seen by examining its effect Hamiltonian H for time t, the propagator is on position and momentum states: h i i −iH t/ ¯ D q G q , 2(x q− q)·xp q|e |q = [ t ] exp [ t ] (18) Rˆ(x p ,xq)|q = e h¯ |2x q − q (12) h¯ and where G[qt ] is the action of the trajectoryqt , which starts at q i − 2(x p − p)·xq and ends at q a time t later [20,21]. Rˆ(x p ,xq)| p = e h¯ |2x p − p, (13) Equation (18) can be reexpressed as a variational expansion which are sketched in Fig. 2. It is thus evident that Rˆ(x p ,xq) around the set of classical trajectories (a set of measure zero) reflects the phase space around x. Note that while we refer that start at q and end at q a time t later. This is an expansion to reflections in the symplectic sense here, and in the rest of in powers of h¯ : the paper, Eqs. (12) and (13) show that they are in fact “an cl. paths xp i i 1 2 inversion around x” in every two-plane of conjugate i and − H t (G[qtj ]+δpG [qtj ]+ δp G[qtj ]+··· ) q |e h¯ |q= D[qtj ] eh¯ 2 , xqi . However, we will keep to the established nomenclature [17]. j An operatorAˆ can now be expressed as a linear combination (19) of reflections: δpG q q ∞ ∞ where [ tj ] denotes a functional variation of the paths tj −n δpG q Aˆ = (2πh¯) dxp dxq Ax (x p ,xq)Rˆ(x p ,xq), (14) and for classical paths [ tj ] = 0. (For further details, we −∞ −∞ refer the reader to Sec. 10.3 of [22].) where Terminating Eq. (19) to first order inh produces ¯ the position state representation of the van Vleck-Morette-Gutzwiller A , Rˆ , †Aˆ , x (x p xq) = Tr ( (x p xq) ) (15) (vVMG) propagator [23–25]: and is called the center representation of Aˆ. Note that while ⎛ 2 ⎞1/ 2 ∂ Gj t (q,q ) Tˆ Rˆ i − Gj t (q,q ) the operators are unitary, the operators are unitary and − H t ∂q∂q i 2 q |e h¯ |q = ⎝ ⎠ e h¯ + O ( h¯ ), self-inverse, and hence also Hermitian. 2π ih¯ This representation is of particular interest to us because j we can rewrite the componentsAx for unitary transformations (20) Aˆ as i where the sum is over all classical paths that satisfy the S(x p ,xq) Ax (x p ,xq) = e h¯ , (16) boundary conditions. n where S(x p ,xq) is equivalent to the action the transformation In the center representation, for degrees of freedom, the Ut p , q Ax produces in Weyl phase space in terms of reflections around semiclassical propagator (x x ) becomes [17] centers x [17]. Thus, S is also called the “center generating” 1 2 ∂ Stj 2 i function. 1 Stj (x p ,xq) Ut (x p ,xq) = det 1 + J eh¯ ∂ 2 For a pure state| , the Wigner function given by Eq. (15) j 2 x simplifies to h2 , ∞ ξ + O ( ¯ ) (21) , πh −n dξ q x (x p xq) = (2 ¯) q xq + S , −∞ 2 where tj (x p xq) is the center generating function (or action) 1 for the center x = (x p ,xq) ≡ [( p,q) + ( p ,q )]. ξq i 2 ∗ − ξq·xp × xq − e h¯ . (17) In general, this is an underdetermined system of equations 2 and there are an infinite number of classical trajectories that The center representation for quantum states immediately satisfy these conditions. The accuracy of adding them up as yields the well-known Wigner function for continuous sys- part of this semiclassical approximation is determined by how tems. The chord representation is the symplectic Fourier separated these trajectories are with respect to h¯ —the saddle- transform of the Wigner function. The center and chord point condition for convergence of the method of steepest representations are dual to each other, and are the Wigner descents. However, some Hamiltonians exhibit a single saddle- and characteristic functions, respectively. Identifying the point contribution and are thus exact at order h¯ 1 [26].

