Abstract
The physical implementation of quantum information processing relies on individual modules—qubits—and operations that modify such modules either individually or in groups—quantum gates. Two examples of gates that entangle pairs of qubits are the controlled NOTgate (CNOT) gate, which flips the state of one qubit depending on the state of another, and the gate that brings a twoqubit product state into a superposition involving partially swapping the qubit states. Here we show that through supramolecular chemistry a single simple module, molecular {Cr_{7}Ni} rings, which act as the qubits, can be assembled into structures suitable for either the CNOT or gate by choice of linker, and we characterize these structures by electron spin resonance spectroscopy. We introduce two schemes for implementing such gates with these supramolecular assemblies and perform detailed simulations, based on the measured parameters including decoherence, to demonstrate how the gates would operate.
Introduction
The inability of conventional computers to solve certain problems efficiently, such as the simulation of quantum systems^{1,2}, is one of the main driving forces for the implementation of quantum computing and quantum information processing (QIP)^{3,4}, which exploit the laws of quantum mechanics. At the theoretical level, several algorithms have been shown to outperform classical computers in certain computational tasks such as factoring large numbers into primes^{5} and searching of unsorted directories^{6}. A variety of quantum systems have shown an excellent performance as the basic units for quantum information (qubits)^{7,8,9,10,11}. However, they are difficult to link controllably into useful arrays producing the twoqubit entangling quantum gates (QGs) crucial for any quantum algorithm. Among the most important entangling QGs are the controlled NOTgate (CNOT) and the gate. The gate brings the twoqubit state to the superposition . The CNOT gate flips the state of the target qubit if, and only if, the control qubit is in the state; this implies that each qubit has to respond inequivalently to an external stimulus. All other quantum gates, however complex, can be constructed from these twoqubit gates and singlequbit gates.
Therefore the major challenge for the physical implementation of QIP is bringing together qubits in an organized, scalable and addressable way to make such QGs^{3,4}. Here we demonstrate that supramolecular chemistry could have a major impact in addressing this challenge.
Molecular nanomagnets have been proposed as qubits^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26} and coherence times of individual molecular qubits have been studied and have improved^{24,25}. Supramolecular chemistry^{27} allows us to bring together, with great control, complex molecules into arrays, and recently this has been exploited to tune twoqubit interactions built from molecular nanomagnets^{28}. Hence supramolecular chemistry is a promising tool to build multiqubit devices; if this could be made scalable this is a competitive route towards the realization of a quantum computer.
Demonstrations of CNOT gates based on molecular nanomagnets have been published^{18,19,23}. The interaction between two dissimilar qubits produces a splitting in the lowlying twoqubit states that allows selective addressing of the transition by means of resonant EPR pulses, while keeping the other components of the wavefunction frozen. In this way, the excitation of the target depends on the state of the control qubit and a CNOT gate is implemented. The drawback of this approach is the permanent direct coupling between the qubits, which makes these proposals hard to scale as with a permanent coupling the state of the qubits experiences an unwanted manybody spontaneous evolution in time (as in NMR QIP schemes^{29}), whose harmful effects increase with the number of qubits. Conversely, a switchable indirect coupling between the two qubits would make the register much more easily scalable as when the switch is in the off state the unwanted spontaneous evolution is suppressed.
We have proposed using {Cr_{7}Ni} heterometallic rings as qubits^{14}. They are twolevel systems (S=1/2 ground state) that have sufficiently long phase memory times to allow many gate operations before state degradation occurs^{25}, and assemblies of {Cr_{7}Ni} rings have been made which show a permanent coupling between the spins^{26}. Here we report twoqubit assemblies which include switchable links, that allow us to propose a fully modular supramolecular design strategy^{27,28} towards quantum computation schemes based on either the CNOT or gate. The strategy uses supramolecular chemistry to tailor the individual components, spatial configuration and hence the properties of the resulting supramolecules. These supramolecules have been studied by electrochemistry, continuouswave and pulsed EPR spectroscopy to understand their static and dynamic spin properties. These measured parameters have been used to perform detailed simulations of the performance of both the CNOT and gates including the effects of decoherence. The two gates we propose are based on either global or local control of the qubit–qubit interaction. The first proposal exploits uniform magnetic pulses to manipulate two inequivalent Cr_{7}Ni qubits in an asymmetric supramolecule, which would implement the CNOT gate. The second exploits the local electric control of a redoxactive linker, which might be addressed by a tip to reversibly switch on and off the qubit–qubit interaction, to implement the gate. The scalability of these approaches is discussed in the last part of the paper.
