A Quantum Cellular Automaton for One-Dimensional QED

A quantum cellular automaton for one-dimensional QED

1, 2, 3, Pablo Arrighi, ∗ CédricBény, † and Terry Farrelly ‡ 1Aix-Marseille Univ, CNRS, LIS and Univ Paris-Saclay, ENS Paris-Saclay, CNRS, INRIA, LSV, France. 2Department of Applied Mathematics, Hanyang University (ERICA), 55 Hanyangdaehak-ro, Ansan, Gyeonggi-do, 426-791, Korea. 3Institut fürTheoretische Physik, Leibniz UniversitätHannover, 30167 Hannover, Germany. We propose a discrete spacetime formulation of quantum electrodynamics in one-dimension (a.k.a the Schwinger model) in terms of quantum cellular automata, i.e. translationally invariant circuits of local quantum gates. These have exact gauge covariance and a maximum speed of information propagation. In this picture, the interacting quantum field theory is defined as a “convergent” sequence of quantum cellular automata, parameterized by the spacetime lattice spacing—encompassing the notions of continuum limit and renormalization, and at the same time providing a quantum simulation algorithm for the dynamics.

I. INTRODUCTION elegantly based on the symmetries of special relativity and the demand that it features local symmetries (e.g., In this work, we propose a discrete spacetime formu- gauge invariance). The result then has to be converted lation of quantum electrodynamics (QED) in one dimen- to a quantum theory, through one of several heuristic sion (the Schwinger model), in terms of quantum cellular quantization processes. Regardless of whether the theory automata (QCA), which are essentially translationally is canonically quantized or quantized via path integrals, invariant circuits of local quantum gates. handling interactions intrinsically requires a form of reg- From a practical point of view, the QCA defines a ularization. This is usually equivalent to a discretization quantum simulation algorithm for the dynamics of an in- in that it involves neglecting small scale features. When teracting QFT (leaving aside the problems of state prepa- one wants to go beyond perturbation theory, recourse to ration and measurements, however). But, from a theoret- numerical simulations is typically necessary. ical point of view it also constitutes a proof-of-principle Thus, a number of discrete space—or spacetime— showing that natively discrete formulations of an inter- formulations of QFT have been studied, understood as acting QFT are possible and elegant. In this picture, approximations designed for numerical simulations. The the QFT is defined as a “convergent” sequence of QCA, main simulation method approximately evaluates the parameterized by the spacetime lattice spacing—echoing path integral in imaginary time (quantum Monte Carlo the notions of continuum limit and renormalization. We [35]). But this only works well in limited cases and, like discuss why we may hope to circumvent some of the tech- all classical simulations of quantum theory, suffers from a nical issues of standard formulations of QFT this way. complexity which is exponential in the number of degrees The construction is intuitive and requires little pre- of freedom. requisites. It leads to a simple, explanatory model of Confronted with the inefficiency of classical comput- QFT based on quantum information concepts. Given ers for simulating interacting quantum particles, Feyn- that QFT can be rather intricate, we believe this also man realized that one ought to use quantum computers constitutes an important pedagogical asset. instead [31]. What better than a quantum system to simulate another quantum system? In the last decades, several such quantum simulation schemes have been de- A. QFT and their quantum simulation vised [34, 36, 40], some of which were experimentally implemented recently [42]. Generally speaking these are arXiv:1903.07007v2 [quant-ph] 4 Feb 2020 Quantum field theory (QFT) is the framework that Hamiltonian based [39], i.e. they are based on a discrete- best describes fundamental particles and their interac- space continuous-time version of the QFT, such as the tions in a relativistic manner [48], though without ac- Kogut-Susskind Hamiltonian [41]. One may then look for counting for gravity. QFT encompasses a heterogeneous quantum systems in nature that mimic this Hamiltonian, set of procedures and techniques, but roughly speaking or that can be tuned into it [54]. Alternatively, one does the first step is always to put together a local action, a staggered trotterization of it so as to obtain unitaries i.e. a way to associate a ‘cost’ to each spacetime history that may be implemented on a digital quantum com- of a classical field. This action is usually derived quite puter. In any case, by first discretizing space alone and not time, these Hamiltonian-based schemes are essentially taking things back to the non-relativistic quantum mechanical setting: Lorentz-covariance is broken. The ∗ [email protected] bounded speed of light can only be approximately recov- † [email protected] ered (e.g., Lieb-Robinson bounds) [26, 46]. Even when ‡ [email protected] the Hamiltonian is trotterized, time steps need remain 2 orders of magnitude smaller than space steps. This also “paths” is not mathematically defined, and hence many creates more subtle problems such as fermion doubling, additional assumptions and techniques have to be used where spurious particles are created due to the periodic in the process of obtaining actual physical predictions. nature of the momentum space on a lattice. This implies that the theory is not entirely specified by An essential aspect of QFT simulations is the prepara- the action. Other formulations such as canonical quanti- tion of the initial state, boundary conditions, and mea- zation (operator formalism) are used in parallel in order surements [36, 37]. This is non-trivial due to the fact that to resolve some of these ambiguities. Moreover, whilst the vacuum of an interacting QFT is not known exactly Lorentz and gauge invariance are manifest in the path- (or if it is, then we usually also have an exact solution for integral formulation, other fundamental properties are the evolution). We do not study this question in detail not, i.e. it cannot directly guarantee the unitarity of the here, except when considering the continuum limit of our dynamics, the existence of a ground state, nor character- QCA in Section IV B. ize the compatibility of observables. Another very important source of ambiguities, which cannot be resolved through the canonical operator for- B. Quantum simulation via QCA malism, comes from the continuity of spacetime. In the classical field theory, this is handled mathematically by From a relativistic point of view, it would be more the assumption that the fields are adequately differen- natural to discretize space and time simultaneously and tiable, and formulating the action as an integral of dif- with the same scale, thereby producing a network of lo- ferential quantities. This does not have an equivalent in cal quantum gates, homogeneously repeated across space the quantum setting. Instead, the quantum theory must and time. Feynman introduced Quantum Cellular Au- be first regularized, typically through an energy cutoff, tomata (QCA) together with the idea of quantum sim- which is essentially equivalent to a discretization of space. ulations of physics [32]. A related model to QCAs was Then the continuum limit must be taken on a per-case also given by Feynman, namely the simple discretized basis. Different theories, or even different boundary con- ‘checkerboard model’ of the electron in discrete (1 + 1)- ditions, each require a different highly non-trivial depen- spacetime [30]. dence of the parameters on the cutoff (renormalization). The one–particle sector of QCAs became known as Hence, although the action-based approach has a con- Quantum Walks (QW), and was found to provide quan- tinuous starting point, it does not actually solve the prob- tum simulation schemes for non-interacting fundamental lem of obtaining a well-defined quantum theory in the particles [7, 15, 44, 53], including in (3 + 1)-spacetime, continuum: the continuum limit needs be taken again in be it curved [3, 5, 6, 23] or not, or in the presence of the quantum setting anyway—under the guise of renor- an electromagnetic field [19]. The sense in which QW malization. Indeed, by itself the action-based approach are Lorentz-covariant was made explicit [13, 16, 18, 20]. only succeeds in yielding genuinely continuous quantum The bounded speed of light is very natural in circuit- theories for (quasi-)free theories. These serve as the based quantum simulation, as it is directly enforced by basis for perturbative solutions to the renormalization the wiring between the local quantum gates. Recently, problem—when a weak interaction gets added. the two–particle sector of QCA was investigated, with We propose that a natively quantum and discrete for- the two walkers interacting via a phase (similar to the mulation of QFT, àla QCA, could bypass a number of Thirring model [22]). This was shown to produce molec- these issues. Since the action-based approach is only ular binding between the particles [1, 17]. apparently continuous anyway, it may be conceptually In the many-particle sector general theorems exist clearer to start from a natively discrete theory. (This showing that simple QCA [8] are able to simulate any may also eventually facilitate a connection with genuinely unitary causal operator [10, 29], and thus arbitrarily com- discrete quantum gravity proposals [2, 50]). Moreover, plex behaviours. Still, the problem of defining a concrete a genuinely quantum description avoids the ambiguities QCA that would simulate a specific interacting QFT had arising when insisting on developing a quantum theory remained out of reach. In this paper we give a QCA de- around a classical action. QCA have some other description of QED in (1+1)–spacetime, i.e. the Schwinger sirable properties: they are manifestly unitary and they model. To the best of our knowledge, this is the first also come with the immediate advantage of having a lo- work relating the many-particle sector of a QCA to an cal formulation, from which an strict limit on the speed interacting QFT. of information propagation can be extracted, as in special relativity. Lorentz-covariance can be handled as in [11, 16, 18, 20]. Gauge-invariance can also be handled as C. Towards QCA formulations of QFT in [12], a technique which is inspired by gauge-invariant quantum walks [4, 24, 25]. This technique allows one The apparent simplicity of the standard, path-integral to construct the interacting theory from the free theory formulation of QFT in fact hides a number of se- and the gauge symmetry requirement, reminiscent of the rious complications. These issues are often summa- way the Hamiltonian of a lattice gauge theory gets con- rized through the fact that the integration measure over structed [49, 52]. 3

