Variational Quantum State Preparation Via Quantum Data Buses Viacheslav V

Variational quantum state preparation via quantum data buses Viacheslav V. Kuzmin1,2 and Pietro Silvi1,2

1Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, A-6020, Innsbruck, Austria 2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria

We propose a variational quantum algorithm to prepare ground states of 1D lattice quantum Hamiltonians specifically tailored for programmable quantum devices where interactions among qubits are mediated by Quan- tum Data Buses (QDB). For trapped ions with the axial Center-Of-Mass (COM) vibrational mode as single QDB, our scheme uses resonant sideband optical pulses as resource operations, which are potentially faster than off-resonant couplings and thus less prone to decoherence. The disentangling of the QDB from the qubits by the end of the state preparation comes as a byproduct of the variational optimization. We numerically simulate the ground state preparation for the Su-Schrieffer-Heeger model in ions and show that our strategy is scalable while being tolerant to finite temperatures of the COM mode. Figure 1: (a) Blue-detuned optical pulse acting on ion j re- alizes interaction Hj , Eq. (1), which couples the internal ion levels to levels of the COM phonon mode. The coupling 1 Introduction rates depend on the state of the system. (b) QDB-MPS circuit in an ion trap generating trial states as in Eq. (6). The Realizing many-body quantum states on quantum de- COM mode (red line), as single QDB, is initially cooled to vices offers an experimental pathway for studying the a low-temperature state ρ0, and ions (black lines) are initialized in pure states. Variational operations exp(−iθ H ) equilibrium properties of interacting lattice models k jk (blue links) are virtually grouped into boxes, parametrized by [1–3], quench dynamics [4–7], or it can be viewed as a corresponding vectors θi = {θk}, as explicitly illustrated for quantum resource, e.g., for sensing [8–11]. Adiabatic the first two boxes. The out-coming ions are measured, while state preparation has been experimentally demon- the state of the QDB is discarded. (c) Mapping of circuit (b) strated, e.g., on trapped ions [12] and atoms in opti- (opt) with the optimized parameters, {θi }, to an MPS diagram cal tweezers [13]. However, this strategy is currently with the qubits state approximating |ψtargi. limited by the finite coherence times of quantum devices, incompatible with the long annealing times required by the adiabatic criterion [14]. An alternative, Mølmer-Sørensen gate [22], or a programmable analog promising pathway towards quantum ground state quantum simulator of the long-range XY spin model arXiv:1912.11722v3 [quant-ph] 22 May 2020 preparation is given by feedback-loop quantum algo- [23]. Both these entangling resources generate ef- rithms, and especially by the Variational Quantum fective interactions between ion qubits by coupling Eigensolver [15–21] (VQE). In VQE, parametrized − them to vibrational modes (phonons). Implement- generally non-universal − quantum resource opera- ing the optical couplings off-resonantly, one effectively tions, available on a given quantum device, are ar- achieves robustness of these resource operations to the ranged in a variational sequence, or circuit ansatz, temperatures of the phonon modes. However, this to generate entangled trial states. The trial states strategy yields relatively slow rates [24, 25], limiting approximate a target ground state as a result of a the performance of VQE, as the quantum processing closed-loop optimization of the variational parame- suffers from decoherence. ters, where an energy cost function is minimized. Here we propose a strategy to improve the efficiency For trapped ion qubits, VQE was recently demon- of VQE on programmable quantum devices where in- strated using as entangling resource either the teraction between qubits is mediated by auxiliary de-

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 1 grees of freedoms: Quantum Data Buses (QDBs). This strategy employs interactions between qubits and QDBs, naturally available in these devices, as variational resource operations in the quantum algorithm. We illustrate our approach on trapped ions, using the acoustic Center-Of-Mass (COM) phonon mode − the axial vibrational mode where all ions os- cillate in phase − acting as a single QDB. We employ resonant optical coupling of ion qubits with the COM mode, yielding faster rates than off-resonant strategies, and thus being less prone to decoherence. Un- like logical gates designed with the resonant coupling, such as the C-phase gate [26], our variational approach is tolerant to finite temperatures of the COM mode, which do not limit the state preparation fidelity. Moreover, our strategy does not require to disentangle the QDB after each resource operation, but Figure 2: (a) Error function Ferr (7) versus size Nions of the only at the end of the state preparation, ultimately states prepared in the circuits QDB-MPS, CSA shown in (b), yielding faster processing. The task of disentangling and CSD-MPS shown in (c) for the fixed number of varia- the QDB from the qubits is carried out by the opti- tional parameters Np = 18. The insets show correlation ma- DMRG mization algorithm itself: As the output qubit state trix Cij (8) (right) with i < j for |ψtargi with Nions = 12 approaches the unique ground state of a target non- obtained by the numerical DMRG method, and deviations DMRG degenerate Hamiltonian, the QDB becomes disentan- ∆Cij = Cij − Cij for Cij obtained for states prepared by gled without ever being measured. the circuits CSA (left) and QDB-MPS (middle); the arrows indicate the corresponding Ferr data-points. The dashed pix- Via numerical simulations, we investigate the scal- els represent arias with negligible absolute value < 0.01. (b) ing and robustness properties of our strategy for 1D CSA and (c) CSD-MPS circuits generating trial states (5). lattice models. As a benchmark goal, we aim to vari- CSA circuit uses global all-to-all qubits operations (3) and z ationally prepare the ground state, |ψ i, of the Su- singe-qubit rotations σi . CSD-MPS circuit uses local entan- targ z Schrieffer-Heeger (SSH) Hamiltonian [27–29] on ion gling MS gates (2) and singe-qubit rotations σi . qubits in a linear trap. Using only blue-detuned sideband optical pulses [30] as resource operations, wards quantum state preparation in ion traps as well we design the variational circuit ansatz shown in as in other quantum platforms that rely on QDBs, Fig.1, which can efficiently realize Matrix Product such as atom qubits coupled via photons in waveg- States (MPS), a class of tailored variational wavefunc- uides [35, 36] or cavities [37–40], or superconducting tions capable of accurately capturing the equilibrium qubits coupled by microwave resonators [41–44]. physics of many-body quantum systems in 1D [31, 32]. The paper is organized as follows. In section2, we This ‘QDB-MPS circuit’ can incorporate various sym- describe the ion trap resource operations which we metries of the target model for enhanced performance, employ in the QDB-MPS circuit. We also review the including approximate translational invariance in the resources used by other VQE strategies in ion traps, bulk. In this paper, we will show that our approach is which we consider for comparison. We briefly intro- scalable, as we can approximate 1D ground states at duce the target SSH model in section3, and we discuss saturating precision in the system size N , without ions in detail the VQE strategies. We exhibit the results increasing the number of variational parameters N p of our numerical simulation in section4. Finally, in [see Fig.2(a)]. Additionally, we compare the results section5, we summarize our conclusions and argue from the QDB-MPS circuit with other VQE strate- future perspectives for our approach. gies, still designed for trapped ion hardware, but using different sets of (coherent) entangling resources: (i) site-filtered Mølmer-Sørensen gates (ii) an analog 2 Quantum resources in trapped ions quantum simulator of a long-range XXZ model [23]. We evaluate the respective accuracies in terms of var- We consider a setup of Nions ions in a linear Paul trap ious figures of merit, including excitation energy, fi- as a relevant quantum platform. The system qubit, or delity, and two-point correlations. We demonstrate spin 1/2, at site j is encoded by two internal electronic that our strategy can realize highly-accurate ground levels of ion j [30], labeled |↓ij and |↑ij. Each ion, states even for QDB initialized at finite temperatures. driven by laser fields in the transition between levels

