Abstract
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 exponentiallygrowing 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 trappedion 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 stateoftheart trappedion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a nearfuture trappedionbased technology.
Introduction
Quantum 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 transistorbased digital computers, the development of quantum chemistry continues to flourish and many powerful methods, such as HartreeFock, configuration interaction, density functional theory, coupledcluster 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 manybody 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}.
Trappedion 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 physicochemical information about quantum chemical problems. The power of trapped ions for quantum simulation is manifested by the highprecision 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 highfidelity 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 stateoftheart in trappedion technologies. The toolkit comprises two components i) First, we present a hybrid quantumclassical variational optimization method, called quantumassisted optimization, for approximating both groundstate energies and the groundstate 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 potentialenergy surfaces (see Fig. 2). Furthermore, we extend the application of the unitary coupledcluster 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 trappedion phonon degrees of freedom not only for quantumgate construction, but also for simulating molecular vibrations, representing a mixed digitalanalog quantum simulation. The phonon degrees of freedom in trappedion systems provide a natural platform for addressing spinboson or fermionbosontype problems through quantum simulation^{23,36,37,38,39,40}. It is noteworthy to mention that, contrary to the continuous of modes required for fullfledged 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 nonlinearly coupled oscillators, e.g., spinboson 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 manybody problem involving electrons and nuclei. Thus, it is very well suited for being simulated with trappedion systems, as we will show below. The full quantum chemistry Hamiltonian, H = T_{e} + V_{e} + T_{N} + V_{N} + V_{eN}, is a sum of the kinetic energies of the electrons and nuclei and the electronelectron , nucleinuclei and electronnuclei 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 secondquantization representation for quantum chemistry. The advantage is that one can choose a good fermionic basis set of molecular orbitals, , which can compactly capture the lowenergy 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 Nelectron system. Denote ϕ_{p} (r) ≡ 〈rp〉 as the singleparticle wavefunction corresponding to mode p. The electronic part, H_{e}(R) ≡ T_{e} + V_{eN} (R) + V_{e}, of the Hamiltonian H can be expressed as follows:
where h_{pq} is obtained from the singleelectron integral and h_{pqrs} comes from the electronelectron Coulomb interaction, . 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 trappedion quantum simulator, one should take into account the fermionic nature of the operators c_{p} and . We invoke the JordanWigner 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 0〉_{p} is represented by the spindown state ↓〉_{p} and an occupied state 1〉_{p} is represented by the spinup state ↑〉_{p}. The exchange symmetry is enforced by the JordanWigner transformation: and , where σ^{±} ≡ (σ^{x} ± iσ^{y})/2. Consequently, the electronic Hamiltonian in Eq. (1) becomes highly nonlocal in terms of the Pauli operators {σ^{x}, σ^{y}, σ^{z}}, i.e.,
Nevertheless, the simulation can still be made efficient with trapped ions, as we shall discuss below.
In trappedion 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 twoqubit gates and detection, are all above 99%^{32}.
The basic interaction of a twolevel trapped ion with a singlemode laser is given by^{32}, , where σ_{±} are the atomic raising and lowering operators, a (a^{†}) is the annihilation (creation) operator of the considered motional mode and Ω is the Rabi frequency associated to the laser strength. η = kz_{0} is the LambDicke parameter, with k the wave vector of the laser and the ground state width of the motional mode. ϕ is a controllable laser phase and Δ the laseratom detuning.
In the LambDicke regime where , the basic interaction of a twolevel trapped ion with a laser can be rewritten as
By adjusting the laser detuning Δ, one can generate the three basic ionphonon interactions, namely: the carrier interaction (Δ = 0) , the red sideband interaction (Δ = −ω_{t}) and the blue sideband interaction (Δ = ω_{t}) . By combining detuned red and blue sideband interactions, one obtains the MølmerSørensen gate^{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.
