F.
Tacchino
a,
A.
Chiesa
bc,
R.
Sessoli
de,
I.
Tavernelli
*a and
S.
Carretta
*bc
aIBM Quantum, IBM Research – Zurich, CH-8803 Rüschlikon, Switzerland. E-mail: ita@zurich.ibm.com
bUniversità di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124 Parma, Italy. E-mail: stefano.carretta@unipr.it
cUdR Parma, INSTM, I-43124 Parma, Italy
dDipartimento di Chimica U. Schiff, Università di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy
eUdR Firenze, INSTM, I-50019 Sesto Fiorentino, Firenze, Italy
First published on 29th July 2021
We show that molecular spin qudits provide an ideal platform to simulate the quantum dynamics of photon fields strongly interacting with matter. The basic unit of the proposed molecular quantum simulator could be realized by synthesizing a simple dimer of spin 1/2 and spin S ≥ 3/2 transition metal ions, solely controlled by microwave pulses. The spin S ion is exploited to encode the photon field in a flexible architecture, which enables the digital simulation of a wide range of spin-boson models much more efficiently than by using a multi-qubit register. The effectiveness of our proposal is demonstrated by numerical simulations using realistic molecular parameters for each of the two ions and the prerequisites delineating possible chemical approaches for the synthesis of suitable platforms are also discussed.
In this respect, chemistry offers a change of perspective which could overcome the aforementioned difficulties. Indeed, molecular spin systems characterized by a sizeable number of accessible levels can be used to encode multi-level logical units (qudits). Each molecular qudit could replace several distinct two-level units (qubits) in various algorithms,9 thus greatly simplifying manipulations of the register. Indeed, since each qudit can host more than two levels, important algorithms can be implemented with a smaller number of computational units, compared to qubit-based realizations. Examples are given by Grover's, Fourier transform or Quantum Phase Estimation algorithms10 or by quantum-error correction schemes recently put forward.11–13 Hence, the qudit-route to the physical implementation of quantum computing appears very promising in the current development stage, where the operations are still noisy and full control of complex quantum devices is hard.
Magnetic molecules are the ideal candidate to implement this alternative architecture.14 Indeed, they are characterized by long coherence times,15–21 which can be further enhanced by chemically designing the molecular structure22,23 or targeting protected transitions.24,25 Moreover, the spin state of these systems can be easily manipulated by microwave or radio-frequency pulses,11 thus implementing single- and two-qubit gates in permanently coupled26–28 or scalable architectures.29–33 Recently, it was proposed to exploit the additional levels typical of these systems for implementing quantum error correction within a single object,11–13 in place of the many qubits required by standard block-codes.34
Here we show how the qudit nature of magnetic molecules could simplify the practical implementation of important quantum simulation algorithms. We focus, in particular, on the simulation of light–matter interaction processes. This problem is generally hard to solve on a classical computer, especially in the ultra-strong coupling regime between light and matter, because it does not generally allow for a perturbative treatment. This class of models is of crucial importance for many fundamental investigations ranging from cavity quantum electrodynamics to photochemistry.35,36 They have mostly been tackled so far with analog37–39 or digital-analog40,41 simulation strategies, i.e. by employing a quantum hardware which directly emulates the target system. Conversely, we pursue here a digital approach, in which the state of the target system is encoded into the qubit or qudit register. This makes the quantum simulator much more flexible with respect to the class of systems which can be simulated, but it implies the necessity to represent radiation modes (characterized in principle by infinite degrees of freedom) as qubit/qudit excitations. This represents a major challenge. Indeed, standard encodings, designed for multi-qubit architectures,42 either use a number of qubits equal to the number of simulated photons43 (i.e. an exponentially large Hilbert space) or reduce the number of qubits at the price of much more complex quantum circuits, involving interactions between distant qubits.42,44
Conversely, here we reduce both the hardware overhead and the complexity of manipulations by mapping each photon mode to a single spin S qudit. The qudit levels are then used to represent the number of photons in each mode (see Fig. 1). Thanks to the power of coordination chemistry, different qudits can be linked together and, e.g., to spin 1/2 units encoding two-level atoms,45–48 in non-trivial molecular geometries. This, together with the capability of manipulating the state of the hardware by resonant and semi/resonant microwave pulses, would allow us to digitally simulate the atom-photon dynamics involving multi-mode fields and/or multiple atoms.43,49
In particular, we show that very simple molecules consisting of dimers of transition metal ions (a spin 1/2 and a spin S ≥ 3/2) can be used to efficiently simulate atom-photon interactions in a non-trivial range of parameters up to strong and ultra-strong coupling.35,36 The same approach can be extended to simulate models involving many field excitations,44 by exploiting the remarkable capabilities of coordination chemistry in synthesizing multi-center molecules with very large total spin,47 or interaction of atoms/spins with any other bosonic field, such as phonons. This allows one to simulate, along the same lines, many other important models, ranging from phonon vibrations,50–53 to lattice gauge theories44 and complex quantum optical setups.54
In the following, we design a sequence of pulses allowing us (i) to determine the ground state of the simulated system using the variational quantum eigensolver algorithm (VQE)3,55–57 and (ii) to follow the time evolution of the system prepared in an out-of-equilibrium initial state. The remarkable performance of the proposed hardware is demonstrated by numerical simulations on hypothetical molecular systems characterized by qudit parameters corresponding to existing coordination compounds,17,25,58–62 weakly coupled to a spin 1/2 unit. The effect of decoherence and the full sequence of pulses needed to implement the algorithms are also included, obtaining nevertheless very large fidelities. These results make the here-proposed molecular quantum simulator very promising and pave the way to forthcoming proof-of-principle experiments.