Finally, we show that the QDB-MPS circuit can be |↓ij ↔ |↑ij, experiences a recoil associated with ab- up-scaled beyond the single-trap limit by being im- sorption or emission of photons. This mechanism cou- plemented in modular ion traps [33, 34]. Overall, ples the internal ion levels with vibrational (phonon) we consider our strategy a viable, eﬃcient route to- modes, either axial or radial, depending on the optical

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 2 setup [30], and allows phonons to be used as bosonic Oﬀ-Resonant Sideband Resources − To address this QDBs for the ion qubits. In ion traps, as well as in hindrance, alternative strategies to realize entangling other quantum hardware using QDBs [35–43, 45, 46], operations were developed, in which couplings of the various strategies have been developed to construct qubits to QDBs are implemented oﬀ-resonantly, and entangling quantum operations between qubits, while the QDBs are only virtually populated [39, 49–52]. ensuring that the QDBs are disentangled from the A prototypical example in ion traps, the Mølmer- qubits by the end of each operation. In what follows, Sørensen (MS) gate [49], can be implemented by cou- we review some of these strategies. pling a subset S of ions to the COM mode by a bichro- Resonant Sideband Resources − One possible ap- matic laser beam with two frequencies, detuned from proach to exploit QDBs in quantum hardware is to the qubit frequency by ± (ν − δ) with δ ν. The transfer information from one qubit to another by resulting Hamiltonian populating QDBs with real excitations [26, 42, 43, S (ηΩ)2 X 45, 46]. In trapped-ion quantum computers, a typ- H = σxσx, (2) MS δ i j ical choice is to elect the axial COM mode as sin- i

Hamiltonian Nions−1 Nions Nions X X Jij X H = σxσx + σyσy + B σz − † + XY 2 i j i j i Hj = iηΩ aσj − a σj (1) i=1 j=i+1 i=1 (3) addressing ion j, where Ω is the Rabi frequency, and −α with power-law Jij ∼ J0 |i − j| long-range cou- achievable rates are of the order ηΩ ∝ 25 kHz [47, {x,y,z} plings, where σ are the Pauli operators act- 48]. In this picture, a is the destruction operator for j ing on qubit j and B is a uniform magnetic field. the COM mode, and σ− = (σ+)† = |↓ih↑| is the j j j Thanks to the off-resonant couplings, the interac- lowering operator for ion j. In Eq. (1), we used a tion (3) is also insensitive to the temperature of the gauge transformation a → eiϕa to highlight the anti- phonon modes. However, typical rates J ∼ 0.05 − unitary symmetry (σ± → σ∓, a → a†, t → −t) in the 0 0.5 kHz [24, 25] are, again, much slower than the rates interaction Hamiltonian. of resonant sideband interactions (1). In Ref. [26], it was proposed to use resonant sideband pulses to realize the C-phase gate in ion traps. However, the main hindrance in using gates built 3 Variational Quantum Eigensolvers out of resonant interactions with a bosonic QDB, as Eq. (1), is the requirement for the QDB to be perfectly VQE enables the experimental realization of ground initialized in a pure state, typically the vacuum |0ia, states of arbitrary Hamiltonians, providing the best where index a refers to the QDB mode. This√ require- approximation for the given resource operations. ment stems from the incommensurate rates l + 1ηΩ Once the optimization is converged, the optimal quan- of the transitions |↓ij |lia ↔ |↑ij |l + 1ia, depending tum state can then be re-prepared on demand. For on the number of excitations l stored in the QDB, example, this state can be used as a resource to ini- as highlighted in Fig.1(a). As a result, the gate tialize the system for digital quantum simulation [55], fidelity of resonant sideband gates in ion traps de- which was recently demonstrated on ion traps in the creases roughly linearly with temperature T of the context of lattice gauge theories [6]. −1/T † initial thermal state ρ0(T ) = (1−e ) exp(−a a/T ) Target model: Su-Schrieffer-Heeger − As an exam- (in units of kB = ~ = 1) of the COM mode. ple, in this work, we consider VQE for the ground

