Toward long-range entanglement between electrically driven single-molecule magnets
Khadijeh Najafi, Alexander Wysocki, Kyungwha Park, Sophia E. Economou, Edwin Barnes
TToward Long-Range Entanglement BetweenElectrically Driven Single-Molecule Magnets
Khadijeh Najafi, ∗ Aleksander L. Wysocki, Kyungwha Park, Sophia E. Economou,and Edwin Barnes
Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A
E-mail: najafi[email protected] a r X i v : . [ qu a n t - ph ] F e b bstract Over the past two decades, several molecules have been explored as possible buildingblocks of a quantum computer, a device that would provide exponential speedups fora number of problems, including the simulation of large, strongly correlated chemicalsystems. Achieving strong interactions and entanglement between molecular qubits re-mains an outstanding challenge. Here, we show that the TbPc single-molecule magnethas the potential to overcome this obstacle due to its sensitivity to electric fields stem-ming from the hyperfine Stark effect. We show how this feature can be leveraged toachieve long-range entanglement between pairs of molecules using a superconductingresonator as a mediator. Our results suggest that the molecule-resonator interaction isnear the edge of the strong-coupling regime and could potentially pass into it given amore detailed, quantitative understanding of the TbPc molecule. Graphical TOC EntryKeywords single molecule magnet, TbPc molecule, long-range entanglement, hyperfine Stark effect,superconducting resonator, strong-coupling limit2uantum information processing (QIP) is based on storing information in quantum two-level systems (qubits) and takes full advantage of key features of quantum mechanics, suchas quantum interference and entanglement, in order to exponentially speed up certain typesof problems. The most well-known example is Shor’s algorithm, which, if implemented in alarge quantum computer, would be able to break the RSA cryptosystem, which is currentlythe predominant system for securely transmitting information. While the requirements forShor’s algorithm are formidable in terms of the necessary number of qubits and level of controlprecision, there also exist important near-term applications that can be implemented witha more modestly sized quantum computer. A notable application is quantum simulation, which would enable the computational modeling of large-scale strongly correlated quantumsystems, with applications in quantum chemistry and medicine. Over the past two decades, several quantum systems have been explored as candidatequbits for QIP. An obvious choice is spin (either electronic or nuclear), as it can be a truetwo-level system and tends to be well isolated from its environment, leading to relatively longcoherence times.
In 2001, Leuenberger and Loss proposed to use the spin of the nanoscalesingle-molecule magnet (SMM) Mn as a qubit, with the control achieved via electron spinresonance pulses. Since then, plausible setups and architectures for quantum computingwith SMMs have been proposed by several groups.
A majority of the proposals arebased on magnetically controlled SMM electron spin qubits, for which the coherence timesare not yet sufficiently long for quantum computing. Magnetic field control also limits thepotential of SMMs for device integration and scalability, as it is extremely challenging toaddress individual qubits this way, and it also tends to yield slow gate operations, limitingthe complexity of algorithms that can be run.Recently, a qubit candidate with remarkable properties was experimentally demonstratedby Thiele et al.: the SMM TbPc , which features a nuclear spin as the qubit, with theattractive and unusual property of being electrically controllable. This combines the bestof both worlds: long-lived qubit coherence with fast controllability. This recent exciting3iscovery opens up the opportunity for the development of scalable SMM-based QIP devices.To develop a quantum information processor, the qubits must fulfill certain criteria. Firstof all, individual qubits must be controllable and measurable. Proof-of-principle demon-strations of these capabilities have been carried out for TbPc SMMs. A second crucialrequirement for QIP is that the qubits must be coupled to each other through some physicalinteraction in order to implement quantum logic gates. The direct dipolar coupling betweennuclear spins is far too weak to achieve significant coupling. Most proposals instead positusing electron spins as mediators of an effective nuclear spin coupling.
However, electronspin dipolar interactions are also weak, and exchange coupling requires the daunting task ofplacing donors or molecules with nanometer precision.A possible way to overcome these challenges is to use a superconducting transmissionline resonator as a ‘bus’ to mediate coupling between TbPc SMMs. The fact that TbPc SMMs are sensitive to electric fields through the hyperfine Stark effect allows for them tocouple to the electric field of the resonator, potentially leading to strong interqubit interac-tions. This approach is also natural given that the splittings between nuclear spin states ina TbPc molecule are on the order of GHz, the typical frequency range of superconductingresonators. Furthermore, this method of coupling has the advantage that it is long-range,enabling direct coupling between pairs of distant qubits, something not possible with nearest-neighbor architectures. Superconducting resonators are widely used to couple qubits basedon superconducting circuits, and have also been employed to couple remote electronspins.
