Digital-analog quantum computation

PHYSICAL REVIEW A 101, 022305 (2020) Digital-analog quantum computation Adrian Parra-Rodriguez ,1Pavel Lougovski,2Lucas Lamata,1,3Enrique Solano,1,4,5and Mikel Sanz1,* 1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 2Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, E-41080 Sevilla, Spain 4IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain 5International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Physics Department, Shanghai University, 200444 Shanghai, China (Received 7 January 2019; accepted 14 January 2020; published 6 February 2020) Digital quantum computing paradigm offers highly desirable features such as universality, scalability, and quantum error correction. However, physical resource requirements to implement useful error-corrected quantum algorithms are prohibitive in the current era of NISQ devices. As an alternative path to performing universal quantum computation, within the NISQ era limitations, we propose to merge digital single-qubit operations with analog multiqubit entangling blocks in an approach we call digital-analog quantum computing (DAQC). Along these lines, although the techniques may be extended to any resource, we propose to use unitaries generated by the ubiquitous Ising Hamiltonian for the analog entangling block and we prove its universal character. We construct explicit DAQC protocols for efficient simulations of arbitrary inhomogeneous Ising, two-body, and M-body spin Hamiltonian dynamics by means of single-qubit gates and a fixed homogeneous Ising Hamiltonian. Additionally, we compare a sequential approach where the interactions are switched on and off (stepwise DAQC) with an always-on multiqubit interaction interspersed by fast single-qubit pulses (banged DAQC). Finally, we perform numerical tests comparing purely digital schemes with DAQC protocols, showing a remarkably better performance of the latter. The proposed DAQC approach combines the robustness of analog quantum computing with the flexibility of digital methods. DOI: 10.1103/PhysRevA.101.022305 I. INTRODUCTION Quantum information science has flourished in recent years as a new paradigm promising to outperform certain classical tasks such as computation, simulations, and communications among others. More specifically, quantum computers (QC) [1–3] are believed to be faster than their classical counterparts in factoring prime numbers [4], or searching databases [5]. From a theoretical computer science point of view, a universal quantum computer can run any algorithm processable by a quantum Turing machine [6], in other words, it can implement an arbitrary unitary evolution. In physical terms, an ideal universal quantum computer can implement an arbitrary Hamiltonian acting on an infinite Hilbert space. However, a realistic quantum computer comprises a finite number of resources, and hence can only perform unitary operations within certain constraints. The complexity of classical simulations of many-body quantum systems typically grows exponentially with the dimension of the system. This was first recognized by Richard Feynman in a seminal paper from 1982 [3], in which he proposed as an efficient solution the simulation of these problems employing another fully controllable quantum system with a similar encoded dynamics [7]. This was the origin of what is now called analog quantum computing (AQC). Later, with the *[email protected] emergence of the first quantum algorithms offering speedups over their classical counterparts [4,5], quantum computing became the most promising application of quantum mechanics and information. The discovery of universal sets of quantum gates and quantum error correction [8–10] provided a clear road map towards a scalable QC mimicking the history of classical computers. This approach is called digital quantum computation (DQC) [11,12], based on an algorithmic sequence of one-qubit and two-qubit gates [6]. However, a practical universal digital quantum computer is considered to be so resource consuming that the implementation of useful applications may be shifted decades into the future. It has been suggested that quantum control techniques [13,14] can be used to improve the fidelity of the quantum gates. However, they require classical optimization algorithms, which are by themselves hard problems to solve [15,16]. Furthermore, they do not consider all possible errors in the system [17] yet. In this context, the ability of quantum systems to solve problems beyond the reach of any current classical computer is known as quantum supremacy [18,19]. This has been proposed for various artisanal problems, such as boson sampling [20,21] and quantum speckle [22,23], and recently experimentally achieved in Ref. [24]. The simplest approach to perform quantum simulations is the use of a controllable quantum system whose effective dynamics is similar to the one of the desired model. Such singlepurpose devices are called analog quantum simulators (AQS). In this sense, there are many proposals ranging from the 2469-9926/2020/101(2)/022305(12) 022305-1 ©2020 American Physical Society ADRIAN PARRA-RODRIGUEZ et al. PHYSICAL REVIEW A 101, 022305 (2020) quantum Rabi model [25–28] and Casimir physics [29–31], to Jaynes-Cummings and Rabi lattices [32–35]. In 1996 [36], Lloyd proved with the