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 3 state preparation of the SSH model on ion qubits. Convergence of the search algorithm can be en- The SSH is an exactly solvable Hamiltonian [27–29], hanced by incorporating symmetries of the target and a free-fermion prototype for symmetry-protected model in the trial states. For the CSA circuit, the topological order in 1D, which reads zero z-magnetization of the initial Néel state |ψini is protected both by the entangler HXY and rotations N −1 z ions exp(−iθjσj ). CP symmetry is protected by HXY , and X j x x y y HSSH = 1 − (−1) t σj σj+1 + σj σj+1 can be enforced on single-qubit transformation ‘layers’ j=1 by using correlated rotations angles θj = −θNions+1−j. + B˜ σz − σz (4) can be approximated as well, by repeating odd-site 1 Nions and even-site rotation angles, θj = θj+2, away from in open boundary conditions. HSSH is real in the the system edges [23]. canonical basis, exhibits z-magnetization symme- While VQE provides the best approximate state P z try [HSSH, j σj ] = 0, and has zero-magnetization preparation for a given set of resources in the pres- ground states with real amplitudes. We added a small ence of actual imperfections and noise [10], it is still staggered magnetic field B˜ = 0.1 at the boundaries, affected by decoherence. By considering a different to lift the four-fold degeneracy arising in the topo- VQE circuit ansatz for trapped ions, which uses the logical phase (t < 0). With this prescription, HSSH high-rate resonant sideband resources of Eq. (1), we is also CP symmetric (invariant under global spin-flip aim to tackle decoherence by speeding-up the prepa- plus spatial reflection), translationally invariant in the ration of trial states. bulk with a period of two sites, and has a critical point VQE with Quantum Data Buses (QDB-MPS)− at t = 0. We design a variational circuit ansatz U(θ) = Q VQE with Closed System Analog resources (CSA) k exp(−iθkHjk ) built on ion-COM mode interaction − In Ref. [23], VQE was performed on trapped ions pulses (1), according to Fig.1(b). Since now the COM using Closed-System programmable Analog resources mode becomes entangled to the ion qubits, the state as in Eq. (3). We report the variational circuit ansatz of the qubits is generally mixed, both during the pro- considered in that work in Fig2(b). This circuit alter- cessing and in the output trial state nates global resource operations, generated by HXY , z ρ (θ) = Tr [U(θ)(|ψ i hψ | ⊗ ρ ) U †(θ)]. (6) and single-qubit rotations exp(−iθσj ). It was shown out COM in in 0 that this strategy is potentially scalable and capable of approximating ground states of 1D lattice mod- Parameters {θk} can be controlled by the dura- els, including lattice gauge theories, with high fidelity. tion of the sideband laser pulses. Similar to the Here, we consider the CSA circuit as a comparison closed-system case, the cost function hHSSHiθ = benchmark for the QDB-MPS circuit in preparing of Tr[ρout(θ)HSSH] is reconstructed from correlation functions evaluated on the trial qubit state, while the ground state of HSSH. We briefly review some details of the CSA strategy to highlight the differences the output state of the COM mode is discarded. As with respect to the QDB-MPS. hHSSHiθ is minimized to approach the ground energy, The variational CSA circuit built with closed- ρout(θ) approximates the unique, pure ground state (r) z |ψtargihψtarg|, and the output state of the COM mode system resource interactions Hk ∈ {HXY , σj } generates (ideally pure) trial states becomes automatically disentangled. In our design, resource operations are arranged such Y (r) that the circuit ultimately generates a variational Ma- |ψout(θ)i = exp(−iθkH ) |ψini , (5) k trix Product State (MPS). In fact, MPS efficiently ap- k proximate ground states of 1D quantum lattice Hamil- where the product of operators is ordered right-to-left tonians, including the SSH model [56, 57]. In contrast for increasing k. The system qubits are initialized in to CSA resources, which are inherently global, blue- a Néel product state |ψ i = |↓i |↑i |↓i ... |↑i sideband operations (1) offer single-site addressabil- in 1 2 3 Nions which can be prepared with high fidelity. For each ity. This property allows us to progressively tailor given set of trial parameters θ, multiple instances of the MPS from one edge to the other, as shown by the |ψout(θ)i are prepared, and projectively measured in Tensor Network diagram [56, 57] in Fig.1(c), in anal- various local bases to reconstruct the energy cost func- ogy to quantum circuits developed in the context of tion hHSSHiθ = hψout(θ)| HSSH |ψout(θ)i from one- quantum machine learning [58]. Previous strategies a a a and two-point correlation functions hσi i, hσi σj i. The for realizing MPS on quantum devices were proposed, cost function hHSSHiθ, function of θ, is evaluated at both for deterministic state preparation [59, 60], as finite precision and minimized with a search algorithm well as for variational state preparation [61] which em- capable of taking into account statistical errors. Upon ployed long-range interactions. In contrast, our vari- convergence, hH i reaches its global minimum, ational circuit requires only local ion-phonon inter- SSH θopt and the optimal set of parameters θopt can be used to actions, and, therefore, can be up-scaled to modular re-prepare the (approximate) non-degenerate ground ion trap architectures (see next section). Addition- state |ψout(θopt)i on demand. ally, we introduce an approximate bulk-translational