Results
Syntheses and structural characterization
We have earlier reported the selective functionalization of [^{n}Pr_{2}NH_{2}][Cr_{7}NiF_{8}(O_{2}C^{t}Bu)_{16}] 1 (ref. 30) with isonicotinate (O_{2}Cpy) to obtain [^{n}Pr_{2}NH_{2}][Cr_{7}NiF_{8}(O_{2}C^{t}Bu)_{15}(O_{2}Cpy)] 2 (ref. 26) (Supplementary Fig. 1a). This synthetic method can be extended to produce [^{n}Pr_{2}NH_{2}][Cr_{7}NiF_{8}(O_{2}C^{t}Bu)_{15}(O_{2}Cterpy)] 3 and [^{n}Pr_{2}NH_{2}][Cr_{7}NiF_{8}(O_{2}C^{t}Bu)_{15}(O_{2}CPhterpy)] 4 (Supplementary Fig. 1b,c), from the controlled reaction of 1 with 4carboxy2,2′:6′,2″terpyridine (O_{2}Cterpy) and 4′(4carboxyphenyl)2,2′:6′,2″terpyridine (O_{2}CPhterpy), respectively (full experimental details are given in the Supplementary Methods)^{31}. 2, 3 and 4 are hereafter abbreviated as {Cr_{7}NiO_{2}Cpy}, {Cr_{7}NiO_{2}Cterpy} and {Cr_{7}NiO_{2}CPhterpy}, respectively. They consist of Cr^{III}_{7}Ni^{II} rings, containing an octagon of metal centres, with the inner rim bridged by fluoride ions and the outer rim by carboxylates: the functionalized O_{2}Cpy 2, O_{2}Cterpy 3 and O_{2}CPhterpy 4 ligands sit on a Cr…Ni edge of the octagon. This provides us with three supramolecular qubits with two different denticities, which allow us to assemble different QGs by appropriate choice of the central node (Fig. 1).
Reaction of equimolar quantities of 2 and 3 with cobalt(II) thiocyanate in a mixture of Et_{2}O/acetone leads to [{Cr_{7}NiO_{2}Cpy}→Co(SCN)_{2}←{Cr_{7}NiO_{2}Cterpy}] 5 in good yield, which has been characterized by Xray single crystal diffraction (Fig. 2a; for all crystallographic information see Supplementary Data 1). Compound 5 contains two inequivalent Cr_{7}Ni qubits coordinated to a central cobalt(II) ion, which has a sixcoordinate CoN_{6} octahedral environment with a cisarrangement of the two thiocyanate N atoms. The CoN bond distances are typical of high spin cobalt(II) ions^{32}, with the bonds to the thiocyanate ligands shorter (average 2.056(12) Å) than those to terpy or pyridine Ndonors (2.137(6) to 2.224(4) Å). Hence the d^{7} Co^{II} site (S_{Co}=^{3}/_{2}) has a ^{4}T_{1g} ground term (using O_{h} symmetry labels)^{33}, which leads to a wellisolated effective S_{eff}=^{1}/_{2} ground state at low temperature (see Characterization of assemblies via EPR spectroscopy below). The cis coordination geometry at the Co^{II} node means the {Cr_{7}NiO_{2}Cterpy} and {Cr_{7}NiO_{2}Cpy}based qubits are arranged in an almost orthogonal orientation. Therefore the two qubits in 5 are symmetry inequivalent, as required for implementing a CNOT gate (Fig. 2b).
Reaction of two equivalents of 2 with a preformed oxocentred pivalatebridged triangular cluster with terminal pyridine groups [Ru^{III}_{2}Co^{II}O(O_{2}C^{t}Bu)_{6}(py)_{3}] 6, hereafter abbreviated as [Ru^{III}_{2}Co^{II}], in acetone gives [{Cr_{7}NiO_{2}Cpy}→[Ru^{III}_{2}Co^{II}O(^{t}BuCO_{2})_{6}(py)]←{Cr_{7}NiO_{2}Cpy}] 7 (Fig. 3a), where two of the terminal pyridine ligands of 6 were replaced by the isonicotinate group of 2. Reaction of two equivalents of 3 or 4 and either cobalt(II) perchlorate or tetrafluoroborate gives [{Cr_{7}NiO_{2}Cterpy}→Co←{Cr_{7}NiO_{2}Cterpy}][X]_{2} [X=ClO_{4}^{−} 8a or BF_{4}^{−} 8b] and [{Cr_{7}NiO_{2}CPhterpy}→Co←{Cr_{7}NiO_{2}CPhterpy}][X]_{2} [X=ClO_{4}^{−} 9a or BF_{4}^{−} 9b] (Fig. 3b,c) (full experimental details are given in the Supplementary Methods). The architecture in 7, 8 and 9 contains two equivalent qubits, separated by a redoxswitchable centre^{34,35}, which makes it suitable for implementation of a ✓iSWAP gate (Fig. 3d).