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Lemma 1. There is only one classical trajectory ( p ,q) → This state describes momentum states [ δp( p − pβ ) in mo- t mentum representation] when β → 0 and position states ( p ,q ) that satisfies the boundary conditions (x p ,xq) = δp(q − qβ ) when β → ∞ . Rotations between these two cases 1 [( p,q) + ( p ,q )] andt under Hamiltonians that are harmonic 2 correspond to Re β 0 and Im β 0. in p and q. = = Gaussians remain Gaussians under evolution by a harmonic Proof. For a quadratic Hamiltonian, the diagonalized Hamiltonian, even if it is time dependent. This can be shown solutions to the equations of motion forn-dimensional ( p,q ) by simply making the ansatz that the state remains a Gaussian are of the form and then solving for its time dependent β , pβ , qβ , and phase pi = α (t)i pi + β (t)i qi + γ (t)i , (22) from the time-dependent Schrödinger equation [22], or just by applying the analytically known Feynman path integral for a q = δp (t)i pi + (t)i qi + η (t)i (23) i harmonic Hamiltonian, which is equivalent to the van Vleck for i ∈ { 1,2, . . . ,n}. Since t is known and ( p,q) can be written path integral, to a Gaussian [28]. in terms of ( p ,q ) by using (x p ,xq), this brings the total Moreover, with the propagator in the center representation number of linear equations to 2 n with 2n unknowns and so known to only have one saddle-point contribution for a there exists one unique solution. harmonic Hamiltonian, it is fairly straightforward to show Since the equations of motion for a harmonic Hamiltonian that this is also true for its coherent state representation are linear, we can write their solutions as (that is, taking a Gaussian to another Gaussian). Applying the propagator to an initial and final Gaussian in the center p p 1 1 = M t + αt + αt , (24) representation q q 2 2 [| β β |Ut | α α|]x (x) where αt is an n vector andM t is an n × n symplectic matrix, ∞ ∞ both with entries in R. In this case, the center generating πh − 3n d d U S , = ( ¯) x1 x2 t (x 1 + x2 − x) function t (x p xq) is also quadratic, in particular, −∞ −∞ i T T T 2 h (x J x2+ x J x+ x J x1) T xp xp × β x (x 2) αx (x 1)e ¯ 1 2 , (29) St (x p ,xq) = α t J + (x p ,xq)Bt , (25) xq xq we see that since Ut is a Gaussian from Eq. (22) (Stj is where Bt is a real symmetric n × n matrix that is related to quadratic for harmonic Hamiltonians) and since the Wigner M t by the Cayley parametrization of M t [27]: representations of the Gaussians, αx and β x, are also known − 1 − 1 to be Gaussians, the full integral in the above equation is a J B t = (1 + M t ) (1 − M t ) = (1 − M t )(1 + M t ) . Gaussian integral and thus evaluates to produce a Gaussian (26) with a prefactor. This is equivalent to evaluating the integral by the method of steepest descents which finds the saddle Since one classical trajectory contribution is sufficient in ∂φ points to be the points that satisfy = 0 where φ is the phase this case, if the overall phase of the propagated state is not ∂x of the integrand’s argument. Since this argument is quadratic, important, then the expansion with respect toh ¯in Eq. (19) can its first derivative is linear and so again there is only one unique be truncated at order h¯ 0. Dropping terms that are higher order saddle point. than ¯h0 and ignoring phase is equivalent to propagating the Indeed, such an evaluation produces the coherent state classical density ρ(x) corresponding to the ( p ,q) manifold, representation of the vVMG propagator [29]. Just as we under the harmonic Hamiltonian and determining its overlap found with the center representation, the absolute value of with the ( p ,q ) manifold after time t. Such a treatment under its prefactor corresponds to the order h¯ 0 term. a harmonic Hamiltonian results in just the absolute value of 1 As a consequence of this single contribution at order 2 1 ∂ Stj 2 0 J h x the prefactor of Eq. (22): | det [1 + 2 ∂x2 ]| . Indeed, this ¯ , it follows that the Wigner function of a state, (x), was van Vleck’s discovery before quantum mechanics was evolves under the operator Vˆ with an underlying harmonic formalized [23]. The relative phases of different classical Hamiltonian by x (M Vˆ (x + α Vˆ / 2) + α Vˆ / 2), where M Vˆ contributions are no longer a concern and the higher-order is the symplectic matric and αVˆ is the translation vector terms only weigh such contributions appropriately. associated with Vˆ ’s action [18]. Here, we are interested in propagating between Gaussian Before proceeding to the discrete case, we note that states in the center representation. In continuous systems, a the center representation that we have defined allows for a Gaussian state in n dimensions can be defined as particularly simple way to express how far the path-integral −n 1 treatment must be expanded in ¯ h in order to describe any β (q) = [π det(Re β )] 4 exp(ϕ), (27) unitary propagation (not necessarily harmonic) in continuous where quantum mechanics. i Reflections and translations can also be described by 1 T 1 0 ϕ pβ (q q ) (q q ) β (q qβ ). (28) truncating Eq. (22) at order h¯ (or h¯ if overall phase is = h · − β − − β − ¯ 2 not important) since they correspond to evolution under a n n qβ ∈ R is the central position, pβ ∈ R is the central harmonic Hamiltonian. In particular, translations are displace- momentum, and β is a symmetricn × n matrix where Re β ments along a chordξ and so have HamiltoniansH ∝ ξ q · p − is proportional to the spread of the Gaussian and Im β ξp · q. Reflections are symplectic rotations around a center x π 2 2 p q p q captures - correlation. and so have Hamiltonians H ∝ 4 [( p − x ) + (q − x ) ].

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From Eq. (14) we see that any operator can be expressed where ⊕ denotes mod-d integer addition. It follows that the as an infinite Riemann sum of reflections. Therefore, since Weyl relation holds again: h1 reflections are fully described by a truncation at order ¯ , Zˆ ˆ Xˆ Z.ˆ it follows that an infinite Riemann sum of path-integral X = ω (34) 1 solutions truncated at order h¯ can describe any unitary The group generated by Zˆ and Xˆ has a d-dimensional irre- evolution. The same statement can be made by considering the ducible representation only if ωd = 1 for odd d. Equivalently, chord representation in terms of translations. Hence, quantum there are only reflections relating any two phase-space points propagation in continuous systems can be fully treated by an on the Weyl phase space “grid” if d is odd [30]. We take infinite sum of contributions from a path-integral approach ω ≡ ω(d) = e 2π i/d [31]. This was introduced by Weyl [32]. truncated at order h¯ 1. d d Note that this means that h¯ = 2π or h = d . For a given , As an aside, in general this infinite sum is not convergent a unit of action [ ¯h] is given by the area in phase space of a and so it is often more useful to consider reformulations that quantum state: d. Hence, for every d, the h¯ normalized by the involve a sum with a finite number of contributions. One way 1 unit of action is 2π . Therefore, since the classical regime is to do this is to apply the method of steepest descents directly on reached as h¯ → 0, our formalism is the same effectiveh ¯ away the operator of interest and use the area between saddle points from the classical regime for all d. as the metric to determine the order of h¯ necessary, instead Another way of interpreting the classical limit in this paper of dealing with an infinity of reflections (or translations). is by considering the appropriately normalized h to be equal This results in the semiclassical propagator already presented, to the inverse of the density of states in phase space (i.e., in but associated with the full Hamiltonian instead of a sum of a Wigner unit cell). As d increases, the phase space increases reflection Hamiltonians. as d2 to accommodate d states, each with area d, and so the In summary, we have explained why propagation between density of states remains constant. Gaussian states under Hamiltonians that are harmonic is By analogy with continuous finite translation operators, 0 simulable classically (i.e., up to order h¯ ) in continuous we reexpress the shift Xˆ and boost Zˆ operators in terms of systems. We will see that the same situation holds in discrete conjugate pˆ and qˆ operators: systems for stabilizer states, with the additional restriction that 2π i q 2π in the propagation takes the phase-space points, which are now Zˆ|n = e d ˆ |n = e d |n (35) discrete, to themselves. and 2π ip Xˆ n e − d ˆ n n . IV. DISCRETE CENTER-CHORD REPRESENTATION | = | = | ⊕ 1 (36) Zˆ We now proceed to the discrete case and introduce the Hence, in the diagonal “position” representation for , center-chord formalism for these systems. It will be useful ⎛1 0 0 · · · 0 ⎞ for us to define a pair of conjugate degrees of freedom 2π i ⎜0 e d 0 · · · 0 ⎜ p and q for discrete systems. Unfortunately, this is not as ⎜ 4π i ⎜ ⎜ e d ⎜ straightforward as in the continuous case since the usual Zˆ = ⎜0 0 · · · 0 ⎜ (37) ⎜. . . . . ⎜ canonical commutation relations cannot hold in a finite- ⎝...... ⎠ 2(d− 1)π i dimensional Hilbert space where the operators are bounded 0 0 0 0 e d (since Tr[p,ˆ qˆ] = 0). We begin with one degree of freedom. We label the and , computational basis for our system by n ∈ {0 1, . . . , d −1}, ⎛0 0 · · · 0 1⎞ for d odd, and we assume thatd is odd for the rest of this paper. ⎜1 0 · · · 0 0⎜ We identify the discrete position basis with the computational ⎜ ⎜ Xˆ = ⎜0 1 0 · · · 0⎜. (38) basis and define the “boost” operator as diagonal in this basis: ⎜. . . . .⎜ ⎝...... ⎠ δpp nδpp Zˆ |n ≡ ω |n, (30) 0 · · · 0 1 0 where ω will be defined below. Thus, We define the normalized discrete Fourier transform oper- d qˆ = log Zˆ = n|nn| (39) ator to be equivalent to the Hadamard gate: 2π i n∈Z/dZ 1 Fˆ = √ ω−mn |mn|. (31) where the logarithm is taken with base (e) and d m,n /d † ∈Z Z pˆ = Fˆ qˆFˆ . (40) This allows us to define the Fourier transform of Zˆ : Therefore, we can interpret the operators pˆ and qˆ as a