The present paper illustrates precisely this point. Our with H, initial states satisfying Gauss’ law also satisfy it point of departure is the simple and well-known Dirac in the future. An alternative is to restrict the set of ob- QCA for free fermions. Based on the demand that it servables instead, and demand that these commute with features a local U(1)–gauge symmetry, a gauge field gets Gauge transformations. introduced that ‘counts’ fermions. Interactions then arise The Schwinger model is phenomenologically very dif- by having fermions pick up a local phase when they move ferent from QED in three dimensions, but it exhibits left or right. Renormalization comes into play in the way many interesting phenomena that arise in other QFT in the QCA parameters are made to depend on the lattice (3+1)–dimensions. The phenomena appearing depend on spacing, leading to renormalization trajectories. whether the fermion field is massive or massless. (Some authors reserve the name Schwinger model for the massless case.) For example, massless fermions suffer con- D. The Schwinger model finement: the effective electric force between particles of opposite charges increases with distance, consequently The Schwinger model, i.e. (1 + 1)–QED, is a useful charges are paired into an effective particle: a massive playground to understand phenomena in QFT in higher boson, whose mass is proportional to the electromagnetic dimensional spaces. Let us give a rapid summary of its coupling strength. main features, based on Ref. [43]. This subsection could be skipped by readers that are only interested in the QCA itself, i.e. not interested in comparing it with standard E. Plan (1 + 1)–QED. The Schwinger model can be formulated covariantly In order to construct the Schwinger QCA we replay the in terms of an action, but let us directly write down a procedure for constructing QED, in a natively discrete Hamiltonian formulation, which will be easier to compare manner. In Section II we start with the one-particle sec- to our QCA. The Hamiltonian formalism requires the tor, namely the Dirac QW, which simulates the Dirac local gauge symmetry to be partially fixed, which can Eq. for a free electron in a simple and elegant man- be done in many ways. In the temporal gauge (with ner, only upgraded to the many non-interacting parti- A0(x) = 0, and writing A(x) = A1(x)), the Hamiltonian cle sector, yielding a Dirac QCA. In Section III we ex- is H = tend the Dirac QCA minimally, so that it acquires a lo- Z cal U(1) symmetry, thereby introducing the electromag- 1 2 dx ψ†(x) [(i∂x + ieA(x)) σz + mσx] ψ(x) + E (x) , netic field—this cellular automata version of the gaug- 2 ing procedure was developed in [12]. In Section IV the (1) QCA is endowed with a simple local phase gate, which where E(x) is the electric field observable at x, and A(x) implements the interaction. Throughout the paper, we is its conjugate momentum, meaning consider both the Schrödingerand Heisenberg pictures [A(x),E(y)] = iδ(x − y). (2) (discussing fermionic creation and annihilation operators, how they evolve, and which are the physical observables). T Here ψ(x) = (ψ1(x), ψ2(x)) is a two component fermion Finally we sketch the continuum limit. Section V sum- field satisfying marizes the results and puts them into perspective.

{ψα(x), ψβ† (y)} = δαβδ(x − y), (3) II. DIRAC QCA The electric charge is denoted by e, and the mass of the fermions is m. A. Schrödingerpicture and qubit representation Choosing the temporal gauge does not completely fix the gauge freedom. One still needs to prevent that the remaining gauge degrees of freedom be observed. One We arrange the Dirac QCA as shown in Fig. 1. We ∈ way of doing this is to demand that physical states com- have two Dirac fermions at each site x = εk, k Z mute with Gauge transformations, making them “block- of a one-dimensional lattice (where the red and black diagonal”. In terms of pure states, this amounts to for- wires cross), which can be thought of as having orthog- bidding superposition across any two distinct eigenspaces onal “chirality”, i.e. that are about to move in opposite of a Gauge transformation. In the present context this directions. There are two ways to describe the QCA: via leads to Gauss’ law: fermions or via qubits (related by the Jordan-Wigner iso- morphism). First, we will give the qubit description, and ∂xE(x) − eψ†(x)ψ(x) |φphysi = f(x)|φphysi. (4) later we will give the QCA in the fermionic picture. We encode the occupation number of each fermion as where the f(x) is fixed. Mathematically it corresponds a qubit, yielding two qubits at each site. The red (resp. to the choice of an eigenspace, physically it can be in- black) wires denote the qubit corresponding to the oc- terpreted as an external, fixed electric field. Notice that cupation number of the left-moving (resp. right-moving) because the operators implementing Gauss’ law commute modes. 4

is the state with the left-mover qubit at position x being in the 1 state with zeroes everywhere else. Then we consider the state at time t given by

