General Methods for Digital Quantum Simulation of Gauge Theories

PHYSICAL REVIEW D 100, 034518 (2019)

General methods for digital quantum simulation of gauge theories

† ‡ Henry Lamm ,* Scott Lawrence, and Yukari Yamauchi

(NuQS Collaboration)

Department of Physics, University of Maryland, College Park, Maryland 20742, USA

(Received 19 April 2019; published 26 August 2019)

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant correlators is detailed. We demonstrate the procedure on small lattices, including the simulation of a 2 þ 1D non-Abelian gauge theory.

DOI: 10.1103/PhysRevD.100.034518

I. INTRODUCTION finite subsets [17,18], Fock-state truncation [12,19,20], dual variables [21–26], optical representations [27], and Quantum simulations are motivated by inherent obstacles the prepotential formulation [28]. A common suggestion is to the classical, nonperturbative simulation of quantum field to limit the register to physical states by gauge fixing, at the theories [1]. Deterministic methods struggle with exponen- cost of increased circuit depth in the time-evolution, tial state spaces. Sign problems stymie Monte Carlo methods although a practical method for non-Abelian theories both at finite fermion density [2,3] and in real-time evolution remains undescribed. [4]. While large-scale quantum computers will greatly Given a register, the next concern is the preparation of enhance calculations in quantum field theory, for the desired quantum states. In strongly coupled theories, how foreseeable future quantum computers are limited to the asymptotic states depend upon the fundamental fields is tens or hundreds of nonerror-corrected qubits with circuit — unknown. Most methods focus on ground state construc- depths fewer than a 1000 gates the so-called noisy tion. Examples include using quantum variational methods intermediate-scale quantum (NISQ) era. Along with hard- [9,29], quantum phase estimation (QPE) [30–32], quantum ware development, theoretical issues impede full use of adiabatic algorithm (QAA) [33,34], and spectral combing quantum computers. Despite these limits, algorithms have – [35]. Some preparation methods require efficient real-time been demonstrated in toy field theories [5 12]. Viable evolution. Classical-quantum hybrid methods [11] and quantum simulations require addressing four interconnected dimensional reduction [36] have been proposed to initial- issues: representation, preparation, evaluation, and propaga- izing thermal states. Simulations of scattering are hindered tion. Proposals in the literature address one or more of these by state preparation [37–42]. topics. For gauge theories, additional complications arise Evaluation of observables represented by a single, which we discuss here. Hermitian operator at a single time is straightforward. This The first hurdle is the representation of a gauge field in a includes most static properties. Other observables, like time- set of qubits, or quantum register. While natural matchings separated composite operators (e.g. parton distribution func- – exist for fermionic fields [13 16], the digitization of a tions) are more complicated. Naively, the first nonunitary bosonic field is nontrivial. This is reminiscent of the early operator collapses the quantum state. One resolution intro- days of classical lattice field theory where memory resour- duces ancillary probe and control qubits [43], which we use ces limited calculations. Ideas include approximating by: here. QPE [30] has been used to compute linear response [6]. Another intriguing idea uses quantum sensors to implement *[email protected] generating functionals [44]. A second problem arises from † [email protected] state contamination. In Euclidean lattice field theory, we can ‡ [email protected] use imaginary-time evolution to separate states overlapping with the same operator. Real-time evolution lacks this Published by the American Physical Society under the terms of separation, so we need novel methods to project onto desired the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to states, perhaps through a distillation-like technique [45]. the author(s) and the published article’s title, journal citation, In this work, we construct a method for mapping the and DOI. Funded by SCOAP3. Hilbert space of a gauge theory onto the Hilbert space of a

2470-0010=2019=100(3)=034518(14) 034518-1 Published by the American Physical Society LAMM, LAWRENCE, and YAMAUCHI PHYS. REV. D 100, 034518 (2019) quantum computer, and simulating its propagation. not depend on whether superconducting qubits or trapped Propagating a system for a time t requires a unitary operator ions are used. UðtÞ¼e−iHt, which cannot be efficiently constructed on a In this section, we describe a set of higher-level con- quantum computer. Instead, one Trotterizes the evolution structs that allows the simulation of a general non-Abelian −iH t N by UðtÞ ≈ ðe N Þ which allows efficient simulation gauge theory. We outline how they might be implemented [37–42,46]. Trotterized versions of UðtÞ exist for elec- in practice, from the lower-level qubit/gateset primitives. tron-phonon systems [12], the Abelian theories [8,10, Hopefully these specifications are helpful to focus the 21–24,27,47–54], and non-Abelian theories [55–62]. implementation of those tools that are most critical. Each depends on the register used. We will simulate a theory with a gauge group G. The Violations of gauge invariance in the time-evolution are central construct is the G-register, analogous to a classical potentially introduced through noisy gates, digitization, variable storing an element of G. The Hilbert space of an and Trotterization. To remedy these, [63] proposed using ideal G-register is HG ¼ CG, the complex vector space oracles to project into the physical subspace. In this work, with one basis element jgi for every element g ∈ G. we construct a Trotterized time-evolution without gauge- (Equivalently, it is the space of square-integrable functions fixing, for arbitrary gauge theories with coupled matter from G to C.) The Hilbert space of multiple G-registers fields. Gauge invariance is exactly preserved after is constructed as a tensor product, so a computer with N H⊗N Trotterization. We connect this procedure to the transfer G-registers has Hilbert space G . matrix formalism [64–67]. The effect of noisy gates on For discrete groups, ideal G-registers are easy to imple- gauge-invariance are hardware-dependent, and we leave ment by virtue of their finite number of elements; for this for future work. The error introduced by field digitiza- instance, an ideal Z2-register requires one qubit. For tion can be treated without impacting the algorithm. continuous Lie groups [e.g. SUð3Þ], ideal G-registers cannot A final, overarching concern is renormalization. Work be made with any finite number of qubits—we must settle in quantum computing [68,69] and elsewhere [70,71] for approximate G-registers (typically with Hilbert space suggests that the renormalization factors may be cheaply H˜ ˜ ∈ G ¼ spanfjgigg∈G˜ for some subset G G) which leads to computed classically, allowing for the matching of the bare intrinsic discretization error. The tradeoff between number of parameters on the quantum computer to renormalized and qubits and the size of discretization error will be crucial to physical ones. In the context of the method considered here, constructing efficient simulations near-term. a mapping between the Trotterization and a Euclidean In this work, we consider only the case of ideal action may allow renormalized couplings to be computed G-registers. Complications arising from the discretization, efficiently on a classical computer with standard lattice which may include the breaking of gauge invariance [63] methods. The transfer matrix formalism naturally provides and difficulties with the definition of the kinetic and Fourier such a mapping. Other quantum-computing based propos- gates are not considered here. A proposal for approximate als exist as well [72,73]. SUðNÞ registers that does not have these problems may be This paper is structured as follows: in Sec. II we describe found in [17,18]. the prerequisite registers and fundamental gates for our For the Hamiltonians we consider, a set of useful construction. With these, Sec. III formulates gauge- primitive gates defined on the G-register is: inversion, invariant UðtÞ for the gauge fields. Section IV links this matrix multiplication, trace, and the quantum Fourier procedure to the transfer matrix formalism, and outlines how transform. The inclusion of matter fields requires additional renormalization may be performed classically. Section V multiplication and Fourier transform gates, and an inner explains how nontrivial correlators may be computed. product gate. A sample implementation for these operations Section VI and VII extend the formulation to include scalar D4 and spinor fields respectively. Section VIII demonstrates of is specified in Appendix A for the -register case. The inversion gate acts on a single register to produce the our method in the non-Abelian D4 theory and the Z2 theory with staggered fermions. We conclude in Sec. IX. register of its inverse group element. This is defined in the fiducial basis by