† conjugate pair similar to conjugate momenta and position Xˆ ≡ Fˆ Zˆ Fˆ . (32) in the continuous case. However, they differ from the latter in that they only obey the weaker group commutation Xˆ Again, as before, we call the “shift” operator since relation δpq ij q/ h ik p/ h q/ h p/ h h Xˆ |n ≡ |n ⊕ δpq, (33) e ˆ ¯ e ˆ ¯ e−ij ˆ ¯ e−ik ˆ ¯ = e −ij k/ ¯ Iˆ. (41)

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This corresponds to the usual canonical commutation relation erator. We can define the discrete reflection operator Rˆ as for the p and q algebra at the origin of the Lie group j( = k = the symplectic Fourier transform of the discrete translation 0); expanding both sides of Eq. (41) to first order in p and q operator we just introduced: yields the usual canonical relation. 2π i T −n (ξp ,ξ q)J (x p ,xq) We proceed to introduce the Weyl representation of opera- Rˆ(x p ,xq) = d e d Tˆ (ξp ,ξ q). tors and states in discrete Hilbert spaces with odd dimension ξp ,ξ q ∈ n d and n degrees of freedom [1,33,34]. The generalized phase- (Z/dZ ) space translation operator (the Weyl operator) is defined as a (45) product of the shift and boost with a phase appropriate to the d-dimensional space: These operators are both unitary and Hermitian, and hence are self-inverse. 2π − 1ξ ξ ξ With this in hand, we can now express a finite-dimensional Tˆ ξ ,ξ −i d 2 p ·ξq Zˆ p Xˆ q , ( p q) = e (42) operator Aˆ as a superposition of reflections:

− 1 d+ 1 2n −n ξp ,ξ q Aˆ = d Ax (x p ,xq)Rˆ(x p ,xq). (46) where 2 ≡ 2 , and ξ ≡ ( ) ∈ (Z/dZ ) and form a discrete “web” or “grid” of chords. They are a discrete xp ,xq ∈ n subset of the continuous chords we considered in the infinite- (Z/dZ ) dimensional context in Sec. II and the fact that there is The reflections Rˆ are Hilbert-Schmidt orthogonal and so the a finite number of chords is an important consequence coefficients Ax (x p ,xq) are given by of the discretization of the continuous Weyl formalism. −n † The translation operators have the important property that Ax (x p ,xq) = d Tr (Rˆ(x p ,xq) Aˆ). (47) † Tˆ ξ ,ξ Tˆ 2n ( p q) = (−ξ p ,−ξ q). x ≡ (x p ,xq) ∈ (Z/dZ ) are centers or Weyl phase-space Aˆ Again, an operator can be expressed as a linear combi- points and, like their (ξp ,ξ q) brethren, form a discrete subgrid nation of translations: of the continuous Weyl phase-space points considered in Sec. II. −n Aˆ = d Aξ (ξp ,ξ q)Tˆ (ξp ,ξ q). (43) Again, the center representation is of particular interest

ξp ,ξ q ∈ to us because for unitary gates Aˆ we can write the function /d n (Z Z ) Ax (x p ,xq) as i The generalized translations are Hilbert-Schmidt orthogonal Ax (x p ,xq) = exp S(x p ,xq) , (48) and the weights are the chord representation of the operatorAˆ: h¯ S , Aˆ −n † where (x p xq) is the action of the operator in the center Aξ ξp ,ξ q d Tˆ ξp ,ξ q Aˆ . ( ) = Tr ( ( ) ) (44) representation. S is called the center generating function. Aside from Eq. (47), the center representation of a state ˆρ ρ When applied to a state ˆ, this is also called the “characteristic can also be directly defined as the symplectic Fourier transform function” of ρˆ [35]. Because the generalized translation of its chord representation ρξ [3]: operators are unitary but not Hermitian for d > 2, the chord 2π i T −n (ξp ,ξ q)J (x p ,xq) representation, and hence the characteristic function, is in ρx (x p ,xq) = d e d ρξ (ξp ,ξ q). general complex valued. ξp ,ξ q ∈ For the operator coefficients to be real valued we require (Z/dZ )n an operator basis that is Hermitian. The discrete center (49) representation provides such an operator basis. As before, the center representation, based on reflections instead of We note again that for a pure state| , the Wigner function translations, requires an appropriately defined reflection op- from Eqs. (47) and (49) simplifies to

