Dipolar Exchange Quantum Logic Gate with Polar Molecules
DDipolar Exchange Quantum Logic Gate with PolarMolecules
Kang-Kuen Ni, abc ∗ Till Rosenband, b and David D. Grimes abc We propose a two-qubit gate based on dipolar exchange interactions between individually ad-dressable ultracold polar molecules in an array of optical dipole traps. Our proposal treats the fullHamiltonian of the Σ + molecule NaCs, utilizing a pair of nuclear spin states as storage qubits. Athird rotationally excited state with rotation-hyperfine coupling enables switchable electric dipolarexchange interactions between two molecules to generate an iSWAP gate. All three states are in-sensitive to external magnetic and electric fields. Impacts on gate fidelity due to coupling to othermolecular states, imperfect ground-state cooling, blackbody radiation and vacuum spontaneousemission are small, leading to potential fidelity above .
99 % in a coherent quantum system thatcan be scaled by purely optical means.
Important progress has been made towards a laboratory quantumcomputer with state-of-the art demonstrations reaching a combi-nation of 5 qubits and 98.3 % CNOT gate fidelity . The criteriafor quantum computation have been identified as (1) a scalablesystem of qubits (2) initialization (3) coherence (4) universal setof qubit gates (5) measurement. Items 2 through 5 have beendemonstrated at sufficient fidelities , showing that computa-tion with many qubits may be possible. But the route towardscalability remains challenging. Here, we focus on the problem ofproducing a high-fidelity two-qubit gate using optically trappeddipolar molecules, with the hope that this physical system allowsfor easier scalability. Recent demonstrations of flexible opticaltweezer arrays show a method by which many qubits couldbe rearranged to implement quantum algorithms.Optically trapped, electrically dipolar neutral molecules havelong been recognized as potential qubits where the dipole-dipole interaction between two molecules mediates a two-qubitgate. However, most proposals rely on static or oscillating dress-ing electric fields to polarize the molecules, where the molecularStark energy is much larger than the dipolar interaction. Thisimposes stringent constraints on field stability.Here we describe concretely how the natural dipolar interac-tion between two molecules can produce an iSWAP gate, withoutthe need for additional polarizing fields, thereby removing an im-portant source of implementation complexity and qubit decoher-ence. This iSWAP gate, together with single qubit rotations, formsa universal set of qubit gates . We exploit the rich molecular a Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Mas-sachusetts, 02138, USA. b Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA. c Harvard-MIT Center for Ultracold Atoms, Cambridge, Massachusetts, 02138, USA. ∗ E-mail: [email protected] internal structure and use NaCs as an example to find molecularqubits that are expected to have long coherence (item 3). Thegate relies on two-qubit interactions that are switched on by driv-ing one qubit state to a third state via a microwave transition. Wefind parameters that allow gates with high fidelity ( F > − − )in 10 ms at a magnetic field of 1 Gauss when light shifts due tothe optical trap are neglected. For an optical trap depth of 600kHz, the same fidelity and duration can be reached for a 35 Gaussmagnetic field. The gate duration could be reduced by applyingshaped pulses rather than the square pulses considered here. It is a well known phenomenon that if two identical systems in-teract weakly, where one system has an energy excitation and theother does not, the excitation eventually transfers. This effect canform the basis for a two-qubit gate and has been discussedin the context of molecule-based quantum simulations of spinmodels . The transfer of excitation via the dipole-dipole in-teraction has been demonstrated for ultracold KRb molecules and atoms with large magnetic dipoles in optical lattices.We rely on the natural dipole-dipole interaction betweenmolecules to enable evolution of the type | e (cid:105) ↔ | e ;0 (cid:105) where | (cid:105) is a sub-level of the rotational ground state, and | e (cid:105) is a sub-levelof the first rotational excited state (see Fig. 1A,B). In the two-particle state, the first position refers to the first molecule, andthe second position refers to the second molecule. This exchangeinteraction leaves the states | (cid:105) and | e ; e (cid:105) unchanged, and theoverall unitary evolution in the basis | (cid:105) , | e (cid:105) , | e ;0 (cid:105) , | e ; e (cid:105) is ˆ U = e − i ˆ Ht / ¯ h = Ω t i sin Ω t i sin Ω t cos Ω t