![]() | (1) |
These requirements are easily fulfilled in coordination compounds containing a spin 1/2 ion coupled to a spin S1 transition metal ion. The latter provides the ideal qudit for the proposed architecture. As shown below, the relatively small number of levels of these qudits (d = 2S1+ 1 ≤ 6) is already sufficient to simulate light–matter interaction from strong to ultra-strong coupling regimes. In addition, transition metal ion complexes with quenched orbital angular momentum ensure significantly long coherence times,17,22,25,58,62 important to achieve a good simulation. We consider, in particular, two paradigmatic cases: CrIII and FeIII ions in a distorted octahedral environment, yielding 3d3 and 3d5 electronic configurations with a single electron per orbital and thus S = 3/2 and 5/2, respectively.60,61 Due to the practically complete quenching of the orbital angular momentum, the spectroscopic tensor g is isotropic and close to the free electron value, while single ion anisotropy is typically in the ∼0.2–0.3 cm−1 range.60,61 As an illustrative example, in the simulations reported below we use d1= 0.24–0.30 cm−1 and g = 1.98 for CrIII, as in ref. 22 and 25 and d1 = −0.30 cm−1 and g = 2.00 for FeIII, as reported e.g. in ref. 58 and 63. By properly adjusting the static field B, the results do not depend on the sign of d1.
These single-ion qudits can be weakly coupled through bond or through space to a spin 1/2 ion, such as CuII in a distorted octahedral ligand cage,17,60,61 typically characterized by g ∼ 2.1–2.3 and in some cases also by remarkable coherence times.17 In the following we assume g2z = 2.3, significantly different from g1z = 1.98–2 to ensure factorization of the system wave-function. For the dipolar interaction we assume Jx,y = 0.008 cm−1, which corresponds to a dipolar coupling (in the point dipole approximation) between ions at a distance ∼6 Å. We remark that using a different form of the spin–spin coupling (for instance a weak isotropic super-exchange interaction) does not alter our conclusions. A more extensive discussion on possible physical implementations is provided in Section 3.
These parameters, combined with a static field of ∼0.3–0.5 T ensure that Δm1,2 = ±1 transitions needed to manipulate the state of the system fall within the 20 GHz range typically explored in coplanar microwave resonators.64,65
Having described in detail the molecular hardware, we now switch our attention to the target model, the object of our simulation, and on how to map it onto the hardware. The target Hamiltonian is the Rabi model:66–68
![]() | (2) |
The molecular processor described by Hamiltonian 1 can be used to compute ground state properties and to mimic the dynamics of the target Hamiltonian 2. To achieve this, we first need to encode the boson field into the spin qudit. Notice that a very good approximation can be obtained by truncating the boson field to a relatively small number of levels. Hence, the d = 2S1 + 1 levels of the qudit are sufficient to encode the radiation field with negligible error, by truncating it to a maximum number nM = 2S1 of bosons. The mapping between Sz1 eigenvalues and number of bosons (n = a†a) is shown in the bottom part of Fig. 1. In parallel, the two-state atom appearing in the Rabi Hamiltonian can be directly encoded on the hardware spin 1/2 degrees of freedom.
Complete control of the hardware is achieved via microwave pulses resonant (or semi-resonant) with specific excitations of the spin 1/2 or of the qudit. In particular, Δm2 = ±1 transitions allow us to rotate the state of the qubit, while Δm1 = ±1 pulses are used to excite the qudit. Moreover, the spin–qudit interaction enables conditioned (entangling) operations.