II. PREREQUISITES −1 U−1jgi¼jg i: ð1Þ Developers of quantum algorithms assume idealized qubits, and certain unitary gates operating on those qubits The other group operation is matrix multiplication, which (a sufficient universal gateset, for example, is the Hadamard requires a two-register gate, defined by π gate, the 8 gate (T), and the controlled-not gate CNOT). This U follows a pattern in classical computing, where certain ×jgijhi¼jgijghi: ð2Þ constructs (e.g., floating-point numbers and arrays) are U assumed to specify an algorithm. The implication of this is Here we have × implementing left multiplication, in that any correct implementation of the primitive constructs the sense that the content of the target register was multiplied will do: the correctness of the quantum algorithm does on the left. This is all that is required here. In general,

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U U the combination of −1 and left-acting × permits The inner product is generally a complex number, but U 1 2 U 1 U 2 U 2 1 right multiplication: ×;Rð ; Þ¼ −1ð Þ −1ð Þ ×ð ; Þ this gate extracts only the real part while remaining U 2 U 1 2 −1ð Þ ×ð ; Þ. unitary. Happily, this is all required for simulation Most quantum gatesets—certainly all gatesets proposed of field theory, where only the linear combination — † † for practical use contain the inverse of every gate as a ϕ2ϕ1 þ ϕ1ϕ2 appear. primitive. This makes it easy, given a circuit implementing Lastly, the scalar Fourier transform gate is defined N one unitary, to obtain a circuit implementing its inverse. analogously to UF above, except for C : we notate this † N Therefore, if U is available, U−1 is readily obtained with UC CN × F . The resulting register is a -register. Since the the assistance of an ancillary register. quantum Fourier transform on CN factorizes into N The trace of a plaquette appears in our gauge C UCN separate transforms on , the implementation of F Hamiltonian, and so to perform this operation we combine C consists of N separate calls to U . with the matrix multiplication gate with a single-register F trace gate: III. PURE GAUGE U θ iθ Re Tr g Trð Þjgi¼e jgi: ð3Þ In this section, we construct a time-evolution operator suitable for the simulation of pure gauge theories on a In our construction, the final gate required on the quantum computer. We take care to exactly preserve gauge – U G-register is the Fourier transform gate [74 76] F. invariance (up to unavoidable gate errors and noise). We This gate acts on a G-register to rotate into Fourier space. take the standard approach, by discussing the Hamiltonian It is defined by formulation of the theory, and then Trotterizing the X X Hamiltonian into a sum of pieces easily implemented via U ˆ ρ ρ F fðgÞjgi¼ fð Þijj ;i;jið4Þ the primitive constructs of Sec. II. This approach will be g∈G ρ∈Gˆ extended in Sec. VI and VII to include scalar and fermionic matter fields, respectively. The second sum is taken over ρ, the representations of G,and A similar approach to non-Abelian gauge simulation has fˆ denotes the Fourier transform of f. This gate diagonalizes been discussed previously [54,59–62,77]. In this section, what will be the “kinetic” part of the Trotterized time- we take care to work only with logical qubits and gates, evolution operator. After application of the gate, the register keeping the algorithm independent from the underlying is no longer a G-register but a Gˆ -register. geometry of the quantum computer. Additionally, we avoid The G-register suffices for the representation of the the need for ancillary G-registers. gauge field. In order to represent scalar matter fields, we In the present work, we use the basis diagonalizing introduce the CN-register, where N is the dimension of the the gauge field operators U, rather than their conjugate fundamental representation1 of G. This register stores a momenta, unlike e.g. [55] in which the fiducial basis is component of CN, and we will define gates allowing chosen to diagonalize the kinetic term of the Hamiltonian. transformation under elements of G. As with the G-register, Our choice simplifies the calculation of plaquettes and it is only possible to implement a ‘truncated” CN register. Wilson lines, as well as the inclusion of matter fields. “ ” (For fermionic matter fields, the CN-register is unneeded, Additionally, simulations done in this position basis and no truncation is necessary.) correspond more closely to calculations performed in classical lattice field theory. Three gates are needed to make the CN-register useful in We begin by reviewing the Hamiltonian formulation a quantum simulation. The fundamental (or adjoint) multi- of lattice gauge theory [78]. Each gauge link has a local plication gate U is defined by ×;f Hilbert space CG. Combined, the L links on our lattice H C ⊗L U ϕ ϕ have Hilbert space ¼ G . ×;fjgij i¼jgijg i: ð5Þ On a lattice with N sites, the space of gauge transformations is GN, where one element of G must be The inner product gate, necessary for the field-theory specified at each site. The transformation rule for a link kinetic term (sometimes called the “hopping” term, to † U from site i to site j is U ↦ V U V . The linear action distinguish it from the quantum mechanical kinetic term), is ij ij j ij i ∈ N ϕ defined by of V G on the Hilbert space is given by ðVÞ:

˜ ˜ † † ˜ ˜ ϕ1 ϕ2 θ ϕ ϕ ϕ ϕ † ϕ ϕ U θ ϕ ϕ δ δ i ½ 2 1þ 1 2 ϕ V U V U V 7 h 1 2j h·;·ið Þj 1 2i¼ ϕ1 ϕ2 e : ð6Þ ð Þj ij i¼j j ij i i ð Þ

1For matter in the adjoint representation, N is the dimension of Gauge invariance demands that two states differing the adjoint representation. only by a gauge orbit be considered physically

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Uð1Þ FIG. 1. Circuits for the propagation of a pure-gauge lattice field theory. The first circuit implements K on four links (in general, Uð1Þ † † L links are needed). The second circuit shows the application of V to a single plaquette Re Tr U13U34U24U12, and must be applied to every plaquette in the theory. Note that in these circuits, we use a doubled line to represent a G-register, rather than a single qubit. equivalent. The physical Hilbert space,2 then, is guidance for how to form the kinetic term of a discrete ⊗L ⊗N — HP ¼ CG =ϕðG Þ. gauge theory this is addressed via the transfer matrix HP can be obtained from the larger space H by a formalism in Sec. IV. gauge-symmetrization operator, which acts as a projection Not only is the Hamiltonian gauge-invariant, but each operator: term is gauge-invariant. This is crucial, as it allows us to Z Z Trotterize without breaking gauge invariance at any order 1 † in Δt. A time evolution operator suitable for implementa- PjU12 i ¼ dV1 dV2 jV2U12V i G N 1 tion on a quantum computer is then j j Z G G 1 Y ϕ −iH Δt −iH Δt ¼ N dV ðVÞjUið8Þ UðtÞ¼ e K e V : ð10Þ jGj GN t=Δt This discussion of the gauge-invariance of physical U Δ states is equivalent to the imposition of Gauss’slaw.An Below, we will use the notation ð tÞ to indicate a single −iH Δt implementation of the operator P for U 1 is given by [63]; Δt step. We now describe how the operators e K and ð Þ − Δ however our method does not require implementing P. e iHV t may be obtained. First, each part (kinetic and We define a Hamiltonian which acts on the entire space potential) of the Hamiltonian consists of mutually commut- H of the gauge theory, implicitly defining a “physical” ing, local terms, and each such term may be treated Hamiltonian on the subspace HP. As will be made clear in individually and sequentially without approximation. later sections, this choice of Hamiltonian is motivated by a That is, desire to recover the Euclidean Wilson action. Y Y − Δ ð1Þ − Δ ð1Þ e iHK t ¼ U ði; jÞ and e iHV t ¼ U ðpÞ: X Y 2 X K V 1 4g 2 hiji p − Uij π H ¼ 2 Re Tr þ ij g a ∈ a p hiji p hiji ð11Þ

¼ HV þ HK ð9Þ Here, again because the factors commute, the order of Uð1Þ The first term, HV, is a sum over spatial plaquettes p and application of the V makes no difference to the final can be thought of as a potential term. The trace is taken in a Uð1Þ result. The plaquette evolution V is constructed by first suitable representation of the gauge group G—for matrix preparing the product of the plaquette in an ancillary groups, this is typically the fundamental representation. register, and then operating on the ancillary with The second term, HK, is the quantum-mechanical kinetic 1 ð1Þ U ð 2 Þ. U is implemented by diagonalizing via U , term describing the mechanics of a free particle moving on Tr g a K F π2 and then applying a diagonal unitary: the surface of G. Therefore, ij is the Laplace-Beltrami π operator on the surface G, and in is the momentum Uð1Þ U U U† operator conjugate to Uij. This prescription gives us no K ði; jÞ¼ F phase F: ð12Þ

2 Uð1Þ Uð1Þ The Gribov ambiguity prevents us from unambiguously Circuits implementing K and V are shown in Fig. 1. assigning a single representative configuration to each gauge U The implementation of phase depends strongly on the orbit; however, we do not take that approach. Physical states in the Hilbert space are not configurations, but formal linear group and the implementation of the G-register, but in all combinations of configurations, and these can be constructed cases it is a diagonal operator. The total gate requirements unambiguously. for the propagation are shown in Table I.