00 0 0 1 , (1) a r X i v : . [ phy s i c s . a t o m - ph ] J u l ig. 1 (A) iSWAP gate based on dipolar exchange between a pair of molecular states with opposite parity. The colored sphere of the | e (cid:105) staterepresents the wavefunction amplitude of the dipole direction for an N = , m N = state, where the quantization axis is horizontal. The states | (cid:105) and | (cid:105) are hyperfine sublevels of the rotational ground state N = . Superpositions of | e (cid:105) and | (cid:105) or | (cid:105) produce an electric dipole moment that oscillates ata frequency corresponding to approximately twice the rotational constant of the molecule and couples to a nearby qubit. The four panels in (A) showthe initial state | (cid:105) evolving through the gate to i | (cid:105) , where horizontal arrows indicate the flow of time. State | e (cid:105) of the storage qubits is light shiftedout of resonance, for individual addressability. (B) includes other basis states and important details of quantum phases. (C) Qubit array based onmolecular hyperfine states. Any pair of qubits can be moved from the storage zone to the gate zone in a flexible array of optical tweezers. During thegate operation, a spatially uniform microwave pulse transfers population from state | (cid:105) to | e (cid:105) in the gate qubits, so that the amplitudes of the | e (cid:105) and | e ;0 (cid:105) states are exchanged. To achieve individual addressability with high spatial resolution, the light that shifts the storage qubits out of resonance(indicated by red shadows) can be produced in a similar way as the tweezer array. where Ω = D / r is the interaction Rabi rate, r is the molecule-molecule distance, and the duration t = π / ( Ω ) produces theiSWAP gate. The factor D (see Eq. 3) depends on the choice ofmolecule, separation direction, choice of states, magnetic field,and light shift. In the Gauss example of Section 5, two NaCsmolecules have t = ms for r = . µ m separation along ˆ x , whichis also the magnetic field direction. For the 35 Gauss example,which includes the effects of a 600 kHz deep optical trap, we usean interaction duration of t = ms with r = . µ m along ˆ x andthe magnetic field direction is ˆ z . While the resolution of opticaltweezers with beam waist below µ m supports smaller separa-tion and gates as fast as t = µ s, two effects place additionalconstraints. 1. Off-resonant population leakage degrades the gatefidelity for short durations (Sec. 6) and 2. smaller molecule sep-aration ( r ) makes the gate more sensitive to motional excitation(Sec. 7).Quantum computing requires two qubit states | (cid:105) and | (cid:105) thatare coherent and couple minimally to the environment and otherqubits. In this proposal, we utilize two hyperfine sublevels of therotational ground state of a molecule as states | (cid:105) and | (cid:105) . Long-lasting coherence of such states has recently been demonstratedin a gas of ultracold NaK molecules . While the hyperfine levelsoffer coherence, they do not produce strong dipole-dipole inter-actions between nearby molecules. To enable this interaction andproduce an iSWAP gate, the | (cid:105) state in two molecules is tem-porarily transferred to the rotationally excited state | e (cid:105) via a mi-crowave π -pulse. Then, after energy exchange, the | e (cid:105) populationis transferred back to | (cid:105) . The propagator in Eq. 1 still applies, but now in the computational basis | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) as shownin Fig. 1(B).The above sequence requires an excited state | e (cid:105) that couples totwo different hyperfine levels ( | (cid:105) and | (cid:105) ) of the ground state viaelectric fields. We rely on the hyperfine interaction term of the in-ternal molecular Hamiltonian that couples molecular rotation tonuclear spin via the nuclear electric quadrupole moment toproduce eigenstates that contain superpositions of different nu-clear spin states. This interaction requires a nuclear spin greaterthan 1/2. Even though external electric fields do not change thenuclear spin directly, they can change the nuclear spins by driv-ing transitions between states with different superpositions .In this manuscript, we use Na Cs as an example because ithas a large permanent electric dipole moment (4.6 Debye), andfull quantum control of individually trapped molecules is beingdeveloped . A similar gate scheme that makes use of inter-nal molecular couplings could also be applied to other ultracoldpolar molecules, including other bialkalis where a single internalquantum state can already be prepared and molecules of Σ electronic structure with spin-rotational coupling . For a quantum computer, a large number of molecular qubitscould be held in an array of optical tweezers for storage (seeFig. 1C). Two arbitrary qubits can be selected for gate operationsby means of a configurable tweezer array and moved so theyare separated from the other qubits and initially far from eachother. Off-resonant light is applied to the array of stored qubits ig. 2