help of the Suzuki-Trotter decomposition [37] that digital quantum simulators (DQS), whose evolutions are decomposed in a universal set of quantum gates acting on a register of qubits, can simulate efficiently any quantum system. Innovative experiments were performed in those lines in superconducting circuits [38–41] and ion traps [42,43]. An alternative paradigm to do quantum simulations, called digital-analog quantum simulations (DAQS), makes use of both digital and analog blocks in order to exploit their intrinsic versatility and complexity [44–51]. Here, we propose the concept of digital-analog quantum computation (DAQC) as a method to reduce the number of gates needed to perform quantum algorithms, in the current spirit of near-term intermediate-scale quantum computation (NISQ). The DAQC paradigm requires the implementation of entangling multipartite evolutions and fast single-qubit gates. Earlier efforts have already proven the universality of such schemes [52–56], extending the ideas triggered by Lloyd [36]. In our DAQC approach, we propose a sequence of entangling time slices, called analog blocks, and fast singlequbit rotations (SQRs), which belong to our class of digital steps. Moreover, we develop constructive protocols for simulating arbitrary inhomogeneous two-body and M-body spin Hamiltonians. An important source of errors when performing quantum algorithms appears by turning on and off interacting Hamiltonians. In order to mitigate these errors, we introduce the concept of banged digital-analog quantum computing (bDAQC)toimprovethestepwiseDAQC(sDAQC).Inthis manner, the proposed algorithms do not require one to halt and activate the analog blocks, while the only required pulses are single-qubit gates. Furthermore, we perform numerical studies of ideal and realistic multiqubit models supporting our theoretical results. II. ANALOG, DIGITAL, AND DIGITAL-ANALOG QUANTUM COMPUTING The concepts of digital and analog quantum computing are broadly used in an informal manner but, to the best of our knowledge, they lack a broadly accepted formal definition, allowing for a classification of algorithms. As the main aim of this manuscript is to describe the DAQC paradigm, proper definitions are of crucial importance. Let us start by introducing the concepts of quantum gates, digital and analog blocks on N-qubit systems. Definition (Quantum gate). A quantum gate is a fixed unitary evolution Un,Un∈B((C2)⊗n). Definition (Digital block). Ak-parametric continuous family of unitary operators Un( φ), with φl∈Il(R) and 1⩽l⩽k, comprises a digital block if it is equivalent to a fixed unitary evolution Unup to a set of local rotations Wi( φ), i.e., Un( φ)=(n iWi( φ))Un. Note that both parameter-fixed entangling quantum gates and single qubit rotations with arbitrary angle are digital blocks. Definition (Analog block). We call analog block akparameter-dependent entangling unitary evolutions Vn( φ) with a semigroup structure Vn( φ)=Vn( φ1)Vn( φ2);  φ= φ1+  φ2.Fork=0, it obviously becomes a quantum gate. Under these definitions, for instance, Un=eiπ 4σ1 zσ2 zis a quantum gate, both Un(φ)=(eiφ1σ1 z⊗eiφ2σ2 z)eiπ 4σi zσj zand Wi(φ)=eiφσ1 zare digital blocks, and Vn(φ)=eiφσi zσj zis an analog block. Let us remark that, unlike the analog block, the entanglement generated by the digital block is the same for all values of the parameters. We call a quantum protocol a digital quantum algorithm, when it makes use only of digital blocks (usually with a small number of quantum gates), whereas an analog quantum algorithm consists of a single analog block for different values of the parameters. Naturally, a digitalanalog protocol contains both digital and analog blocks. In this paper, we will constrain the digital blocks to arbitrary single qubit rotations, such that our total evolution can be written as j(Uj( φj)[iW(j) i( αji)]). For the sake of clarity, Reference [51] and references thereof which use the terminology digital-analog quantum simulations are, based on the aforementioned definitions, purely digital protocols employing multiqubit fixed-phase gates, e.g., Mölmer-Sörensen gates. III. DIGITAL-ANALOG QUANTUM COMPUTATION As it was elegantly proven in Ref. [52], almost any twobody Hamiltonian is universal. In this article, we will focus on two paradigmatic models, ubiquitous in quantum platforms, to exemplify the DAQC paradigm, but it could be straightforwardly extended to other specific situations. We will employ as analog blocks either a homogeneous nearest-neighbor or a homogeneous all-to-all two-body Ising Hamiltonian. Using one of these evolutions, together with single-qubit rotations, we constructively prove their universality, i.e., any unitary can be arbitrarily close simulated employing these resources. We show the protocols, sometimes optimal, to generate increasingly complex families of Hamiltonians with relevance in several fields. The road map towards this complexity comprises the protocols to generate (i) arbitrary inhomogeneous twobody Ising Hamiltonians, (ii) an inhomogeneous two-body Hamiltonian, for which the noncommutation of the terms requires one to superimpose Trotterization to the algorithm, and (iii) an arbitrary M-body Hamiltonian, with polynomial number of resources in the number of spins for fixed M. Finally, we perform several numerical simulations