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TABLE I. Gate requirements for the propagation of a lattice projection with P. In the course of performing this with NP plaquettes and L links, for a time T with time steps of projection, the lattice path integral with the Wilson action size Δt. is recovered. The transfer matrix is an approximation to imaginary- Gate Number time evolution by one unit of time: T ≈ e−H. We construct U 2L T F Δt the transfer matrix in the fiducial basis of H from separate U T phase L Δt kinetic TK and potential TV contributions via the split- U 6 T −1 NP Δt operator approximation T U× 6NP Δ t 1=2 1=2 U T T ¼ T T T ð14Þ Tr NP Δt V K V where the potential term resembles the product of spatial U Uð1Þ plaquettes as appears in the Wilson action Note that V involves multiple V , some of which affect overlapping links. Because all potential operators ˜ X 1 ˜ U12 a0 Uð Þ hU12 jT jU12 i¼δ exp WμνðxÞ : commute (indeed all V are diagonal in the fiducial basis), V U12 ag2 the order they are performed in does not matter. x When preparing an initial state, it is important to ensure ð15Þ that it lies within the physical subspace HP. One procedure for projecting onto the gauge-invariant states for an Abelian Here a is the spatial lattice spacing, and a0 the temporal gauge theory is provided in [63]. spacing; we do not assume an isotropic lattice. We have Although the preparation of the ground state jΩi is borrowed from lattice field theory the Wilson plaquette beyond the scope of this paper, it is worth discussing that common proposals typically require the time-evolution † † WμνðxÞ¼Re Tr½Ux;xþνˆ Uxþνˆ;xþμˆþνˆ Uxþμˆ;xþμˆþνˆ Ux;xþμˆ ; operator UðtÞ as constructed in this section. Assuming such methods are used, nonequilibrium physics can be ð16Þ computed by preparing the ground state with UðtÞ, and then and μ, ν are restricted to spacelike directions. evolving with a different time-evolution operator U˜ðtÞ [11]. For the kinetic term there is no coupling between A typical expectation value of interest, accessible in such a different links, but it is related in lattice field theory to scheme, is plaquettes containing timelike links: Ω U˜ † ˆ U˜ Ω EðtÞ¼h j ðtÞE ðtÞj ið13Þ Y a ˜ † 2 ReTr½U Uij ˜ g a0 ij hU12 jTKjU12 i¼ e : ð17Þ where Eˆ measures the expectation of a plaquette (i.e. the hiji energy density of the gauge field). Other expectation values of interest are discussed in Sec. V below. Note that, like the Trotterized real-time evolution, ½T;P¼0 due to the fact that TV and TK individually IV. THE LATTICE PATH INTEGRAL commute with P. When the transfer matrix is restricted to physical states, it Substantial insight into QCD has been achieved via exactly reproduces the usual lattice path integral, with the Euclidean-spacetime lattice methods on classical com- Wilson action. puters. Accordingly, the Euclidean lattice path integral occupies a dominant place in the QCD literature and the β β Z ¼ TrPT ¼ TrðPTPÞ minds of practitioners. This is in contrast to quantum Z X Y simulations where the Hamiltonian formulation appears 1 ¼ dU12 exp 2 Re Tr Uij : ð18Þ more natural. In this section, we connect the Hamiltonian G g formalism to the Euclidean lattice path integral via a transfer matrix. This implies an important practical conse- Here TrP denotes the trace over only the physical sub- quence: the determination of appropriate bare coupling space HP. The inverse temperature is denoted β,andZ constants may be performed on a classical computer. This is the Euclidean lattice approximation to the partition procedure also guides us in constructing a Hamiltonian for function. a discrete gauge theory (discussed in [79]). Now that we have a transfer matrix corresponding to the Our formalism for the transfer matrix, T, slightly differs Wilson action, we may connect it to the Hamiltonian. This from the usual one [64–67,80] in that T is defined on the is done in detail by Creutz [80] for the gauge-fixed transfer entire space H. The usual transfer matrix is defined only matrix, so we will only summarize the gaugefree result on the physical Hilbert space, and may be obtained by (which is essentially the same) here. The potential part TV

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− of the transfer matrix is exactly e HV . The relationship interest may be obtained. In this section, we show how to between the kinetic part HK of the Hamiltonian and TK is obtain a plaquette-plaquette correlator and the expectation more subtle. It is shown in [80] that, in the limit where the value of an arbitrary, temporally extended Wilson loop. temporal lattice spacing a0 is taken to 0, the kinetic part of Such correlation functions serve as the nonperturbative the Hamiltonian is recovered. input to larger calculations, such as the determination of With a mapping between the Hamiltonian and lattice action parton distribution functions. through the transfer matrix, the bare constants in Eq. (9) for a We accomplish both correlators by the technique set out desired renormalized, physical set of couplings are the same in [43]. To begin, we recall how one can compute an as the lattice action ones. This is particularly important for expectation value of a unitary operator U [81] in a given ensuring that Lorentz invariance is recovered in the con- state jΨi. Introducing a single ancillary qubit, we construct tinuum limit, but also allows us to perform simulations at a controlled unitary operator UC, defined by known values of physical parameters. In particular, in a lattice QCD simulation, the task of determining appropriate bare UCjΨij0i¼jΨij0i and UCjΨij1i¼UjΨij1i: ð20Þ parameters for a desired physical pion mass may be performed cheaply on a classical computer, reserving the (Standard quantum gatesets make the construction of this quantum processor for the difficult real-time evolution. operator straightforward.) Clearly, the Hamiltonian evolution on a quantum com- Generally, the expectation value of U has both real puter is closely related to standard classical lattice simu- and imaginary parts. These must be measured separately. lations. Indeed, the two differ only in: the discretization A measurement of RehΨjUjΨi isgivenbyinitializingthe 1 ≡ pﬃﬃ 0 1 of the gauge group (quantum computers have smaller ancillary qubit to jþi 2 ðj iþj iÞ, performing evolu- G -registers), whether the Trotterization is done in real or tion via UC, and then measuring σx on the ancillary qubit. imaginary time, and the exact form of the kinetic term. The first difference can be removed by performing a classical Ψ ⊗ † 1 ⊗ σ Ψ ⊗ Ψ Ψ ðh j hþjÞUCð xÞUCðj i jþiÞ ¼ Reh jUj i: simulations with reduced precision, and the others are reduced at small temporal lattice spacing. ð21Þ In the case of the Trotterization Δt being equal to a, the Ψ Ψ quantum evolution is simply an analytic continuation of the For Imh jUj i one performs the same process except σ usual isotropic lattice calculations performed in lattice ending with a measurement of y. QCD. This implies that Lorentz invariance is recovered With this procedure in mind, we can show how to in the continuum limit. compute a correlator of the form We conclude by exploiting this procedure to produce a Ψ U − 0 U Ψ Hamiltonian for a discrete gauge theory—that is, a gauge h j ð tÞWμ0ν0 ðx Þ ðtÞWμνðxÞj i: ð22Þ theory where the gauge group is a discrete group (such as In QCD this Wilson plaquette correlator corresponds to a D4, demonstrated below), rather than a connected manifold. In this case, the kinetic part of the Hamiltonian is glueball propagator. The operator in Eq. (22) is clearly not determined not by a Laplacian on a manifold, but can be unitary, so it cannot be directly evaluated by means of the derived from the relevant portion of the transfer matrix by procedure described above. Instead, using the Hermiticity taking a logarithm. of WμνðxÞ, we construct a parametrized family of unitary operators. The derivatives of this family give the operator of interest. Explicitly, this is done via a time-dependent H ¼ HV þ HK perturbation of the Hamiltonian: where HV ¼ − log TV and HK ¼ − log TK; ð19Þ 0 τ ϵ δ τ − 0 0 ϵ δ τ Hϵ1;ϵ2 ð Þ¼H0 þ 2 ð tÞWμ ν ðx Þþ 1 ð ÞWμνðxÞ where TV and TK are defined exactly as in Eq. (14). Note ð23Þ that HK is only well defined if TK is positive-definite: this fact is established for a general discrete group (with the Wilson action) in Appendix B. Time evolving forward in time with Hϵ1;ϵ2 , and back with As a consequence of Schur’s lemma, the kinetic transfer H0, allows us to measure the expectation value Cðϵ1; ϵ2Þ≡ Ψ U − U Ψ gate defined here is diagonal in the Fourier basis. This h j ð tÞ ϵ1;ϵ2 ðtÞj i. Differentiating twice with respect implies that, where HK is obtained in this fashion, UK may to the perturbation yields the plaquette-plaquette correlator be easily implemented from UF. as desired.

