From transistor to trapped-ion computers for quantum chemistry

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

Q uantum chemistry represents one of the most successful applications of quantum mechanics. It provides an excellent platform for understanding matter from atomic to molecular scales, and involves heavy interplay of experimental and theoretical methods. In 1929, shortly after the completion of the basic structure of the quantum theory, Dirac speculated 1 that the fundamental laws for chemistry were completely known, but the application of the fundamental laws led to equations that were too complex to be solved. About ninety years later, with the help of transistor-based digital computers, the development of quantum chemistry continues to flourish, and many powerful methods, such as Hartree-Fock, configuration interaction, density functional theory, coupled-cluster, and quantum Monte Carlo, have been developed to tackle the complex equations of quantum chemistry (see e.g. for a historical review 2 ). However, as the system size scales up, all of the methods known so far suffer from limitations that make them fail to maintain accuracy with a finite amount of resources 3 . In other words, quantum chemistry remains a hard problem to be solved by the current computer technology.
As envisioned by Feynman 4 , one should be able to efficiently solve problems of quantum systems with a quantum computer. Instead of solving the complex equations, this approach, known as quantum simulation (see the recent reviews in Refs. [5][6][7], aims to solve the problems by simulating target systems with another controllable quantum system, or qubits. Indeed, simulating many-body systems beyond classical resources will be a cornerstone of quantum computers. Quantum simulation is a very active field of study and various methods have been developed. Quantum simulation methods have been proposed for preparing specific states such as ground [8][9][10][11][12][13] and thermal states [14][15][16][17][18][19][20] , simulating time evolution [21][22][23][24][25][26][27] , and the measurement of physical observables [28][29][30][31] . Trapped-ion systems (see Fig. 1) are currently one of the most sophisticated technologies developed for quantum information processing 32 . These systems offer an unprecedented level of quantum control, which opens new possibilities for obtaining physico-chemical information about quantum chemical problems. The power of trapped ions for quantum simulation is manifested by the high-precision control over both the internal degrees of freedom of the individual ions and the phonon degrees of freedom of the collective motions of the trapped ions, and the high-fidelity initialization and measurement 32,33 . Up to 100 quantum logic gates have been realized for six qubits with trapped ions 22 , and quantum simulators involving 300 ions have been demonstrated 34 .
In this work, we present an efficient toolkit for solving quantum chemistry problems based on the state-of-theart in trapped-ion technologies. The toolkit comprises two components i) First, we present a hybrid quantumclassical variational optimization method, called quantum-assisted optimization, for approximating both ground-state energies and the ground-state eigenvectors for electronic problems. The optimized eigenvector can then be taken as an input for the phase estimation algorithm to project out the exact eigenstates and hence the potential-energy surfaces (see Fig. 2). Furthermore, we extend the application of the unitary coupled-cluster method 35 . This allows for the application of a method developed for classical numerical computations in the quantum domain. ii) The second main component of our toolkit is the optimized use of trapped-ion phonon degrees of freedom not only for quantum-gate construction, but also for simulating molecular vibrations, representing a mixed digital-analog quantum simulation. The phonon degrees of freedom in trapped-ion systems provide a natural platform for addressing spin-boson or fermion-boson-type problems through quantum simulation 23,[36][37][38][39][40] . It is noteworthy to mention that, contrary to the continuous of modes required for full-fledged quantum field theories, quantum simulations of quantum chemistry problems could reach realistic conditions for finite bosonic and fermionic mode numbers. Consequently, trapped ions can be exploited to solve dynamical problems involving linearly or non-linearly coupled oscillators, e.g., spin-boson models 41,42 , that are difficult to solve either analytically or numerically with a classical computer. Furthermore, we have also developed a novel protocol to measure correlation functions of observables in trapped ions that will be crucial for the quantum simulation of quantum chemistry.

Results and Discussion
Trapped ions for quantum chemistry. Quantum chemistry deals with the many-body problem involving electrons and nuclei. Thus, it is very well suited for being simulated with trapped-ion systems, as we will show below. The full quantum chemistry Hamiltonian, i,j Z j e 2 /jr i 2 R j j potential energies, where r and R respectively refer to the electronic and nuclear coordinates.