to check the advantages in fidelity and time of DAQC with respect to DQC. Our DAQC algorithms consider three basic ingredients that were also required in Ref. [52]. First, we need to exactly evolve for a time twith a Hamiltonian H, such that combined with local rotations we have an evolution with H=RHR†. We will later consider the effect of limited precision in the physical implementation of engineered times. Secondly, we must do a Trotter decomposition e−itH =(e−iH1t/nTe−iH2t/nT)nT+O(t2/nT) in order to simulate Hamiltonians H=H1+H2with noncommuting terms, i.e., [H1,H2]= 0. The error introduced through this approximation is of second order in the time error t. We consider further improved decompositions like the symmetrized Trotter 022305-2 DIGITAL-ANALOG QUANTUM COMPUTATION PHYSICAL REVIEW A 101, 022305 (2020) decomposition, which reduces the error to O(t)3. Finally, given that we can evolve with Hamiltonian Hwe can control the time parameter to evolve with λH, where λ>0isa continuous positive parameter. A. Ising model Let us start illustrating the problem with the implementation of the inhomogeneous all-to-all (ATA) two-body Ising model described with Hamiltonian HZZ =N j<kgijσj zσk z.We use as the elementary analog block the unitary evolution Uzz(t)=eiHzztof the homogeneous ATA two-body Ising model whose Hamiltonian reads Hzz =g N  j<k σj zσk z,(1) with independent time parameter tand a fixed coupling strength g.Weset¯h=1 in the whole article. The only digital blocks employed are single-qubit rotations around the xaxis with the continuous range of phases θ∈[0,2π]. The protocol can be straightforwardly modified to use as an analog block a general fixed inhomogeneous evolution with Hamiltonian ¯ HZZ =N j<k¯gjkσj zσk z. Such Hamiltonians naturally appear in various quantum platforms, e.g., the coupling parameters scale typically as ¯gjk ∝1/|j−k|α, with 0 <α<3inion- trap setups [57]. In effective Hamiltonian models, in which qubits are coupled through linear multimode systems [58,59], more complex coupling distributions naturally emerge and could also be tuned or designed [60,61]. A different decomposition into most entangling Hamiltonians and single-qubit gates has been proven to be a universal quantum machine by Dodd et al. in Ref. [52]. The Ising Hamiltonian is an example of a universal Hamiltonian under the digital-analog paradigm, since it allows us to construct a universal ZZ gate [62] between any two arbitrary qubits, see Appendix B. The target Hamiltonian evolves according to the unitary UZZ =eitFHZZ , where HZZ =j<kgjkσj zσk zand tFthe final time. The task consists of finding a mapping between gjktF and gtnm by slicing the homogeneous time evolution into (at most) N(N−1)/2 analog blocks of different time lengths tnm, sandwiched by local rotations σn xσm x, HZZ = N  j<k gjkσj zσk z=g tF N  j<k N  n<m tnmσn xσm xσj zσk zσn xσm x =g tF N  j<k N  n<m tnm(−1)δnj+δnk +δmj+δmk σj zσk z,(2) as depicted in Fig. 1. Engineering the Hamiltonian reduces the problem of finding the time durations tnm of the analog block evolutions to a matrix-inversion problem. By vectorizing the pairs of indices (n,m)→α=N(n−1) −n(n+1)/2+m and (j,k)→β=N(j−1) −j(j+1)/2+kand write the signs matrix, Mαβ =(−1)δnj+δnk +δmj+δmk ,(3) X1 X2 X1 X2 X1 X3 X1 X3 t1t2 UZZ =¯ HZZ ¯ HZZ FIG. 1. Algorithm to simulate the general inhomogeneous Ising model from a fixed one. Each time step evolution tαis sandwiched by a pair of single qubit gates (Xi≡σi x) applied to qubits (n,m), with α(n,m). Optimal sequences of SQRs can be used to simplify the number of pulses. where the inverted relations are j=1+[β N] and k=β+ 1 2(1 +[β N])(2 +[β N]) −N[β N] and n=1+[α N], m=α+ 1 2(1 +[α N])(2 +[α N]) −N[α N]. The matrix Mαβ has three degenerate eigenvalues, namely, λ1=N(N−9)/2+8, λ2=2(4 −N), and λ3=4 with degeneracies 1, N−1, and N(N−1)/2−N, respectively. Consequently, it is a nonsingular matrix ∀N∈Z−{4}. The corner case N=4 requires the use of a slightly modified set of SQRs, e.g., single σxrotations per site is sufficient for a NN Hamiltonian. The total unitary evolution is UZZ(tF)= eitFβHβ ZZ where Hβ ZZ =gβσj(β) zσk(β) z,(4) and gβ=tαMαβ (g/tF). An example of elements tαMαβ is shown in Table Ifor a three-qubit case. Solving the linear problem, we find the times tα=M−1 αβ gβ(tF/g) required for each analog block to evolve interleaved by the pairs of single-qubit rotations, as shown in Fig. 1. As the matrix Mαβ is invertible, the Rouché-Frobenius theorem ensures that the solution is unique. It is noteworthy to notice that some of the times tαmight be negative. This means that those analog evolutions should be done with inverted coupling signs. However, there is actually a simple method to address this problem consisting in evolving with times ˜ tα=tα+|tmin|(see Appendix Afor further details). To sum up, in this section, we have designed an optimal DAQC protocol to construct an arbitrary inhomogeneous ATA two-body Ising Hamiltonian using as a resource an homogeneous ATA two-body Ising Hamiltonian and xrotations. This protocol, which is quadratic in the total number of qubits, is optimal for a generic Hamiltonian, since it makes use of the same number of resources as degrees of freedom TABLE I. Matrix elements of tαMαβ coupling sign constants for the N=3 qubits case. By performing three pairs of singlequbit