X + t + ✏ |ψ(t)i = ψ (t, x)|x+i + ψ−(t, x)|x−i , (8) x U W with |ψ(t)i normalized to one. Then the unitary dynam- t ics gives us the update rule after one timestep to be + + ψ (t + ε, x) = cψ (t, x − ε) + −isψ−(t, x) + (9) ψ−(t + ε, x) = cψ−(t, x + ε) + −isψ (t, x).

x x + ✏ A rough justiﬁcation of the continuum limit, is simply to expand to ﬁrst order in ε, giving FIG. 1. Dirac QCA: each space-time point, the intersections + + of the dotted lines, (x, t) can be occupied by a left and a ε∂tψ = −ε∂xψ − imεψ− (10) right-moving fermion. All gates are identical and given by + the matrix W , which allows the fermions to change direction ε∂tψ− = +ε∂xψ− − imεψ . (11) with an amplitude that depends on its mass. Because this is Dividing across by ε allows us to write this in terms of a qubit description, each crossing of two fermions triggers a −1 phase, including within the gate W . Pauli matrices to get

∂tψ = −σ3∂xψ − imσ1ψ, (12)

+ T All gate are identical, with components which is the Dirac equation, where ψ = (ψ , ψ−) . More rigorous treatments of the continuum limits of such  1 0 0 0  single-particle dynamics (i.e., quantum walks) can be 0 −is c 0 W =   (5) found in, e.g., [14].  0 c −is 0  0 0 0 −1 C. Heisenberg picture and fermion representation = 1 ⊕ σ1 exp(−imεσ1) ⊕ −1 (6) with c = cos(mε) and s = sin(mε) accounting for the Let us look at the QCA in terms of fermion opera- mass. tors, which will allow us to show that the evolution is The components are ordered so that when the mass consistent with free Dirac QFT. is zero, the particles do not change direction, so that a The quasi-local algebra of operators for free Dirac QFT right-moving mode is transferred from x to x + ε, etc. is generated by the finite products of the operators ax, The minus sign of the bottom-right entry of the ma- bx, ax† and bx† for all x, with ax (resp. bx) the annihila- trix is needed in the qubit representation of two fermions tion operator of a left-moving (resp. right-moving) mode crossing past each other, as we shall see. For the same at each point x. Since the adjoint evolution of the free reason, each crossing of a red and black wire also has a Dirac QFT is an automorphism of the ∗-algebra, it is en- gate tirely characterized by the way it acts upon ax, bx, over an ε–period of time. It naturally arises as the second 1 0 0 0  quantization of Subsection II B: 0 1 0 0 S =   . (7) 0 0 1 0  f(ax† ) = cos(mε) ax†+ε + i sin(mε) bx† (13) 0 0 0 −1 f(bx† ) = i sin(mε) ax† + cos(mε) bx† ε. − This could have been the basis for the definition of a B. Single particle dynamics fermionic QCA [17] in the Heisenberg picture [51]. The relation to the qubit QCA in the previous section can be established as follows. First notice that the f defined To see that this QCA is a discretization of non- by Eqs. (13) is occupation-number-conserving. Now say interacting Dirac particles, let us consider the “one- we had an occupation-number-conserving unitary U such particle” sector, by restricting the QCA to the subspace that f(A) = U AU. We would then have spanned by states with a single fermionic mode occupied. †

In other words, we consider states with all but one qubit f(A)|0i = U †AU|0i = U †A|0i, (14) in the zero state. Deﬁne |x+i to be the state with the qubit corresponding to a right mover at position x being where |0i denotes the vacuum for the modes ax and bx. in the 1 state and zeroes everywhere else. Similarly |x−i This equation would then specify U from f. Since f is 5 local, we can directly deduce W instead, again assuming it is occupation-number-conserving. Indeed, let us write the four possible input states of a gate W as

m n |m, ni = (bx† ) (ax†+ε) |0i. (15) with m being the number of right-moving fermions at x (zero or one, created by bx† ) and n the number of left- moving fermions at x + ε (zero or one, created by ax†+ε). Similarly, we write the four output states as

n m |n, mi0 = (ax† ) (bx†+ε) |0i. (16) FIG. 2. Dispersion relations for the fermion QCA (orange) Occupation number conservation establishes the first col- and for naive fermions in lattice QFT (blue). For small mo- umn of the matrix W †. For the second column, the out- mentum, both dispersion relations look like those for Dirac p 2 2 put state |01i0 = bx†+ε|00i0 must be mapped by W † to fermions E(p) = ± p + m . Notice that the naive fermions also have low energy modes around pε = π. These correspond to doubler modes. f(bx†+ε)|00i = (is ax†+ε + c bx† )|00i = is |01i + c |10i, (17) and similarly for the third column. For the last column, the state |11i0 must be mapped by W † to This is not a problem for free theories. If the initial state has only low momentum modes occupied, then this f(ax† bx†+ε)|00i = (c ax†+ε + is bx† )(is ax†+ε + c bx† ) |00i will not change as the system evolves. In contrast, when 2 2 = (c ax†+εbx† − s bx† ax†+ε) |00i there are interactions, then high momentum, low energy 2 2 particles may be created, even if the initial state has only = −(c + s ) bx† ax†+ε|00i low momentum modes occupied, which would affect, e.g., = −|11i, scattering amplitudes. (18) That was an example of fermion doubling for massive fermions. In the massless case for local Hamiltonians − which justifies the ( 1) of Eq. (6). on lattices, this is unavoidable as a consequence of the Nielsen-Ninomiya theorem [21, 45], without losing local chiral symmetry, or using other complicated tricks, e.g., D. Fermion doubling domain wall fermions [38]. And the naive fermion example here illustrates that the problem can even arise for This subsection is just to discuss well-known technical some discretizations of massive fermions. Let us consider difficulty that sometimes arises when discretizing rela- whether this occurs for our model. tivistic fermions, called the fermion doubling problem. Looking at equations (9), we can write the evolution This occurs when the discrete model has more low en- in the single particle picture as ergy modes than one wants in the continuum [21]. An illustrative example of this is so-called naive fermions in cos(mε)S −i sin(mε) lattice quantum field theory. These are described by the U = , (21) Hamiltonian −i sin(mε) cos(mε)S†