Hyperfine and Zeeman energy levels as a function of magneticfield for the N = (top) and N = (bottom) states of Na Cs ( v = ) inzero electric field. While a number of states are nearly degenerate withthe states of interest | (cid:105) , | (cid:105) , and | e (cid:105) , selection rules prevent them fromparticipating in the interactions. B v = . GHz is the molecularrotation constant. to shift their | e (cid:105) energy level so that the π -pulses have no effect.Individual addressability of qubits requires a spatial light patternwith high contrast ratio between stored and gate qubits, whichcan be generated by similar optics as the array of optical tweez-ers. The light causes only very small differential shifts for thestorage states | (cid:105) and | (cid:105) . A spatially uniform microwave π -pulseof well-defined polarization then transfers | (cid:105) to | e (cid:105) for the twogate qubits. Then the qubits are moved near each other and backapart to allow for the interaction. This movement naturally pro-duces a temporally shaped interaction with gradually rising andfalling strength to reduce off-resonant population leakage. Thefinal π -pulse transfers the | e (cid:105) population to | (cid:105) . An enhanced gatesequence can include a central spin-echo pulse to mitigate slowlydrifting energies (Sec. 6). More advanced dynamical-decouplingsequences can be applied to further reduce decoherence and thesensitivity to qubit motion (Sec. 7). Because the molecule is in a Σ + state, the electronic spin andorbital angular momentum are zero, and do not enter the cal-culation. We identify quantum states in the uncoupled basis bythe quantum numbers N , m N , m , and m , where N is the angularmomentum associated with molecular rotation and m N , m , m areprojections of this angular momentum and the two nuclear spinsonto the magnetic field axis. While the uncoupled basis is conve-nient for calculations, the basis states generally do not coincidewith eigenstates of the molecular Hamiltonian. Nevertheless, welabel Hamiltonian eigenstates as | N , m N , m , m (cid:105) , where we use thequantum numbers of the uncoupled basis state with maximumoverlap. Although this labeling scheme could in principle assignthe same quantum numbers to two different eigenstates, we haveverified that this does not occur for the specific states discussedhere. When two-molecule states are described, they are writtenas | a ; b (cid:105) where a and b are the states of molecules 1 and 2 respec-tively.We solve for the eigenstates associated with the molecular hy-perfine Hamiltonian for NaCs at various magnetic fields andoptical trap depths. The energy levels are shown as a function ofmagnetic field in Fig. 2 and trap depth in Fig. 4. The dipole-dipoleinteraction and electric-field-driven π -pulses both depend on theelectric dipole moments of the molecules. The Hamiltonian asso-ciated with an externally applied electric field to molecule j is ˆ H E = − ˆd j · E (2)where we use the interaction picture and rotating-wave approxi-mation to remove the time-dependence of oscillating fields. Thedipole-dipole interaction is ˆ H DD = πε r (cid:2) ˆd · ˆd − ( ˆd · ˆe r )( ˆd · ˆe r ) (cid:3) (3)where ˆd j is the dipole moment operator of molecule j . Here, ˆe r is the unit vector along the separation direction for the twomolecules and the dot products are evaluated as sums over thethree spatial directions. The dipole moment operators are deter-mined by the rotational part of the energy eigenstates and fora basis state N , m n , this axis has an orientation whose quantumwave function is the spherical harmonic Y Nm . From the eigen-states of the molecular Hamiltonian we calculate the matricesassociated with the dipole operators ˆ d x , ˆ d y , ˆ d z for the laboratorycoordinate system . These matrix elements are diagonal in thequantum numbers m and m of the uncoupled basis and can bereduced to Wigner-3j symbols .For each step ( π -pulse and exchange), we calculate unitary evo-lution according to the time-independent Hamiltonians in Eq. 2and 3. The resonant couplings drive the desired gate behav-ior, while off-resonant couplings to other states cause populationleakage and Stark shifts. For a low magnetic field without light,the most important Hamiltonian terms are listed in Tables 1 and2. The unitary transformation is applied to many test states in thecomputational basis to determine the minimum gate fidelity .The approximate population leakage can also be calculated more rom ( i ) To ( j ) H i j / ¯ h [s − ] H j j / ¯ h [s − ] | e (cid:105) | (cid:105) | (cid:105) | e (cid:105) | (cid:105) | (cid:105) - 117851.0 | (cid:105) | , , / , / (cid:105) | (cid:105) | , , / , / (cid:105) | e (cid:105) | , , / , / (cid:105) | (cid:105) | , , / , / (cid:105) Table 1
Non-zero coupling terms of the | (cid:105) ↔ | e (cid:105) π -pulse Hamiltonian(interaction picture) in the rotating-wave approximation when σ + -polarized radiation at . GHz is applied to a molecule in Gaussmagnetic field. The electric field amplitude is 0.03 V/m, such that the π -pulse duration is 3.01 ms (adjusted for maximum fidelity). Theradiation frequency has been adjusted by 6.3 radians/s to compensatefor dynamic Stark shifts. All terms connecting to | (cid:105) , | (cid:105) , or | e (cid:105) areshown. From To H i j / ¯ h H j j / ¯ h ( i ) ( j ) [s − ] [s − ] | e ;0 (cid:105) | e (cid:105) -390.7 0 | e ;0 (cid:105) | , , / , / (cid:105) -195.6 -82715.8 | e ;0 (cid:105) | , , / , / , , / , / (cid:105) | e ;0 (cid:105) | , , / , / (cid:105) | e ;0 (cid:105) | , , / , / (cid:105) Table 2