The crystal structure of 7 consists of two {Cr_{7}NiO_{2}Cpy} rings linked through the isonicotinate groups to a [Ru^{III}_{2}Co^{II}O(^{t}BuCO_{2})_{6}(py)] triangle (Fig. 3a)^{34,36}. The metal ions are statistically disordered over the three sites within the triangular M_{3}O unit. They have a sixcoordinate, octahedral MO_{5}N environment formed by four carboxylate oxygen atoms from the bridging pivalate ligands in the equatorial plane, with the central oxide and one nitrogen atom from isonicotinate or pyridine groups occupying the axial positions. The three metal ions are positioned at the corners of an isosceles triangle with two distinct intermetallic distances of 3.296 and 3.315 Å. The central oxide lies within the M_{3} plane of the metal atoms, while the six pivalate bridging ligands lie above and below the M_{3}O plane and the pyridine Ndonors rest perpendicular to this plane.
The crystal structure of 8a and 9a are made up of cationic [{Cr_{7}NiO_{2}Cterpy}Co^{II}{Cr_{7}NiO_{2}Cterpy}] and [{Cr_{7}NiO_{2}CPhterpy}Co^{II}{Cr_{7}NiO_{2}CPhterpy}] units together with counterbalancing ClO_{4}^{−} anions (Fig. 3b,c). Both compounds have a central cobalt(II) ion with a sixcoordinate CoN_{6} octahedral environment, with the six CoN bonds in the range expected for a low spin cobalt(II) ion^{37}. The average value of the two Ndonors from the central pyridines of the terpy are shorter (1.923(6) 8a and 1.920(8) 9a Å) than the other CoN contacts (2.118(7) 8a and 1.986(9) 9a Å). This feature leads to an axially compressed octahedral environment for the lowspin d^{7} Co^{II} ions giving a S_{Co}=^{1}/_{2} ground state. Compounds 8a and 9a differ in the shortest Co…M(ring) contact (8.675(2) 8a and 10.979(5) 9a Å) and the staggered 8a or eclipsed 9a arrangement of rings in the assemblies.
The two Cr_{7}Ni qubits in each of 7–9 are linked by a redoxswitchable central node^{34,38}. Cyclic voltammetry on 7–9, and on the central nodes in isolation, show a oneelectron reversible oxidation in each case (measured in CH_{2}Cl_{2}, 0.1 M nBu_{4}NPF_{6}). For 6 and 7 the halfwave potential (E_{1/2})=−0.31 (6), −0.35 (7) V versus Fc^{+}/Fc (Fc=ferrocene)^{39}, and this oxidation is assigned to the [Ru^{III}_{2}Co^{II}] to [Ru^{III,IV}_{2}Co^{II}] couple^{34}. For 8b, 9b and reference complexes [Co(HO_{2}Cterpy)_{2}][BF_{4}]_{2} 10 and [Co(HO_{2}CPhterpy)_{2}][BF_{4}]_{2} 11, E_{1/2}=−0.30 (8b), −0.27 (9b), −0.27 (10) and −0.24 (11) V, which are assigned to the Co^{II/III} couple. The anodic to cathodic peak separation values are similar to those of ferrocene under the same conditions (see Electrochemistry section in Supplementary Methods, Supplementary Fig. 4 and Supplementary Table 3).
Characterization of assemblies via EPR spectroscopy
To study the interactions between the molecular components in the supramolecular structures 5, 7, 8b and 9b we performed magnetometry and multifrequency continuous wave EPR studies. The measured magnetometry data are essentially the sum of the Cr_{7}Ni qubits and the linking nodes (Supplementary Figs 5–9), and therefore are uninformative other than confirming that the interactions are very weak. EPR spectroscopy is much more sensitive to such weak interactions.