rotations sandwiching the interaction base Hamiltonian (1), we derive independent equations for all coupling terms gjk in the simulated Hamiltonian (2). ``````````` α→(n,m) β→(j,k)1→σ1 zσ2 z2→σ1 zσ3 z3→σ2 zσ3 z SQRs 1 →σ1 xσ2 xt1−t1−t1 SQRs 2 →σ1 xσ3 x−t2t2−t2 SQRs 3 →σ2 xσ3 x−t3−t3t3 022305-3 ADRIAN PARRA-RODRIGUEZ et al. PHYSICAL REVIEW A 101, 022305 (2020) ( R θ (1) ( Rθ(4) UXZ ≈ R θ (1) Rθ(4) n T HZZ(g4) HZZ(g1) FIG. 2. Algorithm to simulate the inhomogeneous XZ model from an Ising model. The combination of four different general Ising evolutions and rotations handles the minimum degrees of freedom for each term gμν jk .Rθ(s)rotations can be combined with inner σn xσm x rotations required to implement the inhomogeneous ZZ Ising model into unique blocks of SQRs. the Hamiltonian has. For the case of NN Hamiltonian, the protocol is even simpler, and requires only N−1 SQRs, i.e., one per site. These two protocols show, as a by-product, that the Ising model is universal, since it can be used to construct a ZZ gate between two arbitrary qubits [63]. B. XZ model As the Ising model is a universal Hamiltonian within the DAQC paradigm, we can simulate any other Hamiltonian evolution. In order to do it in a systematic manner, we will make use of a Trotter decomposition on top of the previous algorithm [36]. We illustrate this idea by explicitly constructing an inhomogeneous general two-body XZ Hamiltonian, which is depicted in Fig. 2. The unitary that we want to simulate is UXZ =eitFHXZ , with HXZ =N j<kμ,ν={x,z}gμν jk σj μσk ν. We first perform a Trotter decomposition UXZ ≈(eitF nTHXZ )nT, with nTthe number of Trotter steps, HXZ = N  j<k μ,ν gμν jk σj μσk ν= N  j<k μ,ν,s g(s) jk α(μ,s) jα(ν,s) kσj μσk ν = N  j<k 4  s=1 g(s) jk cos θ(s) jσj z+sin θ(s) jσj x ×cos θ(s) kσk z+sin θ(s) kσk x,(5) with srunning from {1,...,4}, which are the number of combinations of the two types of couplings (for instance, for the XYZ Hamiltonian, it would run from 1 to 9). The implicit definition of parameters is α(x,s) j=sin(θ(s) j) and α(z,s) j=cos(θ(s) j). We decompose the pairs of operators with their coefficients in homogeneous σj zσk zoperators with local rotations Rθ(s) j= (cos(θ(s) j/2)σj z+sin(θ(s) j/2)σj x), Rθ(s) jσj zRθ(s) j=cos θ(s) jσj z+sin θ(s) jσj x, for all pairs of qubits. This rotation is produced by the Hamiltonian Hθ(s) j=(π/2)(−1+cos(θ(s) j)σj z+sin(θ(s) j)σj x). We have to perform the SQRs in all qubits Rθ(s)=⊗ N wRθ(s) w. The total Hamiltonian is reconstructed as HXZ = 4  s=1 Rθ(s)g(s) jk σj zσk zR† θ(s).(6) This equality is only valid in the Hamiltonian and not in the total unitary evolution. In each Trotter time step tT=(tF/nT) FIG. 3. Algorithm to implement the M-body evolution. Generalized rotations R(l)=⊗ N jr(l) jσj x+s(l) jσj y+t(l) jσj zsandwich evolutions of H(l) 0=4 k=1(H(k) 1)(l)+(H(k) 2)(l);seeEqs.(9)and(10)and Appendix Dfor an example with M=4. we must evolve according to H(s) ZZ =g(s) jk σj zσk zbetween a pair of sets of SQRs Rθ(s), UXZ ≈4  s=1 Rθ(s)eitT nT(HZZ )(s)Rθ(s)nT .(7) The problem is reduced to find a set of phases θ(s) wsuch that the system of equations gμν jk =g(s) jk α(μ,s) jα(ν,s) khas independent solutions. A specific set of phases which adequately plays this role is θ(s) w=sπw 2(w+1), with distance between two nearest-neighbor qubit phases scaling polynomially d(s,w)= sπ 2(w2+3w+2). See Appendix Cfor further details. To sum up, in this section, we have proposed a protocol to construct an arbitrary two-body Hamiltonian by exploiting the freedom in the phases of the SQRs and the Trotterization technique. This protocol requires 4N(N−1) (4(N−1)) analog blocks per Trotter step for the ATA (NN) XZ model in total, which is optimal for a generic two-body Hamiltonian in terms of the free coefficients to fix. Similarly, it can be proven that a trivial extension to the general two-body XYZ model requires more angles per rotation summing up to 9N(N−1) (9(N−1)) blocks per Trotter step for the ATA (NN) model. C. M-body Hamiltonian With similar techniques, we can systematically construct the evolution of a completely general Hamiltonian with up to M-body interactions (see Fig. 3). For the sake of clarity, we sketch here the sequence of steps to simulate an arbitrary four-body NN Hamiltonian, explaining how to generalize it to M-body Hamiltonians in the end of the work. We have chosen M=4 to illustrate the protocol, since it is a case of special interest in quantum chemistry and nuclear physics, H4b= jχη gχη (2,j)σj χσj+1 η+ jχηγ gχηγ (3,j)σj χσj+1 ησj+2 γ + jχηγρ gχηγρ (4,j)σj χσj+1 ησj+2 γσj+3 ρ,(8) where the indices run over {χ,η,γ,ρ}∈{x,y,z}and j∈ {1, ..., N}. In the first stage, an inhomogeneous ZZ-Ising Hamiltonian is sandwiched between rotated XX-Ising evolutions and their conjugate transposes, H(k) 1=e−iO1(k) XX H1(k) ZZ eiO1(k) XX ,(9) H(k) 2=e−iO2(k) XX H2(k) ZZ eiO2(k) XX ,(10) with O1(k) XX =(k) 1σ1 xσ2 x+(k) 3σ3 xσ4 x+(k) 5σ5 xσ6 x+... and its translationally shifted O2(k) XX =(k) 2σ2 xσ3 x+(k) 4σ4 xσ5 x+ 022305-4 DIGITAL-ANALOG QUANTUM COMPUTATION PHYSICAL REVIEW A 101, 022305 (2020) (k) 6σ6 xσ7 x+... are built from evolutions of ZZ-Ising models rotated with single-qubit gates in all qubits R=⊗ j(cos(π/4)σj z+sin(π/4)σj x). We would like to remark that operators O1(k) XX and O2(k) XX contain interactions separated by the interaction length L=M/2=2, e.g., O1(k) XX has a term σ1 xσ2 x, but not σ2 xσ3 x. In order to simulate, for example, a three-body (five-body) Hamiltonian, we would need a different decomposition with 3 (5) translationally invariant sets of blocks (see further details in Appendix D). It is straightforward to see that Hamiltonian H(k) 1 contains all two-body and three-body terms with different supports but not all four-body terms, i.e., H(k) 1=h12 +h23 +... +hN−1,N+h123 +h234 +... + hN−2,N−1,N+h1234 +h3456 +h5678 +..., where hij... is an operator acting nontrivially on qubits {i,j, ...