Constructing Clock-Transition-Based Two-Qubit Gates from Dimers of Molecular Nanomagnets

A good qubit must have a coherence time long enough for gate operations to be performed. Avoided level crossings allow for clock transitions in which coherence is enhanced by the insensitivity of the transition to fluctuations in external fields. Because of this insensitivity, it is not obvious how to effectively couple qubits together while retaining clock-transition behavior. Here we present a scheme for using a heterodimer of two coupled molecular nanomagnets, each with a clock transition at zero magnetic field, in which all of the gate operations needed to implement one- and two-qubit gates can be implemented with pulsed radio-frequency radiation. We show that given realistic coupling strengths between the nanomagnets in the dimer, good gate fidelities ($\sim$99.4\%) can be achieved. We identify the primary sources of error in implementing gates and discuss how these may be mitigated, and investigate the range of coherence times necessary for such a system to be a viable platform for implementing quantum computing protocols.

A variety of physical systems have been explored as possible qubits [1], including superconducting devices [2], trapped ions [3], and both electronic and nuclear spin systems [4][5][6][7].The ideal multi-qubit architecture would have an array of independently controlled, long-lived qubits, with adjustable couplings between each pair of qubits.Physical implementations of qubits involve tradeoffs between various important features, such as coherence times, addressability, and scalability.Electronic spin systems have garnered a fair amount of attention in recent years as potential qubits, especially in the context of hybrid quantum architectures, in which spins could fulfill the role of memory qubits [8,9].
Molecule-based spin systems, such as molecular nanomagnets (MNMs), offer several advantages over other types of spin systems.In particular, because they are chemically synthesized, properties such as the spin Hamiltonian and interactions with environmental degrees of freedom can be chemically engineered.A class of heterometallic rings has been extensively studied as possible qubits [10].One of the most-studied of these are the family Cr 7 M, where M is a transition-metal ion [11,12].These systems offer the ability to engineer the total ground-state spin of the system through choice of M. A combination of dilution of the molecules in a nonmagnetic medium and chemical engineering by using different ligands and cations to maximize coherence has yielded T 2 ∼ 15 µs [13].
An important source of decoherence in many qubit spin systems comes about from fluctuations in local electromagnetic fields that change the spin's energy (by, e.g., the Zeeman effect) and thereby induce fluctuations in the phase of the spin's quantum state.One effective technique to ameliorate this mechanism of decoherence is to make use of so-called atomic-clock transitions in which the energy levels of a qubit depend non-linearly on the field in some region (i.e.near an avoided level crossing).In particular, when the transition frequency between levels is independent of field at some field (df /dB = 0), the transition will be immune to field fluctuations to first order, suppressing decoherence from those fluctuations and concomitantly increasing the coherence time T 2 .This technique has been exploited with great effect in superconducting qubits, where it is often referred to as the "sweet spot" [14].Recently, the implementation of clock transitions in a Ho-based MNM has produced a marked enhancement of T 2 in the vicinity of avoided crossings, resulting in T 2 as high as ∼ 8 µs [15], several times higher than the value away from the transitions.Similarly, clock-transition behavior has been observed in the heterometallic ring Cr 7 Mn (S = 1), discussed be-low [12].Such rings can be coupled to each other to form supramolecular dimers, and possibly longer chains, that can be used to build multiqubit systems [16][17][18][19][20][21].
Here we describe a scheme in which MNMs displaying clock transitions (such as Cr 7 Mn) can be joined into dimers in which the resulting states of the coupled system retain the characteristics of clock transitions.Remarkably, although in our scheme the molecular monomers exchange couple to each other, they remain insensitive to field fluctuations, thus enabling single-and two-qubit operations to be implemented while preserving the immunity of the system to field fluctuations.
In zero field, the Hamiltonian for an isolated S = 1 Cr 7 Mn molecule is The D i term represents the system's axial (easy-axis) anisotropy, while the E i term corresponds to the transverse anisotropy.Here the subscript i designates a particular molecule.Figure 1 shows the energy eigenstates for this Hamiltonian as a function of field applied along the z axis.We can identify the S zi eigenstates by their m value: |m = 0 ≡ |0 and |m = ±1 ≡ |±1 .At zero field, the energy eigenstates are to a good approximation The latter two states exhibit an avoided crossing with a "tunnel splitting" of 2E i .These two states constitute an atomic-clock transition, with a significant transition matrix element for the S z operator: +| S z |− = 1.Through variations in synthesis, molecules with different values of parameters (D i and E i ) can be produced, notably the so-called green and purple variants of Cr 7 Mn [20].