Here we demonstrate an implementation of the VQE on the proposed qudit architecture applied to the target Rabi Hamiltonian, eqn (2). We construct the trial wavefunction by designing some basic quantum operations achieved in practice via external control microwave pulses. In particular, as shown in Fig. 2a, we assume a (S1, s2) = (3/2, 1/2) hardware platform and we combine pulses resonant with transitions of s2 (green arrows), implementing rotations of the qubit, with Δm1 = ±1 pulses on the S1 = 3/2 spin (black). To introduce entanglement in the approximate ground state, the operations on the S1 = 3/2 spin are actually conditioned by the state of the spin 1/2, i.e. we rotate each pair of qudit levels by ±θi depending on the sign of m2 (see Fig. 2a). In total, the ansatz contains only 4 free parameters, namely the rotation angles {θi}, and can be implemented with a sequence of microwave pulses that can be as fast as ≃100–200 ns. We also mention that such variational structure, which can be natively realized on our proposed qudit architecture, is closely related to the so-called polaron ansatz, which was recently implemented on superconducting quantum hardware43 through non-trivial decompositions into elementary qubit operations.
In this demonstration, we combine a classical optimization routine (the Nelder–Mead simplex algorithm73), with numerical simulations of the unitary transformations corresponding to every choice of the variational parameters. In fact, each sequence of microwave pulses can be seen as the series of quantum operations reported in the inset of Fig. 2b. Here the black thick (green narrow) line represents the qudit (qubit). Conditioned qubit–qudit operations are depicted with black boxes, while single qubit rotations are shown in green, in direct correspondence with Fig. 2a.
Simulations are performed according to a realistic hardware setup, including all the required external control pulses and molecular parameters discussed above. The effect of a finite spin coherence time T2 is included by simulating the dynamics of the hardware density matrix ρ according to the Lindblad master equation74
![]() | (3) |
In Fig. 2c, we report results of the VQE algorithm simulated by assuming a realistic value of the spin coherence time (10 μs, symbols), compared with exact values (lines) for both the ground state energy and some ground state properties of interest. Notice that, over a wide range of G/Ω values, the proposed ansatz achieves very good approximations of the exact ground state, which in this small-sized example can be computed classically. The limiting factor is essentially the expressibility of the trial wavefunction, i.e. the fact that by using the set of operations reported in Fig. 2a we may not achieve the exact form of the true ground state. This limitation can in principle be overcome by repeating the same basic parametrized structure more than once. It is worth noting that a finite coherence time, similar to small imperfections in the practical realization of quantum gates, only minimally affects the final results. In fact, consistent with the underlying variational principle, noisy ground state energy estimates sometimes converge to values slightly larger than the exact ones.
|ψ(0)〉 → |ψ(t)〉 = e−i![]() | (4) |
In Fig. 3 we show the digital quantum simulation of the Rabi model, eqn (2), realized with the spin qudit encoding described above and for increasingly challenging choices of the G/Ω ratio. Large G/Ω values introduce peculiar features in the dynamics of the target system: the rotating wave approximation fails and the total number of excitations is not conserved. This non-trivial behavior emerges in our simulations below, where we report the time evolution of the average number of photons 〈nphotons〉 in the radiation mode and of the atom population 〈σz〉, assuming an initial vacuum state with zero photons and the atom in its ground state. This vacuum state (with no excitations) would not be subject to any evolution for small G/Ω ratios. Hence, oscillations in 〈nphotons〉 and 〈σz〉 are a direct signature of the ultra-strong coupling regime. In all panels, we compare the reference curves, computed via exact matrix exponentiation, with numerical simulations of a realistic hardware obtained again by integrating eqn (3). A quantitative assessment of the overall quality of the results can be obtained by computing the fidelity between the hardware output ρ and the ideal result |ψid〉 of a digital quantum simulation algorithm realized with the same number of Suzuki–Trotter time steps and the same size of the bosonic Hilbert space. The latter can be obtained with standard matrix algebra.
In the first example, Fig. 3a and b, we show the results of the quantum simulation of the target Hamiltonian S with G/Ω = 0.25 realized with N = 4 (for t ≤ 5) and N = 6 (for t > 5) Suzuki–Trotter steps. Here, the hardware setup is composed of a spin S1 = 3/2, encoding a d = 4 photonic space, and a spin s2 = 1/2 representing the atomic degrees of freedom. The longest pulse sequence requires 1.7 μs, resulting in large average fidelities:
≃ 0.984 for T2 = 50 μs and
≃ 0.951 for T2 = 10 μs.