}. On the other hand, H(k) 2contains (again) all two-body and three-body terms, as well as the complementary four-body terms H(k) 2=... +h2345 +h4567 +h6789 +.... The coefficients of operators hij... are coupled together. For the simulation of an arbitrary four-body NN Hamiltonian with the requested support, it is sufficient to sum four of each of the blocks H0=4 k=1H(k) 1+H(k) 2to disentangle the parameters, generating at least one term operating in each support. Finally, to create all XYZ operators, we need to concatenate 3M= 81 H0blocks interleaved by single-qubit rotations with the estructure R(l)=⊗ N jr(l) jσj x+s(l) jσj y+t(l) jσj z, i.e., H4b= l R(l)H(l) 0R(l),(11) with (r(l) j,s(l) j,t(l) j) fulfilling the constraint |r(l) j|2+|s(l) j|2+ |t(l) j|2=1. The aforementioned construction works for one Trotter step, so it must be repeated nTtimes to approximate the evolution as U4b=e−iH4bt≈(e−iH4bt/nT)nT. The total number of analog blocks, engineered time slices, for the most general simulation with up to M-body interactions and N number of qubits is a(M)N+b(M) with a(M)=9 4(3M−1−3),(12) b(M)=3M−1 23 2−M.(13) In other words, it grows linearly with the number of qubits and exponentially with the number of body interactions. For the four-body system here described, the total number is 117N−306. A more detailed explanation of the algorithm for simulating NN Hamiltonians with general M-body interactions can be found in Appendix D. IV. STEPWISE AND BANGED DAQC An important source of errors in realistic quantum algorithms comes from turning on and off multiqubit quantum gates. In the interest of reducing its effect, we introduce the concept of banged digital-analog quantum computing (bDAQC) as a different way to perform quantum algorithms, in opposition to the stepwise digital-analog quantum computing (sDAQC) previously introduced. FIG. 4. Comparison between a sDAQC and bDAQC protocols. (a) sDAQC digital evolutions with Hamiltonian HRk, and analog block ones evolving with Hamiltonians HIare well separated in time under the sudden approximation. (b) bDAQC: the adiabatic evolution HIis on during the whole time and fast rotations are added (HRk+HI), also under the sudden approximation. The term bang-bang has been routinely used in classical control theory [64,65] and was first introduced in the field of quantum physics as a tool for dynamically decoupling [66,67] a controlled quantum system from its environment. Here, however, we use the term banged to express the fact that the analog evolution is not switched off while the fast SQRs pulses are being turned on. Naturally, the bDAQC introduces an additional digital error in contrast with the sDAQC that can be, in the best case, of third order in time. We argue that this error competes with the experimental error produced by the switching of multiqubit gates and depending on the cases might prove less harmful to the algorithm. A. Stepwise DAQC In a typical sDAQC scenario, a total unitary evolution is built interleaving an evolution of a fixed entangling Hamiltonian HIwith sets of SQRs as UT=...UR2e−iHIt2UR1e−iHIt1UR0,(14) with URna general digital rotation operator acting on any subset of the Hilbert space (e.g., in two qubits); see Fig. 4(a). The sudden approximation states that we can implement such evolution U(t)≈e−itFH(t)with a time-dependent Hamiltonian, H(t)= n=0 HRnt(t−[Tn+nt]) +HITn+1(t−[Tn+(n+1)t]),(15) where HRngives rise to the digital unitary evolution URn= e−iHRnt. We have defined Tn=n r=0tr, assuming t0=0, and the rectangular window function, T(t−ts)=θ(t−ts)−θ(t−(ts+T)),(16) with θ(t)=1fort⩾0 and 0 otherwise. B. Banged DAQC The aforementioned stepwise protocol assumes that we can turn on and off the Hamiltonians HIand HRninfinitely 022305-5 ADRIAN PARRA-RODRIGUEZ et al. PHYSICAL REVIEW A 101, 022305 (2020) fast, something obviously unphysical. The bDAQC protocol consists of implementing the same evolution without turning off the background Hamiltonian HIand performing short (and intense) pulses to implement the single-qubit rotations. Exploiting the symmetrized exponential decomposition for all the SQRs blocks, the ideal evolution of the system in Eq. (14) can be performed without turning off the entangling Hamiltonian, H(t)=HI+ N−1  n HRnt(t−[Tn−t/2]) +HRNt(t−[TN−t]),(17) where again Tn=n r=0tr; see Fig. 4(b). C. Error estimation The additional error per step that we are introducing can be estimated with the aid of a Schatten norm as the difference between evolving the system with the sDAQC protocol with respect to the bDAQC, en= ||1−e−iHIt/2e−iHRnte−iHIt/2ei(HI+HRn)t|| =(t)3 4||[[HI,HRn],HI+2HRn]|| + O((t)4),(18) where we have made use of the Zassenhaus formula. The first and last blocks at the boundaries introduce second-order