POLITECNICO DI TORINO Repository ISTITUZIONALE A Quantum Computation Model for Molecular Nanomagnets Original A Quantum Computation Model for Molecular Nanomagnets / Cirillo, Giovanni Amedeo; Turvani, Giovanna; Graziano, Mariagrazia. - In: IEEE TRANSACTIONS ON NANOTECHNOLOGY. - ISSN 1536-125X. - 18(2019), pp. 1027-1039. [10.1109/TNANO.2019.2939910] Availability: This version is available at: 11583/2761452 since: 2019-12-05T12:10:15Z Publisher: IEEE Published DOI:10.1109/TNANO.2019.2939910 Terms of use: openAccess This article is made available under terms and conditions as specified in the corresponding bibliographic description in the repository Publisher copyright (Article begins on next page) 04 October 2021 1 A quantum computation model for molecular nanomagnets Giovanni Amedeo Cirillo, Student Member, IEEE, Giovanna Turvani, and Mariagrazia Graziano, Member, IEEE Abstract—Molecular nanomagnets can be considered as serious states (encoding 0 and 1) at the same time [2]. Superposition candidates for the definition of a scalable and reliable quantum permits to define an equivalent parallel computation model, computing technology: information is encoded on their spins involving quantum algorithms faster than the corresponding and they ensure relaxation and decoherence times sufficiently long for the execution of tens of quantum operations. Among classical ones, in terms of computational costs. For example, these devices, Cr7Ni supramolecular complexes are extremely of a quantum computer is able to execute in polynomial time interest since they implement a universal set of quantum gates, prime factoring, which requires exponential time in Boolean- made of single-qubit gates and the two-qubit Controlled-phase based computation [3]. gate. A model for analyzing Cr7Ni molecules and a potential Many technologies have been proposed for implementing a quantum computer architecture are proposed: starting from the energy parameters required for spin manipulations, these systems quantum computer based on superconductivity [4]–[7], trapped have been proved to implement elementary quantum algorithms. ions [8], photons [9], [10], technologies (ultracold atoms Within this paper, two, three and four-qubit systems have been in optical lattices, thin-film superconductors, Semiconductor analyzed: the simulation of quantum operations and the analysis Heterostructures) for topological quantum computing [11], of dynamical non-idealities have been done in parallel. Our [12] and spin systems as quantum-dots [13], silicon [14]–[16] approach enables the definition of an operating point - in terms of magnetic driving fields and latency - in which the execution of and molecules [17]–[21], which can be as organic as inorganic elementary quantum algorithms is characterized by negligible [22]. Molecular quantum information is encoded on spin orien- errors. The proposed model has been entirely implemented tations and qubits are manipulated through magnetic resonance in MATLAB, thus obtaining a software infrastructure for the techniques. Many quantum algorithms have been executed on analysis of Cr7Ni molecules that can be extended to quantum apparatus for liquid-state NMR spectroscopy [23]–[30]. The systems with analogous Hamiltonian and behavior. molecules typically employed for NMR quantum information Index Terms—Molecular Nanomagnets, Quantum Computing processing usually encode information on nuclear spins of Architectures, Quantum Algorithms, Innovative Technology hydrogen, carbon and fluorine atoms. The main limitation of this technology is related to the scaling of the number of I. INTRODUCTION qubits, through the synthesis of more complex molecules with more spin qubits; for this reason, the analysis of alternative The high-power consumption due to leakage currents and molecular technologies for quantum computing would permit the technological limits of lithography-based techniques used to overcome these limitations. Among spin-qubit technologies, for fabrication are making the development of Complementary we focused our attention on molecular nanomagnets, that result Metal-Oxide-Semiconductor (CMOS) technology increasingly extremely interesting not only for the capability of scaling harder, with the consequent risk that the computation per- the number of qubits through supramolecular chemistry tech- formance scaling in the next decades cannot be ensured. In niques, but also for their long relaxation and decoherence order to overcome these limitations, two different approaches timescales, enabling the implementation of tens of quantum have been considered: on the one hand, research can focus operations with negligible errors. In particular with this work, on technologies which can inherit the CMOS role of leading Cr7Ni-Co-Cr7Ni molecular nanomagnets, which has been technology for the production of microprocessors based on proved to implement a universal set of quantum gates [31]- deterministic binary information and