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 -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 1014 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 37 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 1424. 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 | 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 , 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 using quantum optics15 and including refs 913 and promises a revolu- liquid-state NMR.25 Recently, the energy

tion in areas such as materials engineering, of another , 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 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

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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.3041 Milestone demonstrations include high-fidelity ini- Figure 1. Calculation of HeHþ molecular energy with NV þ tialization and readout,3033 heralded generation spin register in diamond. (a) HeH , molecule to be simu- of entanglement,3338 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 nucleusnucleus, electron 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 a3 3 matrix in the basis (Ψ1, Ψ6, /21/2(Ψ3 Ψ4)). Each the corresponding classical calculations within chemi- term of the Hamiltonian in the single particle basis 14 cal precision and a deviation of 1.4 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 HartreeFock (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 | 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 Ψ4)) onto its ms = (1, 0, 1) 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 electronnuclear 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).

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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 =10k1 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. The Fourier spectrum of the first iteration (k =1)fixes the energy roughly between 10 and 0 hartree. The precision is then improved iteratively by narrowing down the energy range. In each iteration, the energy range is divided into 10 equal segments. The red area indicates the energy range for the next iteration. After each iteration at least one decimal digit, denoted by the number in the red area, is resolved. Note that the value here is offset by tr(Hsim)/3 (see the Supporting Information for details). (c) The uncertainty of the measured energy as a function of the iteration number.

In the first step, an ansatz state is prepared that has where the trial state is expressible as a superposition

an overlap with the corresponding eigenstate |enæ that of all the Hsim eigenstates |τæ = ∑kak|ekæ.Thereduced decreases at most polynomially as the system size density matrix of the probe register grows. The phase estimation algorithm49 can then be 0 1 2 1 ∑ ja j e iEk t used to project the ansatz state into the exact eigen- 1B k C F (t) ¼ @ k A (3) state with sufficiently high probability. One possible probe 2 iEk t 2 ∑ jakj e 1 approach to realize this requirement is to use adiabatic k 14,25,50 state preparation, the performance of which ff

Publication Date (Web): April 29, 2015 | doi: 10.1021/acsnano.5b01651 obtains a phase shift (o -diagonal elements) that con- depends on the energy gap during the entire evolution Downloaded by HARVARD UNIV on September 3, 2015 | tains the information about the energies. Finally, the process. An alternative approach is to approximate the phase information is transferred to the electron spin for eigenstate with a trial state. In our demonstration, readout by a nuclear spin π/2-pulse, followed immedi- the simulation register is initialized in a trial state ately by selective π-pulses on the electron spin. To τæ ∈ { þ æ æ} | | 1 ,| 1 . Each state is easily found to be close measure the energy precisely, we perform a classical to one eigenstate of the molecular Hamiltonian Hsim by Fourier analysis on the signal for different evolution looking at its matrix representation. In more general times (ts,2ts,...,Lts). This readout method can help to cases such trial states can often be prepared based on 2 resolve the probability |ak| of each eigenstate |ekæ and classical approximate methods. The probe register is approximate the corresponding energy Ek.Wechoose first initialized into state |0æ and then prepared in the 1 ts such that the sampling rate /t >|En|/π.Anexample ψ æ æ þ æ 1/2 s superposition state | (0) = (|0 | 1 )/2 through a for the experimental Fourier spectrum are shown in π /2 pulse to obtain a phase shift in the evolution. Figure 2b. The position of the peak indicates the ff In the next step, a controlled-U(t)gatefordi erent eigenvalue of the molecular Hamiltonian. times t,whereU(t) = exp( iHsim t), is applied on the To enhance the precision of the energy eigenvalues, simulation register to encode the energies into a relative an iterative phase estimation algorithm is performed. phase of the probe register, resulting in the state A central feature of this algorithm includes repeating the unitary operator U to increase readout precision. 1 iEk t jψ(t)æ ¼ pffiffiffi ∑ ak(j0æ þ e j1æ)jekæ (2) Expressing the energy as a string of decimal digits, Ek = 2 k x1, x2,x3 ..., the first digit x1 can be determined by the

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Figure 3. Energy surfaces of the HeHþ molecule. The energy surface of the second excited state can be obtained by subtracting energies of the ground and first excited states from the trace of Hsim and is not shown. All of the measured energies are obtained in five iterations.

first-round phase estimation process. Once x1 is known, the ground and first excited-state energies from the

the second digit x2 can be iteratively determined by trace of Hsim. The potential energy surfaces can be used implementing the unitary operator Up, where p = 10. to compute key molecular properties such as ioniza- For the kth iteration, p =10k1. An increasingly precise tion energies and vibrational energy levels. An impor- energy can be obtained through continued iterations. tant example is the equilibrium geometry: we found However, in practice, the repetitions and therefore the the minimal energy for the ground state, 2.86269 iterations are fundamentally limited by the coherence hartree, at a bond length of 91.3 pm. In addition, time of the quantum system. Moreover, the accumu- we obtained a binding energy of 0.07738 hartree in lated gate errors become a dominant limitation of the our basis. energy precision as the repetitions increase. To avoid To improve the accuracy of our results we would need such shortcomings, in our demonstration the time to simulate the system in a larger basis. Recent advances evolution operators Up are realized and optimized with in this direction show that we can enlarge the size of optimal control theory, which can overcome several the NV spin system by either including more coupled difficult features found when scaling up the register nuclear spins associated with single NV electron size.38 Although it cannot be applied in large registers spin33,39 or correlating several electron spins mediated to generate the quantum gates directly, it can be used through optical photons37 or through magnetic inter- to generate flexible smaller building blocks, ensuring actions between them.36 The former method is able to high-fidelity control in future large-scale applications. provide ∼6 nuclear spins for single NV node, and the In the present case, the method is unscalable because latter method currently is limited to two electron we compute the unitary propagator using a classical spins.36,37 In both hybrid-spin systems the control is computer. However, by using a Trotter-type gate se- shown to be universal.33,36,37,39 Combining these two quence to implement the propagators, e.g.,16 this can methods in principle could enlarge the NV system to