Non-zero coupling terms for | e ;0 (cid:105) of the exchange Hamiltonianwhen two molecules are separated by . µ m along the Gaussmagnetic field. Propagation of the Hamiltonian approximates theexchange evolution (Eq. 1) when applied for a duration 4.02 ms(adjusted for maximum fidelity). The terms | (cid:105) and | e ; e (cid:105) do not coupleto other states. simply via perturbation theory (Appendix A). In addition to the desired evolution in Eq. 1, the iSWAP gate se-quence causes off-resonant coupling to other molecular states.This causes a trade-off between gate speed and fidelity. Althoughtwo NaCs molecules could be brought to a separation below µ m,with an exchange duration of µ s, such an interaction wouldlimit the gate fidelity to . %. At low magnetic field with NaCs,we find that the interactions couple off-resonantly to other stateswith δ > kHz detuning both for the π -pulses and the exchange.For the square pulses considered here, time-energy uncertaintycauses off-resonant population transfer out of the computationalbasis states and a gate error of order ( τδ ) − . Off-resonant cou-pling to other states also causes dynamic Stark shifts which resultin small reproducible phase and frequency shifts. We expect thatthese can be corrected for without loss of fidelity.We have calculated the unitary time evolution for all statesthat could couple via microwave pulses ( ˆ H E ) and dipole-dipoleinteractions ( ˆ H DD ) to the states in Fig 1(B). For low magneticfield (1 Gauss), details of the Hamiltonian terms are given is Ta-bles 1 and 2, where we have chosen the states | (cid:105) = | , , / , / (cid:105) , | (cid:105) = | , , / , / (cid:105) and | e (cid:105) = | , , / , / (cid:105) .We find that a gate with 10 ms duration has a fidelity of F = − . × − . The exchange part of the example gate holds thetwo molecules at a distance of . µ m for a duration τ x = . ms. The π -pulses of duration . ms utilize a σ + polarized microwaveelectric field at a frequency of . GHz and amplitude . V/m.The π -pulse and exchange fidelities due to population leakageare shown separately in Figure 3. Population leakage out of thecomputational basis must be corrected for to support long gate se-quences . Off-resonant coupling can likely be reduced by use ofshaped pulses rather than square ones, to reach the same fidelityin a shorter duration . At a higher magnetic field of 35 Gausswith kHz trap depth, the best states ( F = − × − at 9.4ms) are | (cid:105) = | , , / , / (cid:105) and | e (cid:105) = | , − , / , / (cid:105) , with | (cid:105) asabove. If the molecule is optically trapped, light shifts significantly per-turb the energy eigenstates and transition frequencies. To reachthe wavefunction spread described in Section 7, we assume a kHz trap depth (12.9 kW/cm intensity) with elliptical po-larization . We also assume a trapping laser wavelength of1030 nm. The resulting energy levels are shown in Figure 4 fora magnetic field of 35 Gauss. The higher magnetic field was cho-sen, because the Zeeman splitting reduces off-resonant couplingsduring the | (cid:105) ↔ | e (cid:105) π -pulses that are induced by the light shiftHamiltonian. Figure 5 shows the magnitude of off-resonant cou-pling terms to other molecular states. For interactions that turnon and off sharply, the loss of fidelity due to off-resonant popu-lation leakage is given by a sum of terms involving off-resonantcoupling strengths and detunings (see Appendix A). Overall, wefind that the exchange interaction has reduced leakage comparedto the 1 Gauss case, even for a faster interaction. The | (cid:105) ↔ | e (cid:105) π -pulses cause slightly higher leakage, which may be reduced byshaped or DRAG pulse techniques .For an optical trap with elliptical polarization, the quadraticsensitivity to intensity is calculated to be ∆ f = ( ∆ I / I ) × . kHz,leading to gate errors of × − for relative intensity fluctua-tions of ∆ I / I = × − . A more significant effect is polariza-tion drift. For quartz waveplates with a temperature sensitivityof retardance of − /C, one may expect polarization ellipticityfluctuations of − and light shift fluctuations of × − (2.5Hz). Because these fluctuations are slow thermal effects, spinecho pulses can reduce their impact. If the quadratic light shiftchanges slowly, e.g. due to recoil heating, the effect is also miti-gated. One possible implementation is shown in Table 3 where a | (cid:105) ↔ | e (cid:105) π -pulse is inserted in the middle of the gate sequence.This leads to first-order insensitivity of the gate error with respectto constant light shift errors. The result is equivalent to an iSWAPgate between the qubits, followed by inversion of the individualqubits. Note that this spin-echo example was chosen for simplic-ity, and other dynamical decoupling techniques to compensateslow drifts could also be applied. The | (cid:105) ↔ | e (cid:105) π -pulses, whichare used for the spin-echo have little off-resonant coupling, evenfor a 1 ms duration (see Fig. 5).The differential light shift between two hyperfine ground statesfor NaK was measured as × − Hz/(W/cm ). With the as-sumption that the states | (cid:105) and | (cid:105) in NaCs have similar differen-tial shifts, a beam with kW/cm intensity and 1064 nm wave-length can Stark shift the | e (cid:105) states of the stored qubits by kHz .1 0.5 1 5 1010 - - - (cid:1) - pulse duration [ ms ] Lo ss o ff i de li t y - F (cid:1) - F (cid:1) - - - [ ms ] Lo ss o ff i de li t y - F X Fig. 3