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 4 invariance to the ansatz, to reduce its variational com- plexity: In our circuit, interactions in the bulk can be virtually grouped in ‘boxes’ (drawn in Fig.1(b) as grey rectangles at the edges, and orange and green for the bulk). Each box is defined by a set of variational parameters, which are repeated in the bulk boxes of the same color. At the same time, the maximum achievable bond dimension D ∼ 2l−1 of the generated MPS can be controlled by increasing the size of the bulk boxes l (see AppendixA). Figure1 shows a specific design of the bulk boxes using five operations Figure 3: Energy error related to the excitation gap, ε, (a) per box. Numerical results suggest that the number versus size of the prepared states Nions for fixed number vari- of operations per box is crucial towards the accuracy ational parameters Np = 18 and (b) versus Np for Nions = 12 of the ansatz, while the ordering is marginal (data not in circuits QDB-MPS, CSD-MPS, and CSA, as indicated in shown). AppendixE illustrates our basic strategy to (a). Gray small rounds in (b) give values of εq obtained arrange operations inside a box. in the same QDB-MPS circuit and parameters as the black plot but with different initial pure states of the QDB |qi for As in the CSA case, we again incorporate symme- a q ∈ {0, 1}. tries of the target model in the trial states. First of all, since Hj is fully imaginary, the operation U(θ) is real as well as |ψini and ρ0(T ), and, thus, the generated 0.01 [62, 63]. Other sources of imperfections are not states ρout(θ) have real amplitudes. Moreover, opera- taken into account, and, therefore, in contrast to the tions (1) protect an ‘extended’ magnetization symme- QDB-MPS circuit, simulations of the CSA and CSD- P z † try 1/2 j σj − a a, forcing the output qubits state MPS circuits are free of imperfections. We consider to have well-defined z-magnetization upon qubits and the theoretical limit of unlimited measurement bud- QDB becoming variationally disentangled (see Ap- get [10, 23], i.e., the cost function is the exact expec- pendixB for details). We have analytically verified tation value hHSSHiθ. To optimize parameters θ, we that, in the theoretical limit of infinite-depth circuits, use a gradient-based optimization algorithm (see Ap- the resources given by Eq. (1) are sufficient to gener- pendixD), which can be adapted for noisy cost func- ate ground states of any real, magnetization conserv- tions [64–66]. We emphasize that we consider a low ing Hamiltonian (proof in AppendixC). Controllabil- number of free variational parameters, Np ∼ 10 − 25, ity for an arbitrary lattice model (e.g., without sym- which enables global search methods, such as the di- metries) can be achieved by adding to the resources a viding rectangles algorithm [67, 68], already adopted z resonant carrier operation σj for at least one ion. in VQE [23]. For the analog resource entangler HXY , We stress that resources of the form (1) are typical realistic values α = 1.34 and B = 20 [23] are used. in various experimental platforms with bosonic QDBs, Further details on the circuit ansätze and their simu- such as superconducting qubits connected via a res- lation are given in AppendixE. onator [42, 43, 46]. As such, the numerical analysis To assess the accuracy of the ground state prepara- presented below can be generalized to these platforms. tion, we employ various figures of merit. Besides the CSD-MPS Circuit Ansatz − We also compare our fidelity and the excitation energy (defined later on), QDB-based approach with a variational MPS cir- we also consider a correlation-based error function cuit ansatz realized with Closed System Digital re- Nions Nions X DMRG X DMRG source operations (CSD-MPS circuit), as presented in Ferr = Cij − Cij / Cij , z Fig2(c). This circuit is built with single-qubit σi i=1, j>i i=1, j>i rotations and site-filtered MS gates (2) applied to a (7) local set of ions. Such resource operations protect z- magnetization in the trial state. calculated using (parallel) two-point correlators x x x x Cij = hσi σj i − hσi ihσj i. (8)

4 Results Here Cij is evaluated on the optimized state DMRG ρout (θopt) and Cij is the exact correlator of Performance− We compare the performance of the |ψtargi obtained by DMRG method [69]. We consider variational circuits by numerically simulating VQE this error function in Fig.2(a), to show that the pro- for preparation of the ground state |ψtargi of Hamilto- posed QDB-MPS circuit is able to prepare states with nian (4) in the gapped phase with t = 0.5. Trial states Ferr saturating in Nions at ﬁxed number of variational are obtained by evolving the initial quantum state in parameters Np. In particular, the insets show that terms of density matrixes. For the QDB-MPS cir- these states have correlations Cij accurately capturing DMRG cuit, we consider the COM mode to be prepared in a Cij . By contrast, Ferr arising from the CSA cir- thermal state ρ0(T ) with realistic average number of cuit grows rapidly for this speciﬁc task, and the corre- † 1/T excitations n0(T ) = Tr[a a ρ0(T )] = 1/(e − 1) = lators Cij decay slower than in |ψtargi. We argue that

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 5 this follows from the necessity to variationally remove the strong (power-law) correlations established by the long-range interactions in HXY (3) to recover the ex- ponential decay of correlations in the ground state. Moreover, the SSH ground state exhibits consider- able (short-range) entanglement dimerization, which in turn seems to favor quasi-local resource circuits over global resource circuits. While the results for CSD-MPS and QDB-MPS circuits given in Fig2(a) are compatible, the QDB-MPS is realized with higher- Figure 4: (a) Fidelity and (b) infidelity of the 6 qubit state prepared in the QDB-MPS circuit as functions of number of rate operations, and, therefore, is expected to be less excitations n in ρ (T ) and number of variational parameters prone to decoherence. 0 0 Np in the circuit. In (b), the red line indicates the lowest Scalability − We study the scalability of the cir- number of Np required to preserve infidelity threshold 1−F ≤ cuits in terms of the excitation energy in units of 0.002 at given n0. the energy gap, which bounds infidelity of the prepared states 1 − F ≤ ε = (hHSSHiθopt − E0)/∆ (Ap- pendixF), and is thus a figure of accuracy. Here, F = hψtarg|ρout (θopt) |ψtargi and ∆ = E1 − E0 is the energy gap obtained by DMRG, which converges to a finite non-zero value in the thermodynamical limit Nions → ∞ since the target model is non-critical. Fig- ure3(a) shows that, for the QDB-MPS and CSD-MPS circuits, ε grows linearly with Nions, in contrast to the CSA circuit. This is compatible with the previous observation that the two-point correlation functions carry, on average, an error saturating in Nions, and the energy functional contains a linear amount of such correlators. Figure 5: QDB-MPS circuit in a modular ion trap architec- ture, which generates a state on the ions chain with 4 ions Scaling of the accuracy ε with N , for the various p per trap. The processing is implemented consequently, start- methods and at fixed Nions, is plotted in Fig.3(b). ing from the left trap to the right, and from the bottom to Again, we observe that MPS circuits offer asymptoti- the top in a single trap as presented by the circuits. The red cally better accuracy (or better parametric efficiency) boxes with the arrows indicate the transfer of the edge ions than the CSA global resources. While increasing (or their internal states) to the next trap after the implemen- the number of parameters Np, we observe the CSD- tation of all operations in a trap. MPS circuit reaches slightly better accuracies than the QDB-MPS circuit; we attribute this effect to the