They have high quality factors ( Q ∼ ) and mature fabrication technology.Moreover, the flat structure of the TbPc molecule (Fig. 1) makes the prospects for fabrica-tion with a superconducting resonator promising. A schematic of the envisioned architectureis shown for two molecules in Fig. 1.In this paper, we investigate the feasibility of using superconducting resonators to createentanglement between TbPc qubits. We begin by modeling the single-qubit Rabi oscillationsobserved in Ref. to determine the minimal hyperfine tensor necessary to reproduce these4igure 1: Schematic of TbPc single-molecule transistors coupled to a resonator. (a) Eachmolecule is comprised of a Tb ion sandwiched between two flat Pc ligands that are connectedto a source and drain. Individual molecules are driven via the hyperfine Stark effect bymodulating the voltage across the source and drain. (b) Two molecules coupled to theelectric field of a common cavity resonator via the hyperfine Stark effect. The molecules arelocated at the antinodes of the cavity’s electric field.findings. We use this information to estimate the qubit-resonator coupling, which we findis on the edge of the strong coupling regime defined by the cavity photon loss rate and thespin dephasing time. We then employ these results to construct an effective Hamiltonianfor multiple TbPc qubits coupled to the electric field of a superconducting resonator viathe hyperfine Stark effect. To test the entangling capabilities of this effective interaction,we design two-qubit CNOT gates and determine the fidelities and gate speeds for a rangeof coupling strengths. We find that while fidelities above 99% can be achieved in all cases,pushing gate times to well above the qubit dephasing time will likely require boosting theinteraction strength further. Our results suggest that superconducting resonators may bea promising approach for building quantum processors out of electrically driven SMMs,although further improvements in device designs will likely be needed to reach the strongcoupling regime.The TbPc molecule consists of a Tb ion sandwiched between two flat Pc ligands (seeFig 1). The Tb ion has an electronic configuration of [Xe]4f , which implies a totalorbital angular momentum of L = 3 and a total spin of S = 3 based on Hund’s rules.Therefore, the electronic ground state has total angular momentum J = 6 . An unusually5trong spin-orbit coupling ( ∼ J = 6 multiplet and separates the ground state doublet m J = ± from the first excitedstate doublet m J = ± by an energy gap of 600 K (zero-field splitting). Consequently, atvery low temperatures ( ∼
50 mK), the ground states with m J = ± become well isolated,and the electronic spin becomes Ising-like. In describing the energy units in Kelvin wehave set ¯ h = k b = 1 . Furthermore, the Tb ion contains a nuclear spin of I = 3 / thatcouples to the electronic spin via a hyperfine interaction. This hyperfine coupling (withstrength A = 24 . mK ) lifts the four-fold degeneracy of the nuclear spin, yielding a low-energy manifold of eight non-degenerate electron-nuclear spin states as shown in Fig. 2.The hyperfine interaction also contains a quadrupolar term (with coupling strength P = 0 . mK ) which results in the non-uniform energy spacing evident in the figure. In addition, theoff-diagonal part of the ligand field couples the electronic states m J = ± and thus createsavoided crossings on the order of µ K between states with the same nuclear spin projection(marked with boxes in Fig. 2). These avoided crossings are used to initialize and readoutthe nuclear spin states through quantum tunneling of magnetization. The properties summarized above are captured by the following effective Hamiltonian: H SMM = H Z + H LF + H HF , (1)which includes contributions from Zeeman interactions, the ligand field, and the hyperfineinteraction: H Z = g l µ B J · B , (2) H LF = H DLF + H ODLF , (3)6 HF = A I · J + P (cid:20) I z −