∂2 ϵ ϵ Cð 1; 2Þ 0 V. CORRELATORS − U − 0 0 U ∂ϵ ∂ϵ ¼h ð tÞWμ ν ðx Þ ðtÞWμνðxÞi: 1 2 ϵ1¼ϵ2¼0 Having constructed time-evolution for a pure gauge theory, we now must show how observables of physical ð24Þ

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In practice, this derivative must be obtained by finite extended via tensor product to include D scalar degrees differencing, after computing Cðϵ1; ϵ2Þ for several small of freedom at each site. The new Hilbert space is ϵ ⊗L ⊗N values of . This approach is naturally extended to the H ¼ðHGÞ ⊗ ðHSÞ ,whereHG is the one-link computation of k-time correlation functions, at the cost of Hilbert space defined above, N is the number of sites, requiring the kth numerical derivative. and HS is the one-site Hilbert space. For HS, it is convenient The terms WμνðxÞ added in the perturbation also appear to work in the basis of occupation number, writing in the original Hamiltonian, and are readily implemented HS ¼ spanfjn1 nDi∶ni ¼ 0; 1; …g.Thisisdifferent with the primitive gates defined in Sec. II. from classical lattice field theory where scalars are typically We can use the same method to compute the expectation represented in a basis diagonalizing the field operator or † integrated out, but is more natural for quantum simulations. values of temporally extended Wilson loops, hReTrUijðtÞ U ð0Þi. To do so, we decompose it into terms As before, we first consider how gauge transformations ij act on H. A gauge transformation is still specified by † ba ab 0 ab hReð½UijðtÞ ½Uij ð Þ Þi where a, b are color indices. Vi ∈ G at each site i. The gauge links still transform as We can use this to derive a perturbed Hamiltonian ↦ † Uij VjUijVi , and the scalar fields transform (in the ϕ ↦ ϕ ϕ† ϕ ab † ba ab fundamental) via i Vi i. In this way, j Uij i is Hϵ ðτÞ¼H0 þϵ2δðτ−tÞRe½U þϵ1δðτÞRe½Uij ij gauge-invariant. The action of a gauge transformation V ϵ˜ δ τ− † ba ϵ˜ δ τ ab þ 2 ð tÞIm½Uij þ 1 ð ÞIm½Uij ; ð25Þ on the Hilbert space is

ab ϵ ϵ˜ ϵ ϵ˜ ≡ † which can be utilized in a correlator C ð 1; 1; 2; 2Þ jU12 i⊗ jϕ1 i↦ jðV2U12V1Þi⊗ jðV1ϕ1Þi: ab hΨjUð−tÞUϵ ðtÞjΨi We now take the difference of two ð27Þ second derivatives of the resulting correlator: ∂2 ∂2 The gauge-symmetrization operator is still defined by − − ab ϵ ϵ ϵ˜ ϵ˜ Eq. (8), but acts on the expanded Hilbert space. ∂ϵ ∂ϵ ∂ϵ˜ ∂ϵ˜ C ð 1; 2; 1; 2Þ 1 2 1 2 ϵ¼0 To represent the physical Hilbert space on a quantum U − † baU ab computer, we now make use of the CN-registers trans- ¼ Reh ð tÞ½Uij ðtÞ½Uij i: ð26Þ forming under G. For the scalar fields we consider, the Summing over the a, b yields the desired, gauge-invariant Hamiltonian is trace. The imaginary part can be obtained similarly. X Y X 1 2 For many gauge groups, the different correlators β Uij π † H ¼ Re Tr þ ij ba 0 ab ij ∈p 2 hðUijðtÞÞ ðUijð ÞÞ i are related by gauge symmetry. p h i hiji Consequently, for these gauge groups, the correlator in 1 X 2 X 1 X ∇2ϕ m ϕ†ϕ ϕ − ϕ 2 Eq. (26) need only be evaluated for one particular selection þ 2 i þ 2 i i þ 2 j j Uij ij : of a and b, as all the other terms will be equal. In particular, i i hiji this is true for D4 (simulated below) and SUðNÞ in the ð28Þ fundamental representation. The perturbation introduced, unlike the one used for Although we assume scalar interactions only to order ϕ2, plaquette-plaquette correlators above, is not gauge- the generalization to higher-order couplings is trivial and invariant, meaning that the state during this time evolution introduces no complications into what follows. We con- H does not lie within the physical subspace P. Also unlike struct a Trotterized time evolution operator, just as in the plaquette-plaquette perturbation, this perturbation con- Sec. III, by decomposing into potential and kinetic terms. tains terms not present in the original Hamiltonian, and Uð1Þ requires gates not specified in Sec. II. The gates required The new gates required for the scalar fields are M Uð1Þ are diagonal (in the fiducial basis) phase gates on individual (implementing the mass term), H (implementing the G Uð1Þ -registers. hopping term), and K;scalar. These are defined by This method can also compute a nongauge-invariant 2 † expectation value, e.g. hTrU ðtÞU ð0Þi where j ≠ k. ð1Þ −iΔtðdþm Þϕ ϕ ik ij U ðiÞ¼e 2 i i ; ð29Þ Provided the initial state is correctly prepared, this expect- M ation value must be 0. ð1Þ − Δ ϕ ϕ U i tReð j;Uij iÞ H ði; jÞ¼e ; and ð30Þ

VI. SCALAR FIELDS Uð1Þ −iΔt∇2 K;scalarði; jÞ¼e : ð31Þ Here we extend our Hamiltonian construction to include scalar matter fields which transform in a D-dimensional Note that the mass and hopping gates commute, as both are ϕ†ϕ representation of G. The Hilbert space defined in Sec. III is diagonal in the fiducial basis. The same-site i i terms