In many cases, it is more convenient to work on the second-quantization representation for quantum chemistry. The advantage is that one can choose a good fermionic basis set of molecular orbitals, p j i~c { p vac j i, which can compactly capture the low-energy sector of the chemical system. This kind of second quantized fermionic Hamiltonians are efficiently simulatable in trapped ions 23 . To be more specific, we will choose first M . N orbitals for an N-electron system. Denote w p (r) ; AErjpae as the single-particle wavefunction corresponding to mode p. The electronic part, H e (R) ; T e 1 V eN (R) 1 V e , of the Hamiltonian H can be expressed as follows: where h pq is obtained from the single-electron integral h pq : { ð drw Ã p r ð Þ T e zV eN ð Þ w q r ð Þ, and h pqrs comes from the electron-electron Coulomb interaction, h pqrs : We note that the total number of terms in H e is O(M 4 ); typically M is of the same order as N. Therefore, the number of terms in H e scales polynomially in N, and the integrals {h pq , h pqrs } can be numerically calculated by a classical computer with polynomial resources 9 .
To implement the dynamics associated with the electronic Hamiltonian in Eq. (1) with a trapped-ion quantum simulator, one should take into account the fermionic nature of the operators c p and c { q . We invoke the Jordan-Wigner transformation (JWT), which is a method for mapping the occupation representation to the spin (or qubit) representation 43 . Specifically, for each fermionic mode p, an unoccupied state j0ae p is represented by the spin-down state j#ae p , and an occupied state j1ae p is represented by the spin-up state j"ae p . The exchange symmetry is enforced by Nevertheless, the simulation can still be made efficient with trapped ions, as we shall discuss below. In trapped-ion physics two metastable internal levels of an ion are typically employed as a qubit. Ions can be confined either in Penning traps or radio frequency Paul traps 33 , and cooled down to form crystals. Through sideband cooling the ions motional degrees of freedom can reach the ground state of the quantum Harmonic oscillator, that can be used as a quantum bus to perform gates among the different ions. Using resonance fluorescence with a cycling transition quantum non demolition measurements of the qubit can be performed. The fidelities of state preparation, single-and two-qubit gates, and detection, are all above 99% 32 .
The basic interaction of a two-level trapped ion with a single-mode laser is given by 32 , where s 6 are the atomic raising and lowering operators, a (a { ) is the annihilation (creation) operator of the considered motional mode, and V is the Rabi frequency associated to the laser strength. g 5 kz 0 is the Lamb-Dicke parameter, with k the wave vector of the laser and z 0~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h= 2mv t ð Þ p the ground state width of the motional mode. w is a controllable laser phase and D the laser-atom detuning.
In the Lamb-Dicke regime where g  44 , which is the basic building block for our methods. With combinations of this kind of gates, one can obtain dynamics as the associated one to H e in Eq. (2), that will allow one to simulate arbitrary quantum chemistry systems.
Quantum-assisted optimization. Quantum-assisted optimization 45 (see also Fig. 2) for obtaining ground-state energies aims to optimize the use of quantum coherence by breaking down the quantum simulation through the use of both quantum and classical processors; the quantum processor is strategically employed for expensive tasks only.
To be more specific, the first step of quantum-assisted optimization is to prepare a set of quantum states {jy l ae} that are characterized by a set of parameters {l}. After the state is prepared, the expectation value E l ; AEy l j H jy l ae of the Hamiltonian H will be measured directly, without any quantum evolution in between. Practically, the quantum resources for the measurements can be significantly reduced when we divide the measurement of the Hamiltonian H 5 X i H i into a polynomial number of small pieces AEH i ae (cf Eq. (2)). These measurements can be performed in a parallel fashion, and no quantum coherence is needed to maintain between the measurements (see Fig. 2a and 2b). Then, once a data point of E l is obtained, the whole procedure is repeated for a new state y 0 l È É with another set of parameters {l9}. The choice of the new parameters is determined by a classical optimization algorithm that aims to minimize E l (see Methods). The optimization procedure is terminated after the value of E l converges to some fixed value.
Finally, for electronic Hamiltonians H e (R), the optimized state can then be sent to a quantum circuit of phase estimation algorithm to produce a set of data point for some R on the potential energy surfaces (Fig. 2c shows the 1D case). After locating the local minima of the ground and excited states, vibronic coupling for the electronic structure can be further studied (see Supplementary Material).
The performance of quantum-assisted optimization depends crucially on (a) the choice of the variational states, and (b) efficient measurement methods. We found that the unitary coupled-cluster (UCC) states 35 are particularly suitable for being the input state for quantum-assisted optimization, where each quantum state jy l ae can be prepared efficiently with standard techniques in trapped ions. Furthermore, efficient measurement methods for H e are also available for trapped ion systems. We shall discuss these results in detail in the following sections.