errors e0,N=O((t)2), as the evolutions cannot be symmetrized. The total digital error of the banged protocol is the sum E=nen=Aen, with A∝O(N)orO(N2)given that the total number of analog blocks increases polynomially (linearly or quadratically) with the total number of qubits N for NN or ATA coupling configurations, respectively. V. EXAMPLE: XZ MODEL Let us exemplify this new paradigm of protocols with a numerical simulation of the noncommuting ATA XZ model previously described in Sec. III B. We first compare the exact performance of a purely digital protocol (DQC) with both the sDAQC and bDAQC. We decompose at each Trotter step, i.e., U(tF)≈U(tT)nT,tT=tF/nT, the evolution of the terms in the ATA DQC protocol, eiϕμν jk σj μσk ν=eiπ 4σj yeiπ 4σj μσk νeiϕμν jk σj yeiπ 4σj μσk νe−iπ 4σj y,(19) where ϕμν jk =tTgμν jk into fixed π/4 two-qubit gates. Such evolution might be implemented directly for qubits with strong coupling, typically neighbors. However, such evolution could be decomposed into NN protocols, e.g., with swap gates, protocols that typically improve the performance. Another option for complex systems is the use genetic algorithms to optimize this decomposition [68]. We have used two different fixed entangling ATA ZZ Hamiltonians ¯ HZZ =j<k¯gjkσj zσk zwith (a) ¯gjk =J/|j− k|5/2and (b) ¯gjk =Je−(|j−k|−1)2parameter couplings to simulate Hamiltonian (5) with gμν jk =J/|j−k|1/2couplings where ∀μ, ν ∈{x,z}and ∀j<kand j,k∈{1−5}; see Fig. 5.In Fig. 6, it is plotted the fidelity as a function of Trotter steps between exact and computed states F=|e|c|2of the g13 g14 g15 g12 FIG. 5. Five-qubit spin system. We have simulated the XZ model with coupling parameters of the original system Hamiltonian with polynomial ¯ga jk =J/|j−k|5/2(squares), and exponential ¯gb jk = Je−(|j−k|−1)2(triangles) decay, corresponding to the simulations in Fig. 6. On the other hand, the couplings of the simulated one are gμν jk =J/|j−k|1/2(circles); ∀j<kand j,k∈{1−5}and ∀μ, ν ∈ {x,z}. exact evolution of a five-qubit XZ model for a final time tF=1/J=2, where the initial state has an excitation in the 3 qubit, |(0) = |↓↓↑↓↓. It is clear that fidelities above 90% can be achieved even for the bDAQC protocol taking into consideration different finite times tfor performing the SQRs block where the HI=¯ HZZ is still on, and using two different base entangling Hamiltonians. A. Computational times The times t(s) α=tTM−1 αβ (¯g(s) ββ)−1gβof the analog blocks required to perform the four ZZ-DAQC protocols (both stepwise or banged) might be negative. Here, ¯g(s) ββ is defined as the diagonal matrix with elements ¯g(s) jk in the same β(j,k) ordering. In such case, there is a simple way around the problem of evolving with negative times (or inverted coupling signs) by realizing that the constant vector is an eigenvector of the Mαβ matrix; see demonstration in Appendix A. In Fig. 7(a), we show the times of the two-qubit gates in the DQC protocol in a Trotter step by directly using the ATA ¯ HZZ Hamiltonian or the most efficient protocol that decomposes each σj zσk zinteraction into NN fixed π/4 gates, i.e., the gate decomposition USWAP(j,j+1) =eiπ/4σj xσj+1 xeiπ/4σj yσj+1 y. In virtue of simplicity, we have omitted the times required for performing SQRs to transform σj zσj+1 zgates into σj xσj+1 x or σj yσj+1 y. In opposition, Fig. 7(b) shows the times of the analog blocks in a DAQC protocol, clearly smaller. Both Figs. 7(a) and 7(b) use the same parameters of the simulation in Fig. 6(a). The total sum of times in a Trotter step of the analog blocks (DAQC) or π/4 two-qubit gates (optimized DQC) for the same simulation parameters varying the number of qubits is sketched in Fig. 8. It becomes clear that the the loss of coherence and population is going to affect faster the evolutions decomposed in purely digital gates. B. Experimental error simulation We have performed a stochastic simulation of presumably leading order dephasing errors to compare the DQC and DAQC protocols, using the same parameters of Fig. 6(a). Results are depicted in Fig. 9. In the DQC protocol, we have included two kinds of errors in all two-qubit blocks, (i) a 022305-6 DIGITAL-ANALOG QUANTUM COMPUTATION PHYSICAL REVIEW A 101, 022305 (2020) FIG. 6. Exact Trotter evolution: sDAQC vs bDAQC vs DQC. Fidelity F=|e|c|2between exact and ideal DQC, sDAQC, and bDAQC protocols for an evolution of a five-qubit XZ model as a function of Trotter steps for a final time tF=1/J=2. The couplings of the original system Hamiltonian are (a) polynomially and (b) exponentially decaying; see Fig. 5. The initial state is |(0) = |↓↓↑↓↓. The bDAQC protocol considers different finite times tfor performing the SQRs blocks where the entangling Hamiltonian HI=¯ HZZ is still on. Gaussian phase noise with deviation ξD=N(0,σ D) added to the π/4 phases and (ii) single qubit operators simulating a uniform magnetic field noise Bγ=U(−rUt/2,rUt/2). Here, N(x,y) refers to a Gaussian noise distribution with mean xand deviation yand U(a,b) refers to a uniform noise distribution with range boundaries (a,b). The deviation ratio σD=0.009, compatible with a two-qubit gate fidelity greater than 99%. On the other hand, rU=0.002, a much slower random axis phase noise. In Fig. 9we have chosen a t= 1/500J, corresponding with the orange line in Fig. 6of the bDAQC. For example, an ideal gate transforms into eiπ 4σj ασk β→eiπ 4(1+ξD)σj ασk β+j,γ =x,y,zBj γσj γ.