N 1 X− sin(k) H = ψ† σz + mσx ψk, (19) where S shifts the particle by one lattice step. In mo- k ε ik ipε k=0 mentum space this is S = e− = e− , so we can find the eigenvalues of U for each value of momentum: where ψk = (ak, bk) are the fermion annihilation operators in momentum space and k labels lattice momenta. ± p 2 2 − The dispersion relation is given by Λ (p) = cos(mε) cos(pε) cos(mε) cos(pε) 1. ± (22) p E(p) = ± sin(k)2/ε2 + m2, (20) If we define the quasi-energies by E(p) = i ln[Λ (p)]/ε, then we see from the plot in Fig. 2, that fermion doubling± where p = k/ε corresponds to the continuum momentum does not occur in the spectra for these models. in the continuum limit. This is plotted in Fig. 2 and we Other works have considered similar discrete-time see that, for small k = pε, E(p) is close to the relativistic models in the free case [22, 28, 33], where they reached p dispersion relation E(p) = ± p2 + m2. However, look- the same conclusion, namely that these models do not ing at momenta√ pε = π + δ, the dispersion relation also suffer from the fermion doubling problem. However, a looks like ± δ2 + m2. So there are extra particles that more detailed analysis in the presence of interactions behave like relativistic particles. would be extremely useful. 6

our case—every QCA can be made to have that structure, see [9]. The obtained local condition is that for all ϕ, there exists ϕ0, such that for all x, t + ✏ Rϕ0(x) ⊗ Rϕ0(x+ε) W 0 = W 0 Rϕ(x) ⊗ Rϕ(x+ε) . (24) This is not the case for the Dirac QCA as it stands, U 0 W even when m = 0. Indeed say our input is |10i + |01i, iϕ0(x) we are asking that there be a ϕ0 such that e |10i + t 0 eiϕ (x+ε)|01i be equal to

W eiϕ(x)|10i + eiϕ(x+ε)|01i (25) iϕ(x+ε) iϕ(x) x x + ✏ = e |10i + e |01i, (26)

thereby imposing on ϕ0 the contradictory requirements FIG. 3. (1+1)–QED QCA structure. At x+ε/2 positions lies that for all ϕ and x, ϕ0(x) = ϕ(x + ε) and ϕ0(x + ε) = a wire carrying a state in H representing the gauge field. Its Z ϕ(x). sole role is to count the fermions passing by, and to undergo a phase accordingly: this phase triggers the interaction. In order to fix this, we follow the prescription of lattice gauge theory as follows: we interleave ancillary cells on the edges of the line graph, i.e. at positions x + ε/2. These cells carry an integer l ∈ Z, its Hilbert space is T R R T φ(x) φ(x) φ(x) −φ(x) HZ. This is the so-called gauge field. Let us extend the gauge transformation to become x − /2 x x + ε/2 Gϕ(x) = Tϕ(x) ⊗ Rϕ(x) ⊗ Rϕ(x) ⊗ T ϕ(x) (27) − with T |li = eilϕ(x)|li acting on the left and right edges FIG. 4. The gauge transformation. ϕ(x) at x ± ε/2, see Fig. 4 (now including gray boxes). These Gϕ(x) no longer have disjoint supports, but they do commute, because T ϕ(x) commutes with Tϕ(x+ε). Thus III. IMPOSING GAUGE-INVARIANCE Q − Gφ = x Gφ(x) is well-defined. At x + ε/2, it ends up | i | i il(ϕ(x+ε) ϕ(x))| i A. Schrödingerpicture mapping l into l = e − l . Let us extend W into a W 0 that affects the gauge field, namely We want to enforce a local phase symmetry. The sym-  I 0 0 0  metry group’s elements are specified by a field ϕ : εZ → 0 −isI cV 0 W 0 =   (28) R stating which phase to apply at each point in space, see  0 cV † −isI 0  Fig. 4 (black boxes). The induced, gauge transformation 0 0 0 −I acts on a state ψ according to with V |li = |l − 1i and I the identity. O G ψ = G (x)ψ with G (x) = R ⊗ R This, in turns out, is gauge-invariant with just ϕ0 = ϕ. ϕ ϕ ϕ ϕ(x) ϕ(x) | i x In order to prove it let us focus on the input mln of a gate. When applying Gϕ, this input state will trigger a phase gain ϑ(x, m − l, n + l): Rφ : |0i 7→ |0i mϕ(x) + l(ϕ(x + ε) − ϕ(x)) + nϕ(x + ε) (29) |1i 7→ eiφ|1i = (m − l)ϕ(x) + (n + l)ϕ(x + ε). (30) In other words, both the red and black qubits at x, cor- Now, observe that the numbers (m−l, n+l) are invariants responding to the left and right-moving fermions occu- of W 0, as it takes |mlni into a superposition of the form pation numbers at this discrete position, get multiplied X by the phase φ(x) when they are in the |1i state. αi|m − i, l − i, n + ii. (31) This is the symmetry we are trying to enforce. We i 1,0,1 want to extend the Dirac QCA U into a gauge-invariant ∈{− } 0 QCA U 0, such that for any ϕ, there exists a ϕ0 such that It follows that, when applying Gϕ = Gϕ, the output state will trigger the same phase gain ϑ(x, m − l, n + l). Gϕ0 U 0 = U 0Gϕ. (23) Thus, we restored gauge-invariance at the price of introducing ancillary cells in HZ. We may wonder whether In order to turn this gauge-invariance condition into a finite-dimensional alternatives exist, that would be more local one, we use the fact that U has the quantum circuit in line with the QCA tradition. One argument for this structure of Fig. 1. This simplification is not specific to will given in Subsec. IV A. 7

B. Gauge-invariant observables quasilocal C∗-algebra A generated by the operators Lx together with In a gauge theory, only those observables which are in- ax† ax = ax† ax (38) variant under all gauge transformations are physical. Let † us characterize these observables. This is not strictly nec- bxbx = bx† bx (39) essary for constructing the QCA, but it is instructive to † ax† ax+ε = ax† Vx+ 1 εax+ε (40) see how we could actually define the QCA in the Heisen- 2 berg picture in a purely gauge-invariant manner, via an † † bxbx+ε = bx† Vx+ 1 εbx+ε (41) automorphism of the algebra of physical observables. 2 † Let Ax = ax† ax and Bx = bx† bx be the number operators bxax = bx† ax (42) for the two fermionic modes at the point x. Remember that the gauge field lies at positions x ∈ for all x. Indeed, observe that all even polynomials are 1 generated by order two polynomials. Moreover, for such εZ + ε. On the edge x between vertex x − ε/2 and 2 terms to be local, they must create and annihilate one x + ε/2, we define the operator Lx by Lx|li = l|li. (The “electric field” at site x is E = gL , where g is the fermion. Products of fermionic operators further apart x x can be obtained by anticommuting the above terms. For charge of our particles). Together with Vx, it generates the algebra of operators for the gauge field on that edge, instance: which is defined algebraically by [Vx,Lx] = Vx. {ax† ax+ε , ax†+εax+2ε} = ax† ax+2ε. (43) The operators