Unitary coupled-cluster (UCC) ansatz. The unitary coupled-cluster (UCC) ansatz 35 assumes electronic states jyae have the following form, y j i~e T{T { W j i, where jWae is a reference state, which can be, e.g., a Slater determinant constructed from Hartree-Fock molecular orbitals. The particle-hole excitation operator, or cluster operator T, creates a linear combination of excited Slater determinants from jWae. Usually, T is divided into subgroups based on the particle-hole rank. More precisely, The key challenge for implementing UCC on a classical computer is that the computational resource grows exponentially. It is because, in principle, one has to expand the expressionH:e T { {T He T{T { into an infinity series, using the Baker-Campbell-Hausdorff expansion. Naturally, one has to rely on approximate methods 35,46 to truncate the series and keep track of finite numbers of terms. Therefore, in order to make good approximations by perturbative methods, i.e., assuming T is small, one implicitly assumes that the reference state jWae is a good solution to the problem. However, in many cases, such an assumption is not valid and the use of approximate UCC breaks www.nature.com/scientificreports SCIENTIFIC REPORTS | 4 : 3589 | DOI: 10.1038/srep03589 down. We explain below how implementing UCC on a trapped-ion quantum computer can overcome this problem.
Implementation of UCC through time evolution. We can generate the UCC state by simulating a pseudo time evolution through Suzuki-Trotter expansion on the evolution operator e T{T { 21 . To proceed, we consider an N-electron system with M, where M . N, molecular orbitals (including spins). We need totally M qubits; the reference state is the Hartree-Fock state where N orbitals are filled, and M 2 N orbitals are empty, i.e, jWae 5 j000…0111…1ae. We also define an effective Hamiltonian K ; i (T 2 T { ), which means that we should prepare the state e 2iK jWae.
We decompose K into subgroups , which can be implemented by transforming into spin operators through Jordan-Wigner transformation. The time evolution for each term can be simulated with a quantum circuit involving many nonlocal controlled gates, which can be efficiently implemented with trapped ions as we shall see below.
Here R N (w) is defined as follows: for any m [ N, R N w ð Þ~e +iws z 1 for N 5 4m 6 1, and (ii) R N w ð Þ~e iws y 1 for N 5 4m, and (iii) R N w ð Þ~e {iws y 1 for N 5 4m 2 2. It is remarkable that the standard quantum-circuit treatment (e.g. see Ref. 47 ) for implementing each e {iH l t=n involves as many as 2N two-qubit gates for simulating N fermionic modes; in our protocol one needs only two Mølmer-Sørensen gates, which are straightforwardly implementable with current trapped-ion technology. Furthermore, the local rotation R N (w) can also include motional degrees of freedom of the ions for simulating arbitrary fermionic Hamiltonians coupled linearly to bosonic operators a k and a { k . Probing potential energy surfaces. In the Born-Oppenheimer (BO) picture, the potential energy surface E k R ð ÞzV N R ð Þ associated with each electronic eigenstate jw k ae is obtained by scanning the eigenvalues E k R ð Þ for each configurations of the nuclear coordinates {R}. Of course, we can apply the standard quantum phase estimation algorithm 49 that allows us to extract the eigenvalues. However, this can require many ancilla qubits. In fact, locating these eigenvalues can be achieved by the phase estimation method utilizing one extra ancilla qubit 12 corresponding, in our case, to one additional ion.

Measurement
This method works as follows: suppose we are given a certain quantum state jyae (which may be obtained from classical solutions with quantum-assisted optimization) and an electronic Hamiltonian H e (R) (cf. Eq. (1)). Expanding the input state, jyae 5 X k a k jw k ae, by the eigenstate vectors jw k ae of H e (R), where H e R ð Þ w k j i~E k R ð Þ w k j i, then for the input state j0ae jyae, the quantum circuit of the quantum phase estimation produces the following output state, The corresponding reduced density matrix, of the ancilla qubit contains the information about the weight (amplitude-square) ja k j 2 of the eigenvectors jw k ae in jyae and the associated eigenvalues v k in the off-diagonal matrix elements. All ja k j 2 's and v k 's can be extracted by repeating the quantum circuit for a range of values of t and performing a (classical) Fourier transform to the measurement results. The potential energy surface is obtained by repeating the procedure for different values of the nuclear coordinates {R}.