2π i ξq ξq −n − ξq·xp (d + 1) ∗ (d + 1) x (x p ,xq) = d e d xq + xq − . (50) n 2 2 ξq ∈ (Z/dZ )

It may be clear from this short presentation that the chord tori. This produces and center representations are dual to each other. A thorough ⎧ ⎫ 1 2 review of this subject can be found in Ref. [17]. ⎨ ∂ Stj 2 i ⎬ 1 Stj (x) iθk 2 Ut (x) = det 1+ J eh¯ e +O (h ¯ ), ∂ 2 ⎩ j 2 x ⎭ V. PATH-INTEGRAL PROPAGATION k IN DISCRETE SYSTEMS (51) Rivas and Almeida [18] found that the continuous infinite- where a sum is taken over thej classical center trajectories that dimensional vVMG propagator can be extended to finite satisfy the boundary conditions and an additional average must Hilbert space by simply projecting it onto its finite phase-space be taken over the k center points that are equivalent because

032331-6 SEMICLASSICAL FORMULATION OF THE GOTTESMAN- . . . PHYSICAL REVIEW A 96, 032331 (2017) of the periodic boundary conditions. Maintaining periodicity are harmonic, but in the discrete case there is an additional 2 ∂ Stj θk requirement that they are evaluated at chords and/or centers requires that they accrue a phase [36]. The derivative ∂x2 is performed over the continuous function Stj defined after that take Weyl phase-space points to themselves. Eq. (22), but only evaluated at the discrete Weyl phase-space Just as in the continuous case, the single contribution at 2n h0 points x ≡ (x p ,xq) ∈ (Z/dZ ) . The prefactor can also be order ¯ implies that the propagator of the Wigner function ˆ reexpressed: of states x (x) under gates V with underlying harmonic Hamiltonians is captured by x (M Vˆ (x + α Vˆ / 2) + α Vˆ / 2) for 1 ∂2S 2 M α Vˆ 1 tj d − 1 Vˆ and Vˆ associated with . det 1 + J = { 2 det[1 + M tj ]} 2 , (52) 2 ∂x2 VI. STABILIZER GROUP where M t j is defined in Eq. (24). This is perhaps more pleasing in the discrete case as it does not involve a derivative. Here, we will show that the Hamiltonians corresponding to As in the continuous case, for a harmonic Hamiltonian Clifford gates are harmonic and take Weyl phase-space points H ( p,q), the center generating function S(x p ,xq) is equal to to themselves. Thus, they can be captured by only the single T T α J x + x Bx where Eqs. (26) and (24) hold. Moreover, if contribution of Eq. (55) at lowest order inh ¯. This then implies the Hamiltonian takes Weyl phase-space points to themselves, that stabilizer states can also be propagated to each other by then by the same equations it follows thatM and α must have Clifford gates with only a single contribution to the sum in integer entries. Eq. (51). n This implies that, for m,n ∈ Z , The Clifford gate set of interest can be defined by three generators: a single-qudit Hadamard gate Fˆ and phase-shift p + md + α p / 2 αp / 2 M + gate Pˆ, as well as the two-qudit controlled-not gate Cˆ . We q + nd + α q/ 2 αq/ 2 examine each of these in turn. p + α p / 2 αp / 2 m = M / + α / + dM q + α q 2 q 2 n A. Hadamard gate p The Hadamard gate was defined in Eq. (31) and is a rotation = mod d . (53) π q by 2 in phase space counterclockwise. Hence, for one qudit, it can be written as the map in Eq. (53) where Therefore, phase-space points ( p + md,q + nd) that lie on Weyl phase-space points go to the equivalent Weyl phase-space 0 1 M = , (56) points ( p ,q ). Fˆ − 1 0 Moreover, again if m,n ∈ Z n, and αFˆ = (0,0). We have set t = 1 and drop it from the S(x p + md,xq + nd) subscripts from now on. Since α is vanishing and M has T integer entries, this is a cat map and such maps have been xp + md xp + md xp + md = A + b · shown to correspond to Hamiltonians [37] xq + nd xq + nd xq + nd 2 2 T T H (p,q ) = f (Tr M )[M 12p − M 21q + (M 11 − M 22)pq], xp xp xp xp m = A + b · + d 2 A xq xq xq xq n (57)

T where m m m √ + d A + b · − 1 1 2 n n n sinh x − 4 f (x) = √ 2 . (58) x2 − 4 = S (x p ,xq) mod d, (54) π 2 2 For the HadamardM Fˆ this corresponds toHFˆ = (p + q ), for some symmetric A ∈ Z n×n and b ∈ Z n. Therefore, these 4 a harmonic oscillator. The center generating functionS(xp ,xq) equivalent trajectories also have equivalent actions (since the x ,x B x ,x T 2π i is thus ( p q) ( p q) and solving Eq. (26) finds for the action is multiplied by d and exponentiated). one-qudit Hadamard Hence, there is only one term to the sum in Eq. (51) for harmonic Hamiltonians that take Weyl phase-space points to 1 0 B = . (59) themselves. Moreover, [30] showed that the sum over the Fˆ 0 1 phases θk produces only a global phase that can be factored 2 2 out. Therefore, if we can neglect the overall phase, Thus, SFˆ (xp ,xq) = x p + x q. Indeed, applying Eq. (47) to Eq. (31) reveals that the Hadamard’s center function (up to d 1 Ut (x) = | 2 det[1 + M ]| 2 , (55) a phase) is

2π i 2 2 where the classical trajectories whose centers are (x p ,xq) (xp +x q) Fx(xp ,xq) = e d . (60) satisfy the periodic boundary conditions. As in the continuous case, we point out that this means that Equation (56) shows how to map Weyl phase space to Weyl translations and reflections are fully captured by a path-integral phase space under the Hadamard transformation. Furthermore, treatment that is truncated at order h¯ 1 (or order h¯ 0 if their this map is pointwise, which implies the quadratic form of the overall phase is not important) because their Hamiltonians center generating function obtained in Eq. (60).

