Spin-Based Quantum Computers made by Chemistry: Hows and Whys

Philip C. E. Stamp; Alejandro Gaita-Ariño

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Spin-Based Quantum Computers made by Chemistry: Hows and Whys

Philip C. E. Stamp, Alejandro Gaita-Ariño

arXiv:0807.1986v2Jul 12, 2008quant-phcond-mat.str-el

TL;DR

The review examines how chemistry might build spin-based quantum information systems and identifies decoherence as the fundamental problem. It synthesizes physical descriptions, decoherence mechanisms, suppression strategies, and chemical designs for spin qubits. The paper concludes that insulating molecular or ionic spin qubits offer reduced decoherence and high reproducibility with appropriate care, while large-scale architectures remain a future goal.

Problem
Building a quantum information processing system requires overcoming decoherence, while scalable designs for many entangled qubits remain uncertain.
Method
The review connects computational and physical descriptions of qubits, analyzes spin-bath, oscillator-bath, and dipolar decoherence, and surveys chemistry-based molecular and ionic qubit designs.
Results
Insulating atomic or molecular spin qubits offer reduced decoherence and, with appropriate care, high reproducibility.
Takeaways & Limitations
Chemistry-based spin qubits provide multiple design strategies for controlling decoherence and developing systems of multiply entangled qubits.
Takeaways & Limitations
The problem of decoherence from pairwise dipolar interactions remains open, so the review gives no detailed decoherence-rate results for it.

Abstract

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This introductory review discusses the main problems facing the attempt to build quantum information processing systems (like quantum computers) from spin-based qubits. We emphasize 'bottom-up' attempts using methods from chemistry. The essentials of quantum computing are explained, along with a description of the qubits and their interactions in terms of physical spin qubits. The main problem to be overcome is decoherence - how this works is described, along with ways to suppress contributions from spin bath and oscillator bath environments, and from dipolar interactions. Finally we discuss various strategies for making chemistry-based spin qubits, using both magnetic molecules and magnetic ions.

I. INTRODUCTION

The review asks how chemistry can build scalable spin-based quantum information systems, while emphasizing decoherence as the central obstacle. It connects physical and computational descriptions to motivate chemistry-based approaches.

I. INTRODUCTION: The review’s basic question is how bottom-up chemistry might build a spin-based quantum computer.
I. INTRODUCTION: Decoherence is the fundamental problem because complex entangled states span many subsystems, making the individual component wave-functions lose meaning.
I. INTRODUCTION: Ion traps are among the most successful QIPS designs, but their scalability to many entangled qubits remains unclear.
I. INTRODUCTION: Spin-based systems avoid moving charges, while molecular or ionic subunits can be identical if impurities and defects are eliminated.
I. INTRODUCTION: The review introduces QIPS concepts, chemically realized spin qubits, decoherence mechanisms, and future research directions.

A. Description in Computational Language

Quantum computation generalizes classical computational states and operations to superposed, entangled qubit states. A quantum-walk representation maps these states and their transitions onto a graph governed by a Hamiltonian.

A. Description in Computational Language: Turing machines model computation through a tape, head, program, and register that manipulate discrete states.
A. Description in Computational Language: Quantum Turing machines generalize computational states to quantum superpositions and use unitary transformations to map input states to output states.
A. Description in Computational Language: A qubit is a two-level system that can occupy an arbitrary superposition of its two classical states.
A. Description in Computational Language: For N qubits, the wave-function contains 2N coefficients and 2N relative phases, with typically entangled states that cannot be factorized into independent qubit wave-functions.
A. Description in Computational Language: A quantum walk represents each Hilbert-space state as a graph node and allowed transitions as links between nodes.
A. Description in Computational Language: The quantum-walk Hamiltonian uses node energies and hopping matrix elements, and must be generalized to include internal spin variables for a full QIPS.

B. Physical Description of a QIPS

The paper translates quantum-information descriptions into physical spin-qubit models, using single-molecule magnets such as Fe8 as a concrete example. At low energies, Fe8 can be reduced to a qubit whose effective transverse field drives transitions between localized states.