Jx := Lx+ε/2 − Lx ε/2 − Ax − Bx (32) C. Heisenberg dynamics − for all x generate the group of gauge transformations, at A QCA defined by a map f is gauge invariant, as de- a given time. Specifically, fined in the previous section, if for all ϕ there exists ϕ0 P i ϕ(x)Jx such that, for all observable X, Gϕ = e− x . (33)

f(G† 0 XGϕ0 ) = G† f(X)Gϕ. (44) As in Ref. [43], let us consider the operators ϕ ϕ For X gauge-invariant, this reduces to ax = axΠy>xVy (34) f(X) = Gϕ† f(X) Gϕ (45) b = b Π V (35) x x y>x y for all ϕ, i.e., f(X) must be gauge-invariant too. Hence, (36) this implies that f must be an automorphism of the gauge-invariant algebra. whose adjoints create fermions while increasing the elec- We can obtain such an automorphism f by simply sub- tric ﬁeld to their right. These operators are not local, stituting the new fermionic operators in Eq. (13): but we will use them to build physical local observables. Since the operators Vx commute with each other and f(ax) = c ax+ε − is bx (46) with ax and bx, then ax and bx satisfy the same anticom- f(bx) = −is ax + c bx ε mutation relations as ax and bx, namely, − where c = cos(εm) and s = sin(εm). † {ax, ay† } = {bx, by} = δx,yI But we still have to chose how f acts on Lx. The QCA deﬁned in the previous section, for instance, is compatible † {ax, ay} = {bx, by} = {ax, by} = {ax, by} = {ax† , by} = 0 with Eq.(46), and gives us

f(Lx) = Lx + A 1 − f(B 1 ). (47) Moreover, x+ 2 ε x+ 2 ε ( This has the property that it fixes the gauge genera- αax + βbx if x < y [αax + βbx,Ly] = (37) tors: 0 otherwise f(Jx) = Jx for all x, (48) for any α, β ∈ C. which is how the gauge choice ϕx0 = ϕx manifests in this The point of these new fermionic annihilation opera- picture. To check this equation, it is useful to use the tors is that they commute with all the generators of gauge fact that the number of fermions is locally conserved in transformations Jx. Together with the electric field op- the sense that erators Lx, they generate the set of operators commuting f(Ax) + f(Bx+ε) = Ax+ε + Bx. (49) with the gauge generators Jx for all x. However, we could argue that a much smaller algebra is One could verify that this map f is local by showing physical, if we only allow local operators, and only even that the image of the local generators are local. However order polynomials in fermionic creation or annihilation this follows also from the facts that it is implemented by operators. This suggests restricting observables to the the finite circuit of Fig. 3. 8