032331-7 LUCAS KOCIA, YIFEI HUANG, AND PETER LOVE PHYSICAL REVIEW A 96, 032331 (2017)

B. Phase-shift gate target reversed with respect to the Zˆ basis. This can also seen The phase-shift gate can be generalized to oddd dimensions by looking at its effect in the momentum (Xˆ ) basis: [38] by setting it to † Fˆ Cˆ Fˆ = |j k,kj,k|. (68) (j− 1)j Pˆ = ω 2 |j j |. (61) j,k∈Z/dZ j∈Z/dZ As a result, this gate is described by the map Examining its effect on stabilizer states, it is clear that it is a ⎛1 − 1 0 0⎞ q shear in phase space from an origin displaced by d− 1 1 2 ≡ − 2 0 1 0 0 M = ⎜ ⎜, (69) to the right. This can be expressed as the map in Eq. (53) with Cˆ ⎝0 0 1 0⎠ 1 1 0 0 1 1 M = , (62) Pˆ 0 1 and αCˆ = (0,0,0,0). This corresponds to α 1 , ⎛ ⎞ and Pˆ = (− 2 0). This corresponds to 0 0 0 0 ⎜ 1 ⎜ B 0 0 − 2 0 . 0 0 Cˆ = ⎜ 1 ⎜ (70) BPˆ = . (63) ⎝0 − 0 0⎠ 0 1 2 2 0 0 0 0 Solving Eq. (25) with this BPˆ and αPˆ reveals that 1 1 2 Hence, its center generating function SCˆ (x p ,xq) = −x p xq . S (xp ,xq) = − xq + x . Again, this agrees with the ar- 2 1 Pˆ 2 2 q Again, this corresponds with the argument of the center gument of the center representation of the phase-shift gate representation of the controlled-not gate, which can be found obtained by applying Eqs. (47) to (61): to be 2π i 1 2 π i d 2 (−x q+x q) 2 Px (xp ,xq) e . (64) − d xq xp = Cx (x p ,xq) = e 1 2 . (71) Discretizing the equations of motion for harmonic evolution Therefore, this gate can be seen to be a bilinear p-q coupling for unit time steps leads to between two qudits and corresponds to the Hamiltonian T p p ∂H ∂H H p q , = + J , , (65) Cˆ = 1 2 (72) q q ∂ p ∂q as can be found from Eq. (65) again. where the last derivative is on the continuous function H , As a result, it is now clear that all the Clifford group gates but only evaluated on the discrete Weyl phase-space points. It have Hamiltonians that are harmonic and that take Weyl phase- follows that space points to themselves. Therefore, their propagation can be fully described by a truncation of the semiclassical propagator d + 1 2 d + 1 HPˆ = − q + q. (66) h0 2 2 (51) to order ¯ as in Eq. (55) and they are manifestly classical in this sense. We have obtained the Hamiltonian for the phase-shift gate To summarize the results of this section, using Eq. (46), the by a different procedure than that used for the Hadamard where Hadamard, phase shift, and CNOT gates can be written as we appealed to the result given in Eq. (57) for quantum cat 2π i 2 2 maps. However, as the phase-shift gate is a quantum cat map as − 2 − [− (x +x )−d (xp ξp −x qξq)] ξp ξq Fˆ = d e d p q Zˆ Xˆ , well, we could have obtained Eq. (66) in this manner. Similarly, xp ,xq, the approach we used to find the phase-shift Hamiltonian by ξp ,ξq ∈ discretizing time in Eq. (65) would work for the Hadamard Z/dZ gate but it is a bit more involved since the latter contains both (73) p and q evolution. Nevertheless, this produces Eq. (57) as 2π i 1 2 − 2 − [ (xq−x )−d (xp ξq−x qξp )] ξp ξq well. We presented both techniques for illustrative purposes. Pˆ = d e d 2 q Zˆ Xˆ ,

xp ,xq, C. Controlled-not gate ξp ,ξq ∈ Z/dZ Lastly, the controlled-not gate can be generalized to d dimensions [38] by (74) and Cˆ = |j,k ⊕ j j,k|. (67) 2π i − 4 − [xq xp −d (x p ·ξq− xq·ξp )] ξp ξq j,k∈Z/dZ Cˆ = d e d 1 2 Zˆ Xˆ q xp ,xq, It is clear that this translates the state of the second qudit ξp ,ξ q ∈ by the q state of the first qudit. As a result, as is evident by (Z/dZ )2 examining the gate’s action on stabilizer states, the first qudit (75) experiences an “equal and opposite reaction” force that kicks its momentum by the q state of the second qudit. This is the (up to a phase). This form emphasizes their quadratic nature. phase-space picture of the well-known fact that a controlled- As for the continuous case, there exists a particularly simple not (CNOT) gate examined in the Xˆ basis has the control and way in discrete systems to see to what order in ¯h the path

032331-8 SEMICLASSICAL FORMULATION OF THE GOTTESMAN- . . . PHYSICAL REVIEW A 96, 032331 (2017) integral must be kept to handle unitary propagation beyond in phase space in “steps” such that it always lies along the 2n the Clifford group. We describe this in the next section. discrete Weyl phase-space points (xp ,xq) ∈ (Z/dZ ) . We note that the center generating actions S(x p ,xq) found This Gaussian expression only captures stabilizer states that here are related to the G(q ,q) found by Penney et al. [16], are maximally supported inq space. One may wonder what the which are in terms of initial and final positions, by symmetrized stabilizer states that are not maximally supported in q space Legendre transform [17]: look like in Weyl phase space. Of course, it is possible that some may be maximally supported in p space and so can be q + q G(q ,q,t ) = F , p(q − q) , (76) captured by the following corollary: 2 Corollary 2. If (p) = 0 for all p’s then there exists a n×n n where the canonical generating function θβp ∈ (Z/dZ ) and an ηβp ∈ (Z/dZ ) such that

q + q q + q 2π i T F , p = S xp = p,xq = + p · (q − q), θ ,η ( p) ∝ exp ( p θβp p + ηβp · p) . (83) 2 2 βp βp d