Physical qubit models: A physical QIPS is modeled with qubits, local fields, and time-dependent interactions between qubits.The Pauli operator acts on each qubit, while local fields and interqubit couplings specify the physical dynamics.
Physical qubit models: Solid-state qubit proposals include semiconductor and quantum-dot spins, superconductors, diamond vacancies, single-molecule magnets, and rare-earth spins.
Fe8 example: Fe8 contains eight antiferromagnetically coupled Fe3+ spin-5/2 ions forming a total spin-10 system inside an organic-ligand cage.The molecule is represented using octahedral iron ions, oxo and hydroxo bridges, and tacn ligands.
Fe8 example: The giant-spin approximation reduces the individual transition-metal spins to a single total spin and commonly uses a quadratic biaxial Hamiltonian.For Fe8, the effective model has D/kB = 0.23 K, E/kB = 0.094 K, and omits higher terms of order O(S6).
Model scope: The Fe8 effective Hamiltonian applies below roughly 5−10 K; at higher temperatures, additional states from the general 10-spin Hamiltonian must be considered.
Qubit dynamics: In the two-well regime, the localized |↑⟩ and |↓⟩ states form a qubit, while ∆o acts as an effective transverse field driving transitions between them.Applying a real transverse field changes ∆o, and the two-well system is protected against external transverse-field fluctuations because they have a small effect on ∆o.
Quantum-walk connection: An N-qubit system maps its 2N states onto the corner sites of an N-dimensional hypercube, where the quantum walker moves according to the qubit Hamiltonian.For three qubits, the eight states can be represented both as quantum-walk sites and as qubit states.

C. Realistic Models of QIPS

The paper argues that simplified QIPS models must be extended to include environmental modes and stray interactions. For spin-based molecular qubits, decoherence arises from localized spins, oscillator modes, omitted internal degrees of freedom, and long-range couplings.

Model limitations: Simplified SMM and qubit models omit important degrees of freedom and therefore have several principal shortcomings.
Model limitations: The giant-spin model omits low-energy discrete electronic spin and charge degrees of freedom, including loose spins from structural transformations or weak exchange couplings.Impurities and distorted rogue molecules in multi-SMM systems can also act as loose spins coupled to the qubit.
Environmental modes: SMM qubits couple to nuclear spins, photons, extended and local phonons, and, in conducting systems, electronic excitations.
Environmental modes: Environmental degrees of freedom can also mediate stray couplings between qubits, including slow phonon-mediated interactions with retardation effects.
Realistic modeling: A realistic model proceeds in two stages, beginning by separating localized discrete modes from delocalized modes and representing the former with a spin bath.The spin bath can include defects, nuclear spins, loose spins, and localized phonons.
Spin bath: Spin-bath dynamics are described by qubit-dependent and bath-dependent effective fields, so each bath spin evolves under both influences and reacts back on the qubit.
Spin bath: In Fe8, each molecule couples to over 200 nuclear spins; nearby proton and 57Fe hyperfine couplings are approximately 3−4 mK, while nuclear-nuclear couplings are below 10−7 K.
Oscillator bath: Delocalized modes such as phonons and photons are modeled as an oscillator bath coupled to the qubit through magnetoacoustic operators and coupling functions.For SMMs, the coupling magnitude is typically |g_q| ∼ 10−100 K.

III. DECOHERENCE

Decoherence is the fundamental practical obstacle to building a quantum information processing system, because environmental entanglement destroys phase coherence. The review explains its mechanisms and seeks decoherence-rate results to guide QIPS design.

Decoherence is widely regarded as a fundamental problem blocking the manufacture of quantum information processing systems.
Designing a QIPS requires estimating decoherence because excessive decoherence prevents the system from working.
At low temperatures, most decoherence comes from spin-bath environments of localized modes rather than oscillator-bath models.
The review aims to provide intuitive explanations of decoherence and decoherence-rate results that can help guide QIPS design.
Decoherence arises when a system becomes entangled with its environment, whose unobserved dynamics smear its phase evolution.

A. Spin bath decoherence

Spin-bath decoherence arises because qubit-driven bath-spin motion becomes entangled with the qubit trajectory, smearing its phase dynamics. Precessional decoherence dominates, while topological and noise mechanisms contribute under different conditions.

