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,H =Te+Ve+TN +VN+VeN,
is a sum of the kinetic energies of the electronsTe ≡ −~
2
2m P
i∇2e,i and nuclei TN ≡
−Pi ~
2
2Mi∇ 2
P
j>iZiZje2/|Ri−Rj|, and electron-nucleiVeN ≡ −Pi,jZje2/|ri−Rj|potential
energies, where r andR respectively refer to the electronic and nuclear coordinates. In many cases, it is more convenient to work on the second-quantization represen- tation for quantum chemistry. The advantage is that one can choose a good fermionic basis set of molecular orbitals, |pi = c†p|vaci, which can compactly capture the low-
energy sector of the chemical system. This kind of second quantized fermionic Hamil- tonians are efficiently simulatable in trapped ions [43]. To be more specific, we will choose first M > N orbitals for anN-electron system. Denoteφp(r) ≡ hr|pias the
single-particle wavefunction corresponding to mode p. The electronic part, He(R) ≡ Te+VeN(R) +Ve, of the Hamiltonian H can be expressed as follows:
He(R) = X pq hpqc†pcq+ 1 2 X pqrs hpqrsc†pc † qcrcs, (3.1)
wherehpq is obtained from the single-electron integral
hpq ≡ −
Z
drφ∗p(r) (Te+VeN)φq(r), (3.2)
and hpqrs comes from the electron-electron Coulomb interaction,
hpqrs ≡
Z
dr1dr2φ∗p(r1)φ∗q(r2)Ve(|r1−r2|)φr(r2)φs(r1). (3.3)
We note that the total number of terms in He is O(M4); typicallyM is of the same
order as N. Therefore, the number of terms in He scales polynomially inN, and the
integrals {hpq, hpqrs} can be numerically calculated by a classical computer with poly-
nomial resources [6].
with a trapped-ion quantum simulator, one should take into account the fermionic na- ture of the operatorscp 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 [188]. Specifically, for each fermionic modep, an unoccupied state |0ip is represented by the spin-down state|↓ip, and an occupied state|1ip is repre-
sented by the spin-up state |↑ip. The exchange symmetry is enforced by the Jordan- Wigner transformation: c†p = (Qm<pσmz)σp+ and cp = (Qm<pσzm)σ−p, whereσ± ≡
(σx±iσy)/2. Consequently, the electronic Hamiltonian in Eq. (3.1) becomes highly
nonlocal in terms of the Pauli operators {σx, σy, σz}, i.e.,
He −→ JWT X i,j,k...∈{x,y,z} gijk... σ1i ⊗σ2j⊗σk3... . (3.4)
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 em- ployed as a qubit. Ions can be confined either in Penning traps or radio frequency Paul traps [138], and cooled down to form crystals. Through sideband cooling the ions motional degrees of freedom can reach the ground state of the quantum Har- monic 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 prepa- ration, single- and two-qubit gates, and detection, are all above 99% [95].
The basic interaction of a two-level trapped ion with a single-mode laser is given by [95], H = ~Ωσ+e−i(∆t−φ)exp(iη[ae−iωtt + a†eiωtt]) + H.c., whereσ± are the
of the considered motional mode, and Ω is the Rabi frequency associated to the laser strength. η = kz0 is the Lamb-Dicke parameter, with kthe wave vector of the laser andz0 =
p
~/(2mωt) the ground state width of the motional mode. φis a control-
lable laser phase and ∆ the laser-atom detuning.
In the Lamb-Dicke regime where ηph(a+a†)2i 1, the basic interaction of a two-level trapped ion with a laser can be rewritten as H = ~Ω[σ+e−i(∆t−φ) +
iησ+e−i(∆t−φ)(ae−iωtt+a†eiωtt) + H.c.]
By adjusting the laser detuning ∆, one can generate the three basic ion-phonon in- teractions, namely: the carrier interaction (∆ = 0), Hc=~Ω(σ+eiφ+σ−e−iφ), the red
sideband interaction, (∆ = −ωt), Hr = i~ηΩ(σ+aeiφ−σ−a†e−iφ), and the blue side-
band interaction, (∆ =ωt),Hb=i~ηΩ(σ+a†eiφ−σ−ae−iφ). By combining detuned red
and blue sideband interactions, one obtains the Mølmer-Sørensen gate [174], 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 He in Eq. (3.4), that will allow one
to simulate arbitrary quantum chemistry systems.