13 ( I + 1) I (cid:21) , (4)where, g l = 1 . , µ B is the Bohr magneton, and the ligand field is described in Ref. Theenergy levels shown in Fig. 2 are the lowest eight eigenstates of this Hamiltonian plotted asa function of the magnitude of the external magnetic field, which is chosen to point alongthe z direction. It is clear that far away from the avoided crossings, the energy splittingsbetween states with the same m J are approximately constant. Restricting attention to the m J = − submanifold, we have after diagonalization H DSMM = (cid:88) j =1 ω j | j (cid:105)(cid:104) j | . (5)Away from the avoided crossings, the eigenenergies ω j depend approximately linearly on themagnetic field, with splittings given by (in GHz) ν = 2 . , ν = 3 . , and ν = 3 . (seeFig. 2).Most nuclear spin qubit proposals make use of time-dependent magnetic fields to manip-ulate the spin states. Although very high single-qubit gate fidelities have been achievedwith this approach, the speed of the gates is limited by restrictions on the amplitudeof magnetic pulses. In order to avoid substantial cross-talk and joule heating caused by themicro-coil used to generate the magnetic field, the amplitude must typically be kept below afew mT. To overcome this problem, the manipulation of the nuclear spin state by means ofelectric fields has been proposed for TbPc SMMs and phosphorous donors in silicon. Since the electric field does not directly couple to spin, it is necessary to have an interme-diate interaction which converts an ac electric field into an effective magnetic field. Variousmechanisms have been used to facilitate this conversion, including through spin-orbit cou-pling, magnetic field gradients, and hyperfine interactions. Here, we focus on thehyperfine Stark effect, which was experimentally demonstrated to yield electrically drivenRabi oscillations between nuclear spin states in TbPc . ----------- - ,Q I 6 +� > I 6 } > I 6 } > I +6 � > I 6 � >
I 6 -} > -=--- ----1._----1 _ ___JL...______:� I 6 � >
04 0 I .
0 0 0 IO 2 0 I 3 0 µo
11 ( -3/2 > v =3.63GHz -1/2 > v =3.09 GHz 1-6, + >1-6, +3/2 > (a) (b) Figure 2: (a) Energy level diagram of the lowest eight energy eigenstates of the
TbPc molecule. States are labeled by the electronic and nuclear spin quantum numbers of thenon-interacting states with which they have greatest overlap. The avoided crossings (boxes)are used to initialize and readout nuclear spin states, while gate operations are performedafter tuning away from avoided crossings. One possible choice for the two qubit states isindicated, although this choice is not unique. (b) Zoom-in of four of the levels with energyspacings away from avoided crossings indicated.The hyperfine Stark effect refers to the shift in nuclear spin energy levels caused by anapplied electric field. This effect originates from the dependence of the hyperfine couplingson the shape of the electronic wavefunction, which is of course sensitive to electric fields. Wecan rewrite the hyperfine Hamiltonian as an effective Zeeman interaction, H hf = g N µ N I · B eff ( A, J ) , where B eff is an effective magnetic field felt by the nuclear spin due to a netelectronic spin magnetization. By substituting J = 6 and g N = 1 . , we get B eff = 313 T, showing that the effective magnetic field created by an ac electric field is several ordersof magnitude larger than the actual magnetic fields produced by micro-coils. In Ref. it was found from both experimental results and perturbation theory calculations that thesensitivity of the hyperfine coupling to an applied electric field E is approximately given by ∆ A/A ∼ − for fields on the order of E ∼ mV/nm.Although the experimental demonstrations of Ref. make it clear that electrically drivennuclear spin transitions are enabled by a significant hyperfine Stark effect in this system,8any details have yet to be clarified. Most importantly, the precise form of the hyperfinetensor for TbPc is not yet known, giving rise to uncertainty in precisely how the nuclear spinstates respond to electric fields. This issue is critical not only for improving the quality ofsingle-qubit operations, but also for designing schemes to couple multiple qubits together viaelectrical interactions. Here, we shed some light on the nature of the hyperfine interactionby determining the simplest hyperfine tensor necessary to produce Rabi oscillations.To investigate this matter, we start with the most general form of the hyperfine interaction H HF = (cid:88) αβ I α A αβ J β , (6)where A αβ is a matrix representing the (generally anisotropic) coupling of the electronic andnuclear spins. An applied electric field will shift the hyperfine interaction, which to firstorder in the field yields a second term of the same form: H HF ≈ (cid:88) αβ I α A αβ J β + αE ( t ) (cid:88) αβ I α A αβ J β , (7)where E ( t ) is the electric field, and α is a constant. Our qubit states are defined to be thelowest energy eigenstates of H SMM , which includes the first term in Eq. 7 but not the second.The second term allows us to drive transitions between the different energy eigenstates, andwe thus refer to it as the control Hamiltonian, H c ( t ) . Here, the time dependence reflects thatof the applied electric field. We see that the controllability