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Operating Quantum States in Single Magnetic Molecules: Implementation of Grover's Quantum Algorithm
Clément Godfrin, Abdelkarim Ferhat, Rafik Ballou, Svetlana Klyatskaya, Mario Ruben, Wolfgang Wernsdorfer, Franck Balestro
TL;DR
The paper addresses implementation of Grover’s search algorithm in a multi-level quantum system without entangled qubits or a quantum oracle. It constructs and manipulates a quantum database using a single nuclear spin in a molecular spin transistor, demonstrating state selection and coherent control.
Problem
Grover’s algorithm had previously been demonstrated with entangled qubits, motivating implementation in a multi-level system without entanglement or a quantum oracle.
Method
The experiment uses coherent microwave control of nuclear-spin transitions, a multi-level Hadamard gate to create superpositions, and resonant evolution to amplify a searched state.
Results
The searched nuclear-spin states reached visibilities of 0.9, 0.7, and 0.75, while the resonance map agreed with simulation.
Takeaways & Limitations
The method provides an experimental multi-level implementation of Grover’s algorithm and can be extended to other spin-qubit devices and higher nuclear-spin systems.
Abstract
from arXiv · showhide
Quantum algorithms use the principles of quantum mechanics, as for example quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimisation, simulation and solving large systems of linear equations. Here, we implement Grover's quantum algorithm, proposed to find an element in an unsorted list, using a single nuclear 3/2-spin carried by a Single Molecular Magnet (SMM) transistor. The coherent manipulation of this multi-level qudit is achieved by means of electric fields only. Grover's search algorithm was implemented by constructing a quantum database via a multi-level Hadamard gate. The Grover sequence then allows us to select each state. The presented method is of universal character and can be implemented in any multi-level quantum system with non-equal spaced energy levels, opening the way to novel quantum search algorithms.
I. INTRODUCTION
Quantum algorithms use unitary transformations to encode problem answers in final quantum states. This work focuses on a multi-level implementation of Grover’s search algorithm using a single nuclear spin.
- Quantum algorithms transform an initially prepared quantum state through a finite succession of unitary operations to encode a problem’s answer.
- Grover’s algorithm finds a searched state with high probability after iterations approaching the square root of the database length.
- Unlike earlier demonstrations using entangled qubits, a proposed approach implements Grover’s algorithm in multi-level systems without entangled qubits or a quantum oracle.
- The paper implements this approach with a single nuclear spin I=3/2, treating it as a four-level qudit.
- The experimental Grover implementation uses two successive quantum gates, beginning with a Hadamard gate that creates a quantum directory.
II. READING-OUT NUCLEAR SPIN STATES
The study uses a TbPc2 single-molecule-magnet transistor to manipulate and read out a Tb3+ nuclear spin through electric-field-driven microwave control. Its four nuclear levels have unequal spacing, enabling separate transition addressing and coherent control.
- A bis(phthalocyanine)terbium SMM is contacted by gold electrodes to form the three-terminal transistor used for electric transport measurements.
- The Tb3+ ion carries a nuclear spin I=3/2, whose hyperfine interaction splits each electronic spin state into four nuclear levels.
- The quadrupole term produces unequal energy-level spacing, allowing independent manipulation of the nuclear-spin transitions.
- Rabi oscillations across the three transitions can be tuned from 1.5MHz to 8MHz, with Rabi frequency scaling linearly with microwave amplitude.
- Nuclear states are initialized and read out by sweeping the magnetic field through QTM transitions before and after applying a microwave pulse.
- Transition probabilities are estimated from 1000 repetitions of each pulse sequence using the observed initial-to-final state events.
- At resonance, microwave driving produces coherent population oscillations between nuclear states, while detuning lowers visibility and increases oscillation frequency.
III. GROVER ALGORITHM IMPLEMENTATION
The implementation uses a multi-chromatic microwave pulse to control a multi-level nuclear-spin system, first creating a coherent database superposition and then resonantly amplifying a selected state. Experiments demonstrate state selection with the two-step sequence and establish coherent control and read-out in a molecular spin transistor.
- Pulse design: A multi-chromatic microwave pulse simultaneously drives the transitions of the multi-level system in a generalized rotating frame.The Hamiltonian is treated as time independent under near-resonance and rotating-wave assumptions.
- Quantum database: The Hadamard gate prepares a coherent superposition of all nuclear-spin states, forming the quantum database.The pulse must equalize both the populations and phases of the states; a 24 GHz sampling-rate AWG synthesizes it.
- Quantum database: Coherent superpositions of 2, 3, and 4 nuclear-spin states are created, but the 4-state superposition has different phases for its states.The 2- and 3-state cases use parameters that ensure equal phases when populations are equal.
- Grover search: The second gate creates resonance between the superposed state and the researched state, producing oscillations that transfer population to the selected state after half a period.For an N-element database, the unitary-evolution period has an N dependence, and resonance is set through a specific detuning.
- Grover search: 0.9, 0.7 and 0.75 visibility are obtained for selection of the first, second and third states, respectively.The experimental visibility map agrees with simulation and is maximized at the resonant condition.
- Experimental outcome: The results demonstrate coherent control of a single nuclear spin with nondestructive read-out and provide the first experimental implementation of Grover’s algorithm in a multi-level system.The two-step operation is described as extendable to alternative spin-qubit devices and potentially larger molecular-spin databases.