When coupled together, a pair of molecules with different parameters form a supramolecular heterodimer [18].Interactions between the spins in the dimer can be modeled as a bilinear exchange interaction: We isolate the J zz term here (and implicitly define the J tensor) because it is the only term that directly couples any of the four lowest-energy states to each other.As a consequence, this term is responsible for an error in the implementation of single-qubit rotations, as will be discussed below.It is important to note that molecules within the dimer need not have any simple relative orientation and, thus, each of the principal (easy, medium and hard) axes of the two spins may have any relative orientation.As a consequence, the components of J do not necessarily refer to specific directions in space but to couplings between different axial directions of each spin; e.g.J xz describes the coupling between the hardaxis component of spin 1 and the easy-axis component of spin 2. The total zero-field Hamiltonian for the system is When the D i are much larger than all the other energy parameters (E i , J ij ), the subspace of the four lowestenergy states acts as a system of two coupled effective S = 1/2 spins.For J = 0 and the realistic case of E i ≫ J zz , the lowest and highest energy states in the subspace are to a good approximation |++ and |−− , with energies The two middle-energy states can be represented as where tan 2θ = 2Jzz ∆E , with energies Since the states are constructed from clock states, near zero field all four of these states are barely affected by a magnetic field, as illustrated in Fig. 2, unlike real coupled S = 1/2 spins.For implementation of quantumcomputing protocols we use the energy eigenstates as the logical basis, labelling these with vertical arrows, e.g.|↑↓ .
FIG. 2. Energies vs. field for the four lowest states of a heterodimer of Cr7Mn molecules using known parameters, demonstrating the retention of clock states in the coupled system.
Certain transitions within the four-state manifold are degenerate, e.g.|++ ↔ |↑↓ is degenerate with |↓↑ ↔ |−− .These degeneracies are broken by the J term in Eq. ( 2).For simplicity, we consider the case in which J is diagonal such that (Other forms of J give qualitatively similar results.)To second order in J ⊥ , the |++ and |−− states become, respectively, the states |↑↑ and |↓↓ : and their energies are shifted by ∆E = , respectively, thereby breaking the transition degeneracies.The |↑↓ and |↓↑ states are unchanged by J.Under these circumstances, the four-state system becomes an effective two-qubit system with constant coupling in which standard one-and two-qubit gates can be implemented with pulsed radiation.Transitions between the four states are induced by radio-frequency (rf) radiation: We consider a radiation field dropping terms that correspond to far-off-resonance transitions.We note that since the easy axes of the two spins are not in general parallel, the z components of the radiation fields may correspond to different directions even if B 1 and B 2 are colinear.We simulated our system by solving the Schrödinger equation to find the time evolution under the Hamiltonian H + H rf .The Hamiltonian was transformed into the interaction picture using the operator U int = e −i H t , where After discarding rapidly oscillating terms in the Hamiltonian (rotating wave approximation) as well as dropping an irrelevant constant, one obtains the interactionpicture effective Hamiltonian: where δ = − 2 DJ 2 ⊥ D2 − Ē2 and S 1z,↓↓,↓↑ = ↓↓| S 1z |↓↑ , etc.The radiation coupling, Eq. ( 7), does not provide coupling between any of the four lowest-energy states and any of the higher states, justifying truncating our system to consist of only the four states.
A one-qubit operation changes the state of a single qubit, independent of the state of the other.The field B 1 (B 2 ) will achieve this for qubit 1 (2), provided that S 1z,↓↓,↓↑ = S 1z,↑↓,↑↑ (S 2z,↓↓,↑↓ = S 2z,↓↑,↑↑ ).For J zz = 0, this condition is nearly perfectly fulfilled, with a secondorder error of S 1z,↓↓,↓↑ − S 1z,↑↓,↑↑ ∼ J 2 ⊥ D Ē .The effect of J zz is more severe, resulting in an error ∼ Jzz ∆E .Thus, it is desirable to minimize J zz as much as possible.This may be achievable through chemical engineering of the supramolecule to arrange the relative orientation of the easy axes into a configuration that results in a very small J zz , akin to a "magic angle" effect.Alternatively, one may use a switchable linker in the dimer to turn off the excahnge coupling during the one-qubit operations [16,17].Such an approach requires a fast, local probe to switch the linker state and the ability to measure the state of an individual supramolecule via spin resonance techniques.In contrast, always-on coupling, while potentially leading to single-qubit errors, permit ensemble measurements, a less technically challenging approach.