The EPR spectra of a powder of 5 at 5K (Fig. 4a and Supplementary Fig. 10) have a complex multiplet structure because the S_{eff}=1/2 of the Co(II) node has a very anisotropic effective gtensor and also gives rise to very anisotropic exchange tensors (J) with the {Cr_{7}Ni} rings. This structure is best resolved at Wband (94 GHz; Fig. 4a). The spectra can be simulated^{40} with the effective threespin Hamiltonian (1), with two different anisotropic exchange interactions (J_{12} and J_{23}) between the S_{eff}=½ of the Co(II) ion (S_{2}) and the two S=½ of {Cr_{7}NiO_{2}Cterpy} (S_{1}) and {Cr_{7}NiO_{2}Cpy} (S_{3}) rings, respectively (Fig. 4a).
The multiplets centred at ca. 1, 1.5 and 3.5 T (Fig. 4a) mark the effective g values of the Co(II) ion as g=6.50, 4.25 and ca. 2, while the Cr_{7}Ni rings have well defined g=1.78, 1.78 and 1.74 (the unique value being for the orientation perpendicular to the ring)^{30}. The large difference (Δg) between the g=6.5 and 4.3 orientations of the Co(II) ion and those of the rings, and the weak exchange interactions is useful as it allows us to treat the problem as an ABX spin system. The multiplet structure of the g=6.5 and 4.3 features is then due to weak exchange with two different S=1/2, giving doubletsofdoublets from which we can read the J_{y} and J_{z} components of J_{12} and J_{23}. The high field region is more complicated, due to the much smaller difference between the third Co(II) g value and those of the {Cr_{7}Ni} rings. The remaining parameters (the final components of J_{12}, J_{23} and g_{2}) were obtained by simulation (Fig. 4a). The parameters in Table 1 are labelled to a common reference frame, that is, taking into account the orthogonal orientation of the two {Cr_{7}Ni} rings.
7, 8b and 9b, contain redoxactive linking nodes and their EPR spectra change with the oxidation state. Hence, EPR was performed on both the reduced and oxidized forms at 5K as frozen solutions, allowing cycling of the redox state (Fig. 4b–d). Spectra were measured on the as prepared samples, then solutions were warmed to room temperature and the oxidized forms 7^{ox}, 8b^{ox} and 9b^{ox} generated in situ by addition of [FeCp_{2}](PF_{6}) or AgBF_{4} followed by freezing and measurement. To complete the cycle, the solutions were thawed again and 7, 8b and 9b regenerated by reduction with cobaltocene. We have also measured spectra of the isolated nodes in the paramagnetic Co(II) form, viz. complex 6, and [Co(HO_{2}Cterpy)_{2}](BF_{4})_{2} 10 and [Co(HO_{2}CPhterpy)_{2}](BF_{4})_{2} 11 (Supplementary Figs 2 and 3). Like 5, 6 also contains a ^{4}T highspin Co(II) ion with an S_{eff}=1/2 ground state, giving g=5.61, 4.05, 2.77 (Supplementary Fig. 11), while 10 and 11 have lowspin Co(II) hence S=1/2, giving g=2.047, 2.076, 2.195 (Supplementary Fig. 12) and g=2.022, 2.111 and 2.215 (Supplementary Fig. 13), respectively.
For the supramolecular structures 7, 8b and 9b, the exchange coupling is nicely resolved in K and Qband (24 and 34 GHz) EPR spectra (Fig. 4b–d). In each case there is weak coupling with respect to the difference in Zeeman energies ^{41}. This, and the equivalence of the two Cr_{7}Ni rings, gives AB_{2} spin systems. The {Cr_{7}Ni} ring resonances (the ‘B’ spins) are split into doublets (Fig. 4b–d), giving a direct measure of J (the exchange splittings of the central node resonances are not resolved). The spectra can be simulated with the simple Hamiltonian (2) using an isotropic exchange interaction between the central nodes (S_{2}) and the two {Cr_{7}Ni} rings (S_{1} and S_{3}), even for 7, which has an effective spin 1/2 centre at the node.
Simulations with the isotropic exchange interaction J as the only variable gives J=−0.026, −0.026 and −0.024 cm^{−1} for 7, 8b and 9b, respectively; the g values were fixed to those measured for individual components. The EPR spectra of 7^{ox}, 8b^{ox} and 9b^{ox} are very simple, resembling isolated {Cr_{7}Ni} rings, because the oxidized forms of the central nodes are diamagnetic. Reduction of 7^{ox}, 8b^{ox} and 9b^{ox} back to 7, 8b and 9b regenerates the original spectrum.