Loss of fidelity as a function of duration for the two π -pulses ( − F π − F π ) and exchange ( − F X ). In both cases, the interaction strength (drivestrength or molecule-molecule separation) is adjusted so the operation completes in the nominal duration. The ripples are caused by the square pulseshape whose sinc-function power-spectrum-minima move across off-resonant transitions as the duration is varied. The scaling behavior for themaximum fidelity points is − F π − F π = . × − t − and − F x = . × − t − where t is given in ms. Details of the fidelity calculation are given inSection 4. The solid lines represent the perturbation theory results while the dotted lines correspond to the full Hamiltonian. π ↔ e Ex. π ↔ e Ex. ( − π ↔ e ) | (cid:105) → | (cid:105) → | (cid:105) → −| e ; e (cid:105) → −| e ; e (cid:105) → | (cid:105)| (cid:105) → i | e (cid:105) → i √ | e (cid:105) − √ | e ;0 (cid:105) → √ | e (cid:105) − i √ | e ;0 (cid:105) → | e (cid:105) → − i | (cid:105)| (cid:105) → i | e ;0 (cid:105) → − √ | e (cid:105) + i √ | e ;0 (cid:105) → − i √ | e (cid:105) + √ | e ;0 (cid:105) → | e ;0 (cid:105) → − i | (cid:105)| (cid:105) → −| e ; e (cid:105) → −| e ; e (cid:105) → | (cid:105) → | (cid:105) → | (cid:105) Table 3
Evolution of the computational basis states through the gate, which includes a central spin-echo pulse to cancel the phase evolution fromslowly varying energy shifts between | (cid:105) and | e (cid:105) , such as light shifts. The exchange interaction is split into two parts, where each has one half theduration required for full exchange. - - ( E α / ) [ kHz ] F r eq . d i ff e r en c e [ k H z ] Fig. 4
Differential light shifts of excited rotational states ( N = ) withrespect to the ground state, as a function of trap depth. The large offsetof about 3.48 GHz has been subtracted. States with predominantly ( | m N = − (cid:105) − | m N = (cid:105) ) / √ character with dipole moment along ˆ x areshown in black. States with predominantly ( | m N = − (cid:105) + | m N = (cid:105) ) / √ character with dipole moment along ˆ y are blue. States withpredominantly | m N = (cid:105) character with dipole moment along ˆ z are red.The thick line shows the state | e (cid:105) = | , − , / , / (cid:105) . The polarizationvector is ˆ ε = ˆ x cos γ + i ˆ y sin γ with ellipticity γ = . ◦ , adjusted to nullthe slope of | e (cid:105) at 600 kHz. A magnetic field of 35 Gauss lies along ˆ z . while shifting their | (cid:105) and | (cid:105) states by only . Hz differentially.In this case, the polarization is adjusted for maximum differentialStark shift of | e (cid:105) with respect to | (cid:105) and a spatially patterned beamcan be generated in the same way as the tweezer array. If theStark shifting beam has a relative intensity stability of × − ,the loss of fidelity is − F ≈ × − for a single stored qubit.The large ratio between | (cid:105)−| e (cid:105) and | (cid:105)−| (cid:105) sensitivities makes itpossible to individually address certain molecules by light-shiftingthe | e (cid:105) state of the other molecules. In the ideal gate, both molecules are in the motional ground stateof their optical tweezer, and the exchange interaction strengthis always the same. However, ground-state cooling is imper-fect and the molecules gain kinetic energy due to photon recoil.Given the imaginary polarizability of NaCs Im [ α ] ≈ − (atomicunits) , the total scattering rate for a 600 kHz deep trap is ap-proximately . Hz with a heating rate of . quanta/s axiallyand . quanta/s for each radial direction. Therefore, it is de-sirable for the gate fidelity to exceed the ground-state occupationfidelity. Here we examine the effect of motional excitations wherethe spatial coupling constant of Eq. 3 is modified. For the pro-posed situation where the molecular dipoles lie along ˆ x , this cou-pling scales as ( − θ ) / r where θ is the angle between theseparation axis and ˆ x , and r is the separation distance (2.5 µ mhere). For θ ≈ , a series expansion in terms of spatial coordi- $ e Ex. p $ e Ex. ( p $ e ) | i ! | i ! | i ! | e ; e i ! | e ; e i ! | i| i ! i | e i ! i p | e i p | e ;0 i ! p | e i i p | e ;0 i ! | e i ! i | i| i ! i | e ;0 i ! p | e i + i p | e ;0 i ! i p | e i + p | e ;0 i ! | e ;0 i ! i | i| i ! | e ; e i ! | e ; e i ! | i ! | i ! | i Table 3
Evolution of the computational basis states through the gate, which includes a central spin-echo pulse to cancel the phase evolution fromslowly varying energy shifts between | i and | e i , such as light shifts. The exchange interaction is split into two parts, where each has one half theduration required for full exchange. - - - - - - - ( (cid:1) ij ) [ MHz ] | (cid:2) ij / (cid:1) ij | i = | e i or i = | e ;0 i (A) | e i $ | e ;0 i , ms p i = . ⇥ (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) - - - - - ( (cid:4) ij ) [ MHz ] | (cid:5) ij / (cid:4) ij | (B) | i $ | e i , . ms i = | i , p i = . ⇥ i = | e i , p i = . ⇥ i = | i , p i = . ⇥ (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:2)(cid:2)(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2) (cid:2) (cid:3)(cid:3) (cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) - - - - - ( (cid:4) ij ) [ MHz ] | (cid:5) ij / (cid:4) ij | (C) | i $ | e i , . ms i = | i , p i = . ⇥ i = | e i , p i = . ⇥ i = | i , p i = . ⇥ Fig. 5
