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Molecular Spin Qudits for Quantum Algorithms
Eufemio Moreno-Pineda, Clément Godfrin, Franck Balestro, Wolfgang Wernsdorfer, Mario Ruben
TL;DR
Quantum hardware requires material platforms that can satisfy the requirements for quantum operations. This review examines TbPc2 as a molecular nuclear-spin qudit and concludes that it meets these requirements, including enabling Grover’s algorithm.
Problem
Quantum-information hardware requires suitable material platforms, while some non-molecular systems face complex fabrication, decoherence, and entanglement challenges.
Method
The review evaluates TbPc2’s molecular nuclear-spin qudit properties against quantum-information requirements, including initialization, manipulation, readout, coherence, and gates.
Results
TbPc2 nuclear-spin qubits meet practically all essential characteristics for quantum operations, culminating in the realization of Grover’s algorithm.
Takeaways & Limitations
TbPc2 is presented as a plausible molecular hardware platform, with a single molecule embedded in scalable circuits for spin readout and information processing.
Takeaways & Limitations
The review focuses on TbPc2, and molecular systems with different readout characteristics require further study and alternative readout schemes.
Abstract
from arXiv · showhide
Presently, one of the most ambitious technological goals is the development of devices working under the laws of quantum mechanics. One prominent target is the quantum computer, which would allow the processing of information at quantum level for purposes not achievable with even the most powerful computer resources. The large-scale implementation of quantum information would be a game changer for current technology, because it would allow unprecedented parallelised computation and secure encryption based on the principles of quantum superposition and entanglement. Currently, there are several physical platforms racing to achieve the level of performance required for the quantum hardware to step into the realm of practical quantum information applications. Several materials have been proposed to fulfil this task, ranging from quantum dots, Bose-Einstein condensates, spin impurities, superconducting circuits, molecules, amongst others. Magnetic molecules are among the list of promising building blocks, due to (i) their intrinsic monodispersity, (ii) discrete energy levels (iii) the possibility of chemical quantum state engineering, and (iv) their multilevel characteristics, leading to the so called Qudits (d > 2), amongst others. Herein we review how a molecular multilevel nuclear spin qubit (or qudit, where d = 4), known as TbPc2, gathers all the necessary requirements to perform as a molecular hardware platform with a first generation of molecular devices enabling even quantum algorithm operations.
Introduction
Quantum algorithms exploit superposition and entanglement for computational tasks beyond classical computers, motivating new quantum-information hardware. Magnetic molecules, especially TbPc2, offer chemically engineerable, atomically precise multilevel spin systems for encoding, coupling, and readout of quantum information.
- Quantum algorithms: Quantum algorithms use superposition and entanglement to perform computational tasks not achievable with classical computers.Shor’s factoring algorithm and Grover’s search algorithm exemplify early quantum-information developments, while quantum simulation later validated Feynman’s proposal.
- Quantum hardware: Quantum-mechanical hardware requires new material classes, including defect-based solids, quantum dots, and molecular quantum magnets.The introduction identifies molecular quantum magnets, also termed single-molecule magnets, as promising systems because of their magnetic characteristics and chemical manipulability.
- Molecular devices: Molecular devices can encode and process quantum information in long-lived nuclear spin states while electronic states provide coupling and readout.Chemical manipulation can modify ligand fields of spin carriers and their interactions with other units.
- TbPc2: TbPc2 is presented as a prototype molecular qubit whose chemical control, engineered quantum states, and defect-free atomically precise production support quantum-algorithm applications.These properties are described as making magnetic molecules plausible candidates for performing quantum algorithms.
Elementary unit for Quantum Information Processing
Quantum information uses qubits whose superposition states enable operations beyond classical deterministic computation. Viable physical platforms must satisfy criteria including well-defined levels, coherence, initialization, universal gates, and readout, while multilevel qudits can support algorithms through their interconnected states.
- Quantum bits and states: A qubit is a two-state quantum information unit that uses |0⟩ and |1⟩ states and permits superposition absent in classical computation.A general state is represented as |𝜓⟩= 𝑎0|0⟩+ 𝑎1|1⟩ with |𝑎0|2 + |𝑎1|2 = 1.
- Quantum gates: Quantum gates differ from classical Boolean gates by allowing superposition of the states involved during an operation.The classical NOT gate produces a deterministic single-bit output, whereas the single-qubit Hadamard gate creates superposition.
- Qubit requirements: A viable qubit platform requires five core properties: scalable well-defined levels, long coherence times, initialization, universal quantum gates, and readout.These requirements address state definition, preservation during operations, preparation, algorithmic manipulation, and controlled measurement.
- Qubit requirements: Multilevel qudits can mimic intrinsically interconnected qubits and exploit multilevel characteristics for quantum algorithms.A Hadamard operation can act within a single multilevel qudit without entanglement, and Grover’s algorithm relies solely on state superposition.
- Qubit requirements: DiVincenzo’s additional communication requirements concern interconversion between stationary and flying qubits and exact transmission of flying qubits.These requirements are relevant to non-local qubits and photon-based entanglement transmission.
Quantum Algorithms
Quantum algorithms illustrate the computational power of quantum information processing through speedups for integer factorisation and database search, while molecular spin systems have been proposed as qubits for Grover’s algorithm without requiring entanglement.