(20) The bDAQC errors include (i) Gaussian coherent noise in analog block times t(s) α→t(s) α+δb, where δb=N(0,rbt) plus single-qubit gate errors modeled as those in the DQC. We have chosen a deviation ratio rb=0.9ofthettime required to perform a SQR. The total evolution of an analog block in FIG. 7. Times in a Trotter step to perform the DQC and DAQC protocols in terms of the simulated two-body interaction term index βand the analog block index α, respectively, for the simulation in Fig. 6(a). (a) (Black) ATA DQC protocol improves considerably time resources when it is decomposed into NN SWAP gates (green). (b) However, both of them still perform worse than the DAQC protocol, implemented through four ZZ blocks of times t(s) α(blue points), in which the most negative times |t(s) min|have been highlighted (red points). the bDAQC becomes ei¯gjktασj zσk z→ei¯gjk(t(s) α+δb)σj zσk z+j,γ =x,y,zBj γσj γ.(21) Finally, the sDAQC analog blocks are transformed analogously to those in the bDAQC but the two-qubit Gaussian and coherent phase noise has a stronger value capturing the FIG. 8. Total times (sums of points in Fig. 7) for each Trotter step as a function of number of qubits. The (blue) DAQC protocol requires (polynomially with N) less time than the (green) DQC protocol as the number of qubits increases. This systematic protocol used is not valid for N=4 qubits. 022305-7 ADRIAN PARRA-RODRIGUEZ et al. PHYSICAL REVIEW A 101, 022305 (2020) FIG. 9. Trotter evolution with errors: sDAQC vs bDAQC vs DQC. The DQC protocol (blue) is simulated with a Gaussian phase noise with deviation ξD=N(0,σ D) added to the π/4 phases, where σD=0.009. sDAQC (black) and bDAQC (red) have Gaussian time noise in the analog blocks δs=N(0,rst)andδb=N(0,rbt), respectively, and rb=0.9=rs/2. All simulations include random axis magnetic field noise Bγ=U(−rUt/2,rUt/2), with rU= 0.002. The rest of the parameters are equal to Fig. 6(a) where t= 1/500J(orange line for bDAQC). switching on-off errors expected in an experiment, i.e., t(s) α→ t(s) α+δs, with δs=N(0,rst) and rs=2rb. Stochastic evolutions of the fidelity averaged over 1000 runs show a competition between sDAQC and bDAQC methods, both of them clearly above the DQC protocol. We emphasize here that the DQC simulation assumes that one can perform directly each of the ATA terms, i.e., a NN decomposition would perform worse in terms of dephasing than the blue line in Fig. 9as it would require an increasingly linear number of two-qubit gates. VI. QUANTUM ARCHITECTURES The DAQC paradigm proposed in this article is intended to be implemented on NISQ [19] architectures where digital computation based on quantum error correction will still be beyond reach for several years or even decades. Different physical architectures are currently being investigated to perform quantum processing tasks, of which, those based on superconducting circuits and trapped ions are leading in performance and have potentially the brightest future. As stated above, the main requirements that the physical implementation must meet is the (simultaneous) single-qubit addressing to perform random-phase rotations, and a global entangling dynamics. Superconducting qubits. One of the most successful implementations of qubits is based on superconducting circuits with Josephson junctions, nonlinear systems that play the role of artificial atoms. Different configurations of junctions, e.g., grounded (transmon qubits [69]) or in loops (flux qubits [70]), can be coupled directly, or indirectly through transmission line resonators, and effectively modeled with Ising Hamiltonians with transverse fields [71], HSQ/¯h≈ i ωi 2σi z+ ij gijσi xσj x.(22) Here ωiare the energies of each qubit system and gij are the coupling parameters. Extra single-qubit drivings can be added by either coupling the qubits (i) magnetically (in SQUID configurations) to external feed lines for σzrotations, or (ii) electrically to the transmission line resonators as effective terms of capacitive couplings for σxand σyrotations. Going to an interaction picture, one can turn on and off the local terms to apply the single-qubit rotations when required [51], to match the requirements of the above Sec. III. Trapped ions. In the Lamb-Dicke regime, a trapped-ion chain is also modeled with the Ising Hamiltonian [57]. Additional single-qubit addressing can be implemented by passing laser light through splitters and acousto-optic modulators to focus on single ions and engineer a.c.-Stark shifts [72], HIons/¯h≈ i (Bi+Wi(t))σi z+ ij Jijσi xσj x.