ARTICLE Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register Ya Wang,*,† Florian Dolde,† Jacob Biamonte,*,‡ Ryan Babbush,z,§ Ville Bergholm,‡ Sen Yang,† z ) Ingmar Jakobi,† Philipp Neumann,† Ala´ n Aspuru-Guzik, James D. Whitfield, and Jo¨ rg Wrachtrup*,† †Third Institute of Physics, Research Center Scope and IQST, University of Stuttgart, 70569 Stuttgart, Germany, ‡ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy, zDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138 United States, §Google, 150 Main Street, Venice Beach, California 90291, United States, and Department) of Physics, Vienna Center for Quantum Science and Technology, University of Vienna, Boltzmanngasse 5, Vienna 1090, Austria ABSTRACT Ab initio computation of molecular properties is one of the most promising applications of quantum computing. While this problem is widely believed to be intractable for classical computers, efficient quantum algorithms exist which have the potential to vastly accelerate research throughput in fields ranging from material science to drug discovery. Using a solid-state quantum register realized in a nitrogen-vacancy (NV) defect in diamond, we compute the bond dissociation curve of the minimal basis helium hydride cation, HeHþ. Moreover, we report an energy uncertainty (given our model basis) of the order of 10À14 hartree, which is 10 orders of magnitude below the desired chemical precision. As NV centers in diamond provide a robust and straightforward platform for quantum information processing, our work provides an important step toward a fully scalable solid-state implementation of a quantum chemistry simulator. KEYWORDS: quantum simulation . electronic structure . molecular energy . diamond crystal . nitrogen-vacancy centers uantum simulation, as proposed with the cost scaling linearly in propagation by Feynman1 and elaborated by time.6 There is now a growing body of Q 2 3À7 ffi Lloyd and many others, ex- theoretical work proposing e cient quan- ploits the inherent behavior of one quan- tum simulations of chemical Hamiltonians, tum system as a resource to simulate e.g., refs 14À24. Publication Date (Web): April 29, 2015 | doi: 10.1021/acsnano.5b01651 another quantum system. Indeed, there In contrast, experimental realizations of Downloaded by HARVARD UNIV on September 3, 2015 | http://pubs.acs.org have been several experimental demon- quantum simulations of quantum chemistry strations of quantum simulators in various problems are still limited to small-scale architectures including quantum optics, demonstrations and are only performed in trapped ions, and ultracold atoms.8 The liquid-state NMR and photonic systems. importance of quantum simulators applied First experiments demonstrated the sim- to electronic structure problems has been ulation of the electronic structure of molec- detailed in several recent review articles ular hydrogen using quantum optics15 and including refs 9À13 and promises a revolu- liquid-state NMR.25 Recently, the energy tion in areas such as materials engineering, of another molecule, the helium hydride * Address correspondence to drug design, and the elucidation of bio- cation, was calculated in a photonic system [email protected], chemical processes. using a quantum variational eigensolver [email protected], [email protected]. The computational cost of solving the full algorithm.26 Besides the electronic struc- Schrödinger equation of molecular systems ture, simulation of chemical reaction dy- Received for review March 17, 2015 using any known method on a classical namics on an eight-site lattice was per- and accepted April 23, 2015. 27 computer scales exponentially with the formed in NMR. Published online April 23, 2015 number of atoms involved. However, it has Recently, the nitrogen-vacancy (NV) cen- 10.1021/acsnano.5b01651 been proposed that this calculation could ters in diamond attracted significant atten- be done efficiently on a quantum computer, tion due to its unique optical and spin C 2015 American Chemical Society WANG ET AL. VOL. 9 ’ NO. 8 ’ 7769–7774 ’ 2015 7769 www.acsnano.org ARTICLE properties.28 The NV center consists of a substantial nitrogen atom at the carbon site and an adjacent vacancy. Its negative charge state forms a spin triplet ground state, with ms = 0 and ms = ( 1 sublevels that are separated by a zero-field splitting of D ≈ 2.87 GHz. This spin system can be initialized and read out via optical pumping and spin-dependent fluorescence. The NV center therefore does not suffer from signal losses with increasing system size like NMR and avoids challenges such as the need for high-fidelity single- photon sources and detectors that are still beyond the present capabilities in optical quantum systems.29 Progress to date demonstrates that the NV centers are among the most accurate and most controllable candidates for quantum information processing.30À41 Milestone demonstrations include high-fidelity ini- Figure 1. Calculation of HeHþ molecular energy with NV þ tialization