Quantumassisted optimization
Quantumassisted optimization^{45} (see also Fig. 2) for obtaining groundstate 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 quantumassisted optimization is to prepare a set of quantum states {ψ_{λ}〉} that are characterized by a set of parameters {λ}. After the state is prepared, the expectation value E_{λ} ≡ 〈ψ_{λ} H ψ_{λ}〉 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 into a polynomial number of small pieces 〈H_{i}〉 (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_{λ} is obtained, the whole procedure is repeated for a new state with another set of parameters {λ′}. The choice of the new parameters is determined by a classical optimization algorithm that aims to minimize E_{λ} (see Methods). The optimization procedure is terminated after the value of E_{λ} 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 quantumassisted optimization depends crucially on (a) the choice of the variational states and (b) efficient measurement methods. We found that the unitary coupledcluster (UCC) states^{35} are particularly suitable for being the input state for quantumassisted optimization, where each quantum state ψ_{λ}〉 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 coupledcluster (UCC) ansatz
The unitary coupledcluster (UCC) ansatz^{35} assumes electronic states ψ〉 have the following form, , where Φ〉 is a reference state, which can be, e.g., a Slater determinant constructed from HartreeFock molecular orbitals. The particlehole excitation operator, or cluster operator T, creates a linear combination of excited Slater determinants from Φ〉. Usually, T is divided into subgroups based on the particlehole rank. More precisely, T = T_{1} + T_{2} + T_{3} + … + T_{N} for an Nelectron system, where , and so on.
Here creates an electron in the orbital a. The indices a, b label unoccupied orbitals in the reference state Φ〉 and i, j label occupied orbitals. The energy obtained from UCC, namely is a variational upper bound of the exact groundstate energy.
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 expression into an infinity series, using the BakerCampbellHausdorff 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 Φ〉 is a good solution to the problem. However, in many cases, such an assumption is not valid and the use of approximate UCC breaks down. We explain below how implementing UCC on a trappedion 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 SuzukiTrotter expansion on the evolution operator ^{21}. To proceed, we consider an Nelectron system with M, where M > N, molecular orbitals (including spins). We need totally M qubits; the reference state is the HartreeFock state where N orbitals are filled and M − N orbitals are empty, i.e, Φ〉 = 000…0111…1〉. We also define an effective Hamiltonian K ≡ i (T − T^{†}), which means that we should prepare the state e^{−iK} Φ〉.
We decompose K into subgroups K = K_{1} + K_{2} + K_{3} + … + K_{P}, where P ≤ N and . We now write e^{−iK} = (e^{−iKδ})^{1/δ} for some dimensionless constant δ. For small δ, we have . Since each K_{j} contains N^{j}(M − N)^{j} terms of the creation c^{†} and annihilation c operators, we will need to individually simulate each term separately, e.g., and , which can be implemented by transforming into spin operators through JordanWigner 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.
Implementation of UCC and simulation of time evolution with trappedions
Our protocol for implementing the UCC ansatz requires the simulation of the smalltime t/n evolution of nonlocal product of Pauli matrices of the form: , where for i, j, k ∈ {x, y, z}. Note that for any Nspin interaction, the terms are equivalent to through local spin rotations, which are simple to implement on trapped ions. Such a nonlocal operator can be implemented using the multiparticle MølmerSørensen gate^{23,39}: U_{MS}(θ, φ) ≡ exp [−iθ(cos φS_{x} + sin φS_{y})^{2}/4], where is a collective spin operator. Explicitly,
Here R_{N}(ϕ) is defined as follows: for any , for N = 4m ± 1 and (ii) for N = 4m and (iii) for N = 4m − 2.
It is remarkable that the standard quantumcircuit treatment (e.g. see Ref.^{47}) for implementing each involves as many as 2N twoqubit gates for simulating N fermionic modes; in our protocol one needs only two MølmerSørensen gates, which are straightforwardly implementable with current trappedion technology. Furthermore, the local rotation R_{N}(ϕ) can also include motional degrees of freedom of the ions for simulating arbitrary fermionic Hamiltonians coupled linearly to bosonic operators a_{k} and .
Measurement of arbitrarilynonlocal spin operators
For any given state ψ〉, we show how to encode expectation value of products of Pauli matrices , where i, j, k ∈ {x, y, z}, onto an expectation value of a single qubit. The idea is to first apply the unitary evolution of the form: , which as we have seen (cf Eq. 3) can be generated by trapped ions efficiently, to the state ψ〉 before the measurement. For example, defining , we have the relation
which equals for θ = π/4. Note that the application of this method requires the measurement of one qubit only, making this technique especially suited for trapped ion systems where the fidelity of the measurement of one qubit is 99.99%^{48}.
This method can be further extended to include bosonic operators in the resulting expectation values. For example, redefine and consider θ → θ (a + a^{†}) in Eq. (4). We can obtain the desired correlation through the derivative of the singlequbit measurement: . Note that the evolution operator of the form can be generated by replacing the local operation R_{N}(ϕ) in Eq. 3 with . This technique allows us to obtain a diverse range of correlations between bosonic and internal degrees of freedom.
Probing potential energy surfaces
In the BornOppenheimer (BO) picture, the potential energy surface associated with each electronic eigenstate ϕ_{k}〉 is obtained by scanning the eigenvalues 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.