(77) Proof. This can be shown following the same methods ∂F employed by Gross [3] but in the discretep basis. for p(q − q) given implicitly by ∂ p = 0. Applying this to the actions we found reveals that Unfortunately, it is easy to show that Corollaries 1 and 2 do not provide an expression for all stabilizer states (except for GFˆ (q ,q,t ) = q q, (78) the odd prime d case, as we shall see shortly) as there exist stabilizer states for odd nonprime d that are not maximally d + 1 2 GPˆ (q ,q,t ) = (q − q ), (79) p q 2 supported in or space or any finite rotation between those two. To find an expression that encompasses all stabilizer and states, we must turn to the Wigner function of stabilizer states. G q ,q , q ,q ,t , An equivalent definition of stabilizer states on n qudits Cˆ (( 1 2) ( 1 2) ) = 0 (80) is given by states Vˆ |0 ⊗n where Vˆ is a quantum circuit which is in agreement with [16]. consisting of Clifford gates. We know that the Clifford circuits are generated by the Pˆ, Fˆ , and Cˆ gates, and that the D. Classicality of stabilizer states Wigner functions x (x) of stabilizer states propagate under Vˆ x M α / α / In this section we show that stabilizer states evolve to as ( Vˆ (x + Vˆ 2) + Vˆ 2), it follows that the Wigner function of stabilizer states is stabilizer states under Clifford gates, and that it is possible to describe this evolution classically. For oddd 3, the positivity δp ·M ·x,r , (84) of the Wigner representation implies that evolution of stabilizer 0 Vˆ 0 states is noncontextual, and so here we are investigating in where = 0 0 and r = (0,0). We have therefore detail what this means in our semiclassical picture. 0 0 I n 0 To begin, it is instructive to see the form stabilizer states proved the next theorem: take in the discrete position representation and in the center Theorem 2. The Wigner function x (x) of a stabilizer state d n δp n representation. Gross proved the following [3]: for any odd and qudits is · x,r for 2n × 2 matrix and n n Theorem 1. Let d be odd and ∈ L 2[(Z/dZ ) ] be a state 2 vector r. vector. The Wigner function of is non-negative if and only As an aside, Theorem 2 allows us to develop an all- if is a stabilizer state. encompassing Gaussian expression for stabilizer states for Gross also proved the following [3]: the restricted case that d is odd prime. In this case, the Corollary 1. Given that (q) = 0 ∀ q, a vector is a following corollary shows that a “mixed” representation is stabilizer state if and only if it is of the form always possible, where each degree of freedom is expressed in either the p or q basis: π i 2 T Corollary 3. For odd prime d, if is a stabilizer state for θ ,η (q) ∝ exp (q θβ q + η β · q) , (81) β β d n qudits, then there always exists a mixed representation in n×n n position and momentum such that where θβ ∈ (Z/dZ ) and q,ηβ ∈ (Z/dZ ) . Applying Eq. (50) to (81), the Wigner function of such 1 2π i T θ ,η (x) = √ exp (x θβ x + ηβ · x) , (85) maximally supported stabilizer states can be found to be βx βx d d x x θ ,η (x p ,xq) β β x where xi can be either pi or qi . π i −n 2 Proof. We begin with a one-qudit case. We examine the d exp ξq (ηβ xp 2θβ xq) . ∝ d · − + equation specified by · x = r : ξq ∈ n Z/dZ αq1 + βp 1 = γ (86) (82) for α, β, and γ ∈ Z/dZ . If α = 0 then (q1) is maximally Therefore, one finds that the Wigner function is the discrete supported and if β = 0 then (p1) is maximally supported δp θ q p Fourier sum equal to ηβ − xp + 2θβ xq . For β = 0 the state is a since the equation specifies a line on Z/dZ in 1 and 1 momentum state at xp . Finite θβ rotates that momentum state respectively. If α = 0 and β = 0 then the equation can be

032331-9 LUCAS KOCIA, YIFEI HUANG, AND PETER LOVE PHYSICAL REVIEW A 96, 032331 (2017) rewritten as

q1 + (β/α )p1 = γ /α, (87) and it follows that q1 covers all of Z/dZ as p1 is varied since d is odd prime. Therefore, (q1) is maximally supported. However, it is also possible to reexpress the equation as:

p1 + (α/β )q1 = γ /β. (88)