Implementing interesting quantum algorithms requires qubits with long phase memory times, such that they can be manipulated many times without errors. Individual {Cr_{7}Ni} heterometallic rings and simpler paramagnetic centres have shown long enough phase memory times (T_{M})^{21,22,24,25} to perform coherent electron spin manipulations before state degradation occurs. To check that this key property is preserved when they are incorporated in the supramolecular assemblies, we have performed pulsed EPR measurements at resonances corresponding to both the central nodes and the {Cr_{7}Ni} rings (Supplementary Figs 14–26). For the latter we find similar values for all compounds, and in both oxidation states where relevant (T_{M}=683, 749, 767, 790, 984, 750 and 1,031 ns for 5, 7, 7^{ox}, 8b, 8b^{ox}, 9b and 9b^{ox}, respectively, measured at Qband and 3K; Supplementary Table 4), demonstrating that the phase memory times are not strongly influenced by structural or magnetic differences in the linkage of the rings. In addition, values of T_{M} measured at resonances fields corresponding to the central node (in the paramagnetic state) are also similar for all compounds (500–700 ns at Qband and 3K; Supplementary Tables 5–12). Hence, the phase memory time of the central node does not represent a limitation (see also the Discussion below) for the applicability of our schemes for implementing a universal set of quantum gates. These results are used in the simulations below, and are extremely promising, as these T_{M} times are sufficiently long for spin manipulation even when they are integrated in a supramolecular assembly.
CNOT gate with uniform magnetic pulses
In the following, we introduce two quantum computation schemes, based on either local or nonlocal control of the interqubit interaction employing the structures described and the parameters obtained by EPR spectroscopy. First, we show that compound 5 is suitable for a CNOT gate, using uniform magnetic pulses as the only manipulation tool^{42}. As we are treating three interacting inequivalent doublets this produces 2^{3} energy levels. We define the computational basis within the lowenergy subspace where Co^{II} is frozen into its S_{z}=−1/2 state, which corresponds to the four lowest levels shown in red in Fig. 5a. The four levels correspond to arrangements of the spins of the two qubits, S_{1} and S_{3}, having the relative orientations ↓↓>, ↓↑>, ↑↓> and ↑↑>, respectively, which we label as 00>, 01>, 10> and 11> in Fig. 5a.
In a field of a few Teslas the eigenstates are factorized, that is, the eigenfunctions of the rings and the Co^{II} are not entangled. Hence it is possible to implement highfidelity singlequbit rotations by EPR pulses resonant with lowenergy gaps (see, for example, the shorter arrow in Fig. 5a). The combination of the inequivalent and anisotropic ringCo exchange interaction and of the perpendicular arrangement of the two rings makes the two qubits significantly inequivalent. In particular, there is a sizeable gap (about 33 μeV in a field of 5 T) between the 00>→01> and 00>→10> transitions, which enables independent singlequbit rotations. We note that a distribution of the spin Hamiltonian parameters arising from a reasonable g or J strain (∼1%, see ref. 43) would yield a broadening of the energy levels with s.d. ∼5 μeV, thus keeping the two transitions distinguishable in realistic conditions, possibly with the help of narrow and/or composite pulses^{44,45}.
A controlled phaseshift (C_{φ}) gate is obtained by a pulse resonant with the transition corresponding to the longer arrow in Fig. 5a where the cobalt(II) ion is temporarily in its S_{z}=+1/2 state (blue energy levels), followed by a repetition of the same pulse that would bring the state back with an additional phase φ. The value of φ is controlled by the phase difference between the first and the second pulse. For a pulse with field along the y direction, the implementation of this gate is fast, owing to the large value of g_{y} for Co^{II} ion. Moreover, the relatively large exchange interaction in equation (1) allows us to employ large oscillating fields (50 G), since the desired transition is spectroscopically well resolved from all the others. Consequently, the C_{φ} gate can be performed in only about 12 ns, with fidelities close to 99.99 % (see Computational Details section in Supplementary Methods).
The CNOT gate can then be obtained by the sequence of gates R_{y}(π/2) C_{Z} R_{y}(−π/2), where R_{y}(α) is a rotation of the target Cr_{7}Ni qubit by an angle α around the y axis (a singlequbit operation) and where C_{Z} is the phaseshift gate described above with φ=π. We have numerically solved the timedependent Schrödinger equation for the Hamiltonian (1) in presence of this pulse sequence. Results are reported in Fig. 5b and show that this twoqubit gate can be obtained with very high fidelity in only 30 ns. As noted above, the two qubits are significantly inequivalent, even if the g tensor of the rings is nearly isotropic. This makes this complex well suited for the quantum simulation of antisymmetric Hamiltonians (Supplementary Figs 28 and 29 for an illustrative example).
gate with local electric control
The redox properties of the central node in 7, 8 and 9 can be exploited to perform the universal gate on the {Cr_{7}Ni} qubits, whose effect on the computational basis is given by:
and .