This method works as follows: suppose we are given a certain quantum state ψ〉 (which may be obtained from classical solutions with quantumassisted optimization) and an electronic Hamiltonian H_{e}(R) (cf. Eq. (1)). Expanding the input state, , by the eigenstate vectors ϕ_{k}〉 of H_{e}(R), where , then for the input state 0〉 ψ〉, the quantum circuit of the quantum phase estimation produces the following output state, , where . The corresponding reduced density matrix,
of the ancilla qubit contains the information about the weight (amplitudesquare) α_{k}^{2} of the eigenvectors ϕ_{k}〉 in ψ〉 and the associated eigenvalues ω_{k} in the offdiagonal matrix elements. All α_{k}^{2}'s and ω_{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 trappedion resources needed to simulate, e.g., the prototypical electronic Hamiltonian as described in Eq. (1), for the specific case of the H_{2} molecule in a minimal STO3G basis. This is a twoelectron system represented in a basis of four spinorbitals. 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 nonlocal interactions. Each of the nonlocal terms can be done as a combination of two MølmerSørensen (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 π/2 MS gates can be done in ~ 50 μs and local rotations can be performed in negligible times (~ 1 μs)^{22,32}, the total simulation time can be assumed of about 800 μs for the n = 1 protocol, 1.6 ms and 2.4 ms for the n = 2 and n = 3 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 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 − 〈Ψ_{S}Ψ_{E}〉^{2} of the simulated state Ψ_{S}〉 versus the exact one Ψ_{E}〉, 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 n = 1, 2, 3 in each plot, considering a protocol with an error per Trotter step of (a), (b) and (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. Trappedion twoqubit gates are predicted to achieve in the near future infidelities of 10^{−4}, 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 ↑↑↓↓〉 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 trappedion 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 trappedion 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 SuzukiTrotter method described in the previous section. Denote this choice of the cluster operator as T^{(0)} and other choices as T^{(k)} with k = 1, 2, 3, …. The corresponding energy 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 . Then, the expectation value of the energy can be obtained by measuring components of the second quantized Hamiltonian, . Recall that the coefficients are all precomputed and known.
In order to obtain measurement results for the operators , we will first convert the fermion operators into spin operators via JordanWigner transformation; the same procedure is applied for creating the state ϕ_{1}〉. 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, NelderMead method, or quasiNewton methods. For completeness, we summarize below the application of gradient descent method to our optimization problem.
First we define the vector to contain all coefficients in the cluster operator T^{(k)} at the kth step. We can also write the expectation value E (T^{(k)}) ≡ 〈ϕ_{k} H ϕ_{k}〉 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)}), −∇E (T^{(k)}). Therefore, the (k + 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 finitedifference method to approximate the gradient. However, numerical gradient techniques are often susceptible to numerical instability. Alternatively, we can invoke the HellmanFeynman theorem and get, e.g., , which can be obtained with a method similar to that for obtaining E(T^{(k)}).
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 and eventually E (T^{(k)}) converges to a minimum value for large k. Finally, we can also obtain the optimized UCC quantum state.
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Acknowledgements
We thank J. Whitfield for insightful discussions. The authors acknowledge funding from Basque Government IT47210 Grant, Spanish MINECO FIS201236673C0302, Ramón y Cajal Grant RYC201211391, UPV/EHU UFI 11/55, SOLID, CCQED, PROMISCE and SCALEQIT European projects. M.H.Y. and A.A.G. acknowledge support from the Defense Threat Reduction Agency under grant HDTRA11010046DOD35CAP as well as the National Science Foundation under grant 1037992CHE, Sponsored by United States Department of Defense. M.H. Y. also acknowledges the funding from the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301 and the National Natural Science Foundation of China Grant 61033001, 61061130540. A. A.G. also acknowledges support from the Air Force Office of Scientific Research under award FA95501210046. J.M. is supported by the DOE Computational Science Graduate Fellowship under grant number DEFG0297ER25308. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the U.S. Government.
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M.H.Y., J.M. and A.A.G. are responsible for the parts involving quantum chemistry. J.C., A.M., L.L. and E.S. are responsible for the parts involving trapped ions. All authors contributed to the writing of the paper.
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Yung, MH., Casanova, J., Mezzacapo, A. et al. From transistor to trappedion computers for quantum chemistry. Sci Rep 4, 3589 (2014). https://doi.org/10.1038/srep03589
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DOI: https://doi.org/10.1038/srep03589
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