finite temperature of the COM mode we employed shows that, for fixed Np, F decreases linearly with for the simulations. This argument is supported by n0 (similarly to the C-phase gate [26] built with side- the gray curves in Fig.3(b), showing relative excita- bands), but with a slope that flattens for increasing tion energies εq obtained with the same QDB-MPS Np. Therefore, to achieve a fidelity threshold, Np circuit and the same parameter values as the black must be increased with n0. Remarkably, for the case plot but with different initial pure states of the COM under consideration, it seems that the Np required mode |qia for q ∈ {0, 1}. For n0 = 0.01, we have to obtain a static fidelity threshold scales logarithmi- 2 q ε = ε0p0+ε1p1+O(n0), with pq = hq|aρ0(T )|qia ∼ n0. cally with n0. This behavior is illustrated in panel (b), The plot shows that the two quantum trajectories are where the density plot shows the infidelity of the opti- both variationally optimized in such a way that their mized state as a function of n0 and Np. The red curve error contributions are commensurate, ε0p0 ∼ ε1p1. highlights a fixed fidelity threshold 1 − F ≤ 0.002. Tolerance to temperature − Here, we further in- This property sets our QDB-based VQE apart from vestigate tolerance of the QDB-MPS approach to the approaches using sideband interactions to con- finite temperatures T of the COM mode. As struct logical gates [26], in which imperfect cooling T increases, more amplitudes pq are populated in imposes a hard bound on the attainable accuracies. P ρ0(T ) = pq(T ) |qia hq|, and, therefore, more equa- Scalability on a modular ion trap architectures − tions U (θopt) |ψini |qia ∼ |ψtargi |qia need to be si- The proposed QDB-MPS circuit1(a) relies on inter- multaneously satisfied, requiring higher Np to achieve actions between individual ions and the COM mode. the same accuracies. In Fig.4, we consider the fidelity However, the number of ions which can be coupled of the optimized output state of Nions = 6 qubits, as to the COM mode in a single trap is limited by a function of the average number of excitations n0(T ) ∼ 10 − 100 [34]. To further scale the ion trap quan- and number of variational parameters Np. Panel (a) tum processors, modular ion trap architectures were

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 6 scalability of the QDB-MPS approach for VQE, as it can treat hardware interfaces as impurities, and eﬃ- ciently compensate for them with small variations of the circuit ansatz.

5 Outlook

Figure 6: Energy error related to the excitation gap versus We proposed an approach for VQE which uses qubit- size Nions of the optimized state in a single trap using Np = QDB interactions, available in currently-existing pro- 18 variational parameters and in Ntraps traps with 6 ions per grammable quantum devices, as variational resource trap using Np = 18 + 2 . operations. In our strategy, the QDB is disentangled from the optimized states only at the end of the state preparation, as a byproduct of the optimization. For proposed [33, 34]. Following these proposals, we de- trapped ions, we discussed how to realize our strategy scribe ground state preparation with QDB-MPS cir- with resonant blue-sideband pulses, potentially yield- cuit across multiple traps, as shown in Fig.5. We ing faster state-preparation rates compared to exist- consider coherent shuttling of few ions [33, 70] (or ing strategies, and thus helping to mitigate decoher- their internal states [34]) as a quantum communica- ence effects. The variational ansatz circuit we de- tion channel among the traps, and we assume it to be signed is tailored to efficiently prepare MPS, leading error-free. We consider each trap containing a small to a potential improvement in scalability over current number of ions, such that each ion is addressable via VQE strategies. blue sideband pulses, realizing interactions as Eq. (1) with the COM mode of the corresponding trap. For We numerically simulated the ion trap implemen- simplicity, we assume that the initial thermal states, tation of VQE based on the QDB-MPS ansatz circuit, and showed strong evidence of scalability of our ap- ρ0, of the COM modes are identical in each trap, as well as and their effective coupling to the ions. proach, even when it is realized in modular ion traps. The variational circuit ansatz is constructed se- While the ions-COM mode sideband interactions op- quentially along the network of traps, i.e., by ap- erate in an extended Hilbert space, we demonstrated plying operations in trap k + 1 after all operations that these resource operations can incorporate sym- in trap k are completed. Since each trap has an in- metries in the variational circuit, similar to the closed- dependent COM mode as a bosonic QDB, we intro- system VQE approaches. Although here we demonstrated robustness of the method to the initial tem- duce a variational ‘interface’ operation UI (θ5), constructed with resource interactions (1), which is ap- perature of the COM mode, further investigations are plied to a set of l − 1 ions (with l the size of the required to study other realistic imperfections, includ- bulk boxes) as the last operation for trap k. This ing (i) qubits decoherence, (ii) fluctuations of the con- operation can be optimized in order to disentangle trol parameters, and (iii) shot noise resulting from a the COM mode of trap k, effectively implementing finite budget of measurements to reconstruct the cost † function. UI ρqubits,kUI = ρqubits⊗ρ0, with ρqubits,k the common state of ions and the COM mode. After transferring The developed QDB-based VQE can be readily the set of l − 1 ions to trap k + 1, we apply the uni- used in other currently available experimental plat- † tary UI to the same ions plus the COM mode of trap forms, such as quantum dots [71] or atoms coupled k + 1 prepared in state ρ0, effectively reconstructing by (chiral) waveguides [72]. Moreover, we expect that † the state ρqubits,k+1 = UI (ρqubits ⊗ ρ0)UI = ρqubits,k. the scaling and parametric efficiency of our results, Ultimately, this operation removed the ‘impurity’ in- found for the MPS circuit ansatz, extend to other troduced by the interface between the two traps. efficiently-contractible tensor network states [58, 61], In Fig.6, we compare the excitation energies, from such as MPS2 [73, 74], thus allowing us to accomodate optimizing the QDB-MPS circuit, with a single trap the entanglement content required to explore higher and with a modular trap network. Both circuits dimensions. Also, by including simultaneous opera- have the same number of variational operations in tions on multiple qubits, it is possible to build many- the boxes in the edges and in the boxes of the bulk, body wavefunction ansätze beyond tensor networks, and the interface operation UI contains 2 variational ultimately enabling the efficient preparation of quan- sideband operations. The error emerging from the tum states that can not be numerically simulated. Fi- optimization of the ion-trap network deviates only nally, we envision the possibility of including control- slightly from the error in the single trap case, and lable dissipative elements to the QDB resources [75] we recover linear scaling of the error with the sys- in order to construct optimized cooling algorithms be- tem size. This result is an even stronger indicator of yond stabilizer codes [76, 77].

