Efficient quantum gates and algorithms in an engineered optical lattice

In this work, trapped ultracold atoms are proposed as a platform for efficient quantum gate circuits and algorithms. We also develop and evaluate quantum algorithms, including those for the Simon problem and the black-box string-finding problem. Our analytical model describes an open system with non-Hermitian Hamiltonian. It is shown that our proposed scheme offers better performance (in terms of the number of required gates and the processing time) for realizing the quantum gates and algorithms compared to previously reported approaches.


STRUCTURE OF THE PHYSICAL SYSTEM A. Brief review of the experimental-setup of engineered lattice
Optical lattices, whcich are artificial crystals of light, are considered to be one of the key potential techniques for quantum simulation of matters. The optical lattice (or crystal) can be created by applying counter-propagating laser lights to ultracold atomic gas at low temperature, see Fig.(S1a). The laser beams produce a lattice-like potential that is with periodic spatial dependence. In this work, we consider the laser beams applied from four directions, and thus, a square Fig. S1. Simplified layout of the experimental-setup to create an artificial square lattice or crystal, and then the mott-insulator phase. Following, it performs the Raman beams to this lattice. two dimensional (2D) lattice is created, see Fig.(S1a). Such lattice can be conceived ideal with no impurities (or vibrations) by employing laser beams with highly quality and stability. The periodic potential is given by: L(y, z) = ∑ r=y,z L 0 sin 2 ( 2πr λ l ), where L 0 is the maximum potential depth of the laser and λ l = c ν l refers to the laser wavelength, c is the speed of light and ν l is the photon frequency. Consequently, the atoms are trapped in a yz-plane and frozen along the whereâ vk refers to the lowering operator of bosonic (or fermionic) atom of spin state k at site v, n v↑ =â † v↑â v↑ is the number operator for the boson (or fermion) atoms, ℘ refers to the sites number, µ k indicates the chemical potential of the interacting atoms, g k = dr * vk (r)[p 2 /2m + L(r)] jk (r) denotes the tunneling matrix element between adjacent sites v, j, jk is the Wannier state, p = (p y ,p z ) is the momentum operator vector, and Ω kk = (4π sh 2 /m) dr| vk | 2 | vk | 2 expresses the interaction strength (or the repulsion) of two atoms of an individual site between species k, k = (↑, ↓), and s is the scattering length between the atoms. Given a short-range interactions, compared to the space of lattice, the interaction strength between atoms can be described by Ω kk . Here, the first term of Eq.(S1) (i.e.,n vk (n vk − 1)) vanishes for fermion atoms. In (S1), we note that, g * ↑ = g ↓ , g * ↓ = g ↑ , µ ↑(↓) = +(−)µ, Ω ↑↑ = Ω ↑↓ , Ω ↓↓ = Ω ↓↑ , Ω ↑↓ = Ω ↓↑ ,k = −k,â −v↑ = −â v↓ andâ −v↓ = −â v↑ .