Increasing the values of the target G/Ω ratio, Fig. 3c–f, yields larger oscillations in the average number of photons and atom populations. To capture these features we need, on the one hand to increase the number of digital steps (N), and on the other hand to enlarge the bosonic space (nM). This last step is fundamental to correctly capture the system dynamics at significant G/Ω, as clearly shown in panels (g) and (h), where we compare the time evolution obtained by truncating the number of photons to 3 or 5, for G/Ω = 0.7. Indeed, by slightly increasing nM, we practically obtain the exact dynamics (continuous line). Given nM = 2S1, on the synthetic side, this simply translates in changing the qudit spin from 3/2 to 5/2.
Conversely, increasing N (and hence the length of our manipulations) requires larger T2 or faster pulses. These can be applied with high fidelity only in the presence of well resolved transitions, a condition which can be achieved, e.g., by properly engineering the molecular spectrum. In this respect, the large degree of chemical flexibility represents a valuable resource. In particular, it is helpful to replace the s2 = 1/2 with a spin s2 = 1 system. A promising candidate ion is for example NiII, for which coherence times in the regime of microseconds were reported.62 While only two consecutive levels, e.g. m2 = 0,1, are used for the actual encoding of the target model, the presence of an additional zero-field splitting term d2sz22 in the hardware Hamiltonian greatly improves the frequency resolution of the relevant transitions, thus allowing for larger operation fidelities with reasonably fast control pulses. For NiII, d2 can be in the 0.1–1 cm−1 range (in the octahedral ligand field).60,61 In Fig. 3c and d we report a digital simulation for G/Ω = 0.5, obtained with N = 7 on a (S1, s2) = (3/2, 1) model hardware. Here, the pulse sequences last approximately 0.9 μs on average, resulting in average fidelities ≥ 0.92 also for T2 = 10 μs. Finally, we achieve in Fig. 3e and f a digital simulation well above the ultra-strong coupling threshold (G/Ω = 0.7, N = 8) with a model hardware (S1, s2) = (5/2, 1) (i.e. with a bosonic space truncated at d = 6). More demanding pulse sequences are required in this case, with an average duration of ∼1.6 μs and average fidelity around
≃ 0.84 for the shortest T2.
To identify a suitable molecular platform, we need to combine requirements on the different units discussed in the previous sections. As already illustrated, a prototypical hardware could consist of a dimer of transition metal ions, respectively with spins S1 ≥ 3/2 and s2 ≥ 1/2. In order to ensure factorization of the two-ion wave-function, the two ions should be weakly interacting (no matter if through space, through bond or via super-exchange) and characterized by g factors significantly different along a given direction. Single-ion anisotropy on both S1 and s2 (if the latter is ≥1) could help to better resolve different transitions. In particular, the difference between the system energy gaps should be larger than the typical line-width measured in pulse EPR experiments on molecular complexes, which can be in the ∼0.001 cm−1 range for spin 1/2 systems (see, e.g. ref. 76).
Such single constraints do not appear so stringent. For instance, CrIII and CuII have sufficiently different g values gCr = 1.98, gCu = 2.10–2.3 to allow factorization of the wavefunction. At the same time, the individual spin resonance frequencies are both accessible in the same resonator. d1 values in the order of the tenth of cm−1 characterize ions that have half-filled valence orbitals, like MnII, FeIII, or GdIII, as well as half-filled t2g orbitals in an octahedral ligand field, such as CrIII. It must be said that in this case the rhombicity and principal directions of the magnetic anisotropy are difficult to predict and control synthetically, but they are not crucial for the feasibility of our scheme.