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 7 Acknowledgments

We warmly thank M. Hettrich, R. Kaubruegger, C. Kokail, M. Meth, T. Monz, P. Schindler R. van Bijnen, and P. Zoller for private discussions. Authors kindly acknowledge support by the EU Quantum Technol- ogy Flagship project PASQUANS and the QuantEra project QTFLAG, the Austrian Research Promotion Agency (FFG) via QFTE project AutomatiQ, the US Air Force Office of Scientific Research (AFOSR) via IOE Grant No. FA9550-19-1-7044 LASCEM, and by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-15- 2-0060. The views and conclusions contained in this document are those of the authors and should not Figure 7: (a) QDB-MPS circuit with the measurement of be interpreted as representing the official policies, ei- the bosonic QDB, which results in probabilistic generation of ther expressed or implied, of the Army Research Lab- MPS |Φi in qubits. (b) Tensor network diagram of |Φi. oratory or the U.S. Government. The U.S. Gov- ernment is authorized to reproduce and distribute Appendix reprints for Government purposes notwithstanding any copyright notation herein. The computational results presented have been achieved (in part) using A Maximum bond dimension the HPC infrastructure LEO of the University of Inns- bruck. Source codes used for our results are avail- In the section, we study the maximum bond dimen- able at https://github.com/Viacheslav-Kuzmin/ sion, DM , of MPS generated with the QDB-MPS cir- quantum_bus_vqe, released under a MIT license. cuit using a bosonic QDB and built with resource operations (1) described by the Anti-Jaynes-Cummings Hamiltonian. Here we distinguish two sets of MPS accessible by the circuit. The first set contains all pure states which can be generated by the QDB-MPS circuit. The states of this set can be generated probabilistically [59] after measuring (and therefore disentangling) the QDB at the end. It defines the search space for VQE and, therefore, is desired to be limited. The second set — a subset of the first set — contains deterministically generated translationally invariant MPS, which are disentangled from the QDB by the end of the circuit without measuring the EM. This set, in opposite to the first one, is required to be big enough to include a good approximation of the target state.

Set of probabilistically generated MPS We start with the set of MPS generated probabilistically. We characterize this set of MPS by considering a possible range of their bond dimensions. Bond dimension DM of MPS of size N is the maximum of bond dimensions Dn between two blocks of qubits [1, n] and [n + 1,N]. Dn corresponds to entanglement entropy between the blocks [57], such that the Schmidt decomposition of MPS |Φi in the basis of two (i) (i) blocks {|φ[1,n]i} and {|φ[n+1,N]i} is

Dn X (i) (i) |Φi = ai|φ[1,n]i|φ[n+1,N]i. i=1

The entanglement entropy of the blocks is S[1,n] = −Tr ρ[1,n]log(ρ[1,n]) ≤ logDn, with ρ[1,n] the state

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 8 of qubits [1, n]. Set of deterministically generated translationally invariant MPS Using singular value decomposition, one can show In order to generate MPS with translationally invari- that, for an arbitrary state of size N, it is valid ant Dn in the bulk, the virtual ancilla has to own min(n,N+1−n) that Dn ≤ 2 , and, therefore, DM ≤ dimension which repeats itself with a target period 2bN/2c. By using a not universal set of resource along the state preparation. This includes the an- operations (9) we limit DM and, therefore, restrict cilla dimension after realization of the last box in the the set of achievable states. As presented in Ap- bulk, after which the edge box is aimed to disentan- pendixB, interactions (1) preserve an extended sym- gle the QDB from l − 1 ancilla qubits as required by PN z † deterministic preparation of a pure state. This can metry Z = 1/2 j=1 σj − a a. In the following, we l−1 be done only if the eﬀective dimension of the ancilla explain that this symmetry imposes Dn bn/2c·2 , / l−1 where l is the size of the bulk boxes in the QDB-MPS does not exceed dimension 2 of the qubits. Hence, circuit, and limits the maximum bond dimension as the limit of the bond dimension of the translationally l−1 invariant MPS deterministically generated in the con- DM b2/3 · Nc 2 . Therefore, in comparison with / l the general case, the bond dimension of the MPS gen- sidered QDB-MPS circuit is ∼ 2 . Here, we deﬁned erated by the QDB-MPS circuit is exponentially sup- the limit not exactly, since the actual bond dimension pressed. can vary within one period of the bulk.

Figure7, demonstrates entanglement distribution B Symmetries in the QDB-MPS circuit between the qubits by a subsystem, highlighted by orange color, which includes In this section, we discuss the symmetries protected the QDB and l − 1 qubits shared by neighbor virtual by the operations boxes. Such a subsystem can be considered as a vir- ˜ − † + tual ancilla that sequentially interacts with the qubits Hj = iΩ aσj − a σj , (9) distributing entanglement among them [59]. The ef- ˜ fective dimension of this ancilla (dimension of the pop- with Ω the Rabi frequency, a is the destruction operator for the bosonic QDB, and σ− = (σ+)† = |↓i h↑| ulated Hilbert space) bounds bond dimension Dn. j j j the lowering operator for qubit j, with {|↓ij , |↑ij} the internal logical states of the qubit. Let us consider the QDB initially prepared in state Magnetization − A number symmetry, forming an