of the TbPc nuclear spin qubitis determined by the matrix elements of H HF with respect to the lowest-energy eigenstatesof the full Hamiltonian H SMM . We find that all of these matrix elements, taken with respectto the states depicted in Fig. 2, vanish identically if A αβ is purely diagonal. Thus, in order todrive Rabi oscillations between states in the low-energy manifold, it must be the case that atleast one off-diagonal entry of A αβ is nonzero. Furthermore, we want to choose the z axis tobe along the easy anisotropy axis so that the ground state doublet is J z = +6 and J z = − .Thus, we consider the simplest case where only one off-diagonal component is nonzero, and9e take this to be A xz . Given that Rabi oscillations have been demonstrated experimentally,we know that such a term must be present. The presence of A xz or A yz terms reflectsa deviation from the 4-fold axis symmetry. This can be caused by the transverse electricfield or by deviations of the molecular structure from the ideal D h symmetry. Determiningthe precise nature of this anisotropy requires detailed ab-initio calculations that we leave tofuture work.Taking the diagonal entries of A αβ to be the same for simplicity (all equal to A ) andretaining only A xz from the off-diagonal entries, we arrive at the following form for thecontrol Hamiltonian, H c ( t ) = ηA cos( ω p t ) n · I , (8)where n = sin( θ )ˆ z + cos( θ )ˆ x , θ = arctan A/A xz , ω p is the frequency of the oscillating electricfield, and η is a constant that depends on α , J , m J , and the magnitude of the electricfield. To arrive at Eq. 8, we have projected the electronic angular momentum J onto the m J = − submanifold since our focus will be on driving transitions between states withinthis manifold.Now that we have established a form for the control Hamiltonian, we proceed to in-vestigate the controllability of the TbPc qubit as a function of the hyperfine anisotropyparameter θ . We focus on the lowest energy states |− , +3 / (cid:105) and |− , +1 / (cid:105) as our qubitstates, which are separated in energy by ν = 2 . GHz, although the same analysis couldbe applied for any two nuclear spin states. Notice that in this two-level subspace we caneffectively make the replacements I x → √ σ x and I z → σ z where σ x and σ z are Pauli ma-trices. Initially, we consider the case of resonant driving, for which the detuning vanishes: ∆ = ω p − ν = 0 . Solving the time-dependent Schrödinger equation for the evolution opera-tor U with the Hamiltonian from Eq. 8, we obtain the transition probability as a function oftime as shown in Fig 3. Fig. 3(a) shows the resulting Rabi oscillations for several different10alues of θ . As expected, only the component in the x direction is capable of driving tran-sitions between the states, while the population transfer is zero for θ = π/ . Importantly,we see that for any finite amount of anisotropy, it is possible to completely transfer thepopulation from one state to the other. Moreover, while the transfer becomes slower as theanisotropy is reduced, the transfer time increases slowly with increasing θ . This indicatesthat the performance of single-qubit gates is relatively insensitive to the precise form of thehyperfine tensor.The fact that Eq. 8 yields a σ z term in addition to σ x makes the present control problema bit different from the standard Rabi problem. Thus, it is worth checking the extent towhich the usual Rabi behavior applies here. For a general detuning ∆ , and without ignoringthe fast oscillating field in the interaction representation (no rotating wave approximationapplied) the Rabi frequency is given by the formula Ω R / π = (cid:113) (∆ / π ) + ( √ g N µ N B x /h ) ,where B x is the transverse component of the effective magnetic field. When ∆ = 0 , this leadsto the following expression for the Rabi period: T R = 2 π/ Ω R = 4 π/ ( √ ηA cos( θ )) , whichagrees well with the numerical results shown in Fig. 3(a). We note that our numerical resultsfor the Rabi frequencies are also compatible with the reported experimental values, whichare on the order of a few µ s. Fig. 3(b) shows the behavior of the Rabi oscillations for off-resonant driving, ∆ (cid:54) = 0 . As is the case for the standard Rabi problem, the Rabi frequencyis minimal at resonance and increases as one tunes away from resonance. We conclude thatby adjusting the driving time and detuning, it is possible to create any single-qubit gate forwhich the rotation axis is in the xz plane. All other single-qubit gates can be obtained byconcatenating these operations using standard composite pulse