B. Synthesis of SOC and ZF mechanisms inside lattice
In this section, the Hamiltonian of 2D crystal lattice, that incorporates spin-orbit coupling (SOC) and 2D Zeeman field (ZF) mechanisms, is developed . Herein, the SOC mechanism indicates the intrinsic interaction between the neutral atoms spin and its motion. Experimentally, the SOC and ZF phenomena are realized by applying external Raman laser to the confined atoms in 2D lattice (yz-plane), see Fig.(S1e). One of the components of 2D ZF is perpendicular to SOC, and the other is parallel to SOC. Such experiment indicate that the Raman beams can be employed to dress atoms with spin-dependent momentums [3]. Here, each lattice site is a sample of two hyperfine ground states that can be coupled with an excited state by utilizing Raman beams with proper Rabi frequencies [4][5][6]. Three to thirteen Raman lasers of uniform plane waves are used for generating the required SOC and ZF dynamics. The two degenerate dark states in Refs. [4][5][6] can be defined here in the form of {| ↑ , | ↓ } states. Using the same procedure in [4][5][6], the coefficients of the subspace of the dark states can be determined via the Raman laser parameters to obtain the effective low-energy Hamiltonian. Upon projecting into the subspace of dark states, the resulting Hamiltonian is given by:Ĥ mec = α 1 (p yσz −p zσy ) + α 2σx − α 3σy , where σ = (σ x ,σ y ,σ z ) is the vector of Pauli matrices, and α 1 ∼¯h Λ R 2m √ 3 , α 2 ∼ 3hω 2 0 ∆ R and α 3 ∼¯h p ∆ R denote the strengths of SOC, perpendicular ZF and parallel ZF, respectively, [4][5][6]. Here, Λ R is the Raman laser wavevector, ω 0 is the Rabi frequency magnitude, and ∆ R is the detuning of the Raman laser. The parameter p is a known experimental parameter [4,6]. In the most cases, Λ l analogous with Λ R (i.e., Λ R Λ l ). Thus, the systemĤ mec in the continuous space is described as the pairing between the spin-flip and the neighboring sites hopping, as in the following: whereφ(r) can be approximate in case of single-band by: ∑ j,kâjk jk (r). Hence, the Hamiltonian for any two-site v and j in the optical lattice can be given by: 1σ z ) and ε vj = (0, 1, 1) is the vector from a site j at position q j to a site v at position q v .m The values of ζ r 1 =hα 1 dr * vk (r) ∂ ∂r jk (r), ζ 2 = α 2 dr * vk (r) jk (r) and ζ 3 = α 3 dr * vk (r) jk (r) can be tuned by employing the coherent destructive tunneling methods [10]. Here, we assume that the parameters ζ y 1 , ζ z 1 and ζ 2 are much less than Ω kk . Using the systems (S1,S3), the Hamiltonian of the spin- 1 2 fermions (or bosons) atoms is given by: whereĤ 0 andĤ 1 are given by: For each atom occupying a site, say v, that is governed by the system (S4), the possible states (for pseudo-spin basis) can be given by: | ↑ = |n v↑ = 1;n v↓ = 0 and | ↓ = |n v↓ = 1;n v↑ = 0 . We also have:

C. Formulation of non-Hermitian Hamiltonian
Typically, in engineered lattices, a number of two or more atoms are located at the same site. Hence, significant collision coupling can take place. However, for weak tunneling transitions, the average number of atoms at a site can be conceived constant. Consequently, on having the interaction strength between any two atoms with different internal states (| ↑ and | ↓ ) significantly larger than the SOC and ZF effects, virtual transitions between levels take place and the degeneracy in spin configurations can be ignored. Accordingly, this degeneracy can be removed and the system evolution can be described by an effective Hamiltonian. To this end, we apply a non-unitary transformation to the open quantum systems Hamiltonian (S4). The action of this transformation is inspired from the concept of the Schrieffer-Wolff transformation [7]. The proposed transformation includes a non-Hermitian generator,Ô ± , so the system (S4) in the new gauge reads: HereÔ + is for Ω ↑↑ = Ω ↓↓ = Ω ↑↓ for boson atoms, andÔ − is for Ω ↑↑ , Ω ↓↓ → ∞ for fermion atoms. These are defined as in the following: The generator satisfies [Ô ± ,Ĥ 0 ] = −Ĥ 1 , and thus: where [Ô ± , [Ô ± ,Ĥ 1 ]] ∼ = 0 for the case of g k , ζ z(y) 1 , ζ 2 , ζ 3 Ω ↑↓ . On calculating the required commutators, and using the iso-spin operators ( S v ), the Hamiltonian (S5) can be expressed as in the following: The values of f z and J z coefficients are not equal as shown in previous studies [8,9]. Also, in the case of Ω ↑↓ = 2µ, we note that is produced through the generation of Dzyaloshinskii-Moriya interaction (DMI). This term has no effect on the following results of our work and can be omitted.

ENGINEERING CONTROL TO REALIZE QUANTUM GATES
The SOC and ZF mechanisms can be engineered to implement quantum gates. The SOC effect can be used to optimize the gates' cost. This approach of using the SOC effect for cost optimization has not been previously considered.