|p0ia = |0ia, and the qubits prepared in state |ψini = Abelian U(1) group of symmetric transformations |↓i1 |↑i2 |↓i3 ... |↑iN . Then, because of the preserved W (ϕ) = exp(−iϕZ), is generated by the integer op- symmetry, the maximum possible population of the erator QDB after interaction with n ions can not be higher N than |bn/2ci . Multiplying the growing dimension of X a Z = 1/2 σz − a†a = M − a†a, (10) the QDB by the dimension of l−1 qubits in the ancilla, j l−1 j=1 we obtain that Dn / bn/2c · 2 . From the other side, as explained in [59], Dn has to decrease as n P z where N is the number of qubits, M = 1/2 j σj , approaches to N to guarantee that the ancilla (with z and σj = |↑i h↑|j − |↓i h↓|j. It is straightforward to QDB measured in |p i ) is decoupled by the end of N a check that Z is a symmetry since [Hj,Z] = 0 for every the state preparation. Therefore, there exists some site j ∈ {1..N}. Z thus defines a quantum number, nM , for which DnM achieves its maximum. To obtain which is conserved during the controlled variational nM , we study how Dn can decrease while the ancilla dynamics. Consider the initial state of the system of interacts with the qubits after qubit n . Intuitively, M qubits plus the QDB as |Ψini = |ψini ⊗ |sia, with this can be done other way around by considering |ψini = |↓i1 |↑i2 |↓i3 ... |↑iN the state of the qubits the growth of bond dimension D˜ if MPS would † N−n and |sia the Fock state of the QDB, a a|sia = s|sia, be generated from the other side of the qubits chain, s ∈ N. This state has well-defined quantum num- starting from n = N. Since the measured output ber Z |Ψini = s |Ψini. Due to the symmetry of state |pN ia can contain number of excitations pN ≥ 0, Eq. (10), the evolution of this state will preserve its the number of excitations in the QDB can not only quantum number, and the state will always be of increase but also decrease as the QDB interacts with P the form |Ψouti = q,m,t δm,q−scm,t|m, tmi ⊗ |qia, the qubits, populating Hilbert space with the twice where the qubit states are labeled by the magneti- bigger dimension than if the QDB would be initially zation m (eigenvalues of M) and degeneracy t . At ˜ m prepared in |0ia. Therefore, DN−n ≈ 2Dn˜ n˜=N−n / the end of the variational quantum state preparation, l−1 ˜ (N − n)2 . Solving DnM = DN−nM we obtain that, after optimization of the process, the output state after nM / b2/3 · Nc, Dn has to decrease with n. will be disentangled between phonon and qubits, and l−1 P Thus, we find that DM = DnM / b2/3 · Nc 2 . reads |Ψopti = ( t cm,t|m, tmi) ⊗ |m + sia. The final

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 9 + † − qubit state will thus have a well-defined magnetiza- starting from the bare resources Hj = i(σj a − σj a). tion m ∈ N. This requirement identifies the class of We start by considering the commutator i[Hj,Hj0 ] = − + + − 0 target Hamiltonians H targetable with the chosen re- i(σj σj0 − σj σj0 ), for sites j 6= j , which is an en- sources: which must contain the magnetization sym- domorphism of magnetization sector m. Moreover, z z metry: [H,M ] = 0. it is easy to show that i[Hj, i[Hj,Hj0 ]] = σj Hj0 , for Conjugation − A second symmetry is nested within j 6= j0. By recursion and locality, it follows that we our controls, and it becomes evident after gauging can generate any string operator of the form the resource Hamiltonians as in Eq. (1). In this − + + − Y z qk format, the resource Hamiltonians are purely imag- i(σj σj0 − σj σj0 ) (σ ) (11) inary Hj = −H¯j when expressed in the canonical k6=j,j0 t basis, and thus antisymmetric Hj = −Hj and trace- less Tr[H ] = 0. It follows that the unitary oper- for any binary string qk. We run one additional j − + + − z − + −iHj t/ commutator, namely i[(σ σ 0 − σ σ 0 )σ , (σ σ − ators Uj(t) = e ~ of time evolution under Hj j j j j k k k0 + − − + + − − + + − must be real, as they are imaginary exponentials of σk σk0 )] = 2i(σj σj0 − σj σj0 )(σk σk0 + σk σk0 ) where imaginary matrices, and have determinant det(Uj) = all sites are different. By recursion, we conclude that e−itTr[Hj ] = 1. This means that, with the chosen re- we can generate operators H¯ which are tensor prod- − + + − sources, we can only perform transformations in the uct of one operator i(σj σj0 − σj σj0 ) at a pair of d 0 z special orthogonal group SO(2 ), a subgroup of the sites j 6= j , any amount of σ` , and any amount of d − + + − unitaries U(2 ), which, in turn, limits the set of states (σk σk0 + σk σk0 ) (each one acting on separate sites). we can prepare to real states (vectors with real coeffi- We remark that these operators form a basis for the cients in the canonical basis). In conclusion, the class imaginary Hermitian endomorphisms of the symme- of models targetable with our variational scheme is try sector m. To see this, consider the canonical basis restricted to models expressed by real, or symmetric, i(|q1q2q3 ...ihr1r2r3 ... |−H.c.), where ~q and ~r are any Hamiltonians, which always possess real eigenbases. two-bit strings with total magnetization m. This op- We stress that the complex conjugation symmetry can erator can be easily decomposed in operators of the ¯ be relaxed by adding a detuning δj = ν ± to one or form H: for bits j that do not flip (qj = rj = ±1) just z more of the control lasers, which can be helpful if we insert a (1±σj ) in the tensor product string, while for T − + + − aim to quantum simulate a non-real Hamiltonian H . pairs of bits that flip-flop, insert a (σk σk0 + σk σk0 ), − + + − except one, which is substituted by i(σj σj0 − σj σj0 ). This concludes the proof. C Controllability We now discuss the controllability problem and show D Parameters optimization that, within the constraints set in the symmetry section, the selected controls are theoretically able to In the section, we describe the method used for the op- prepare any quantum state, in the limit of infinite timization of variational parameters θ in the numeri- available resources. First of all, we argue that start- cal simulations of VQE with the circuits given in the ing from the state |ψini ⊗ |qia it is possible to shift main text. Since the purpose of the paper is to study the system qubits state to any magnetization sector. the achievable accuracy of the proposed approaches, This is shown recursively, by highlighting that the we employ the exact value of cost function hHSSHiθ state |↓i |qi under the action of H for a time equal j a√ j in the optimization. The expectation value hHSSHiθ to t = π/(Ω˜ q + 1) is completely mapped into the is obtained by simulating the circuits in terms of den- state |↑ij |q + 1ia, ultimately increasing the magne- sity matrixes without considering shot noise caused tization by +1. Secondly, we show that our controls by the finite number of measurements of the trial ions Hj provide strong controllability within each magne- states. To optimize parameters θ we use a gradient- tization sector m (of dimension dm) of real states, based optimization algorithm, which relies on numer- meaning that the Lie algebra generated by {Hj} con- ically accessible approximate values of ∇θi hHSSHiθ = tains the Lie algebra of imaginary, hermitian matri- (hHSSHiθ |θi→θi+∆ −hHSSHiθ)/∆|∆→0. To be more ces onto the m sector, generators of SO(dm), as we precise, we use the basin-hopping method [78, 79], will show later in this section. From this observa- which is a two-phase method that combines a global tion, it follows that any two real states |ψ1i|qia and stepping algorithm with local minimization at each |ψ2i|qia can be dynamically connected, i.e., |ψ2i = step. As a local minimization algorithm, we use the −iH0t/ e ~|ψ1i for some t, simply because the effec- quasi-newton method of Broyden, Fletcher, Goldfarb, 0 0 0 tive Hamiltonian H = i|ψ1ihψ | − i|ψ ihψ1|, with and Shanno (BFGS) [80–83]. 0 |ψ i ∝ |ψ2i − |ψ1ihψ1|ψ2i, can be generated with our To decrease noise in the data presented in the result resources. section, we ensured that hHSSHiθ is monotonically de- We now prove that, by means of linear combination creases with decrease of temperature T of the COM and commutation, we can generate the whole algebra mode (or equivalently number of excitations n0), with of imaginary Hermitian matrices for the sector m by decrease of state size Nions, and with growth of the