sequences. While we have seen that it is possible to perform any single-qubit gate on isolated TbPc qubits, this is not guaranteed to remain true when we start coupling two or more qubitstogether. For example, we need to ensure that it is possible to address each qubit individuallywithout disturbing the rest. This is achievable by taking advantage of the dc stark effectinduced by applying a dc gate voltage. Such a voltage will shift the energy levels of the11 = θ = π / θ = π / θ = π / τ [ μ s ] ( | 〈 + | U | + 〉 | ) (a) Δ = ���� Δ = ���� Δ = ���� Δ = ���� Δ = ���� ��� ��� ��� ��� ��� ��� ��� ��������������������� τ [ μ � ] ( | 〈 + � � | � | + � � 〉 | ) � (b) Figure 3: Rabi oscillations. An ac electric field drives transitions between the lowest twonuclear spin states | − , +3 / (cid:105) and | − , +1 / (cid:105) . (a) The transition probability as a functionof driving time for several different values of the hyperfine anisotropy parameter θ . The Rabiperiods obtained from the formula in the main text are T R [ θ = 0] = 2 . µs and T R [ θ = π/
6] = 2 . µs , which agree well with numerical results. (b) The transition probability as afunction of driving time for five different values of the detuning. The values of detuning ∆ are in MHz, and we have set θ = π/ . We set η = 0 . in both panels.nuclear spin states, allowing us to adjust the qubit resonance frequency at will. Thus, whenwe perform an operation on one qubit, we can first tune it away from the other qubits toavoid driving them. As we will see later, this ability to shift the resonance frequency is alsocrucial to achieving high-fidelity entangling gates. In Refs., shifts in the nuclear spinresonance frequency of ∆ ν exp = 1 . MHz and . MHz were measured and compared withperturbation theory for gate voltages of V g = 10 mV and mV, respectively. These valuesin turn correspond to shifts of the hyperfine constant on the order of ∆ A/A = 5 . × − and ∆ A/A = 2 . × − . To check these findings, we have used the perturbation theory results where we included the hyperfine constant shifts in our numerical simulation and computedthe resulting frequency shifts in each case, which yielded slightly different numerical values: ∆ ν num = 1 . MHz and ∆ ν num = 7 . MHz for V g = 10 mV and mV, respectively. As wecan see from Fig 3(b), the larger voltage offsets should be sufficient to decouple the qubitfrom the driving field. Next, we use both the experimental and numerical results for thefrequency shifts to estimate the coupling strength between a TbPc qubit and the electricfield of a microwave resonator and to design high-fidelity two-qubit entangling gates.12ntangling gates are a requirement for any universal quantum computer and a basic in-gredient for all quantum algorithms of interest. Creating entanglement on demand requiressufficiently strong, controllable interactions between qubits. However, this is notoriously dif-ficult to achieve for qubits based on the spin of a donor atom or molecule. This is because thetwo main options for spin-spin coupling, namely dipolar couplings and exchange interactions,are either too weak or diminish too quickly with distance and thus require the capability toplace the spins in close proximity to each other with high accuracy. The latter is due to thestrong confinement of the electronic wavefunction around the donor or molecule. Longer-range spin-spin couplings mediated by resonators have been proposed previously, but these schemes are normally based on magnetic interactions that are again too weak(10-100 Hz) to be practical for achieving coherent interactions between individual spins. Toovercome this issue, we can instead consider using the electric field of a superconductingtransmission line resonator to mediate interactions between TbPc qubits. While this ap-proach was originally developed to couple superconducting qubits, recently, it has beensuccessfully implemented to create long-distance coupling between electron spins in semicon-ductor quantum dots and between electron spins and superconducting qubits. A similarapproach has also been proposed for qubits based on the nuclear spin of phosphorous donorsin silicon. To obtain a stronger coupling between a TbPc SMM and a resonator, we can againleverage the hyperfine Stark effect to couple the molecule to the electric field of the cavityinstead of its magnetic field. For this purpose, resonators based on NbTiN nanowires areparticularly promising. These are microwave-frequency resonators that possess a large ki-netic inductance, and they have already been successfully coupled to spins in semiconductorquantum dots.