IV. INTERACTING QCA Hence, we do propose to modify the QCA by simply interleaving a step resulting from the integration of the 1 P 2 2 The QCA defined in the previous section is gauge- Hamiltonian 2 ε x g Lx (where ε comes from the dis- invariant. However at this stage the addition of the gauge cretizing of the integral in Eq. (1)), yielding the new gate field has no dynamical effect on the electrons. This is best   seen in the Heisenberg picture, where the the “dressed” I 0 0 0 0 −isI cV 0 i ε2g2L2   2 creation operators a† and b† generate the same algebra, W 00 = e (50) x x  0 cV † −isI 0  and evolve according to the same automorphism as the 0 0 0 −I free Dirac QCA. Hence its continuum limit is also identical to that of the free Dirac QFT, but in terms of dressed with L|li = l|li. This is clearly still gauge-invariant, since creation operators, which, in turn, is identical to the the operators Lx are gauge-invariant observables. Schwinger model defined by Eq. (17) of Ref [43], with It is interesting to note that if L2 = 4π/g2ε2 exactly, zero charge: g = 0. the phase wraps up around 2π. Since the spectrum of This equivalence with the free Dirac QFT holds also L is in Z, this can only happen if ε2 = (4π/g2)/k with for non-zero mass. Ref. [7] rigorously proves that the k an integer. But, if we restrict ourselves to values of continuum limit of the Dirac quantum walk, which is the ε such that this is the case, then we no longer need the one-particle sector of the QCA, yields the Dirac equation. whole Hilbert space HZ to represent the gauge field at The same is true for the full QCA [28], and so we can each point : we can replace it by a k-dimensional Hilbert consider that our dressed QCA is a reformulation of the space instead. This, however, still requires that k −→ ∞ Schwinger model defined by Eq. (17) of Ref [43], with as ε −→ 0. This idea of restricting the gauge field to zero charge. finite-dimensions labelling roots of unity is not new and What is missing compared to the full Schwinger model has been evaluated in [27]. is the local Hamiltonian term proportional to the square We write |0i for the vacuum of the modes ax and bx, 2 2 2 of the electric field: Ex = g Lx. It is this term that tensored with any fixed joint eigenstate of the field oper- makes the model effectively interactive. ators Lx, for all lattice sites x ∈ εZ. It is easy to see that the states created by applying any number of time the “dressed” fermionic creation op- A. Schrödingerpicture erators ax† and bx† to |0i, for any x, are eigenstates of i ε2g2L2 e 2 . Hence, when considering these states, our par- Can we guess how to modify our QCA, so that the ticles still move as free particles, dragging the associated missing local Hamiltonian term shows up in continuum electric field with them. However, they now take a phase limit? Inspired by the many time-discretizations of whose dynamics depends on the energy stored in that Hamiltonian systems, one may be tempted to use the electric field. operator–splitting method and simply interleave an ε– To observe the effect of the interaction, we need to period of evolution under the extra local Hamiltonian consider wavefunctions that are smooth with respect to term, in-between any two time steps. For this to be jus- the lattice spacing. For instance, consider an external tified by the Trotter-Kato formula, however, the free evo- electric field in the shape of a step function, i.e. let |0i lution over an ε–period ought to be approximately equal be such that L |0i = 0 for all x < 0, and L |0i = −|0i to the identity, too. x x for x ≥ 0, so that if we create a particle at any position Let us examine whether this is the case. The gate x 0 with a , then the electric field is still zero far away W as defined is parameterized by the space/time lattice x 0 on both sides of space. spacing ε so as to yield free Dirac QFT in the limit ε → 0. Let us consider a single particle wavepacket Observe that ε only appears in the mass term, so that in the limit, W simply tends to its massless version, which X 0 |ψi := eipxf(x) a |0i, (51) is the swap. Hence the gate does not tend to the identity x† x ε in the continuum limit, per se. ∈ Z Crucially, for the continuum limit to make sense, one for some smoothly varying envelop f satisfying f(0) ' 0. must also make states dependent on the space-time spac- Then the expectation value of the momentum operator ing ε, in such a way that they are smooth at the scale of ∂ −i ∂x is approximately equal to p. A typical choice of the lattice. In the single particle setting, this corresponds envelop is the gaussian amplitude to bounds on the derivatives [7]. For many-body states, 2 one may require that they be approximately invariant 1 (x − x0) f(x) = 1 exp − 2 (52) under the permutation of nearby fermions. (2πσ2) 4 4σ Hence, for a given value of ε, we may consider only those states |ψi which are approximately invariant un- with x0 0 the mean and σ > 0 the standard devia- der the swap gate involved in the free Dirac dynamics, tion chosen so that 1/σ p and yet x0 + σ 0. For i.e. W 0|ψi ≈ |ψi so that W 0 ≈ Id in the appropriate this choice of envelop and for an adequate superposition subspace. of chiralities, [47] shows that in the non-relativistic limit 9 p m, the wavepacket’s expected position will move spanned by acting on the vacuum only with A the quasi- under the free Dirac equation, with constant velocity local algebra generated by Eqs. (38)–(42) (i.e. the bal- v = p/m. Because W 0 implements free Dirac QFT, this anced even polynomials of ax, bx and Lx), it follows that should still hold. But now W 00 will precede each step by of our states look far away just like the vacuum does. i ε2g2L2 e 2 , whose effect is to add a phase Moreover, the bosonic commutation relations and Klein- Gordon dynamics are true only weakly with respect to X i 2 ipx 2 εg x |ψ0i = e − f(x) ax† |0i, this Hilbert space, i.e., the equations hold only in terms x ε of expectation values with respect to these states. ∈ Z (53) X 0 We will not attempt to given a rigorous proof of the = eip xf(x) a |0i. x† continuum limit here. We just sketch the argument. Our x ε ∈ Z approach is inspired by the derivation of the solution in 1 2 Ref. [43]. Thus, p0 = p − 2 εg the new expected momentum after an ε–period of time. This means that the particle is Let us therefore consider the massless case: m = 0, accelerating, according to an external force equal to gE, and the QCA with W 00 as defined in Eq. (50). At zero 1 coupling (charge) g = 0, The Heisenberg equations of mo- where E = − 2 g is the effective electric field that it feels given our boundary conditions. tions are exactly those of the free Dirac QFT, except for the discrete restrictions of the space and time variables. In the continuum, however, the Hilbert space is usually B. Continuum limit restricted to the Fock space of the following annihilation operators: In order to prove that we recover the Schwinger model, ( ( ap, if p ≥ 0 bp† , if p ≥ 0 we need to take a continuum limit of our system. This cp = and dp = (54) is a hard problem for any interacting many-body system, bp, if p < 0 ap† , if p < 0 as it usually requires one to be able to “solve” the theory. This difficulty also appears at the core of standard quan- where tum field theories, where a finite scale (minimal length X 1 ipx or maximal energy, called a regulator) usually has to be ap := ε √ axe (55) ε introduced in order to obtain finite predictions. Then x the parameters of the theory have to be made dependent X 1 b := ε √ b eipx, (56) on this regulator in such a way that the predictions con- p x x ε verge as the regulator tends to zero or infinity (whichever corresponds to the continuum limit). This procedure is and p takes value on the circle Tε of circumference 2π/ε. known as renormalization. But doing this exactly would So that, for instance, require being able to fully integrate the theory, so as to find the dependence of the predictions on the regulator. ap† aq + aqap† = 2πδ(p − q)I (57) In the standard QFT formalism, this is done by defining the theory perturbatively, starting from a solvable where model (typically a gaussian, i.e., a quasi-free theory de- ε X δ(p) = eixp (58) fined by a Hamiltonian that is quadratic in the canonical 2π variables). The continuum limit is then taken indepen- x dently for each order in perturbations. is the Dirac delta over the circle: This means that we first need to establish the continuum limit for a set of parameters where we expect Z Z π/ε dp δ(p)f = dp δ(p)f = f . (59) to arrive at an exactly solvable QFT. Thankfully, the p p 0 Tε π/ε Schwinger model can be solved when the fermions are − massless, provided that one works with the Hilbert space The inverse Fourier transform is formed by creating a finite number of fermions over a spe- Z 1 dp ipx cial “vacuum” state, which is Dirac’s half-filled electron √ ax = e− ap. (60) sea. ε Tε 2π However, this “solution” takes the form of an entirely Also, the Kronecker delta δ defined for x ∈ ε is different theory: a theory of free bosons with mass equal x Z to g . Since that theory is solvable in the sense that we Z √π dk ik(x y) can compute any n-point functions, all we need to do is δx = ε e − . (61) T 2π to figure out which operators in the algebra of our QCA ε are those effective bosonic operators, and to show that It is easy to check that one step of the QCA defined by they behave that way. W 0 (or W 00 with g = 0) yields the Heisenberg dynamics One may think of the choice of vacuum as a type of iε p iε p boundary condition: since we consider the Hilbert space cp 7→ e− | |cp dp 7→ e− | |dp. (62) 10

Also, we have any shape provided that it is bounded by a ball of radius ( that is a fixed multiple of Λ, and is symmetrical with αax + βbx, if y > x respect an exchange of p and q, so as to yield a self-adjoint [αax + βbx,Ly] = (63) 0, else operator. Proper convergence may also require replacing the sharp region by a smearing function with coefficient for any α, β ∈ C. decaying rapidly beyond Λ, such as a Gaussian. Let us consider the density of the left-moving fermion Let us consider the coarse-grained density of both modes: types of fermions: 1 1 ρΛ = AΛ + BΛ. (71) Ax = a† ax = a† ax (64) q q q ε x ε x We will also refer to the non-smeared density Clearly, [Ax,Ly] = 0, so that the interaction (g =6 0) has no bearing on its evolution at all. Instead, let us consider ρq = Aq + Bq. (72) a smeared version of it, by cutting off hight momentum modes. If f denotes the Heisenberg automorphism for the QCA First, consider its Fourier transform with no coupling (g = 0), then