Accepted in Quantum 2020-05-19, click title to verify. Published under CC-BY 4.0. 10 Q (r) (r) |ψout(θ)i = i exp{−iθiHi } |ψini, with {Hi } the resource operations. Afterward, the required correlation functions can be obtained from |ψout(θ)i. Since this method requires numerical operation with the full quantum state, the number of the qubits in the CSA circuit in our study is limited by 12. On the con- trary, one can see from Fig.8 that, in order to obtain correlation functions of the states generated by the QDB-MPS and CSD-MPS, the full state is not required to be kept, but only a reduced density matrix of at maximum l qubits plus the QDB, with l the size Figure 8: Strategy for increasing of the number of the varia- of the bulk boxes. The qubits which do not more par- tional parameters in circuits: (a) QDB-MPS, (b) CSD-MPS, ticipate in the evolution until the end of the circuit and (c) CSA. The numbers indicate the order in which the can be measured in the required bases, and the next new variational operations are inserted. The operations with- qubit can be added to the remained qubits state by out the numbers represent the minimum set. In (a), blue links using the Kronecker product operation. in the left box represent interactions between QDB (red line) and qubits (black lines) and, for simplicity, are replaced by the blue squares in the following boxes. F Energy bounds on ﬁdelity and purity

In this section, we show that the final energy hHi = number of variational parameters Np. We begin by in- Tr[Hρ(θ)], in respect to a target Hamiltonian H, mea- dependently optimizing parameters for all data points sured on the optimized variational quantum state ρ(θ) (defined by one or two values of n0, Nions, and Np), imposes a bound on its fidelity F = hψtarg|ρ(θ)|ψtargi, each of which starts from several sets of initial pa- with |ψtargi the exact ground state. Establishing this rameters θ with values ∼ 0.001 − 0.1. Then, the opti- bound requires a good estimate of the two lowest en- mized parameters of the best result in one data point ergy levels E0 and E1, which can be estimated on the is used as the initial parameters set for the neighbor quantum device via a subspace expansion technique data point, and the obtained result is compared with for mixed states, illustrated in detail in the Supple- the existing ones. In the case when the neighbor data mentary Information of Ref. [84]. This analysis pro- point requires less initial parameters, the extra pa- vides self-verification of the variational quantum state rameters are removed from the inserted parameters preparation algorithm [23]. Alternatively, for 1D lat- set; otherwise, parameters with 0 can be inserted. To tice Hamiltonians, E0 and E1 can be obtained using obtain the presented data, we iteratively re-optimize numerical method DMRG. the parameters from one side of data points range to Let us consider the following inequality another and, afterward, in the reverse direction until convergence. hHi − E0 = Tr[Hρ(θ)] − E0 X X = hej|(H−E0I)ρ(θ)|eji = (Ej−E0)hej|ρ(θ)|eji ≥ j≥0 j≥1 E Gates sequence in the circuit X ≥ (E1 − E0) hej|ρ(θ)|eji In the section, we consider in detail the sequences j≥1 of variational operations, Fig. (8), used to construct = (E1 − E0) (1 − he0|ρ(θ)|e0i) = (E1 − E0) (1 − F) circuits considered in our work. The number of varia- (12) tional operations in the edge boxes of the QDB-MPS and CSD-MPS circuits is fixed to 2 and 3 correspond- stating that the state fidelity F is bound from below ingly, as shown in Fig. (8). In these circuits, the by the ratio E − hHi number of variational parameters is increased by pro- F ≥ 1 , (13) gressively inserting operations in the first bulk box E1 − E0 (orange) or second (green), alternately. In the QDB- delivering an actual (positive) bound only when the MPS circuit, we increase this number by 1, arranging variational state energy is below the first excited en- the operations such that the last operation in one box ergy, i.e. hHi < E1. A lower bound on the fidelity, in and the first in the next box do not act to the same turn, infers a bound on the purity Tr[ρ2]. In fact qubit. In the CSD-MPS circuit, we add a single layer, consisting of a local MS operation and σz rotations X i Tr[ρ2] = he |ρ2|e i ≥ he |ρ2|e i at each ion. To increase the number of variational j j 0 0 j≥0 operations in the CSA circuit, we add a single layer X z = he |ρ|e ihe |ρ|e i ≥ he |ρ|e i2 = F 2. (14) consisting of HXY operations and σi per ion. 0 j j 0 0 0 The trial states in the CSA circuit are obtained as j≥0

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