Publication Date (Web): April 29, 2015 | doi: 10.1021/acsnano.5b01651 be designed with polynomially scaling. a 10-qubit quantum processor which is enough to

Downloaded by HARVARD UNIV on September 3, 2015 | Figure 2b,c show the results of such an iterative perform more complicated tasks. In these large systems, process in the case of internuclear distance R =90pm the simulated propagators can be implemented using with trial state |þ1æ. As the iterations increase, more Trotter sequences and should be accompanied by error precise decimal digits of the ground-state energy correction. Optimal control methods, as we have de- are resolved. After 13 repetitions, the molecular en- monstrated here, should prove necessary to perform

ergy, with an offset tr(Hsim)/3, is extracted to be these tasks with satisfactory precision. 1.020170538763387 ( 8 1015 hartree, very close CONCLUSION to the theoretical value, which is 1.020170538763381 hartree, with an uncertainty of (1.4 1014 hartree. We have demonstrated the most precise quantum To the best of knowledge, our results are four times simulation of molecular energies to date, which repre- more accurate than the previous record.25 sents an important step toward the advanced level of Once the energies have been measured, we can control required by future quantum simulators that will obtain the potential energy surface of the molecule outperform classical methods. The energies we ob- by repeating the procedure for different distances R tained for the helium hydride cation surpass chemical (Figure 3). The ground-state energy surface is obtained precision by 10 orders of magnitude (with respect to with the trial state |þ1æ, and the first excited state the basis). The accuracy of our results can be increased energy surface is obtained with the trial state |1æ.We by using a larger, more flexible single-particle basis set, obtain the remaining eigenenergy (of the second ex- but this will require a larger quantum simulator that cited state) without further measurement by subtracting eventually will require error correction schemes.18

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Our study presents evidence that quantum simula- we took was based on iterative phase estimation47 and tors can be controlled well enough to recover increas- optimal control decompositions38;these will form ingly precise data. The availability of highly accurate key building blocks for any solid-state quantum simu- energy eigenvalues of large is presently far lator. Even more generally, this study would suggest out of reach of existing computational technology, and that the techniques presented here should be em- quantum simulation could open the door to a vast ployed in any future simulator that will outperform range of new technological applications. The approach classical simulations of electronic structure calculations.

METHODS. and the Corning Foundation. J.W. acknowledges support by the Computation of Molecular Hamiltonians. The full configuration EU via IP SIQS and the ERC grant SQUTEC as well as the DFG via interaction Hamiltonian is a sparse matrix, and each matrix the research group 1493 and SFB/TR21 and the Max Planck element can be computed in polynomial time. The N-electron Society. J.D.W. thanks the VCQ and Ford postdoctoral fellow- Hamiltonian is asymotpically sparse. For a basis set with M ships for support. We thank Mauro Faccin and Jacob Turner for orbitals, there are M4 terms in the Hamiltonian but the providing valuable feedback regarding the manuscript. Hamiltonian is of size ((M!)/(N!(M N)!)) ≈ MN which is exponential as the number of electrons grow. To generate the Supporting Information Available: Detailed information Hamiltonian, we fix the nuclear configuration and then compute about the initialization of spin register and realization of con- the necessary one- and two-body integrals which parametrize trolled gate are provided. The Supporting Information is avail- the FCI matrix at each fixed bond length in the standard STO-3G able free of charge on the ACS Publications website at DOI: basis,51 using the PSI3 electronic structure package.52 The mini- 10.1021/acsnano.5b01651. mal basis HeHþ system has two spatial orbitals which we denote as g(r)ande(r) and two spin functions denoted as R(σ)andβ(σ) REFERENCES AND NOTES which are eigenstates of the Sz operator. We combine these to χ R χ β σ χ R σ form four spin orbitals, 1= g(r) (r), 2 = g(r) ( ), 3 = e(r) ( ), 1. Feynman, R. P. Simulating Physics with Computers. Int. J. χ β σ and 4 = e(r) ( ). There are six possible two-electron Slater Theor. Phys. 1982, 21, 467–488. Ψ A χ χ Ψ A χ χ Ψ A χ χ Ψ determinants, 1 = ( 1 2), 2 = ( 1 3), 3 = ( 1 4), 4 = 2. Lloyd, S. Universal Quantum Simulators. Science 1996, 273, A χ χ Ψ A χ χ Ψ A χ χ ( 2 3), 5 = ( 2 4), and 6 = ( 3 4). More explicitly 1073–1078.

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