Off-resonant coupling terms | W ij / d ij | that cause populationleakage during the gate steps. Here, W ij is the coupling Rabi ratebetween source state i and leakage state j , and d ij is the frequencydifference in radians/s. For each step, the probability of populationleakage p i is shown (see Appendix A), where the summation is over allcoupled states j . p i can be used to estimate off-resonant populationleakage without calculating the full unitary time evolution. (A) Exchangeinteraction with separation of . µ m along ˆ x for a 2 ms exchangeduration. The fidelity, calculated from unitary time evolution is F = . ⇥ . (B) | i $ | e i p -pulse with electric field amplitude . V/m along ˆ z for a 3.12 ms pulse duration. The fidelity for a pair of p -pulses, calculated from the unitary time evolution is F = . ⇥ . (C) | i $ | e i p -pulse with electric field amplitude 0.0157 V/m along ˆ x fora 1.233 ms pulse duration. The state | i can be neglected if itspopulation has already been transferred to | e i . where | m i = | n , x , n , y , n , z ; n , x , n , y , n , z i is the state of two-molecule motion, W = D / r is the interaction Rabi rate with-out motion of Eq. 1, n j , k is the motional excitation number formolecule j in direction k x , y , z } , s k = p ¯ h / ( m w k ) are the zero-point wave function spreads, f x = , f y , z = , and the approxi-mation includes terms up to second order in position. The abovetrap depth corresponds to motional frequencies of w x , y = p · . kHz and w z = p · . kHz for NaCs, where a Gaussian beam radiusof µ m is assumed. Because the loss of fidelity is proportionalto the square of the Rabi rate deviation from the mean, F cancalculated from the variance of the Rabi rate, which can be ex-pressed in terms of mean motional quantum numbers ¯ n j , k for athermal distribution of motional states: F ⇡ p  j , k f k ( ¯ n j , k + ¯ n j , k )( s k / r ) . (5)If the molecules are near the motional ground state with excita-tions dominated by imperfect cooling of Na , the mean excita-tion numbers of each molecule along x , y , and z are .019, .024,and .024 respectively. Then motional effects cause a fidelity lossof ⇥ for the above trap strength. The effect of motion onthe two-qubit gate can be reduced by implementing a BB1 se-quence . Because the sign of f z is opposite f x , it is possible tochoose a separation direction that makes the interaction Rabi ratefirst-order insensitive to n j , z . This occurs for ˆ r = ˆ x cos q + ˆ z sin q with q = ( / ) cos ( / ) , and the interaction strength is re-duced by 20 %. An optical lattice in the weakly confining direc-tion would also reduce the effects of axial motion. Note that theterm ¯ n j , k s k in Eq. 5, which limits fidelity for ¯ n j , k > , is indepen-dent of trap intensity when ¯ n j , k is dominated by recoil heating.The motional state also affects the average light shift, becausethe tweezer intensity drops away from the trap center. For a Gaus-sian beam trap with thermally excited molecules whose meanquantum numbers are ¯ n k , we find the relative intensity variance var ( I / I ) ⇡  k g k ( ¯ n k + ¯ n k ) where g k = ( s k / w ) for the radial di-rections, g z = ( s z l / ( p w )) and w is the beam radius. The as-sociated standard deviation in relative intensity is ⇥ for theresidual motion after ground-state cooling described above. Thiscorresponds to the intensity stability assumed in Section 6. Effects from blackbody radiation and spontaneous emission aresmall for the system we have outlined above. The dominant effectat room temperature is due to blackbody radiation and for NaCsthe vibrational transition absorption rate is . ⇥ s whilethe rotational transition rate is negligible . The spontaneous (cid:43)(cid:80)(cid:86)(cid:83)(cid:79)(cid:66)(cid:77)(cid:1)(cid:47)(cid:66)(cid:78)(cid:70)(cid:13)(cid:1)(cid:60)(cid:90)(cid:70)(cid:66)(cid:83)(cid:62)(cid:13)(cid:1) (cid:60)(cid:87)(cid:80)(cid:77)(cid:15)(cid:62) (cid:13) Fig. 5
Off-resonant coupling terms | Ω ij / δ ij | that cause populationleakage during the gate steps. Here, Ω ij is the coupling Rabi ratebetween source state i and leakage state j , and δ ij is the frequencydifference in radians/s. For each step, the probability of populationleakage p i is shown (see Appendix A), where the summation is over allcoupled states j . p i can be used to estimate off-resonant populationleakage without calculating the full unitary time evolution. (A) Exchangeinteraction with separation of . µ m along ˆ x for a 2 ms exchangeduration. The fidelity, calculated from unitary time evolution is F = − . × − . (B) | (cid:105) ↔ | e (cid:105) π -pulse with electric field amplitude . V/m along ˆ z for a 3.12 ms pulse duration. The fidelity for a pair of π -pulses, calculated from the unitary time evolution is F = − . × − . (C) | (cid:105) ↔ | e (cid:105) π -pulse with electric field amplitude 0.0157 V/m along ˆ x fora 1.233 ms pulse duration. The state | (cid:105) can be neglected if itspopulation has already been transferred to | e (cid:105) . nates yields a change in the interaction Rabi rate Ω per motionalquantum of ∆ (cid:104) m | Ω | m (cid:105) / ∆ n j , k ≈ f k Ω ( s k / r ) (4)where | m (cid:105) = | n , x , n , y , n , z ; n , x , n , y , n , z (cid:105) is the state of two-molecule