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 quan- tum Hamiltonians specifically tailored for pro- grammable quantum devices where interac- tions 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 prepara- tion for the Su-Schrieffer-Heeger model in ions and show that our strategy is scalable while be- ing 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 cir- cuit 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 ini- tialized 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 de- vices, incompatible with the long annealing times re- quired 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 algo- rithm. 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 strate- gies, 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 ap- proach is tolerant to finite temperatures of the COM mode, which do not limit the state preparation fi- delity. Moreover, our strategy does not require to dis- entangle 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 us- ing 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].