and readout,30À33 heralded generation spin register in diamond. (a) HeH , molecule to be simu- of entanglement,33À38 implementation of quantum lated. It consists of a hydrogen and a helium nucleus and two electrons. The distance (bond length) between the 38,42,43 40 control, ultralong spin coherence time, non- nuclei is denoted by R. Dot-dashed line, straight line, and volatile memory,41 quantum error correction,33,39 as dotted arrows indicate the nucleusÀnucleus, electronÀ À well as a host of metrology and sensing experi- nucleus, and electron electron Coulomb interactions, re- spectively. (b) Nitrogen-vacancy center in diamond, used as 44,45 ments. Several proposals to scale up the size of a quantum simulator. The electron spin is used for simulation NV systems currently exist, e.g., refs 38 and 46. This and the nuclear spin as the probe qubit for energy readout. makes the NV center an ideal candidate for a scalable (c) Energy level diagram for the coupled spin system formed by the NV electron spin and associated 14N nuclear spin. quantum simulator. Optical transitions between ground and excited state are Here, we demonstrate the quantum simulation of an used to initialize and measure the electron spin state. electronic structure with the NV center at ambient conditions. We use a quantum phase estimation contracted Gaussian orbitals. After taking symmetries algorithm47 to enhance the simulations precision. into account, the Hamiltonian can be represented as 1 Our experimentally computed energy agrees well with a3Â 3 matrix in the basis (Ψ1, Ψ6, /21/2(Ψ3 ÀΨ4)). Each the corresponding classical calculations within chemi- term of the Hamiltonian in the single particle basis À14 cal precision and a deviation of 1.4 Â 10 hartree. (e.g., Æχi|(Te þ VeN)|χjæ) is precomputed classically at Furthermore, we obtain the molecular electronic po- each internuclear separation R using the canonical spin tential energy surfaces by performing the simulation orbitals found via the HartreeÀFock (HF) procedure for different distances of the atoms. which often scales as a third order polynomial in the number of basis functions. RESULTS AND DISCUSSION After obtaining Hsim through this (typically) efficient Publication Date (Web): April 29, 2015 | doi: 10.1021/acsnano.5b01651 The chemical system we consider in this paper is the classical computation, we perform the quantum simu- þ Downloaded by HARVARD UNIV on September 3, 2015 | http://pubs.acs.org helium hydride cation, HeH (Figure 1a), believed to be lation of this molecule on a single-NV register, which the first molecule in the early universe.48 While HeHþ is consists of an electronic spin-1 and an associated 14N isoelectronic (i.e., has the same number of electrons) nuclear spin-1 forming a qutrit pair (Figure 1b). The with the previously studied molecular hydrogen, the electronic spin-1 of the NV system acts as the simula- reduced symmetry requires that we simulate larger tion register through mapping the molecular basis 1 subspaces of the full configuration interaction (FCI) (Ψ1, Ψ6, /21/2(Ψ3 ÀΨ4)) onto its ms = (1, 0, À1) states. Hamiltonian Hsim. Specifically, we consider Such a compact mapping is more efficient in that the states of simulated system and of the simulation H ¼ T þ W þ V (R) þ E (R) (1) sim e ee eN N system are simply enumerated and equated. The 14N in a minimal single particle basis with one site per nuclear spin-1 is used as the probe register to read out atom. Here, Te and Wee are the kinetic and Coulomb the energies using the iterative phase estimation 47 operators for the electrons, VeN is the electronÀnuclear algorithm (IPEA), as shown in Figure 1c. The simula- interaction, and EN is the nuclear energy due to the tion is realized by three steps: (i) preparation of the Coulomb interaction between the hydrogen and the system into an ansatz state |ψæ, which is close to an helium atom. The last two terms depend on the inter- eigenstate of the simulated Hamiltonian Hsim; (ii) evo- nuclear distance R. lution of the simulation register under the molecular In this work, we consider the singlet (S = 0) sector of Hamiltonian Hsim to generate phase shift on the probe the electronic Hamiltonian in a minimal single-electron register; and (iii) readout of the phase shift on the probe basis consisting of a single site at each atom given by register to extract the molecular energy (Figure 2a). WANG ET AL. VOL. 9 ’ NO. 8 ’ 7769–7774 ’ 2015 7770 www.acsnano.org ARTICLE Figure 2. Energy readout through quantum phase estimation algorithm. (a) Experimental implementation of the IPEA algorithm. The controlled gate U0 is realized using optimal control. The x, y phases in the last π/2 pulse measure the real and imaginary parts of the signal, respectively, which yield the sign of the measured energy. The number of repetitions N =10kÀ1 depends on the iteration k. (b) Experimental results of iterative phase estimation algorithm to enhance the precision of measured energy for the case of R = 90 pm.