034518-7 LAMM, LAWRENCE, and YAMAUCHI PHYS. REV. D 100, 034518 (2019) generated by the hopping term of the Hamiltonian have above for scalar fields. Also as before, the representation of been absorbed into the mass gate, assuming a square lattice fermionic fields on the quantum computer will not match of dimension d. These gates, together with the pure-gauge that on a classical computer, where they are typically Uð1Þ integrated out. In the implementation of spinor fields on a V;K, are combined to form quantum computer, while the register is trivial to specify Y Y Y ð1Þ ð1Þ ð1Þ and exact, a significant technical difference emerges: the U ¼ U ðpÞ U ðiÞ U ði; jÞð32Þ V V;gauge M H fundamental operators provided on most digital quantum p i i;j computers are bosonic, meaning that operators acting on U ⨂Uð1Þ ⊗ ⨂Uð1Þ different qubits will commute with each other. For spinor K ¼ K;gaugeði; jÞ K;scalarðiÞ: ð33Þ hiji i fields, we want fermionic operators, which anticommute at different physical sites. We will translate commuting With these, the full time-evolution operator for a single operators into anticommuting operators via the Jordan- time step is Wigner transformation, paying the price that local fermionic operators are generated by nonlocal bosonic

UðΔtÞ¼UVUK ð34Þ operators. More efficient, albeit complex, schemes, relating local fermionic operators to more local bosonic operators A single time step, therefore, consists of five steps on the have been devised [14–16]. quantum computer: The Hilbert space of a gauge theory with d-component ⊗L ⊗dN (1) For each plaquette, compute the plaquette product fermions can be written H ¼ðHGÞ ⊗ ðHFÞ ,where U Δ 2 U H 0 1 and apply Trð t=g aÞ. This implements V;gauge. there are L links, N sites, and F ¼ spanfj i; j ig is the 3 Uð1Þ Hilbert space of a single fermion on a site. Note that, as a (2) For each link, apply the scalar hopping gate H . 1 consequence of the exclusion principle, H can be put on the (3) For each site, apply the scalar mass gate Uð Þ. F M quantum computer without approximation or truncation. Uð1Þ (4) For each link, apply the kinetic gate K;gauge. Writing the Hamiltonian using operators that obey the Uð1Þ (5) For each site, apply the scalar kinetic gate K;scalar. standard fermionic anticommutation relations is easily This is the general algorithm for time-evolving gauge fields accomplished in the absence of gauge fields via the in the presence of matter. Before moving on to discuss Jordan-Wigner transformations [13]: each fermionic oper- scalar field correlators, we note that the transfer matrix ator pair is given an index from 1 dN, and the ith discussion of Sec. IV could be repeated here, including the † operators are constructed from bosonic operators ai and ai : scalar fields, to connect the evolution of the quantum computer to a classical, Euclidean spacetime path integral. ðzÞ ðzÞ † ðzÞ ðzÞ † ψ ¼ σ1 σ −1a and ψ ¼ σ1 σ −1a : ð35Þ Thus, renormalization calculations again may be performed i i i i i i classically. In the presence of gauge fields, we must ensure that the Steps 1 and 4 above are the same as for a pure-gauge U fermionic operators obey the appropriate gauge transfor- simulation. Step 2 requires two calls to ×;f, two calls to mation law. To be concrete, consider when the fermions U U −1, and one call to ð·;·Þ. Step 3 is a register-implemen- transform in the fundamental of SUð3Þ. Then at each site i, tation dependent diagonal phase gate. Step 5, analogous to for each spinor index α, there are three annihilation and Uð1Þ 2 UC the implementation of K;gauge, entails N calls to F and creation operators, which must transform under gauge N diagonal phase gates per site, where N is the dimension transformations as of the scalar’s representation. 3 The procedure for computing plaquette correlators is the X ψ a ↦ ψ b same as in Section V. αi gab αi ð36Þ 1 The construction of Sec. V may be generalized to obtain the b¼ gauge-invariant two-point function of a fermion, connected where a, b are color indices. There are now three times as ψ¯ ψ 0 by a Wilson line: h ðy; tÞW ðx; Þi.However,forageneral many fermion operators, as for each site and each spinor, Wilson line, this construction requires one derivative to be there must be three different fermionic degrees of freedom. taken at every time slice. As the derivatives must be taken via This may be accomplished if the bosonic operators a; a† finite differencing, this is not a plausible method for comput- from which the fermions are constructed transform in the ing these correlators, even in the far future. Constructing a fundamental of SUð3Þ. Thus each operator has a color more efficient method remains an open problem. a b index, and a ↦ gaba . Crucially, no change to the Jordan- Wigner procedure is needed since the σ-matrices are VII. FERMION FIELDS

Here we consider the inclusion of spinor matter fields, 3As a consequence of writing the fermionic space as a tensor which can be done in the same manner as was discussed product, the fermionic operators will be nonlocal.

034518-8 GENERAL METHODS FOR DIGITAL QUANTUM SIMULATION … PHYS. REV. D 100, 034518 (2019) independent of the gauge group (however the σ matrices do inherit the color index). With fermionic operators constructed, the time-evolution operator is constructed in a similar fashion to that from Sec. VI above.

Uð1Þ¼UVUK Y Y FIG. 2. The lattice geometry used for the D4 gauge simulation. 1 1 † † † † U Uð Þ ð Þ The plaquettes are given by U2U0U3U0 and U3U1U2U1. Dash where V ¼ V;gaugeðpÞ UV;spinorði; jÞ p i;j lines are used to indicate repeated links due to the periodic boundary conditions. U ⨂Uð1Þ ⊗ ⨂Uð1Þ and K ¼ K;gaugeði; jÞ K;spinorðiÞ; ð37Þ hiji i degrees of freedom, and 2 ancillary qubits. Note that, for The key difference is that the spinor kinetic and potential brevity, we have broken with the notation of previous operators are derived from the Dirac equation. The gov- sections, in referring to a link not by the source and sink erning Hamiltonian is now sites, but instead with a single direct index 0…3. Y We define a trace on D4 (not uniquely specified by the X X 2 1 2 group structure) by embedding D4 into Uð Þ, and defining β Uij π H ¼ Re Tr þ ij ∈ 2 the trace via the fundamental representation of that Lie p hiji p hiji X X group. The embedding of D4