A. Dissipative one-atom gates
In this sucbsection, we consider one-atom gates with dissipation.Consequently, when the tunneling and interaction between atoms are vanishing (Ω kk = 0, g k = 0 and ε vj = 0), the Hamiltonian in Eq.(S4), for Raman lasers coupled to single-atom a site v, is given by: The chemical potential in the mott-insulator phase and non-interacting atoms represented by µ ↑ = µ 0 and µ ↓ = −µ 0 . Accordingly, the system (S7) can be introduced in term of the iso-spin operators ( where γ = ∨ y is the damping (or dissipating) parameter of the one-atom system. The Hamiltonian (S8) is non-Hermitian. The dissipation effect contributes on state | ↑ decaying to the state | ↓ . This dissipation leads to imperfect achievement of the gates. For the system (S8), the constant term i∨ y 2 does not include any damping and has no effect during the gates' realization. Hence, this term is omitted for the forthcoming analysis.

A.1. Proposed one-atom gates
Experimentally, in (S8), one can get ζ x = ζ k = ζ µ = ζ. It then follows that, the atom that undergoes 2D ZF and 1D SOC can be used to realize √ −PYZ gate. This is a novel gate that is a combination of √ iPY and √ iPZ gates. The solution of the governing equation when θ 2 = 2π, in the subspace, spanned by {| ↑ , | ↓ }, is described by the following dynamical evolution: can provide the dissipative √ −PYZ gate, as in the following: with ζΓ 1 = Γ a and Γ p = πΓ 1 √ 27 . The Raman beams can be applied to any individual atom in the crystal to generate 1D ZF along x-axis and 1D SOC along z-axis. Thus, the Hamiltonian (S8) with p → 0 and θ 2 = π, can be expressed as: The temporal evolution of the system (S11) with proper timing can be used to realize the new one-atom gate √ −I 2 (called the square root of the minus identity operator). Particularly, when the configuration is designed such that ζ x = ζ µ = ζ, the solution of Eq.(S11) is given by: leads to the damped √ −I 2 gate, as in the following: where . For a non-defect system or for vanishing 1D SOC, such that γ = 0, one can obtain the ideal gates of √ −PYZ and √ −I 2 . Hence, such ideal gates in the computational space {| ↑ , | ↓ }, are given by: The new gates (S10,S13) will be later employed to implement new circuits for controlled not (CN) gate and controlled controlled not (CCN) gate.

A.2. Other one-atom gates
For coexisting atoms at a site with Raman lasers applied to individual atoms, the atoms can produce 1D SOC around z-axis with 1D ZF around any direction. Hence, one can realize extra dissipative one-atom gates for proper time pulses during the evolution of the atom. Specifically, one can generate 1D SOC and 1D ZF along x-direction when ζ k , ζ µ , ∨ z → 0 and ζ x = −ζ, and also 1D SOC and 1D ZF along y-direction when ζ x , ζ µ , ∨ z → 0 and ζ k = ζ. Thus, the time evolution of (S8) for the two cases is given by: where Γ 3 = ζ 2 − γ 2 . Also, in the cases of ζ x , ζ k , ∨ z → 0 and ζ µ = ζ, the system (S8) reads: The temporal evolution of system (S16) is given by: We observe that after performing specific pulses through the evolution (S15,S17), the dissipative gates of the imaginary Pauli operators and their square roots can be accomplished as in the following:Û In previous studies, one-atom gates were employed using auxiliary gates to build some of the various circuits for CN-gate. These circuits can be accomplished through the evolution (S15,S17), for γ = 0, as in the following: Also, for γ = 0, we have: This means that the cost of H-gate is two gates. The phase-rotation gate can be realized by the specific angle Ph(ϑ R ) = exp(iϑ R /2)I 2 , which is employed as an auxiliary gate for the CN circuits. Moreover, by taking into account other rotation gates, such as R n 1 ( π 3 ) and R n 2 ( π 3 ) about specific axes n 1 = 1 √ 3 (1, 1, −1) and n 2 = 1 √ 3 (−1, 1, 1), the total time to accomplish these gates can be estimated to be τ 1 .