When the central node is in the diamagnetic state ([Ru^{III,IV}_{2}Co^{II}] 7 or Co^{III} 8–9), the two qubits are decoupled and only singlequbit operations can be performed. Conversely, when the paramagnetic oxidation state, [Ru^{III}_{2}Co^{II}] 7 or Co^{II} 8–9, is present the system behaves as a trimer described by the Hamiltonian (2). Previously, Lehmann et al.^{16} have proposed that two spins S=1/2 connected by a redoxactive unit can be exploited for the implementation of the gate by switching the redox unit with a scanning tunnelling microscope (STM) tip at an appropriate potential. In this way one electron can be added or removed from the redox unit very quickly. It was demonstrated that particular sets of parameters of the trimer Hamiltonian lead to a pure evolution of the two qubits after specific time intervals. However, high fidelity for the gate is guaranteed only for fixed ratios between the qubit–qubit exchange (J_{qq}) and the qubitredox unit exchange (J_{qr}). For the measured parameters of 7, 8 and 9 (J_{qr}≡J and J_{qq}=0 in equation (2)) the fidelity of the gate would be low.
Here we propose a more flexible scheme for the gate based on our measured parameters, which works if the two qubits have the same Zeeman energy but one different to that of the switch in the ‘on’ state. This is the case here, where the g values of the qubits and switches are very different, g_{1z}=g_{3z}=1.74 for {Cr_{7}Ni} rings and g_{2z}=2.77 (7), 2.195 (8) and 2.215 (9) for the central node. This difference in the g values means that in magnetic fields of a few T, the ringcentral node exchange J is small compared with the difference between the Zeeman energies of the ring and the central node. Hence, the spin state of the central node is nearly frozen in the S_{z}=−1/2 state and has only tiny virtual fluctuations that lead to an effective interaction between the two {Cr_{7}Ni} qubits given by
where the field is along z, S_{1} and S_{3} refer to the spins of the first and second Cr_{7}Ni qubits and
The feasibility of the scheme relies only on the equivalence of the qubits and on the hierarchy of the interactions (see Supplementary Methods for more details).
Therefore when the central node is paramagnetic, the state of the two qubits evolves according to (3), with negligible entanglement with the central node. For specific times, this evolution coincides with the gate apart from singlequbit rotations along z due to the second term in (3).
This perturbative picture is confirmed by the results of detailed calculations for compound 9 using the full Hamiltonian (2) (Fig. 6). Starting from the 10> logical state, we report in Fig. 6a the time evolution of the trimer wavefunction that would implement the gate. In a magnetic field of 3 T, after 4 ns the wavefunction has equalweight contributions from 10> and 01>, which is the gate, while after 8 ns the states of the two spins are exchanged, that is, we have the 01> state. An extremely good fidelity F (larger than 0.99) for compound 9 is obtained for fields of the order of about 2.5 T or larger, after a suitable gating time t_{f} of the order of a few nanoseconds (Fig. 6b). For such fields the perturbative picture of equation (3) holds very well, and t_{f} is proportional to B (Fig. 6b), which is consistent with the form of the effective qubit–qubit coupling Γ∝1/B.
Analogous or better fidelities can be obtained with the parameters derived for compounds 7 and 8. Indeed, larger values of g_{1z}−g_{2z} (as in 7) increase the validity of the perturbative picture (3), even for smaller values of B. Hence, we can exploit the modular strategy and optimize the performance of the gates by targeted chemical manipulations.
Discussion
Our initial simulations above did not include decoherence. To gain a deeper insight into the performance of the proposed quantum gates, we have performed further simulations that include the effect of both relaxation and pure dephasing, allowing for the finite and measured values of T_{1} and T_{M}, respectively. For the values of T_{1} measured by pulsed EPR (>10 μs in all compounds, see the Supplementary Tables 4–12), the effect of relaxation is found to be completely negligible.
Pure dephasing is accounted for by numerically solving the Lindblad master equation for the system density matrix ρ (refs 46, 47):
Here the commutator describes the coherent evolution induced by the full system Hamiltonian H, while the second term describes pure dephasing mechanisms. The subscript k=1, 2 and 3 labels the spins of the qubits and of the switch, while and are spin 1/2 raising and lowering operators.