034518-9 LAMM, LAWRENCE, and YAMAUCHI PHYS. REV. D 100, 034518 (2019) XL−1 1 −1 i XL ðxÞ ð Þ ðzÞ i H ¼ σ þ χ¯ 1σ χ þm ð−1Þ χ¯ χ 2 iðiþ1Þ 2 iþ iðiþ1Þ i i i i¼1 i¼1 ð42Þ

where the index i denote lattice sites, the sum is taken over all pairs of adjacent sites, and χ is a one-component fermionic operator. Using the Jordan-Wigner procedure, we translate from the fermionic operators χi into a set of bosonic operators σi: χ σðzÞ σðzÞ σ σ i ¼ 1 i−1 i, where i is the lowering operator for site i. In one-dimension without periodic boundary con- FIG. 3. Simulation of a D4 gauge theory with two plaquettes. The expectation value of one of the plaquettes is shown as a ditions, this results in a local bosonic Hamiltonian. function of t. The exact result is shown in black; sampled data with Δt ¼ 0.2 (Δt ¼ 0.5) are shown in red (blue). Differences XL m H − −1 iσðzÞ σðxÞ between the simulated quantum calculation are due to sampling ¼ 2 ð Þ i þ iðiþ1Þ error (estimated with error bars) and Trotterization. i¼1 1 þ ð−1ÞiσðzÞ ðσðxÞσðxÞ þ σðyÞσðyÞ Þ : ð43Þ 4 iðiþ1Þ i iþ1 i iþ1 For the same lattice gauge theory, we also demonstrate the procedure described in Sec. V for determining the There are two distinct families of bosonic operators: the σi expectation value of a Wilson loop with temporal extent t: which live at a site and correspond to fermions, and the † σ 1 hRe TrU0ðtÞU0ð0Þi. Simulated results are shown in iðiþ1Þ on the link from site i to site i þ , corresponding to Fig. (4). For the finite differencing, we perform simulations the Z2 gauge field. The Trotterization of this Hamiltonian with ðϵ1;ϵ2Þ¼ð0.1;0.0Þ;ð0.0;0.1Þ;ð0.1;0.1Þ, and the same takes four steps, corresponding exactly to the four terms in for the parameters ϵ˜1; ϵ˜2. As before, a Trotterization time Eq. (43). In this way full translational invariance (neglect- step of Δt ¼ 0.2 is used. While smaller ϵ is preferred ing the boundary) is maintained even in the approximated (smaller deformations improve the finite differencing evolution. calculation), there is a tradeoff because the resulting smaller The starting state of this Z2 simulation is obtained by finite difference requires a larger number of samples to be projecting the j0000000i state to the gauge-invariant space, collected, scaling like ϵ−2. and then applying the gauge-invariant excitation operator † † ðzÞ In order to test our technique with matter fields, we χ1χ2σ12 . The number density of the leftmost site as a simulated staggered fermions transforming in a Z2 gauge function of t is shown in Fig. 5 where the simulated results field, in 1 þ 1 dimensional spacetime. A similar model was are compared to the exact values. Since we do not begin also simulated on a quantum computer in [10]. The lattice in an eigenstate of the Hamiltonian, this is not constant. Hamiltonian for this model is The difference between the two is due to the Trotterization, and as with the D4 simulation may be made systematically

FIG. 4. Expectation value of a temporal Wilson loop in D4 FIG. 5. Simulation of Z2 gauge theory with staggered fermions gauge theory, as a function of the time extent of the loop. For each on four sites. The expectation value of the number density at the data point, a total of 2 × 105 measurements were collected. The leftmost site as a function of t. A simulated quantum calculation systematic errors (not shown) present are Trotterization and finite is shown in red. Differences with the exact result are due to differencing. sampling error (100 samples per data point) and Trotterization.

034518-10 GENERAL METHODS FOR DIGITAL QUANTUM SIMULATION … PHYS. REV. D 100, 034518 (2019) smaller by reducing the lattice spacing. This simulation The D4 register is implemented with three qubits. The uses 8 qubits: 4 fermion sites, 3 gauge links, and a single state jabci corresponds to the matrix ancillary. For this lattice, 36 CNOT gates are used per timestep, putting the simulation (slightly) out of reach of 01 a i 0 2bþc present-day processors. A time step of Δt ¼ 0.6 is used : ðA2Þ 10 0 −i for T ¼ 30.

IX. DISCUSSION Thus, the two least-significant bits specify an amount of rotation, to be followed by a flip if the most-significant bit In this work, we have given a general procedure to is 1. This establishes the isomorphism needed between CD4 construct a unitary time-evolution operator and timelike and the three-qubit Hilbert space on a quantum computer. separated correlation functions for use on an idealized It remains to describe the inversion, multiplication, trace, universal quantum computer. This construction ensures that and Fourier transform circuits. The inversion and multipli- the Trotterization never mixes the physical and unphysical cation circuits are both equivalent to classical circuits (as they states within the Hilbert space, provided that the requirements consist exclusively of controlled-not gates), and are shown of Sec. II are implemented exactly. Further, we show how in Fig. 6. The trace circuit is a three-qubit controlled phase gauge-invariant correlators (including a temporal Wilson U θ 000 −2iθ U θ 010 gate, defined by Trð Þj i¼e and Trð Þj i¼ loop) are constructed. We have demonstrated the efficacy e2iθ (with all other states picking up no phase). of the procedure by the implementation of two gauge theories Finally, the quantum Fourier transform which diagonal- on a simulated quantum computer. Additionally, we have izes the D4 momentum operator is given by discussed the classical procedure for determining the bare couplings to be used in a quantum simulation. 0 1 1 1 1 1 1 1 1 1 pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi Critically, we have defined a set of primitive constructs 8 8 8 8 8 8 8 8 B 1 1 1 1 −1 −1 −1 −1 C which may be used in the construction of gauge theory B pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi pffiffi C B 8 8 8 8 8 8 8 8 C simulation. The practical and efficient implementation of B 1 1 1 1 1 1 1 1 C B pffiffi − pffiffi pffiffi − pffiffi pffiffi − pffiffi pffiffi − pffiffi C such registers and gates is an important open question in the B 8 8 8 8 8 8 8 8 C B C literature. Finally, future work must address the issue of B 1ffiffi 1ffiffi 1ffiffi 1ffiffi 1ffiffi 1ffiffi 1ffiffi 1ffiffi C B p − p p − p − p p − p p C how the behavior of different noisy gates affects the mixing F ¼ B 8 8 8 8 8 8 8 8 C: B 1 1 1 1 C of the physical and unphysical state, whether this can be B 0 − 0 0 − 0 C B 2 2 2 2 C mitigated by special constructions or error-mitigating codes B 0 − 1 0 1 0 1 0 − 1 C B 2 2 2 2 C within our construction, and indeed whether such mitiga- B C @ 1 1 1 1 A tion is necessary. 0 2 0 − 2 0 2 0 − 2 1 0 − 1 0 − 1 0 1 0 ACKNOWLEDGMENTS 2 2 2 2 ðA3Þ H. L., S. L., and Y. Y. are supported by the U.S. Department of Energy under Contract No. DE-FG02- Rather than attempt to determine a quantum circuit by 93ER-40762. The authors thank P. Bedaque and A. hand (although general methods do exist [85–87]), we Alexandru for conversations on the Hamiltonian formu- perform a classical simulated annealing search to find a lation of gauge theories, Z. Davoudi for a quantum short circuit that implements exactly this unitary within computing seminar, J. Stryker for helpful comments on π the QISKIT gate set using Hadamard, CNOT, and -gates (T). this paper. Finally, the authors would like to thank C. 8 Bonati, M. Cardinali, G. Clemente, L. Cosmai, M. D’Elia, A. Gabbana, S. F. Schifano, R. Tripiccione, and D. Vadacchino for pointing out errors in our manuscript.