motion, Ω = D / r is the interaction Rabi rate with-out motion of Eq. 1, n j , k is the motional excitation number formolecule j in direction k ∈ { x , y , z } , s k = (cid:112) ¯ h / ( m ω k ) are the zero-point wave function spreads, f x = , f y , z = − , and the approxi-mation includes terms up to second order in position. The abovetrap depth corresponds to motional frequencies of ω x , y = π · . kHz and ω z = π · . kHz for NaCs, where a Gaussian beam radiusof µ m is assumed. Because the loss of fidelity is proportionalto the square of the Rabi rate deviation from the mean, − F cancalculated from the variance of the Rabi rate, which can be ex-pressed in terms of mean motional quantum numbers ¯ n j , k for athermal distribution of motional states: − F ≈ π ∑ j , k f k ( ¯ n j , k + ¯ n j , k )( s k / r ) . (5)If the molecules are near the motional ground state with excita-tions dominated by imperfect cooling of Na , the mean excita-tion numbers of each molecule along x , y , and z are .019, .024,and .024 respectively. Then motional effects cause a fidelity lossof × − for the above trap strength. The effect of motion onthe two-qubit gate can be reduced by implementing a BB1 se-quence . Because the sign of f z is opposite f x , it is possible tochoose a separation direction that makes the interaction Rabi ratefirst-order insensitive to n j , z . This occurs for ˆ r = ˆ x cos θ + ˆ z sin θ with θ = ( / ) cos − ( / ) , and the interaction strength is re-duced by 20 %. An optical lattice in the weakly confining direc-tion would also reduce the effects of axial motion. Note that theterm ¯ n j , k s k in Eq. 5, which limits fidelity for ¯ n j , k > , is indepen-dent of trap intensity when ¯ n j , k is dominated by recoil heating.The motional state also affects the average light shift, becausethe tweezer intensity drops away from the trap center. For a Gaus-sian beam trap with thermally excited molecules whose meanquantum numbers are ¯ n k , we find the relative intensity variance var ( I / I ) ≈ ∑ k g k ( ¯ n k + ¯ n k ) where g k = − ( s k / w ) for the radial di-rections, g z = − ( s z λ / ( π w )) and w is the beam radius. The as-sociated standard deviation in relative intensity is × − for theresidual motion after ground-state cooling described above. Thiscorresponds to the intensity stability assumed in Section 6. Effects from blackbody radiation and spontaneous emission aresmall for the system we have outlined above. The dominant effectat room temperature is due to blackbody radiation and for NaCsthe vibrational transition absorption rate is . × − s − whilethe rotational transition rate is negligible . The spontaneousemission rate from the N = rotationally excited state is of order − s − .The rate of decoherence due to scattering of optical tweezerphotons has not been calculated. However, where long coherencebetween hyperfine ground states of NaK was observed, the deco-herence was attributed to spatial intensity variation in the optical ↔ e Rot. − π ↔ e | (cid:105) → | (cid:105) → α | (cid:105) − β | e (cid:105) → α | (cid:105) + β | (cid:105)| (cid:105) → −| e (cid:105) → γ | (cid:105) − δ | e (cid:105) → γ | (cid:105) + δ | (cid:105) Table 4
Single qubit rotation sequence where the π -pulses areperformed as Y -rotations, and the “Rotation" step acts on the states | (cid:105) and | e (cid:105) . trap rather than scattering . A calculation of scattering ratesshould distinguish between internal-state-preserving (Rayleigh)and internal-state-changing (Raman) scattering, as the rate of Ra-man scattering can be several orders of magnitude lower thanRayleigh scattering .Magnetic field fluctuations cause dephasing of quantum statesif their energies have an unequal slope with respect to fieldchanges. For both the 1 Gauss and 35 Gauss examples, we findthat the relative sensitivities of the | (cid:105) , | (cid:105) , and | e (cid:105) states are be-low 1 kHz/Gauss. While this field sensitivity is small, the as-sociated loss of fidelity grows with a factor N if maximally en-tangled states such as ( | ··· (cid:105) + | ··· (cid:105) ) / √ with N qubits arestored in the qubit array. The same scaling applies to light thatdifferentially shifts the phase of | (cid:105) and | (cid:105) (see Section 6). Suchcommon-mode dephasing errors can be reduced by use of dynam-ical decoupling or decoherence-free subspaces . Single qubit X and Y rotations are less complex than two-qubitgates and can be accomplished by individually addressing onlyone molecule and using a simplified sequence without dipole-dipole exchange. It may be advantageous to first perform a π -pulse from | (cid:105) to | e (cid:105) , then a rotation between | (cid:105) and | e (cid:105) andfinally a π -pulse from | e (cid:105) to | (cid:105) as shown in Table 4. The | (cid:105) ↔ | e (cid:105) π -pulses can have less off-resonant coupling than the previouslydiscussed | (cid:105) ↔ | e (cid:105) pulses (see Fig. 5), and high fidelity can beachieved for a shorter gate duration than in the two-qubit case.Rotation about the Z axis can be accomplished by a | (cid:105) ↔ | e (cid:105) π -pulse with one oscillator phase, followed by a second | (cid:105) ↔ | e (cid:105) π -pulse with a different oscillator phase.