- Shor’s algorithm: O(log N)^3 time enables Shor’s quantum factorisation algorithm, contrasting with the best classical time t = exp(O(log N)^1/3(log log N)^2/3)).Shor’s result exposes the vulnerability of RSA systems to quantum attacks.
- Experimental demonstrations: Shor’s algorithm was experimentally demonstrated using seven nuclear-spin qubits in a (2,3-13C)hexafluorobutadiene molecule manipulated by nuclear magnetic resonance pulse sequences.The reported demonstrations factorised N = 15 and 21.
- Grover’s algorithm: N = 2^n database items require O(N) classical operations but only O(√N) permutations with Grover’s algorithm.The quadratic speedup arises from coherent quantum states.
- Grover’s algorithm: In 2001, Loss and Leuenberger proposed single-molecule magnets as qubits for Grover’s algorithm using multilevel resources without requiring entanglement.This proposal uses the multilevel character of single-molecule magnets as a computational resource.
- Other algorithms: Other quantum algorithms include Deutsch–Jozsa, which distinguishes balanced from unbalanced functions, while quantum information processing can simulate quantum systems beyond current computers.These examples broaden the scope of quantum algorithms beyond factorisation and database search.
Qubit’s Materials
Molecular materials offer chemically tailored, monodisperse qubits, while multilevel qudits can process quantum information in parallel within a single unit. The review focuses on TbPc2, a robust molecular unit combining a central Tb3+ ion with two phthalocyanines and satisfying implementation requirements.
- Molecular materials: Molecular qubits offer tailored chemical control and monodispersity, alongside potential advantages over non-molecular systems.Their implementation in hybrid devices could speed information processing through interactions with other qubits.
- Multilevel systems: Multilevel qudits can realize Hadamard gates without entanglement and process quantum information in parallel within a single unit.Qudits create superpositions of many-level states and can reduce error rates relative to two-level qubits.
- Electron spin qubits: Electron spins are highly susceptible to spin-bath interactions, with performance depending strongly on ligand and solvent characteristics.Results obtained in large crystal ensembles also make initialization challenging because of inhomogeneity effects.
- Nuclear spin qubits: Nuclear spins provide environmental shielding, extremely long coherence times, and very low error rates, but their small magnetic moments make integration, read-out, and manipulation difficult.Read-out and manipulation have nevertheless been demonstrated for nuclear spins.
- TbPc2 molecular qubit: TbPc2 comprises a central Tb3+ ion between two parallel phthalocyanines in a square-antiprismatic D4d geometry.The molecule is robust and can be deposited on multiple substrates at high temperatures while conserving its molecular, electronic, and magnetic characteristics.
The TbPc2 Molecular Qubit: A scalable well-defined system
TbPc2 provides a four-level nuclear-spin qudit with experimentally demonstrated initialization, manipulation, single-molecule read-out, and coherence lasting up to 0.32 ms. Its multilevel structure has also enabled the first experimental demonstration of a quantum algorithm on a single-molecule magnet.
- The TbPc2 Molecular Qubit: A scalable well-defined system: Strong hyperfine coupling splits the Tb3+ ground doublet into four nuclear-spin states, mI = ±3/2 and ±1/2, forming a four-level qudit.The four avoided level crossings arise from coupling between the Tb3+ electronic states and its I = 3/2 nuclear spin.
- The TbPc2 Molecular Qubit: A scalable well-defined system: Nuclear-spin read-out was demonstrated through transport measurements on TbPc2 suspended on carbon nanotubes and trapped between gold junctions.Low-field QTM changes the electronic magnetic moment while preserving the nuclear spin, enabling electrical detection.
- The TbPc2 Molecular Qubit: A scalable well-defined system: At low fields and temperatures, all four nuclear-spin states could be initialized, including controlled manipulation from |+3/2⟩ to |+1/2⟩.Initialization used repeated magnetic-field sweeps between ±60 mT until the desired QTM transition was observed.
- The TbPc2 Molecular Qubit: A scalable well-defined system: 0.32 ms was the measured maximum coherence time of the nuclear-spin qubit, determined from exponentially decaying Ramsey fringes.The coherence time represents the duration of the quantum superposition and was measured using two π/2 microwave pulses separated by an interpulse delay.
- The TbPc2 Molecular Qubit: A scalable well-defined system: TbPc2 enabled the first experimental proof of a quantum algorithm on a single-molecule magnet through population oscillations following implementation of the Hadamard gate.The proposed Grover algorithm uses the molecule’s multilevel states without requiring inter-qubit entanglement.
Conclusions
TbPc2 can be embedded in scalable electronic circuits with individual spin read-out, while its long-lived nuclear spin states encode and process information. Molecular multilevel nuclear spin qubits provide isolation, initialisation, read out, long coherence times, and manipulation, enabling Grover’s algorithm, with alternative read-out methods envisioned for molecules lacking the radical.
- Conclusions: A single TbPc2 unit was embedded in scalable electronic circuits, enabling individual spin read-out through ligand-based read-out dots.The spin-dot containing long-lived nuclear spin states is used to encode and process information.
- Conclusions: Molecular multilevel nuclear spin qubits meet the essential characteristics for quantum operations: isolation, initialisation, read out, long coherence times, and manipulation.These characteristics ultimately led to the realisation of Grover’s algorithm.
- Conclusions: For molecules lacking the π-radical, read-out could use cavity-transmission differences, photon-emitter coupling, or transport measurements.The read-out scheme will depend entirely on the characteristics of the system.