X ikx iε(q p) Ak = ε e Ax f(ap† aq) = e− − ap† aq, (73) x iε(q p) f(b† bq) = e − b† bq. (74) Z dp dq p p = δ(p + k − q)ap† aq (65) Tε Tε 2π Hence Z × dp Λ iεk Λ = a† ap+k f(A ) = e− A , (75) 2π p k k Tε Λ iεk Λ f(Bk ) = e Bk . (76) We could define a smeared density such as Using the finite difference Z Λ ˜Λ dk ikx Ax = e− Ak (66) 2 Λ 1 2 Λ Λ Λ Λ 2π ∆t ρq := 2 (f (ρq ) + ρq − 2f(ρq )), (77) − ε for some cutoff Λ < π/ε. But this clearly still commutes we obtain with Ly. 2 Λ 2 Λ Instead, let us first smear the field operators them- ∆t ρq = −q ρq + O(ε). (78) selves, and then use those to define the charge density. For instance, we could use To lowest order in ε, this is the massless Klein-Gordon equation. Z Λ + dp ipx The error term still has to be appropriately bounded to ψΛ (x) := ap e− , (67) Λ 2π show convergence, but we will leave this for further work. − Instead, let us consider how this equation changes in the which is meant to tend to an actual Dirac field opera- presence of an interaction g =6 0. If our system behaves tor in the continuum. We then define the corresponding like the Schwinger model, we should see an effective mass fermion density operator term appear in the Klein-Gordon equation. P ipx Let Lq = ε L 1 e (where the sum is over x ∈ Λ + + x x+ 2 ε A := (ψ (x))†ψ (x). (68) x Λ Λ εZ), then, Λ Similarly, we define define ψΛ−(x) and Bx from bx. 3/2 X X iqy ipx Λ [ap,Lq] = ε e e ax Let us expand Ax in momentum to see how it differs x y x ˜Λ ≥ from Ax . We have X X = ε3/2 eiqz ei(p+q)xa (79) Z x Λ dk Λ z 0 x A = A , (69) ≥ x 2π k Tε = θq ap+q, with Fourier transform P iqy with θq := ε y 0 e . It follows that Z ≥ Λ dp dq Ak = δ(p + k − q) ap† aq. (70) 2 2π [ap,Lq†Lq] = |θq| ap + θ q Lqap q + θq L qap+q, (80) BΛ − − − where the double momentum integral is over the region where we used that fact that Lq† = L q since Lx is self- − BΛ = [−Λ, Λ] × [−Λ, Λ]. Observe that this is just Ax adjoint. We obtain the same equation substituting bp for if BΛ = Tε. More generally, the region BΛ could be of ap. 11

The electric ﬁeld Hamiltonian is corresponding term stemming from the equivalent ma-

Z nipulations on ap† kL kaq in the expression above. 1 2 X 2 2 dk − − HE := ε g Lx = g L† Lk. (81) We have, where all integrals are over dp, 2 2π k x Tε Z Z Z Then ap† L kap+l+k = ap† L kdp†+l+k + ap† L kcp+l+k − − − Z dk p+l+k<0 p+l+k 0 2 ≥ [ap† aq,HE] = g θkap† L kaq+k − θ kap†+kLkaq . − − Z Tε 2π ≈ ap† L kdp†+l+k (82) − p+l+k<0 Let us consider the evolution of the coarse-grained elec- Z Z = dpL kdp†+l+k + cp† L kdp†+l+k tron density given by Eq. (71). We have − − p+l+k<0 p+l+k<0 Z π/ε Z p<0 p 0 Λ 2 dk dpdq ≥ [Al ,HE] = g θk δ(p + l − q) Z π/ε 2π BΛ 2π (83) − ≈ dpL kdp†+l+k (84) − × ap† L kaq+k − ap† kL kaq p+l+k<0 − − − p<0 (We applied the transformation k → −k on the second Z = L kdpdp†+l+k − θkap†+kdp†+l+k term, which does not change the value of the expression). − If there is no restriction on the integral in p and q, this p+l+k<0 p<0 is zero because the first and second term differ only by a Z symmetry p → p−k of the integration domain. However, = −L kdp†+l+kdp − θkap†+kdp†+l+k − this argument fails thanks to the limited domain BΛ. p+l+k<0 We can simplify this expression by carefully choosing p<0 the dependence of Λ on ε, and constraining the states to be sufficiently ”smooth”. For instance, suppose we consider states in the Fock space of ax, bx for all x. Moreover, let us consider a state |ψi in which only modes p with If k + l =6 0, we can just anti-commute dp†+l+k and dp, to |p| < δΛ are occupied, where 0 < δ < 1. Then the expec- get tation value of the integrand is zero for all |k| < (1−δ)Λ. Z Z The terms in the integral with larger k can be made to ap† L kap+l+k ≈ θkap†+kdp†+l+k vanish in expectation if we also impose that the states − p+l+k<0 of the electric field are smooth in that they contain no p<0 mode with k above this bound. Consequently, the whole Z Z commutator is again zero in expectation value. = θkdp+kdp†+l+k + θkcp†+kdp†+l+k That is, as we tend to the continuum by decreasing p+l+k<0 p+l+k<0 p<0, p+k<0 p<0, p+k 0 ε and increasing Λ proportionally to π/ε, we also con- ≥ (85) sider fermion wavefunctions and electric field wavefunc- Z ≈ θ d d† tions which are progressively smoother. (We leave open k p+k p+l+k for now the question of whether this constraint is pre- p+l+k<0 p<0, p+k<0 served by the dynamics). Z To have an example where the commutator simplifies = −θkdp†+l+kdp+k ≈ 0, without entirely vanishing in this continuum limit, let us p+l+k<0 instead do this for the Fock space of the electron and p<0, p+k<0 positron annihilation operators cp and dp, as mentioned in the beginning of this section. where we also assumed l =6 0. To see which terms of the commutator survive, we need It is when k + l = 0 that a non-trivial term pops up, to commute operators such that either cp or dp are on the namely right side of monomials (or their adjoint on the left-side), so that they annihilate the state |ψi if the modes p are Z π/ε Z 0 Λ 2 dk not occupied. In the derivation below, we assume that, as [Al ,HE] ' g θk dp δ(k + l)L k π/ε 2π 1/Λ − above, the cutoffs Λ, ε and |ψi are such that this allows − − Z π/ε Z k such term to be translate by k without affecting their 2 dk − g θk dp δ(k + l)L k, expectation values once integrated. π/ε 2π 1/Λ − Let us focus on the first term, and abbreviate the inte- − − Z π/ε gral over p as a sum for a more compact notation. Below, 2 dk = g θk k δ(k + l)L k the symbol ≈ indicates that we removed a term using the π/ε 2π − − fact that it can be translated by k and cancelled with the (86) 12

Hence, with respect the Fock space with cutoﬀ Λ, we Finally, obtain g2 2 2 Λ ' 2 Λ − Λ g fg (ρq ) f (ρq ) (Aq + Bq), (97) [Aq ,HE] ' q θ qLq (87) π 2π − to lowest order in ε. Moreover, in expectation with re- iεHE for q =6 0. Since [Lq,HE] = 0, then, given U = e , spect to our “smooth” states, we expect that