More demanding is the control of the interaction between the spin qubit and the qudit. In general, weak spin–spin interactions are poorly characterized and hard to predict exactly, only based on the molecular geometry. Indeed, while dipolar couplings can be easily computed and controlled, exchange interactions can depend critically on the choice and orientation of the linkers (see below). It is important to note that this does not hinder the implementation of our scheme. In fact, once the complex has been characterized and even in the presence of an interaction slightly larger than expected, it is possible to tune the external field in order to ensure factorization of the wave-function. This could require the experimental setup to be adapted to work at larger frequencies than commercial resonators, as demonstrated for instance in ref. 65, where superconducting coplanar resonators operating up to 50 GHz were reported. These superconducting resonators could also employ high-Tc superconductors to support large magnetic fields.77
We now discuss chemical strategies to obtain the suitable range of interactions. The short distance (∼6 Å) associated with the dipolar coupling used in the simulations reported above (0.01 cm−1) is compatible with compact linkers like oxalate, cyanide, azide etc. These bridging ligands are very efficient in also transmitting exchange interactions, and thus are not ideal for single spin addressing. The optimal choice falls on very weak exchange interactions that are expected to be almost ubiquitous when the two spin centers are embedded in the same molecular scaffold. Single spin addressability in weakly coupled heterometallic species has been recently achieved by combining copper porphyrin with a carboxylate-coordinated TiIIICp2 unit.78 However, even for the shorter bridge the qubit–qubit interaction is too weak to allow the implementation of the quantum simulator proposed here. An elucidating example of the wide range of achievable interactions is the case of condensed CuII porphyrin complexes, which have been attracting increasing interest for the relatively long and robust coherence combined with semiconducting properties and convenient processability.18,79–81 In a recent study, electron–electron double resonance has been used to investigate the spin–spin interactions in edge-fused coplanar CuII dimers and in meso–meso singly linked dimers.82 In the latter, the Cu-porphyrin rings are mutually orthogonal and exchange interaction fully suppressed, significantly smaller than the dipolar interaction, estimated to be 0.0028 cm−1. On the contrary, the planarity imposed by the triple link between the two units boosts the antiferromagnetic exchange interaction to −2.64 cm−1. These results demonstrate the wide tuneability of the molecular approach.
The choice of the linkers between the two magnetic ions should also fulfill other constraints. In particular, we need to control the decoherence of the system. A coherence time T2 above 10 μs at low temperatures can be observed for S = 1/2 transition metal ions, especially if the first coordination sphere is nuclear spin free, e.g. oxygen, sulphur, or carbon donor atoms, and if total or partial deuteration of the ligand is affordable. This requires nitrogen from the first coordination sphere and aliphatic CHn groups in the molecular scaffold to be eliminated, thus reducing the available library of molecular candidates. Nevertheless, we notice that remarkably good results are already obtained using T2 in the 10 μs range and even smaller values of a few μs could already be sufficient to achieve the first proof-of-principle demonstrations. Concerning S > 1 metal complexes, coherence times have been less investigated. In the case of CrIII, T2 up to 8 μs have been observed in a CrIII complex with the N,N′-dimethyl-N,N′-dipyridine-2-yl-pyridine-2,6-diamine ligand.83 Interestingly, the longer T2 compared to other complexes with nuclear spin free donor atoms, such as [Cr(C3S5)3]3− (see ref. 25) or [Cr(C2O4)3]3− (see ref. 15), has been attributed to the weaker single ion anisotropy, di = 0.18 cm−1, a value very close to that employed in our simulations. di values in the range 0.15–0.25 cm−1 characterize chloro-tetraphenylporphinatochromium(III) complexes with a weakly bound neutral ligand,84 suggesting that a heterometallic conjugated porphiryn dimer might satisfy the requirements to implement the protocol described here.
As a final reminder, efficient operation of the simulator requires that the qudit–qubit pairs are well isolated, while still retaining control over the molecular orientation. An isostructural diamagnetic matrix is thus mandatory. While this is usually accessible for single qubits, in the case of a two-spin architecture the co-crystallization of the para- and dia-magnetic molecules must occur without metal scrambling. This can be easily avoided using inert d3/d6 ions, as in the case of CrIII and low spin CoIII. Metal scrambling is however much more common for labile d1/d9 ions, such as CuII, requiring the use of polydentate linkers, such as the porphyrin ligands mentioned above, in the design of the molecular architecture.
The scheme is flexible and allows one to simulate a wide range of interesting models, thanks to the chemical tunability of the proposed hardware. Indeed, although we have focused here on very simple single ions, much larger S1 can be obtained by exploiting the total spin ground multiplet of multi-nuclear complexes with tailored interactions.47,85 With larger S1, one could for example include more photons in the simulations, thus enabling the treatment of more exotic regimes such as the deep strong coupling for light-matter interactions86 or fundamental models such as lattice gauge theories.44,87,88 The latter require a large number of boson modes and excitations for a detailed description in arbitrary dimension, thus representing a challenging task for both classical devices and near time qubit-based architectures.44 Additionally, models involving multiple two-level atoms or boson modes49 can be simulated by chemically engineering the structures in order to link together several qudit and/or qubits.45,46
In conclusion, it is worth stressing that an effort to synthesize molecules satisfying the conditions highlighted in this work would place molecular nanomagnets among the most promising platforms for the realization of effective quantum simulators.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1tc00851j |
This journal is © The Royal Society of Chemistry 2021 |