Numerical investigation. In order to show the feasibility of our protocol, we can estimate the trapped-ion resources needed to simulate, e.g., the prototypical electronic Hamiltonian H e~X h pq a { p a q z 1=2 ð Þ X h pqrs a { p a { q a r a s as described in Eq. (1), for the specific case of the H 2 molecule in a minimal STO-3G basis. This is a two-electron system represented in a basis of four spin-orbitals. The hydrogen atoms were separated by 0.75 Å , near the equilibrium bond distance of the molecule. The Hamiltonian is made up of 12 terms, that include 4 local ion operations and 8 non-local interactions. Each of the non-local terms can be done as a combination of two Mølmer-Sørensen (MS) gates and local rotations, as described in Table 1. Therefore, to implement the dynamics, one needs 16 MS gates per Trotter step and a certain number of local rotations upon the ions. Since p/2 MS gates can be done in , 50 ms, and local rotations can be performed in negligible times (, 1 ms) 22,32 , the total simulation time can be assumed of about 800 ms for the n51 protocol, 1.6 ms and 2.4 ms for the n52 and n53 protocols. Thus total simulation times are within the decoherence times for trappedion setups, of about 30 ms 32 . In a digital protocol performed on real www.nature.com/scientificreports SCIENTIFIC REPORTS | 4 : 3589 | DOI: 10.1038/srep03589 quantum systems, each gate is affected by an error. Thus, increasing the number of Trotter steps leads to an accumulation of the single gate error. To implement an effective quantum simulation, on one hand one has to increase the number of steps to reduce the error due to the digital approximation, on the other hand one is limited by the accumulation of the single gate error. We plot in Fig. 3a, 3b, 3c, the fidelity loss 1 2 jAEY S jY E aej 2 of the simulated state jY S ae versus the exact one jY E ae, for the hydrogen Hamiltonian, starting from the initial state with two electrons in the first two orbitals. We plot, along with the digital error, three horizontal lines representing the accumulated gate error, for n51, 2, 3 in each plot, considering a protocol with an error per Trotter step of~10 {3 (a),~10 {4 (b) and~10 {5 (c). To achieve a reasonable fidelity, one has to find a number of steps that fits the simulation at a specific time. The vertical lines and arrows in the figure mark the time regions in which the error starts to be dominated by the digital error. Trapped-ion twoqubit gates are predicted to achieve in the near future infidelities of 10 24 , thus making the use of these protocols feasible 50 . In Fig. 3d we plot the behavior of the energy of the system for the initial state j""##ae for the exact dynamics, versus the digitized one. Again, one can observe how the energy can be retrieved with a small error within a reduced number of digital steps.

Conclusions
Summarizing, we have proposed a quantum simulation toolkit for quantum chemistry with trapped ions. This paradigm in quantum simulations has several advantages: an efficient electronic simulation, the possibility of interacting electronic and vibrational degrees of freedom, and the increasing scalability provided by trapped-ion systems. This approach for solving quantum chemistry problems aims to combine the best of classical and quantum computation.

Methods
To implement the optimization with the UCC wavefunction ansatz on a trapped-ion quantum simulator, our proposal is to first employ classical algorithms to obtain approximate solutions 35,46 . Then, we can further improve the quality of the solution by searching for the true minima with an ion trap. The idea is as follows: first we create a UCC ansatz by the Suzuki-Trotter method described in the previous section. Denote this choice of the cluster operator as T (0) , and other choices as T (k) with k51, 2, 3, …. The corresponding energy E 0~W h je T 0 j i of the initial state is obtained by a classical computer.
Next, we choose another set of cluster operator T (1) which is a perturbation around T (0) . Define the new probe state w k j i:e T k In order to obtain measurement results for the operators w 1 h jc { p c { q c r c s w 1 j i, we will first convert the fermion operators into spin operators via Jordan-Wigner transformation; the same procedure is applied for creating the state jw 1 ae. The quantum measurement for the resulting products of Pauli matrices can be achieved efficiently with trapped ions, using the method we described.
The following steps are determined through a classical optimization algorithm. There can be many choices for such an algorithm, for example gradient descent method, Nelder-Mead method, or quasi-Newton methods. For completeness, we summarize below the application of gradient descent method to our optimization problem.
First we define the vector T k T to contain all coefficients in the cluster operator T (k) at the k-th step. We can also write the expectation value E (T (k) ) ; AEw k j H jw k ae for each step as a function of T (k) . The main idea of the gradient descent method is that E (T (k) ) decreases fastest along the direction of the negative gradient of E (T (k) ), 2=E (T (k) ). Therefore, the (k 1 1)-th step is determined by the following relation: where a k is an adjustable parameter; it can be different for each step. To obtain values of the gradient =E (T (k) ), one may use the finite-difference method to approximate the gradient. However, numerical gradient techniques are often susceptible to numerical instability. Alternatively, we can invoke the Hellman-Feynman theorem and get, e.g.,    Finally, as a valid assumption for general cases, we assume our parametrization of UCC gives a smooth function for E (T (k) ). Thus, it follows that E T 0 and eventually E (T (k) ) converges to a minimum value for large k. Finally, we can also obtain the optimized UCC quantum state.