There are several reasons for choosing this particular type of resonator.First of all, it has a high critical magnetic field (B ∼
350 mT) that is well above the fields usedin TbPc experiments. Second, the high kinetic inductance of the nanowires leads to anincrease in the characteristic impedance up to Z r ∼ ¯ h Ω , which is two orders of magnitude13arger than what is typically achieved in coplanar waveguides. Furthermore, the increasein impedance leads to an increased resonator vacuum rms voltage of V RMS ∼ µ V, whichin turn produces larger shifts in the TbPc hyperfine coupling. We now give an estimateof the resulting SMM qubit-resonator coupling. Earlier we noted that a dc gate voltageon the order of 10 mV produces a shift in the qubit resonance frequency on the order of1-2%., as reported in Ref. 16. There is currently not enough experimental data on how theresonance frequencies depend on gate voltage to extrapolate this finding to other voltages.This dependence factors critically into the effective qubit-resonator coupling, highlightingthe need for further experimental work along these lines. To proceed with our estimate ofthe coupling, we instead rely on the perturbation theory result of Ref. 16, which gives alinear dependence of the hyperfine interaction on the gate voltage, implying that a resonatorvacuum rms voltage of 1-20 µ V can produce a frequency shift of up to 0.002%. Combiningthis with the value for the hyperfine constant, A = 518 MHz, we estimate the qubit-resonatorcoupling to be g/ (2 π ) ∼ × − | m J | A ∼ kHz. While this value is well above couplingsgenerated by magnetic interactions, it lies below that of other systems where an electricalspin-resonator coupling of order 1-10 MHz has been achieved. To determine whetherthis estimate can be considered to lie within the strong coupling regime, we must compareit to the resonator decay rate and the spin dephasing time. Assuming a resonator qualityfactor of Q ∼ and a cavity frequency on the order of - GHz, the cavity decay rateis κ/ (2 π ) = ω c /Q (cid:39) kHz, a little below our estimated qubit-resonator coupling. Thespin dephasing rate is given by γ/ (2 π ) = 1 /T ∗ (cid:39) kHz, where we have used the measuredvalue of the dephasing time: T ∗ (cid:39) . ms. These numbers suggest that the TbPc qubit-resonator system is currently near the edge of the strong coupling regime ( g > γ, κ ) . It maybe possible to increase coherence times further by using better substrates to eliminate sourcesof noise and by employing dynamical decoupling schemes (taking advantage of the fact thatcharge noise—the dominant type of noise in this system—is concentrated at low frequencies)to make T (cid:29) T ∗ , rather than T ∗ , the relevant timescale. On the other hand, improving14he quality factor of resonators much beyond may not be feasible. However, it may bepossible to enhance the hyperfine constant itself and/or its sensitivity to electric fields. Bothdepend on the ligand field, which in turn could likely be influenced by external factors suchas the choice of substrate. Determining the extent to which the electrical coupling can beincreased first requires a deeper understanding of what determines the hyperfine constantand how the TbPc molecule responds to its environment.To obtain a better understanding of how much stronger the qubit-resonator couplingneeds to become, we now investigate the performance of two-qubit entangling gates as afunction of the interaction strength. We begin by writing down a Hamiltonian that describesmultiple TbPc qubits coupled to a common resonator mode: H = ω c a † a + (cid:88) n,j ω n,j | n, j (cid:105)(cid:104) n, j | + ( a + a † ) (cid:88) n,j (cid:15) n,j | n, j (cid:105)(cid:104) n, j | (9) + (cid:88) n,j (cid:0) ξ − n,j a † | n, j (cid:105)(cid:104) n, j + 1 | + ξ + n,j a | n, j + 1 (cid:105)(cid:104) n, j | (cid:1) , where ω c denotes the resonator frequency, ω n,j indicates the j th energy level of qubit n (herewe consider n = 1 , ), (cid:15) n,j ≡ ηA sin( θ n ) , ξ ± n,j ≡ ηA cos( θ n ) (cid:112) I ( I + 1) − j ( j ± , and I = 3 / is the total spin of the nucleus. Here, we have made the rotating wave approximation inwhich we remove counter-rotating terms under the assumption that ω n,j +1 − ω n,j ∼ ω c . Thisis essentially a Jaynes-Cummings-type Hamiltonian, but with an additional (cid:15) n,j term whichimplements energy-level tuning in addition to the inter-level transitions generated by theusual Jaynes-Cummings ξ ± n,j terms.In order to perform the maximally entangling two-qubit gates needed for quantum com-puting algorithms, it is sufficient to electrically drive a single SMM qubit that is resonator-coupled to a second qubit. The most