g2 Λ Λ Λ Aq + Bq = ρp ' ρ . (98) U †Aq U ' Aq + iε q θ qLq. (88) p 2π − Λ Λ Λ Similarly Since also fg(ρq ) = f(ρq ), then ρq approximately satisfies the different equation 2 Λ g [Bq ,HE] ' − q θ qLq. (89) g2 2π − ∆2 ρΛ ' −p2ρΛ − ρΛ, (99) t p p π p Λ Λ Λ We see that, since ρq = Aq + Bq , √ which is the Klein Gordon equation with mass g/ π, as Λ [ρq ,HE] ' 0. (90) expected for the (massless) Schwinger model. Clearly, this derivation remains informal, but has the Which is what we expect from the Schwinger model. virtue of indicating the assumptions that may be required In the Heisenberg picture, one step of the QCA defined to obtain a rigorous continuum limit. Another step that by W 00 is fg(A) = f(U †AU) with U defined as above, and we will not take here, would be to check that perturbation f is the QCA step defined by W 0. of this solution for a small mass also carries over to the One step of the full QCA is continuum limit. Λ iεq Λ iεq Λ Λ fg(ρq ) ' e− Aq + e Bq = f(ρq ) (91) V. CONCLUSION We can compute a second step easily using the above results, and we obtain Summary of achievements. Fig. 3 describes a quantum 2 Λ i2εq Λ i2εq Λ fg (ρq ) ' e− Aq + e Bq circuit made by infinitely repeating the local quantum 2 (92) gate given by (50). This is a quantum cellular automa- g iεq iεq + iε q θ qf(Lq)(e− − e ). ton (QCA), or rather a family of QCA, since the lat- 2π − tice spacing and local quantum gates are parameterized We observe that by ε. We have argued, in two ways, that this family of QCA constitutes an alternative formulation of Schwinger iεq iεq X iq(y ε) iq(y+ε) (e− − e ) θ q = ε (e − − e ) model, i.e. (1 + 1)–dimensional quantum electrodynam- − y 0 (93) ≥ ics (QED). We first did this by taking the limit when iqε 2 ' ε(1 + e− ) = 2ε + O(ε ), ε → 0, and studying perturbations over suitable choices of ‘smooth’ vacuum states, which allowed us to recover provided that it multiplies operators which vanish at x → the main features of (1+1)–QED. Namely, electron accel- ∞ when evaluated on states of our Fock space. eration under an electric field in (53),√ and the emergence Also, of free bosons of effective mass g/ π in (99) over Dirac’s half-filled electron sea. The second argument is based on X iqx † f(Lq) = ε e (Lx 1 ε + ax† ax − bx εbx ε) the fact that the very construction of the QCA was justi- − 2 − − x (94) fied on the same grounds that the construction of (1+1)– = Lq + ε(Aq − Bq). QED is justified on. I.e. we started from an established QCA for the free theory (Fig. 1 with (6)), restored U(1)– But all states in our Fock space are gauge-invariant, gauge invariance by introducing the gauge field (Fig. 3 i.e., Jx|ψi = 0. Hence, with (28), which satisfies (23)), and eventually gave this gauge field a simple gauge-invariant dynamics. X iqx Lq = ε e L 1 QCA formulation of QFT. x+ 2 ε The path-integral formula- x tion of QFT suffers from a number of issues related to the X iqx quantization of the action, a process which breaks conti- = ε e (Lx 1 ε + Ax + Bx) (95) − 2 x nuity, introduces ambiguities, and jeopardizes unitarity. iqε We hope that the (1+1)–QED QCA hereby presented will = e L + ε(A + B ). q q q serve as an illustration that QCA formulations of QFT Therefore, are feasible and advantageous in these respects. Taking a natively quantum and discrete dynamics as the starting − iqLq = Aq + Bq + O(ε). (96) point directly shortcuts all of these issues. Getting back 13 to the continuum is no extra work: renormalization had proach, the same strategy should also work for QCA. The to be worked out in the path-integral formulation any- idea is to start from a known “non-interacting vacuum”, way. We showed that gauge-invariance could be handled and simulate an evolution where an interaction param- straightforwardly, at least within the temporal gauge. In eter is slowly (adiabatically) turned on with time. This principle, Lorentz-covariance could also be verified using works only if there is a sufficiently large energy gap above the approach form Ref. [11, 16, 18, 20]. Altogether the the ground state, which needs be unique for every value construction is therefore simple and pedagogical. The re- of the parameter. Typically, this parameter would be the sult is more in line with the discrete spacetime formalisms charge, but in the specific example at hand the fermionic introduced in Quantum Gravity proposals [2, 50]. mass seems a good candidate. Indeed, in that case an en- Quantum simulation. Again, this (1 + 1)–QED QCA ergy gap is suggested by the mass of the effective bosons, is but a quantum circuit, see Fig. 3. Each W 00 can be in turn controlled by the electric charge. expressed in terms of standard universal gates such as For measurements, the prescription to measure local CNot, Hadamard, Phase. Thus, the QCA is directly charge densities via phase estimation ([36] section 4.5) interpretable as a digital quantum simulation algorithm, should carry over to our framework. Electric field oper- to run on a Quantum Computer. Moreover, this quan- ators are also local and again simple to measure. tum simulation algorithm is efficient, in the sense that it Perspectives. Regarding the (1+1)–QED QCA hereby requires an O(st/ε2) gates in order to simulate a chunk presented, we hope that more detailed analysis will fol- of space of size s, over t time steps, with ε the spacetime low that will anchor the connection with the Schwinger resolution—which for free fermions is known to control model on firmer ground. Whilst recovering the full path the precision in || · ||2–norm linearly [7]. Of course the integral formulation theory from the QCA seems out of output produced is a quantum state, and so one may then reach (due to a lack the mathematical techniques for do- need to repeat the process several times in order to ob- ing so), a more thorough derivation of its phenomenology tain useful statistics about it. Classically however, just is at hand. We also leave open the interesting question the state space itself is already of size an O(exp(s/ε)), of determining the vacuum for a QCA, since there is no as it grows exponentially with the number of quantum Hamiltonian. Here we chose it based on knowledge of the systems to be simulated—the classical time complexity target continuum theory. However, given only a QCA, is therefore at least an O(exp(s/ε)t/ε). Clearly, the how would one go about arguing for a specific choice of exponential gain here is due to the fact that that the vacuum? Eventually, the work needs to be extended in (1 + 1)–QED QCA simulates multi-particle systems, just the obvious directions: Lorentz-covariance, higher spatial like in Hamiltonian-based multi-particle quantum simu- dimensions, and Yang-Mills theories. Plenty of fascinat- lation schemes. QW-based quantum simulation schemes, ing questions lie ahead. on the other hand, are by definition in the one-particle sector, and thus can only yield polynomial gains. State preparation and measurement. When simulating quantum field theories, the preparation of states, and Acknowledgments the identification of observables which match those rel- evant to a specific experiment, are challenging matters. CB was supported by the National Research Foun- One central issue is the fact that the nature of the vac- dation of Korea (NRF-2018R1D1A1A02048436). PA uum for the full theory (lowest energy state) is not known would like to thank Pablo Arnault, Nathanaël Eon´ and a priori. This question of the state preparation for quan- Giuseppe Di Molfetta for helpful discussions. TCF would tum simulation was considered specifically for fermions like to thank Tony Short and Tobias Osborne for useful in [36]. Although this work uses a continuous-time ap- discussions.

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