10 Conclusion
We have described a room-temperature scheme for quantum com-puting based on iSWAP gates performed by dipolar molecules,individually trapped in optical tweezers. Calculations indicate apotential gate fidelity above 0.9999 with low decoherence, al-though the rate of Raman scattering of optical tweezer photonsremains to be determined. A modest magnetic field is used, andelectric fields or field gradients are not needed. Scaling to manyqubits would require an equal number of optical dipole trapsin a movable pattern. For this gate to be realized, individualneutral ground-state molecules must still be produced, and im-proved state-measurement is needed. We expect that long gatesequences will require the mitigation of population leakage andrecoil heating, e.g. by periodically teleporting the state of usedmolecules onto new ones . While this proposal utilizes two hy-perfine ground levels of NaCs, thirty other ground levels exist,which might allow each molecule to contain several qubits. Although the proposed iSWAP gate is slower than quantumgates in other systems, decoherence effects can be small due tothe fact that the qubit states are isolated from the environmentby the symmetry of Σ + states. The longer gate duration also re-duces the noise bandwidth of actively stabilized parameters. Forthe foreseeable future, experimental quantum computing will aimto increase the number of available qubits and gate fidelity, andmolecules have the potential to advance both goals. References
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We wish to calculate the population leakage during gate interac-tions. In this case, there are two “main" states (here | a (cid:105) and | b (cid:105) )and leakage states | j (cid:105) to which the main states connect with largedetuning and/or weak coupling. For simplicity, we assume thatthe Hamiltonian associated with the interaction turns on and offinstantaneously to produce time-independent coupling. In a realimplementation, it will be important to ramp the interactions upand down smoothly to minimize off-resonant coupling.One approach for calculating leakage is to generate the fullHamiltonian matrix and diagonalize it to compute the unitarytime evolution. While straightforward, this is computationallyexpensive when there are many leakage states. Complementaryto the full calculation of unitary time evolution, we use first-orderperturbation theory below to calculate the population leakage asa simple sum. A comparison of the techniques can be seen inFig. 3. Note that this does not account for coherent populationbuildup, which may develop if several interactions are combinedwithout phase randomization.During interactions between degenerate primary states | a (cid:105) and | b (cid:105) with coupled leakage states | j (cid:105) , the Hamiltonian can be writ-ten in terms of “desired" coupling ˆ H and “leakage" couplings ˆ H (cid:48) s ˆ H = ˆ H + ˆ H (cid:48) (6) ˆ H / ¯ h = Ω | a (cid:105)(cid:104) b | + Ω | b (cid:105)(cid:104) a | + ∑ j δ j | j (cid:105)(cid:104) j | (7) ˆ H (cid:48) / ¯ h = ∑ i ∈{ a , b } , j ( Ω i j | i (cid:105)(cid:104) j | + Ω i j | j (cid:105)(cid:104) i | ) . (8)Here, the basis state phases have been chosen to make the coeffi-cients Ω and Ω i j real and positive. The base Hamiltonian ˆ H haseigenvectors | + (cid:105) , |−(cid:105) , | j (cid:105) with eigenvalues ¯ h Ω , − ¯ h Ω , ¯ h δ j and | + (cid:105) = ( | a (cid:105) + | b (cid:105) ) / √ (9) |−(cid:105) = ( | a (cid:105) − | b (cid:105) ) / √ . (10)According to first-order perturbation theory, the perturbed eigen-states (assuming that different | j (cid:105) states don’t couple to one an-other) are | + (cid:48) (cid:105) = | + (cid:105) + ∑ j α j | j (cid:105) (11) |− (cid:48) (cid:105) = |−(cid:105) − ∑ j β j | j (cid:105) (12) | j (cid:48) (cid:105) = | j (cid:105) − α j | + (cid:105) + β j |−(cid:105) . (13)where α j = Ω aj + Ω bj √ ( Ω − δ j ) and β j = Ω aj − Ω bj √ ( Ω + δ j ) . These states approxi-mately diagonalize the time-evolution operator and can be usedto estimate the transitions from the initial state | a (cid:105) into states | j (cid:105) due to unitary time evolution ˆ U ( t ) = e − i ˆ Ht / ¯ h : (cid:104) j | ˆ U ( t ) | a (cid:105) ≈ ∑ k (cid:48) (cid:104) j | k (cid:48) (cid:105)(cid:104) k (cid:48) | a (cid:105) e − it (cid:104) k (cid:48) | ˆ H / ¯ h | k (cid:48) (cid:105) (14) ≈ √ e − it δ j (cid:0) β j − α j (cid:1) + √ e − it Ω α j − √ e it Ω β j (15) Where the Eq. 14 sum is over all perturbed states. Simplifica-tion from Eq. 14 to Eq. 15 utilizes unperturbed energies in theexponential terms. In our case for any j , due to selection rules,only one of Ω a j and Ω b j is ever non-zero. We call the non-zerovalue Ω j . If one makes the assumption t = π / ( Ω ) ( π -pulse), thepopulation leakage is (cid:12)(cid:12) (cid:104) j | ˆ U ( t ) | a (cid:105) (cid:12)(cid:12) ≈ | Ω j | (cid:16) δ j − δ j Ω sin (cid:0) δ j t (cid:1) + Ω (cid:17)(cid:16) δ j − Ω (cid:17) ≈ (cid:12)(cid:12)(cid:12)(cid:12) Ω j δ j (cid:12)(cid:12)(cid:12)(cid:12) (16)where the first approximation is due to the use of perturbationtheory and the second approximation is valid when Ω (cid:28) δ j . Iden-tical expressions are found for |(cid:104) j | ˆ U ( t ) | b (cid:105)| . Equation 16 de-scribes the leakage out of states | (cid:105) and | e (cid:105) during the | (cid:105) ↔ | e (cid:105) π -pulse, states | (cid:105) and | e (cid:105) during the | (cid:105) ↔ | e (cid:105) π -pulse, and states | e (cid:105) and | e ;0 (cid:105) during the | e (cid:105) ↔ | e ;0 (cid:105) exchange. To treat leakagefrom | (cid:105) during the | (cid:105) ↔ | e (cid:105) π -pulse, let Ω = . Then 15 simplifiesto (cid:12)(cid:12) (cid:104) j | ˆ U ( t ) | (cid:105) (cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) Ω j δ j (cid:12)(cid:12)(cid:12)(cid:12) ( − cos ( δ j t )) . (17)A careful choice of π -time reduces this leakage term, equivalentto minimizing leakage via alignment of power-spectrum zeros.To evaluate the fidelity of each interaction step, we calculatethe total population leakage probability from each nominally-populated state | i (cid:105) as p i = ∑ j (cid:12)(cid:12) (cid:104) j | ˆ U ( t ) | i (cid:105) (cid:12)(cid:12) (18)using Eq. 16 or 17, as appropriate. The minimum fidelity is then F ≈ −
12 max i p i . (19)(19)