significant source of gate errors in this case is leakage tonuclear spin states outside the logical subspace or to excited resonator states. Our strategyto address this leakage is based on pulse designs using analytical approaches. To implement15igh-fidelity two-qubit entangling gates, we employ a recently developed formalism knownas the SWIPHT protocol. This method enables a speedup of the two most common entan-gling gates (CZ and CNOT) that can range from a factor of two to more than one order ofmagnitude while maintaining high fidelities and using only smooth pulses given by analyticalexpressions. In this paper, we focus on the well-known two-qubit entangling CNOT gate inwhich the state of one qubit is flipped conditionally on the state of the other qubit.Diagonalizing the two-qubit-resonator Hamiltonian given in Eq. 9, we obtain the inter-acting dressed states. We define our logical qubit states to be the four dressed states thathave the largest overlap with the non-interacting two-qubit states | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) . Wedenote the logical states as (cid:103) | (cid:105) , (cid:103) | (cid:105) , (cid:103) | (cid:105) , (cid:103) | (cid:105) . Here, we consider identical qubit-resonatorcouplings for both qubits, however, our analysis can be adapted straightforwardly to the caseof non-identical couplings as well. A microwave electrical pulse drives transitions betweenthe logical states as described by the following control Hamiltonian: H p = Ω( t ) cos( ω p t ) (cid:104) λ c ( a + a † ) + (cid:88) n,j λ n ( (cid:15) n,j | n, j (cid:105)(cid:104) n, j | + ξ − n,j | n, j (cid:105)(cid:104) n, j + 1 | + ξ + n,j | n, j + 1 (cid:105)(cid:104) n, j | ) (cid:105) , (10)where Ω( t ) is the amplitude of the pulse and ω p its frequency. We included the parameters λ c , λ n to indicate which components of the tripartite system are driven by the pulse. Here,we consider the case with λ c = λ = 0 , λ = 1 , which means we are only driving the secondqubit.We can implement a CNOT gate by performing a π -rotation on the target transition (cid:103) | (cid:105) ↔ (cid:103) | (cid:105) while avoiding all other transitions in the spectrum. For qubit-resonator cou-plings that are not too strong, there is typically only one other driven transition that isnearby in frequency: the transition (cid:103) | (cid:105) ↔ (cid:103) | (cid:105) . In the absence of a qubit-resonator cou-pling, this unwanted transition would be degenerate with the target transition since they bothcorrespond to driving the second qubit between its two states, and without the inter-qubit16oupling, this would not depend on the state of the first qubit. When the qubit-resonatorcoupling is switched on, the two transitions remain nearly degenerate. Typically one wouldneed to resort to long, spectrally selective pulses to avoid exciting this unwanted transition,which leads to slow gates. However, the SWIPHT formalism allows one to avoid long pulsesby letting the pulse drive the unwanted transition also, in such a way that it undergoes cyclicevolution and acquires a trivial π phase, so that a CNOT gate is still achieved. It was shownin Ref. that pulses which perform a SWIPHT-based CNOT gate can be constructed usinga systematic recipe. This approach is based on the fact that, for a driven two-level system,both the driving field Ω( t ) and the time evolution operator U ( t ) can be expressed in termsof a single real function χ ( t ) : Ω( t ) = ¨ χ (cid:113) δ − ˙ χ − (cid:114) δ − ˙ χ cot(2 χ ) , (11) U ( t ) = e − i π σ y cos χe iψ − sin χe − iψ + − sin χe iψ + cos χe − iψ − , (12)where ψ ± = (cid:82) t dt (cid:48) (cid:113) δ − ˙ χ csc [2 χ ( t (cid:48) )] ± arcsin( χδ ) , and δ is the pulse detuning. In order tohave a valid solution, χ must satisfy the constraint | ˙ χ | ≤ | δ | along with the initial conditions χ (0) = π/ , ˙ χ (0) = 0 . This formalism has been used to design two-qubit entangling gates insuperconducting qubits and quantum dots with fidelities exceeding 99%. This methodhas also been experimentally demonstrated in the case of superconducting qubits. We can use this construction to create a CNOT gate on two SMM qubits by taking thetwo-level system to be the two states of the second qubit. In this case, we can impose that itsevolution be cyclic by requiring that χ ( τ ) = π/ and ˙ χ ( τ ) = 0 , where τ is the duration of thepulse. At the same time, we also need to ensure that the pulse performs a π rotation on thefirst qubit, as required for a CNOT. If we take the pulse to be resonant with this qubit, thenthis is tantamount to requiring that the area of the pulse be equal to π/ : (cid:82) τ dt Ω( t ) = π/ .17he following ansatz