B. Idealistic two-atom gates
The purpose of this section is to realize new two-atom gates (including gates with no dissipation). On tuning the Raman beams to be freely projected anywhere in the lattice during the realization of SOC and ZF, it possible to achieve interacting system with no defects. Consequently, the imaginary part of the proposed Hamiltonian vanishes. On adjusting the strengths of the tunneling, repulsion, ZF and SOC effects so that J z is identical to 5J y , and D x and D y are eliminated, the effective model (S6) for θ 1 = π can be given by : where ϕ = arctan( D z J y ). The above rotation in (S23) produces an antiferromagnetic or ferromagnetic Heisenberg chain, and it gauges away the DMI effect. For the case of D z J y 1, the spiral magnetic fields rapidly rotate, with minimum at D z J y 0. The last term of Eq.(S23) has no effect on the gates realizations and, thus, can be eliminated by using the time-dependent transformation exp(iℵÎt). Using the canonical transformationˆ vj = exp[ iϕ 2 (Ŝ z j −Ŝ z v )], the Hamiltonian (S23) can be rewritten as in the following: The DMI can be weaker than the exchange couplings as estimated in [11]. By properly manipulating the experimental-setup of the system, we can tune the parameters of SOC and ZF to be larger or smaller than the tunneling parameters.Different solid-state experiments [12] suggest that the value of D z J y is restricted to the interval [−5, 5]. Here, for our ultracold atomic gas scheme, we consider D z J y in this interval. The constraints of D z ≷ J y help us to realize some of the quantum gates. The atoms interaction that is governed by the system (S24) include atoms at the same site, atoms at neighboring sites, pair atoms with nearest neighboring, and pair atoms with next-to-nearest neighboring (i.e., between atom pairs 1-2, 2-3, 1-3 and so on). We focus our attention to only two atoms pairs. The time evolution of the system (S24) for two interacting atoms across site 1 and 2 (℘ = 2) can be expressed as in the following: . For Zeeman parameter f z equal J y , the evolution (S25) becomes: (S26)

B.1. Proposed two-atom gates
According to [13], there are two forms of the −SWCZ gate: √ −SWCZ = ±i √ SWCZ. One of them is define and generated in Ref. [14], while both will be generated in the current work. For J y = 4hθ 5 and J y = 2hθ 5 , through the first and second branch of Eq.(S26), respectively, the system (S26) can be given by: After the action of a π/4−pulse (π/4 = θt), two new gates of the two portraits can be generated: This means that the required cost to achieve i √ SWCZ or −i √ SWCZ is one gate. In the computational space {| ↑ , | ↓ } ⊗ {| ↑ , | ↓ }, the ±i √ SWCZ gates can be expressed as in the following:

B.2. Other two-atom gates
For strong DMI effect ( J y =¯h θ 4 ), the evolution is governed by (S26). By applying 2π-pulse (2π = θt), a controlled-z (CZ) gate can be formed, as detailed in following: Each branch of Eq.(S29) leads to CZ-gate. This means that the cost to achieve CZ is three gates.
On the other hand, to realize the square root of CZ-gate and its inverse, some non-dissipative rotation gates (of one-atom type around the z-axis with temporal evolution (S26)) are needed. Thus, if we assume that the exchange coupling is set to be equal to 3hθ 4 and¯h θ 4 with pulse π, we can achieve these gates as in the following: (S30) Furthermore, using the present system, we can implement previously implemented two-atom gates for CN-gate realization with lower cost. For instance, for weak SOC, D z < J y , and robust SOC, D z > J y , we have: Also, for f z = − 25J y 4 and J y = 4hθ 5 ; f z = − 15J y 2 and J y = 2hθ 5 , we obtain By implementing the above two-atom gates, either for strong DMI or weak DMI, we can realize √ SWAP gate with the evolution (S26) for the cases of f z = − and J y = 6hθ 25 , respectively. In other words, such a gate can be achieved by a series of robustness and weakness of the DMI with specific pulses during the time evolution of the system (S24). Specifically, this gate can be realized as in the following: Similarly, when f z = − Thus, we can conceive the square root of CN ( √ CN) gate with DMI effect as in the following: From Eq.(S30), we can note that the cost of √ CZ gate is three gates, and thus the cost of √ CN gate is five gates.