APPENDIX A: IMPLEMENTATION OF A D4-REGISTER

The group D4 of isometries of the square is treated as a subgroup of Uð2Þ generated by the matrices i 0 01 and : ðA1Þ 0 − 10 i FIG. 6. Classical-inspired quantum circuits for D4 inversion π (top) and multiplication (bottom). The least-significant qubits are The first matrix represents a 2 rotation, and the second shown on the top. For the multiplication circuit, the target register represents reflection. is in the bottom three qubits.

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positive-definite to define a Hamiltonian, a fact used in Sec. IV. Unlike prior proofs, we are concerned with positive-definiteness on H, not just HP. TK is the tensor product of single-link operators FIG. 7. Quantum circuit implementing the Fourier transform on ⊗ ð1Þ a D4 register. The least-significant qubits are on the top. Here, H TK ¼ TK ði; jÞ; ðB1Þ π hiji and T are the Hadamard gate and 8 gate, respectively.

ð1Þ and is therefore positive-definite as long as TK is. The The circuit found is shown in Fig. 7. The annealing single-link transfer matrix is defined by algorithm will be detailed in a future work. Since the size of F does not scale with volume, this method does not 1 † hg˜jTð Þjgi¼eβRe Tr½ρ ðg˜ÞρðgÞ; ðB2Þ suffer from an exponentially large physical Hilbert space. K After being diagonalized by the Fourier transform, where g, g˜ are the elements of G, ρ is a faithful repre- the kinetic gate corresponds simply to a phase gate on sentation, and β is the inverse coupling. Therefore, our the most-significant qubit, along with a phase gate on the ð1Þ state j000i: proof need only establish that TK is positive definite. Ψ ð1Þ Ψ 0 It is sufficient to show that h jTK j i > for all † iθ1χa b c 0 iθ2a FUKðθÞF jabci¼e ¼ ¼ ¼ e jabci: ðA4Þ jΨi ≠ 0 in CG. We must establish that for any function fðgÞ, θ θ β 0 8 The constants 1 and 2 are coupling-dependent. For ¼ . , X 1 thecouplingusedinthispaper,wehaveθ1 ≈ 1.263 and ˜ ˜ ð Þ f ðgÞfðgÞhgjTK jgi θ2 ≈ 0.409. ˜∈ g;g G X ˜ βRe Tr½ρ†ðg˜ÞρðgÞ 0 APPENDIX B: POSITIVE-DEFINITENESS ¼ f ðgÞfðgÞe > : ðB3Þ OF THE TRANSFER MATRIX g;g˜∈G

Here we provide a proof that the kinetic part TK of We work order-by-order in β to see that each term is the transfer matrix for a discrete gauge theory is bounded below by 0:

X X∞ Xn n X † β fðg˜ÞfðgÞeβRe Tr½ρ ðg˜ÞρðgÞ ¼ C fðg˜ÞfðgÞTr½ρ†ðg˜ÞρðgÞiTr½ρðg˜Þρ†ðgÞn−i ðB4Þ 2nn! n i g;g˜∈G n¼0 i¼0 g;g˜∈G where nCi are binomial coefficients. Each term indexed by n, i is non-negative: X X X i X n−i ˜ ρ† ˜ ρ i ρ ˜ ρ† n−i ˜ ρ† ˜ ρ ρ ˜ ρ† f ðgÞfðgÞTr½ ðgÞ ðgÞ Tr½ ðgÞ ðgÞ ¼ f ðgÞfðgÞ ðgÞab ðgÞba ðgÞcd ðgÞdc ˜∈ ˜∈ g;g G g;g G a;b c;d X X 2 f g ρ g ρ g ρ g ρ g ≥ 0 ¼ ð Þ ð Þb1a1 ð Þbiai ð Þc1d1 ð Þcn−idn−i fag;fbg;fcg;fdg g∈G ðB5Þ

… … Ψ ð1Þ Ψ 0 where faðbÞg ¼ ðaðbÞ1; ;aðbÞiÞ and fcðdÞg ¼ ðcðdÞ1; ;cðdÞn−iÞ. It remains only to show that h jTK j i¼ is impossible. For the inequality to be saturated, the following must hold for any n, i and any combination of matrix components in fag; fbg; fcg; fdg. X f g ρ g ρ g ρ g ρ g 0 B6 ð Þ ð Þb1a1 ð Þbiai ð Þc1d1 ð Þcn−idn−i ¼ ð Þ g∈G

As the representation ρ is faithful, the space of polynomials on G forms a complete basis, and thus Eq. (B6) establishes that the inner product of fðgÞ with any vector in CG vanishes, implying that fðgÞ¼0. ρ ð1Þ Therefore, for faithful representations (such as the fundamental representation), TK is positive-definite.

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