(23) The coupling term can be engineered to have a polynomial decay Jij =Jmax/|i−j|αwith 0 <α<3. In practice, however, the parameter is typically set between 0.5<α<1.8, for experimental issues regarding heating of motional harmonic modes and an increase of decoherence [73,74]. For example, in [74], local static Biand dynamic Wi(t) transverse terms were engineered. Other experimental schemes achieve simultaneous single-qubit addressing by putting the ions in magnetic field gradients and using global microwave radiation [75–77]. VII. CONCLUSIONS We have shown how to construct digital-analog quantum algorithms based on a combination of single-qubit rotations and multiqubit entangling dynamics. With this DAQC approach, we have explicitly provided protocols to construct generic Ising, two-body, and M-body Hamiltonians. Furthermore, we have analyzed a banged approach, bDAQC, as an alternative possibility to the stepwise protocol, sDAQC. In this way, to improve the fidelity of the global quantum dynamics, one can reduce the effect of switching on/off the entangling evolution. Numerical simulations performed in an XZ model with five qubits suggest the advantage of DAQC over DQC protocols in terms of fidelity and computational time, both in ideal and realistic scenarios. This alternative paradigm paves the way for the implementation of useful quantum algorithms beyond the current state-of-the-art in the NISQ era. ACKNOWLEDGMENTS We thank I. Arrazola and J. Casanova for very fruitful and encouraging discussions. We thank Giancarlo Gatti, Ana Martin, and Takahiro Tsunoda for critical reading of the manuscript. The authors acknowledge support from Spanish Government PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE) and Basque Government IT986-16. The authors also acknowledge support from the projects QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on Quantum 022305-8 DIGITAL-ANALOG QUANTUM COMPUTATION PHYSICAL REVIEW A 101, 022305 (2020) Technologies, as well as EU FET Open Grant Quromorphic (828826). This material is also based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advance Scientific Computing Research (ASCR), under field work Proposal No. ERKJ333. APPENDIX A: NEGATIVE TIMES IN THE ISING MODEL Some of the times tα=M−1 αβ gβ(tF/g) of the analog blocks to implement the Ising model can be negative. In a sDAQC protocol, one solution would be to do such analog evolutions with inverted coupling signs. However, in the bDAQC paradigm, we must keep untouched the base entangling Hamiltonian ¯ HZZ. There is a simple way to mimic such behavior by using a tantamount set of times that produces the same evolution. Given that the vector of times tα=(t1,t2, ..., tmin, ...) with tmin <0 and tmin <tα∀α, there is an equivalent evolution with N(N−1)/2−1 blocks of times ˜ tα=tα+|tmin|e1and an extra analog block with time |tmin|λ1, where λ1is the eigenvalue of a matrix Mαβ with corresponding constant eigenvector e1= γ(1,1,1...)T. Let us first focus on the properties of the matrix Mαβ defined in Eq. (3), created from the signs of applying sequentially gates σn xσm xbefore and after the analog block evolutions. This matrix has an eigenvector e1=γ(1,1,1...)T which corresponds to the eigenvalue λ1=N(N−9)/2+8, that is, negative for N∈{3,5,6}and positive thereafter N∈ Z⩾7. Notice that we omit the case for N=4 as the matrix is noninvertible and different set of gates must be performed. For N<7 we observe the following identity, Mαβ (tα+|tmin|e1−|tmin|e1) =Mαβ (tα+|tmin|e1)−λ1|tmin|e1 =Mαβ ˜ tα+|λ1tmin|e1,(A1) which corresponds to an evolution with N(N−1)/2−1 blocks of times ˜ tα. One block has zero time (tmin +|tmin|= 0), and there is an extra analog block of time |λ1tmin|not sandwiched by any SQR. The evolution decomposes into UZZ(tF)=e(iβgMαβ ˜ tασj zσk z)e(i|λ1tmin|βgσj zσk z),(A2) where the dependence of j,kon βhas not been explicitly written and is the same as in Eq. (4). APPENDIX B: UNIVERSALITY We briefly recall that a machine able to implement singlequbit rotations and an entangling gate, e.g., a controlledphase gate, has the ability to perform universal quantum computation efficiently. It can be easily proven that a twoqubit ZZ gate, ZZij()=e−iσi zσj z, combined with singlequbit rotations can be used to implement a controlled-phase gate CZij(φ)=(Zi(−φ)⊗Zj(−φ))ZZij(φ/2) (up to a general phase) where Z(φ)=10 0eiφ,CZ(φ)=⎛ ⎜ ⎝ 100 0 010 0 001 0 000e−i2φ ⎞ ⎟ ⎠.(B1) APPENDIX C: INDEPENDENT COUPLING PARAMETERS IN XZ MODEL The implementation of the XZ model with the protocol of Sec. IIIB requires the solution of the following linear system: α(μ,s) jα(ν,s) k ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ g(1) 12 g(2) 12 g(3) 12 g(4) 12 g(1) 13 . . . g(4) N−1,N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ gXX 12 gXZ 12 gZX 12 gZZ 12 gXX 13 . . . gZZ N−1,N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(C1) with the matrix, α(μ,s) jα(ν,s) k= N  j<k ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S1 θjS1 θkS2 θjS2 θkS3 θjS3 θkS4 θjS4 θ2 S1 θjC1 θkS2 θjC2 θkS3 θjC3 θkS4 θjC4 θk C1 θjS1 θkC2 θjS2 θkC3 θjS3 θkC4 θjS4 θk C1 θjC1 θkC2 θjC2 θkC3 θjC3 θkC4 θjC4 θk ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (C2) where we have defined the parameters Ss θj=sin(θ(s) j)=α(x,s) j and Cs θj=cos(θ(s) j)=α(z,s) j. This matrix is invertible for a dense set of phase values, that is, the sets of phases that make it singular has measure zero. From a practical perspective, we do not want eigenvalues close to zero, because after inversion we would have long simulating times. One useful and well-behaved array of phases is θ(s) w=sπ(w) 2(w+1), with distance between two nearest-neighbor qubit phases scaling polynomially d(s,w)=|θ(s) w−θ(s) w+1|= sπ 2(w2+3w+2). FIG. 10. Sets of engineered interactions Os XX for creating allsupport interactions. (Green) Two sets of generalized Hamiltonians required to create interactions in all supports for M⩽4. (Red) Three sets of generalized Hamiltonians required to create all interactions with support M⩽6. 022305-9