C. Dissipative two-atom gates
As some defects may occur inside the lattice system due to the arising heat effect during the application of Raman lasers, dissipation takes palce. On the other hand, such defects may be arisen due to other sources (mentioned above). To take disspation into account, we obay the same transformations detailed above (S24) while the dissipation parameters are incorporated. It then follows that the dissipative Hamiltonian (S6) for any two interacting atoms (say at sites 1 and 2) can be expressed as in the following: where 1 = (hθ − 2iγ 2 cos ϕ), 2 = (2γ 2 sin ϕ + iγ 1 + f z + 2.5J y ), 3 = (4iγ 2 + 2.5J y ) and 4 = (hθ + 5J y − 2iγ 2 e iϕ + f z ). To find the damped form, we compute the time evolution of the total unnormalized state of the wave function that is governed by the non-Hermitian Schrodinger equation, yielding: where and ∅ 3 = 4 h . By using appropriate values: f z ≡ f ± J y , D z ≡ D ± J y , ( f + , D + ) = (− 25 4 , 3 4 ) and ( f − , 2 ), the dissipative evolution (S37) reads: where and ϕ ± = arctan(D ± ). Analytically, we can show thatÛ ± ( π 4θ , γ 1 , γ 2 ) can be used to from dissipative ±i √ SWCZ gates. On the other hand, on having D + = 15J y and f z = f + J y for all D ± , the evolution (S37) yields: . By functioning the rotation gates of Eq.(S17) [which arê U z,v ( 31πh 16ζ ) ,Û + ,Û z,v ( 21πh 16ζ ) andÛ − ] we can realize the damped √ CZ gate and its inverse.

D. Constraints to realize quantum gates
The chosen constraints to realize the proposed gates with SOC and ZF mechanisms are reached under several considerations. These include: (i) for other constraints than the above, say f z = ± 1 J y and J z = 2 J y (0 < 1 < 1, 2 < 5), the realization for each two-atom gate requires larger number of gates. (ii) for other constraints of J y , say J y = 3h θ with 3 > 1, the ratio D z J y will be with imaginary value. Also, for J y with 3 < 1, larger number of gates are required and the DMI effect becomes an extremely strong. (iii) the gate ±i √ SWCZ can not be realized by taking other values for J y . On the other hand, previous studies of the optical lattice without SOC and ZF mechanisms (which are governed byĤ ± = ∑ 2 v,j=1 (±J 1σ require the value of J 2 to be larger than or equal the coupling J 1 to achieve the XXZ gate. Therefore, the time evolution ofĤ ± requires a series of gates to realize CZ-gate. For instance, this evolution needs at least one auxiliary gate of the two-atom kind (controlled-phase gate) and two auxiliary gates of the one-atom kind for two-atom gates of Eqs.(S30,S31,S33,S34). Moreover, the evolution ofĤ ± requires two gates of the one-atom type to achieve ±i √ SWCZ gates. Consequently, our two-atom gates require lower cost than the two-atom gates in the absence of the DMI or SOC effect. Hereby, it is shown that our realization of Eqs.(S28-S34) is more efficient than their previous realization.

DETAILS ON THE ALGORITHM TO SOLVE SP
Overview: We know that the idealistic Simon problem (SP) is a polynomial-time algorithm and it can distinguish between two complexity classes by employing a polynomial-time function. The first-class is the bounded error probabilistic polynomial-time, which can be solved by a Turing machine. The second-class is the bounded-error quantum polynomial-time, which may be solved by a quantum computer with error probability less than first class. Besides, if the function of the Turing machine is used as an oracle or circuit in quantum, then it requires an exponential number of oracle evaluations to solve the problem. Thus, the solution to the problem is difficult for the classic conception while it will be exponentially faster with the quantum conception.