boolean logic [1] (the have been analyzed. Starting from these properties, we present so-called classical computers); on the other, new computing here a simulation and validation model, entirely developed paradigms - alternative to the classical one - can be analyzed. in MATLAB, for the analysis of the execution of quantum Quantum Computation belongs to this set: its unit of in- operations and algorithms with these molecules. This bottom- formation, named qubit, is physically mapped on a quantum up methodology can be further exploited for the investigation state (e.g. a spin) instead of a classical state as a voltage. of other molecular technologies that can be candidates for the Since quantum states are characterized by superposition, the definition of a future quantum computer architecture. After an qubit has a non-null probability of being in two different overview of quantum computation fundamentals and the state of the art for quantum computing with molecular nanomagnets, The authors are with the Electronics and Telecommunications Department of Politecnico di Torino, Torino, 10129, Italy (e- the Cr7Ni-qubit model is presented, focusing on the universal mail: giovanni [email protected]; [email protected]; maria- set of quantum gates of this technology. In SectionVa [email protected]). potential quantum computer architecture is proposed, with the Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other other purposes must be Cr7Ni complex behaving as a processing unit. An overview obtained from the IEEE by sending a request to [email protected]. of the implemented instructions, exploitable for simulating 2 elementary quantum algorithms up to four qubits, is reported. the azimuth angle. The two basis states j0i (θ = 0) and j1i In SectionVI non-idealities are analyzed, in order to define (θ = π) - corresponding to the states 0 and 1 of a classical a quasi-optimal operating point for the molecular processor. bit - lie at the poles of the sphere; in these cases the value of In Section VIII the results obtained from the simulation of a φ is irrelevant. Points on the same latitude have the same 2 2 three-qubit algorithm are reported. Future perspectives for the probabilities jc0j and jc1j . technology and modeling infrastructure are reported in Section Operations on a qubit correspond to moving the state vector, IX. thus eventually changing its probabilities of being j0i or j1i, through rotations of the Cartesian axes described by matrices II. BACKGROUND Rx(θ), Ry(θ), Rz(θ) [2]. They are related to the well-known The quantum unit of information is the qubit; it is encoded Pauli matrices σx, σy and σz - typically employed for the on a quantum physical system with two basis states - tipically description of two-level quantum systems - by the following named j0i and j1i - implemented in many ways, as spin equations: states of electrons, atomic energy levels, polarization states −i θ σ θ θ R (θ) = e 2 x;y;z = cos I + i sin σ ; of photons, or paths in an interferometer. In quantum circuit x;y;z 2 2 x;y;z and information theories the qubit is abstracted from physical θ θ π i details. The main difference between a qubit and a classical (σx;y;z) = e 2 Rx;y;z (θ) ; (2) bit is that the first is not confined to one of the states j0i and 1 2 where I is the identity matrix. For example, (σx) = j1i, but can be found in arbitrary superposition states π π i 4 π i 2 e Rx 2 and σz = e Rz(π). It must be observed that 1 0 c0 j i = c0 j0i + c1 j1i = c0 · + c1 · = (1) quantum gates implemented by real quantum computers are 0 1 c1 typically characterized by phase shifts of type eiφ that do where c0; c1 2 C are the probability amplitudes, each one as- not affect the final result. In fact, if a generic quantum gate sociated to one qubit basis state. The square magnitude of each eiφM is applied on the quantum state j i, resulting in a final 0 iφ iφ probability amplitude is equal to the probability of finding the state j i = e M j i = e j M i, the scalar term does 2 0 2 qubit in the corresponding state, thus jc0j = P (j i = j0i), not affect any probability j of the final state since 2 2 2 2 2 2 jc1j = P (j i = j1i) and jc0j + jc1j = 1. For these reasons, 0 iφ 2 j = je j · Mj = Mj . the qubit can be in two different states (encoding 0 and 1) According to the vector description of qubit, any quantum gate at the same time. The one-qubit model can be generalized involving k qubits can be described by a 2k ×2k unitary matrix to a quantum register of N qubits with 2N basis states N which modifies the probability amplitudes of a state vector of P2 −1 2 k j i = i=0 ci j ii, with jcij = P (j i = j ii) and length 2 . N P2 −1 2 i=0 jcij = 1. Superposition is the property permitting to quantum bits of being more powerful than their classical III. QUANTUM COMPUTING WITH MOLECULAR equivalents, since a higher degree of computing parallelism NANOMAGNETS can be exploited by employing qubits, given the same number N of information units.