We perform simulations using the established Hamiltonian parameters for the (1) green and (2) purple variants of Cr 7 Mn: D 1 = 21 GHz, D 2 = 16.5 GHz, E 1 = 1.9 GHz, E 2 = 2.6 GHz [22,23].In addition, we take J ⊥ = 100 MHz and g = 2, while choosing different values of J zz as discussed below.J ⊥ and J zz can be controlled during synthesis [16,17,19].A basic one-qubit gate is a π/2 rotation, implemented by setting either B 1 or B 2 to 10 G for a sufficient time (∼ 18 ns).Different kinds of rotations (X i , Y i , . ..) are achieved by setting the phase φ i of the corresponding radiation field.(The gate "direction", e.g.X, does not correspond to a physical principal axis, e.g. x.) We characterize the gate performance by applying it to the 20 states comprising all the mutually unbiased bases of a four-state system [24], determining the fidelity F = | φ|ψ | 2 from the simulated (|ψ ) and ideal (|φ ) output states for each input state and then averaging the 20 fidelities.For J zz = 0, we obtain an average one-qubit gate fidelity F = 99.98%.With J zz = 50 MHz, the fidelity drops to F = 99.92% while for J zz = 100 MHz, the fidelity is reduced to F = 99.7%,illustrating the importance of J zz in the error of the single-qubit gates.
Implementing two-qubit gates follows protocols developed for NMR-based quantum computing [5,25].Such gates rely on the J ⊥ coupling to entangle the states of the two qubits.To demonstrate a CNOT gate, we follow a standard implementation protocol represented by )Y 2 (ignoring irrelevant phase factors), where X i , etc., indicate π/2 rotations about the given axis for the ith spin.Pairs of single-qubit gates enclosed in square brackets can be implemented simultaneously using two-tone pulses [26,27].The process denoted U J (t π/2 ) indicates a period of free evolution (B 1 = B 2 = 0) that entangles the states of the two qubits.The duration of this process is t π/2 = π/2δ = 924 ns, for the parameters of our simulations.Small adjustments in the timing of each gate are made to optimize the performance of the CNOT.Simulating the CNOT gate and evaluating average fidelity as described above yields the following results.For J zz = 0, we obtain F = 99.94%, for J zz = 50 MHz, F = 99.8%, and for J zz = 100 MHz, F = 99.4%.The reduction in fidelity with increasing J zz is almost entirely attributable to the accumulated errors from single-qubit gates.Figure 3 shows a comparison between the density matrices for a simulated CNOT gate and that of an ideal gate using a representative input state for J zz = 50 MHz.
The results presented above do not include any effects of decoherence.To include the effects of decoherence, we adopt a model of "pure dephasing" given that T 1 >> T 2 in this system [22] and define a Lindblad collapse operator We simulate the time evolution of our system using the Lindblad master equation: (12) Numerically solving this equation using our optimized CNOT pulse sequence and γ 1 = γ 2 = 1/T 2 , allows us to compute fidelity F = φ|ρ|φ .Figure 4 shows F as a function of T 2 .Near zero field we expect that decoherence in a Cr 7 Mn dimer due to field fluctuations will be suppressed because all of the transitions are clock transitions, leading to an increase in T 2 .The results in Fig. 4 show that an order of magnitude increase in T 2 , like the increase reported in the Ho-based MNM [15], can lead to a substantial enhancement in the CNOT gate fidelity.Since T 2 for the molecular Cr 7 Ni rings has been found to be ∼15 µs under optimized conditions [13], use of clock transitions to enhance the coherence further could appreciably impact gate fidelities in Cr 7 Mn heterodimers.
These results are encouraging for implementing quantum computing protocols in realistic supramolecular systems, such as those that have already been synthesized and characterized.All the necessary pulses can be readily implemented using existing ESR techniques.Thus, by using clock transitions to enhance coherence, one should in principle be able to implement basic quantum computing protocols in supramolecular structures with good fidelity.

FIG. 1 .
FIG. 1. Energy level diagram for a single Cr7Mn molecule, showing the zero field avoided crossing between |m = ±1 states, creating the |± clock states.Inset: Molecular structure of Cr7Mn.

FIG. 3 .
FIG. 3. Density matrices for a CNOT gate applied to the test input state (|↑↑ + i |↓↓ ) √ 2. The vertical axis represents the amplitude and the color represents the phase.(a) The ideal final state.(b) The simulated final state using Jzz = 50 MHz.For this example, the simulation yields a calculated fidelity of 99.65%.