for χ ( t ) can be used to satisfy all these criteria: χ ( t ) = C ( t/τ ) (1 − t/τ ) + π/ . (13)This ansatz automatically obeys the initial and final conditions on χ . Moreover, we can tunethe parameters C and τ until the pulse area constraint is also satisfied. We find numericallythat the values C = 138 . and τ = 5 . / | δ | achieve this. Note that since the pulse isresonant with the first qubit, δ is equal to the difference in resonance frequencies of thetarget and unwanted transitions. This frequency difference determines the gate time of theSWIPHT pulse, as is clear from the above formula for τ .To evaluate the performance of the resulting CNOT gate, we numerically solved thetime-dependent Schrödinger equation in the interaction picture defined with respect to H to obtain the evolution operator for the two-qubit system. We define our gate in the interac-tion picture so that it is created purely from the applied control pulse and does not includeadditional phases coming from free evolution. We performed this calculation using the pulseobtained from Eqs. 11 and 13 and for a range of coupling strengths g and resonator fre-quencies ω c . In each case, we computed the fidelity of the gate using the standard formula F ≡ (Tr[ U U † ] + | Tr[ U † CNOT ] | ) , optimized over single-qubit gates on both qubits. Oursimulations include a frequency shift of MHz on the second qubit, which can be obtainedby applying a dc gate voltage on the order V = 90 mV. We found that this produces betterperformance in terms of both fidelity and gate speed.The results are summarized in Fig. 4, which shows the infidelity − F and the gate time asa function of the coupling g and for three different resonator frequencies. First, it is evidentthat the CNOT fidelity remains above 99% and is largely insensitive to the qubit-resonatorcoupling over the full range of couplings considered. In fact, the fidelity remains essentiallyconstant for couplings below 20 MHz. This is true for all three resonator frequencies weconsidered. In contrast, the gate time is very sensitive to the coupling strength: for couplings18n the range 40-50 MHz, the gate times are on the order of a few µs , while for couplings onthe order of a few MHz, the gate times approach milliseconds to seconds. Furthermore, wenotice that as the resonator frequency is tuned further from the qubit frequencies, the gatetime increases further. In the case where the resonator frequency is closest to the qubits, ω c = 2 . GHz, the coupling would need to be at least MHz to get the gate time below thedephasing time of T ∗ ∼ . ms. Although alternative gate designs such as the cross-resonancegate may lead to shorter gate times, this result highlights the importance of finding waysto further enhance the coupling between TbPc qubits and microwave resonators.Figure 4: Infidelity − F and gate time τ of the two-qubit entangling CNOT gate as functionsof the qubit-resonator coupling g for three different values of the resonator frequency ω c (inGHz). The gate was generated by the voltage pulse defined by Eqs. 11 and 13. Other systemparameters were chosen as in Fig. 3.In conclusion, we investigated the possibility of using superconducting resonators toachieve strong coupling between TbPc nuclear spin qubits by leveraging the hyperfine Starkeffect. To better understand the nature of this effect, we examined single-qubit Rabi os-19illations, where we found that anisotropy in the electron-nuclear hyperfine interaction isnecessary to electrically drive transitions between the nuclear spin states. This anisotropymust be present since such transitions have been demonstrated experimentally. With thisresult, we then estimated the qubit-resonator interaction, finding that it lies close to the edgeof the strong coupling regime. To understand the implications for entanglement creation,we constructed a Hamiltonian that describes two TbPc qubits coupled by a resonator andshowed that it is possible to perform high-fidelity two-qubit entangling gates with this archi-tecture. However, we find that in order to reduce gate times sufficiently, it may be necessaryto increase the qubit-resonator coupling through improved device designs. Acknowledgement
K. N. would like to thank George Barron and Fernando Calderon-Vargas for helpful discus-sions. This work was supported by DOE grant no. de-sc0018326.
Supporting Information Available
The following files are available free of charge.The following files are available free of charge.• Hamiltonian of TbPc molecule, nuclear state of TbPc molecule as qudit, coherentmanipulution of nuclear state References (1) Nielsen, M. A.; Chuang, I. L.
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