It follows that p1 can also take all values onZ/dZ — (p1) is also maximally supported. Therefore, one can always choose FIG. 3. The two classes of stabilizer states possible for odd prime d 7 in terms of support: (a) maximally supported inp or q and (b) either ap or q basis such that the state is maximally supported = 1 1 maximally supported inp and q. The central grids denote the Wigner and so is representable by a Gaussian function. function x (p,q ) of a stabilizer state with d = 7. The projection In general, for n qudits, there are 2n variables and n equa- of this state onto p space is shown in the upper right [ | (p)|2] and tions specified by the Wigner function’s associated equation the projection onto q space is shown in the upper left [| (q)|2]. · x = r . So there are at least n independent variables and indeed ( | r) always has rank n, which is preserved under The form of Eq. (85) is more general than Eq. (81) because symplectic Clifford transformations. Moreover, for quditi , pi it does not depend on the support of the state. As we saw in and qi cannot both be independent variables in this linear system of equations. To see this, we can assume that our the proof, this representation is generally not unique; for every qudit i that is not a position or momentum state,xi can be either stabilizer state has pi s and qi s both as dependent variables for m p q pi or qi . However, if it is a position state, then xi = p i and if qudits such that j and j are both independent variables for x m other qudits. Considering that Clifford operations preserve it is a momentum state, then i = q i ; position and momentum the symplectic area, it is always possible to start from the states must be expressed in their conjugate representation in |0 · · ·0 state, whose Wigner function has support onnd points order to be captured by a Gaussian of the form in Eq. (85) (which we refer to as its symplectic area through a slight abuse instead of Kronecker deltas. of notation), and apply Clifford gates to get the state with them The reason this mixed representation does not hold for p q nonprime oddd is that the coefficients above can be (multiples independent variables j s and j s, which has symplectic area d p n m d md2 m of) prime factors of and so no longer produce “lines” in i ( − 2 ) + + . Since symplectic area is conserved q under Clifford operations, this area must be equal to the initial or i that cover all of Z/dZ . nd n m d md 2 m An example of the different classes of stabilizer states that symplectic area: = ( − 2 ) + + . This implies d that d = 1 and is thus not possible. Therefore, m must be are possible for odd prime , in terms of their support, is shown equal to zero and so there must be exactly one independent in Fig. 3. There, it can be seen that a stabilizer state is either maximally supported in pi or qi , and is a Kronecker delta variable pi or qi for each qudit i . We now consider adding another qudit to our one-qudit function in the other degree of freedom, or it is maximally case such that the Wigner function of the state becomes supported in both. On the other hand, for odd nonprimed, we see in Fig. 4 that x (p1,p 2,q1,q2). There are now two equations specified by · x = r and it follows that it is always possible to combine another class is possible: stabilizer states that are maximally supported in neitherpi or qi . In fact, rotating the basis in any of the two equations such thatp1 and q1 are in only one equation and written in terms of each other (and generally the second the discrete angles afforded by the grid still does not produce a degree of freedom): basis that is maximally supported. Notice, also, that Fig. 4(b) shows that it is no longer true that a state that is maximally αq1 + βp 1 + γ q2 + δpp2 = , (89) supported in only qi or pi is automatically a Kronecker delta when expressed in terms of the other. for α, β, γ , δpand ∈ Z/dZ . It will turn out that theγ q2 + δpp2 In summary, stabilizer states have Wigner function δp· x,r term is irrelevant. We can rewrite the above equation as: and, for odd prime d, are Gaussians in mixed representation p , q q1 + (β/α )p1 + (γ /α)q2 + (δp/α)p2 = /α, (90) that lie on the Weyl phase-space points (x x ). Heuristically, they correspond to Gaussians in the continuous case that p if α = 0. Since there is no other equation specifying 1, this spread along their major axes infinitely. The only reason that is an equation for a line on Z/dZ and so is is maximally they are not always expressible as Gaussians in the mixed q supported on 1. Otherwise, rewriting the above equation as: representation is that they sometimes “skip” over some of the p + (α/β )q + (γ /β)q + (δp/β )p = /β, (91) discrete grid points due to the particular angle they lie along 1 1 2 2 phase space for odd nonprime d. if β = 0 shows that is maximally supported on p1. Both Wigner functions x (x) of stabilizer states propagate α,β = 0 since p1 or q1 must be an independent variable. The under Vˆ as x (M Vˆ (x + α Vˆ / 2) + α Vˆ / 2), and this preserves same procedure can be performed to find if p2 or q2 produce the form of the state. In other words, Clifford gates take a maximally supported state and so on for more qudits. stabilizer states to other stabilizer states, as expected, just Therefore, it follows that every degree of freedom (corre- like in the continuous case Gaussians go to other Gaussians sponding to a qudit) is maximally supported in either thep or under harmonic evolution. It is also clear that stabilizer state q basis and so Eq. (85) always describes stabilizer states for propagation under Clifford gates can be expressed by a path odd prime d. integral at order h¯ 0.

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α 1 , and Tˆ = (− 4 0). This corresponds to 0 0 BTˆ = 1 . (94) 0 − 4 Thus, the center function

2π i 1 2 − (xq−x ) Tx (xp ,xq) = e d 4 q , (95) corresponding to the phase-shift Hamiltonian applied for only half the unit of time. The operator can thus be written

2π i 1 2 − 2 − [ (xq−x )−d (xp ξq−x qξp )] ξp ξq Tˆ = d e d 4 q Zˆ Xˆ . (96)

xp ,xq, ξp ,ξq ∈ Z/dZ Although this operator is quadratic, it no longer takes the h0 FIG. 4. The four classes of stabilizer states possible for odd Weyl center points to themselves. This means that the ¯ limit nonprime d = 15 in terms of support: (a), (b) maximally supported of Eq. (55) is now insufficient to capture all the dynamics in p or q, (c) maximally supported inp and q, and (d) not maximally because the overlap with any |q will now involve a linear supported in p or q. The central grids denote the Wigner function superposition of partially overlapping propagated manifolds. x (p,q ) for a stabilizer state with d = 15. The projection of this It must therefore be described by a path-integral formulation 1 state onto p space is shown in the upper right [| (p)|2] and the that is complete to order h¯ . In particular, 2 q q 2π i 1 2 projection onto space is shown in the upper left [| ( )| ]. − 1 − (xq−x ) Tˆ = d e d 4 q Rˆ(xp ,xq), (97)