Using the values of T_{M} of the Cr_{7}Ni qubits measured by pulsedEPR spectroscopy (between 700 and 1,000 ns, see above) we still find very high fidelities: 99.3% for the implementation of the CNOT gate on compound 5 and 99.6% for the gate on 9. As expected, the fidelities are only marginally affected by decoherence because gating times are much shorter than T_{M}.
It is also worth discussing the role of the decoherence of the switch in the present schemes. As far as the CNOT gate scheme (compound 5) is concerned, the contribution of the central Co switch to decoherence is similar to that of the rings, because the switch is temporarily excited to the S_{Z}=+1/2 state during the CNOT gates. Conversely, only virtual excitations of the switch are involved in the implementation of the gate exploiting the redoxactive linker. Hence, the fidelity remains very high even for dephasing time of the switch of the order of the gating time (some ns). This difference is shown by the two colour maps reported in Fig. 7. They show the dependence of the fidelity of a (left) and of a CNOT gate (right) on the dephasing times T_{M} of the rings and of the switch. It is evident that the fidelity of the is practically independent on whereas its dependence on is much more pronounced. Conversely, the implementation of the CNOT on compound 5 leads to a similar dependence of the fidelity on and .
The chemistry described above can be extended to make onedimensional (1D) chains incorporating the twoCr_{7}Ni supramolecules, linked by either single cobalt sites or oxocentred metal triangles. Such 1D chains have already been made involving single Cr_{7}Ni units and [2] and [3]rotaxanes and the principles established, especially for 1D chains of rotaxanes^{48}, should work well. The key steps are to include two functional groups per Cr_{7}Ni ring, which has already been done for isonicotinate acid, and by functionalizing the termini of the central organic thread of rotaxanes. Inclusion of two different functionalities is more taxing but entirely feasible (see the detailed schemes in Supplementary Figs 30 and 31). Therefore the chemistry is potentially scalable.
While this challenging chemistry proceeds, it is possible to show that the schemes are theoretically scalable. The extension of a twoqubit QG to a multiqubit register raises important issues concerning the propagation of errors. In such a register, we can identify two main sources of errors, whose effect increases with the number of qubits: pure dephasing and imperfect operation of the switch arising from a residual interqubit interaction still present in the ‘off’ state.
Errors induced by pure dephasing (decoherence) increase with the overall computational time. However, a finite chain of qubits with interposed switches would allow simultaneous (parallel) manipulation of nonoverlapping parts of the register, which drastically reduces the computation time, and hence decoherenceinduced errors, with respect to a serial implementation. For instance, a setup based on compound 5 can be manipulated in parallel for interesting classes of quantum simulation algorithms^{17}. This requires inclusion of two distinct switches in the structure—Switch_{A} and Switch_{B}—with different responses to external stimuli, such that the 1D chain has as a repeat pattern QubitSwitch_{A}QubitSwitch_{B}.
For compounds 7, 8 and 9, the scheme requires a local addressability of individual switches on the chain, which are separated by about 3 nm. For a first proofofprinciple experiment with a single assembly, an STM tip could be used to provide the best control. A parallel implementation of gates would be possible if different Co switches could be addressed individually at the same time. To achieve a scalable structure, a molecular chain of qubits might be layered onto a surface and addressed by means of a crossbar architecture similar to that proposed in refs 16, 49. It is worth noting that Cr_{7}Ni rings have already been deposited on surfaces, without significant modification of their magnetic properties^{50}.
We now examine the scaling of the errors with the number of qubits in the register. As a first step, we consider the effect of pure dephasing on a set of noninteracting qubits subject to Lindblad (Markovian) dynamics. It can be shown (see ref. 51 and Supplementary Information) that the decoherence error ɛ=1−F^{2} on N qubits scales at most as , valid for t_{f}<<T_{M} (where t_{f} is the optimal gating time). In the parallel quantumsimulator implementation considered in ref. 17, manipulating a chain of N qubits takes the same time as manipulating the shortest chain that contains the two distinct switches (qubitSwitch_{A}qubitSwitch_{B}qubit). Therefore t_{f} is limited by the value it assumes for a chain of three qubits, while in a serial scheme t_{f} increases linearly with N. By chemical engineering of Cr_{7}Ni qubits, a T_{M} of 15 μs has been obtained^{25}; this should allow the implementation of around twenty 2qubit gate operations on a chain consisting of 10 qubits, while keeping the error below 0.1.