xp ,xq ∈ VII. DISCRETE PHASE-SPACE REPRESENTATION (Z/dZ ) OF UNIVERSAL QUANTUM COMPUTING where Rˆ should be substituted by its path integral. A similar statement to the one we made in Sec. II, i.e., Note that this does not imply efficient classical simulation that any operator can be expressed as an infinite sum of path- of quantum computation but quite the opposite. Indeed, for n n integral contribution truncated at order h¯ 1, can be made in qudits, there are d2 terms in the sum above. While every discrete systems. However, there is an important difference in Weyl phase-space point has only a single associated path the number of terms making up the sum. when acted on by Clifford gates, this is no longer true in To see this, we can follow reasoning that is similar to that any calculation of evolution under the T gate. Equation (97) employed in the continuous case. Namely, from Eq. (46) we expresses the T gate as a sum over phase-space operators (the see that any discrete operator can also be expressed as a linear reflections) evaluated on all the phase-space points. Thus, it combination of reflections, but unlike the continuous case, can be interpreted as associating an exponentially large number this sum has a finite number of terms. Since reflections can of paths to every phase-space point instead of the single paths be expressed fully by the discrete path integral truncated at found for Clifford gates. Therefore, any simulation of the T order h¯ 1, as discussed previously, it follows that any unitary gate naively necessitates adding up an exponential large sum operator in discrete systems can be expressed as a finite sum of over paths and so is comparably inefficient. contributions from path integrals truncated at orderh ¯1. Again, Subsequent to the work presented here, Dax et al. found the the same statement can be made by considering the chord equivalent conclusion that gates from universal gate sets have representation in terms of translations. associated actions that are polynomials of degree greater than Hence, quantum propagation in discrete systems can be two [19]. fully treated by a finite sum of contributions from a path- 1 integral approach truncated at order h¯ . To gather some VIII. CONCLUSION understanding of this statement, we can consider what is necessary to add to our path-integral formulation when we The treatment presented here formalizes the relationship complete the Clifford gates with theT gate, which produces a between stabilizer states in the discrete case and Gaussians universal gate set. in the continuous case, which has often been pointed out [3]. The T gate is generalized to odd d dimensions by Namely, only Gaussians that lie along Weyl phase-space points directly correspond to Gaussians in the continuous world in (j− 1)j terms of preserving their form under a harmonic Hamiltonian, Tˆ = ω 4 |j j |. (92) an evolution that is fully describable by truncating the path j∈Z/dZ integral at order h¯ 0. Furthermore, we showed that the Clifford This gate can no longer be characterized by anM with integer group gates, generated by the Hadamard, phase-shift, and entries. In particular, controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order. We found 1 1 that this was because their Hamiltonians are harmonic and M = 2 , (93) Tˆ 0 1 take Weyl phase-space points to themselves. This proves

032331-11 LUCAS KOCIA, YIFEI HUANG, AND PETER LOVE PHYSICAL REVIEW A 96, 032331 (2017) the Gottesman-Knill theorem for Clifford gate propagation. i center generating function multiplied byh¯ . This is very similar The T gate, needed to complete a universal set with the to the form of the vVMG path integral in Eqs. (21) and Hadamard, was shown not to satisfy these properties, and (51). However, in Eqs. (14) and (46), reflections serve as so requires a path-integral treatment that is complete up to the prefactors measuring the reflection spectral overlap of a h¯ 1. The latter treatment includes a sum of terms for which propagated state with its evolute and the center generating the number of terms scales exponentially with the number of actions provide the quantal phase. Thus, this formulation can qudits. be interpreted as an alternative path-integral formulation of the We note that our observations pertaining to classical vVMG, one consisting of reflections as the underlying classical propagation in continuous systems have long been very trajectory only, instead of the more tailored trajectories that well known. In the continuous case, the Wigner function result from applying the method of steepest descents directly of a quantum state is non-negative if and only if the state on an operator. is a Gaussian [39] and it has also long been known that The fact that any unitary operator in the discrete case quantum propagation from one Gaussian state to another can be expressed as a sum consisting of a finite number of only requires propagation up to order ¯h0 [15]. Indeed, it has order ¯h1 path-integral contributions has the added interesting been shown that this is a continuous version of “stabilizer implication that uniformization—higher-order h¯ corrections state propagation” in finite systems [40], and is therefore, in to the “primitive” semiclassical forms such as Eq. (21)—is principle, useful for quantum error correction and cluster-state not really necessary in discrete systems. Uniformization is quantum computation [41,42]. It is also well known in the characterized by the proper treatment of coalescing saddle discrete case that quadratic Hamiltonians can act classically points and has long been a subject of interest in continuous and be represented by symplectic transformations in the study systems where “anharmonicity” bedevils computationally effi- of quantum cat maps [18,30,43] and linear transformations cient implementation. It seems that this problem is not an issue between propagated Wigner functions [44]. Interestingly, in the discrete case since a fully complete sum with a finite though, this latter work appears to have predated the discovery number of terms, naively numbering d2 for one qudit, exists. that stabilizer states have positive-definite Wigner functions As a last point, there is perhaps an alternative way to [3] and, therefore, as far as we know, has not been directly interpret the results presented here, one in terms of “resources.” related to stabilizer states and theh ¯0 limit of their path-integral Much like “magic” (or contextuality) and quantum discord formulation, which is a relatively recent topic of particular can be framed as a resource necessary to perform quantum interest to the quantum information community and those operations that have more power than classical ones, it is familiar with the Gottesman-Knill theorem. Otherwise, this possible to frame the order in ¯h that is necessary in the claim has been pointed out in terms of concepts related to underlying path integral describing an operation as a resource positivity and related concepts in past work [2,45]. necessary for quantumness. In this vein, it can be said that We also note that our exploration of continuous systems is Clifford gate operations on stabilizer states are operations that not meant to explore the highly related topic of continuous- only require ¯h0 resources while supplemental gates that push variable quantum information. Many topics therein apply to the operator space into universal quantum computing require our discussion here, such as the continuous stabilizer state h¯ 1 resources. The dividing line between these two regimes, propagation we mentioned above. However, our intention in the classical and quantum world, is discrete, unambiguous, introducing the continuous infinite-dimensional case was not and well defined. to address these topics, but to instead relate the established continuous semiclassical formalism to the discrete case, and ACKNOWLEDGMENTS thereby bridge the notions of phase space and dynamics between the two worlds. The authors thank Professor A. Ozorio de Almeida for very There is an interesting observation to be made of the weights fruitful discussions about the center-chord representation in of the reflections that make up the complete path-integral discrete systems and B. Drury for his help proofreading and formulation of a unitary operator. Namely, as is clear in bringing [16] to our attention. This work was supported by Eqs. (14) and (46), the coefficients consist of the exponentiated AFOSR Award No. FA9550-12-1-0046.

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