Finally, we analyse the consequences of an imperfect operation of the switch and their effect on scalability. For an ideal gate the interqubit exchange interaction would be completely turned off when the central node is in its diamagnetic state. Double electron–electron resonance measurements (Supplementary Fig. 27) reveal a very weak, residual interaction between the qubits. The resulting oscillations are on a time scale (ca. 0.2 μs) much longer than our gating times (a few ns) and could be corrected by means of refocusing techniques^{29}. In the case of the CNOT scheme (compound 5), the always ‘on’ exchange interaction between the rings and the Co ion yields a small residual qubit–qubit coupling (Supplementary Methods). As a consequence, an unwanted slow evolution of the qubits occurs on a timescale T_{UE} of the order of a few hundreds of ns. Although this is longer than the CNOT gate time (about 30 ns), larger values of T_{UE} would be needed for performing sequences of many gates. This can be obtained by modifying the molecule to decrease the Coring exchange interaction. For instance, T_{UE} can be increased by a factor of 50 by reducing the Coring exchange interaction by a factor of three, which could be achieved chemically by adding an extra phenyl group between the ring and the central node. A similar reduction of the residual qubit–qubit coupling can be obtained by increasing the static magnetic field.
In summary, we have described two different schemes for universal quantum information processing, based on either local or global control of the qubit–qubit interaction. We have demonstrated that the flexibility of molecular {Cr_{7}Ni} qubits makes them suitable for the implementation of each of these two schemes, if properly functionalized and linked by means of a supramolecular design strategy. The twoqubit units can be controlled either magnetically or electrically, and implement either the CNOT or perfectly entangling gates. Our realistic simulations, based on experimental parameters and including decoherence, show that these gates can be performed with remarkably high fidelity. The modular strategy proposed here offers a degree of control in terms of the magnitude of the coupling between molecular spin qubits, the spatial orientation of the modules and the possibility to have a switchable interaction that represents a significant step forward with respect to previous achievements on the assembly of qubits.
Since future developments of quantum technology are unpredictable, we emphasize the importance of pursuing both these two parallel roads towards the actual realization of a quantum computer.
Additional information
Accession codes: The Xray crystallographic coordinates for structures reported in this Article have been deposited at the Cambridge Crystallographic Data Centre (CCDC), under deposition number CCDC 1029608–1029613 and 1415380–1415383. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.
How to cite this article: FerrandoSoria, J. et al. A modular design of molecular qubits to implement universal quantum gates. Nat. Commun. 7:11377 doi: 10.1038/ncomms11377 (2016).
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Acknowledgements
This work was supported by the EPSRC (UK), the European Commission (Marie Curie IntraEuropean Fellowship to J.F.S. (622659) and A.F. (300402)). E.M.P. thanks the Panamanian agency SENACYTIFARHU for funding. R.E.P.W. thanks the Royal Society for a Wolfson Merit Award. We also thank EPSRC (UK) for funding an Xray diffractometer (grant number EP/K039547/1) and the National EPR Facility. We thank Diamond Light Source for access to synchrotron Xray facilities. A.C., S.C. and P.S. acknowledge financial support from the FIRB Project No. RBFR12RPD1 of the Italian Ministry of Education and Research.
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J.F.S., E.M.P. and A.F. synthesized the majority of the compounds discussed; S.A.M. synthesized compound 6. G.T. was involved in design of the synthetic strategies. E.M.P. and I.J.V.Y. carried out the Xray single crystal diffraction studies. E.M.P., J.F.S. and F.T. performed the EPR spectroscopic studies, and J.F.S. carried out the electrochemical studies reported here. E.J.L.M. helped interpret the EPR spectra. A.C., S.C. and P.S. designed the schemes for quantum gates and A.C. carried out their detailed simulations. R.E.P.W. oversaw the project and wrote the paper with J.F.S., S.C. and P.S., with significant input from all other authors.
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Supplementary Information
Supplementary Figures 131, Supplementary Tables 112, Supplementary Methods and Supplementary References (PDF 3345 kb)
Supplementary Data 1
Cif files for compounds 3 to 11 (ZIP 8485 kb)
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FerrandoSoria, J., Moreno Pineda, E., Chiesa, A. et al. A modular design of molecular qubits to implement universal quantum gates. Nat Commun 7, 11377 (2016). https://doi.org/10.1038/ncomms11377
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DOI: https://doi.org/10.1038/ncomms11377
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