Mechanism: Qubit-dependent bath-spin trajectories entangle the qubit with its environment, converting definite qubit dynamics into decoherence when unobserved bath histories are averaged.The bath-spin path depends largely on the qubit field trajectory, with only a small perturbation from other bath spins.
Precessional Decoherence: Precessional decoherence is the strongest spin-bath mechanism and smears qubit phase through bath-spin precession without energy exchange.Its rate is roughly proportional to the average solid angle swept by the bath spins.
Precessional Decoherence: For Fe8, Γφ is zero at H0 = 0, peaks near hk ≃ωk, and decreases again at high field as the qubit-induced bath-field orientations become nearly parallel.The maximum occurs when bath spins precess strongly as the qubit field flips between orientations separated by approximately 90°.
Topological Decoherence: Topological decoherence comes from inelastic bath-spin transitions during finite-time qubit flips, but it is very small in Fe8 because ωk ≪Ω0.For Fe8, Ω0 is approximately 5 K while most ωk are below 1 mK, even across roughly 200 nuclear spins.
Noise Decoherence: Freezing the qubit with a longitudinal bias also largely freezes the spin bath, producing a roughly hundredfold increase in T1 and T2 for off-resonant Mn12 molecules.Without the qubit’s time-varying field, only weak bath-spin interactions drive independent bath dynamics.
Noise Decoherence: Standard T1/T2 phenomenology is inappropriate when spin-bath decoherence dominates because the resulting singular spectral peaks are not associated with dissipation.The review emphasizes that precessional decoherence has no classical analogue or fluctuation-dissipation connection.

B. Oscillator Bath Decoherence

Oscillator-bath decoherence is modeled through spin-boson dynamics and differs from spin-bath decoherence by retaining a quantum-classical correspondence and fluctuation-dissipation relation. It renormalizes qubit motion and causes dissipative relaxation, with phonons providing the relevant bath in insulating molecular systems.

Model: The spin-boson model couples a qubit to an oscillator bath whose decoherence and dissipation are related through a fluctuation-dissipation theorem.This contrasts with spin-bath decoherence, which lacks that correspondence.
Effects: The oscillator bath slows qubit flips by renormalizing ∆ to ˜∆ with ˜∆ < ∆ and causes relaxation through spontaneous or stimulated emission.The bath can excite an oscillator from its ground state or scatter an existing excitation.
Spectral Models: Ohmic models are not generally appropriate for qubit experiments because spin baths are non-Ohmic and insulating systems couple to phonons with temperature-dependent, higher-than-linear spectral functions.The review notes that the Ohmic form is reliably valid only for simple conductors.
Molecular Systems: For an SMM, acoustic-phonon coupling has J(ω) ∼ω^3, and both quantum-relaxation and decoherence rates have been calculated for this interaction.The rate expression contains a squared spin-phonon matrix element and a density-of-states factor proportional to ∆^3.
Design Implication: Ohmic oscillator-bath decoherence is typically large, motivating insulating systems that avoid moving electrons, although spin-bath decoherence remains.The review contrasts this with a golden-rule-like result for another bath regime.

C. Pairwise Dipolar Decoherence

Unwanted long-range interactions can decohere many qubits simultaneously, especially through dipolar couplings and retarded phonon-mediated interactions. The review identifies error correction and reduced-dimensional architectures as the main ways to mitigate this problem.

Interaction Types: Long-range decoherence arises from instantaneous dipolar or inductive interactions and from retarded interactions mediated by slow particles such as phonons.The distinction depends on whether the mediating interaction can be treated as instantaneous on the QIPS timescale.
Challenge: These interactions are dangerous because they affect many qubits at once, and the problem remains open because simplified and detailed models yield different results.The review therefore gives no detailed decoherence-rate results for pairwise long-range interactions.
Architectural Responses: The principal mitigation strategies are error correction and lower-dimensional geometries that reduce the number of distant qubits capable of resonant interaction.The review favors combining both strategies where possible.

IV. DESIGN CRITERIA: POSSIBLE ARCHITECTURES

Chemistry-based bottom-up architectures seek controllable, addressable spin qubits while exploiting molecular reproducibility, small size, and the absence of moving electrons. Their central constraints are weak individual signals and potentially harmful dipolar interactions.

Design Goals: A QIPS requires accurately controlled or fixed and tunable qubit interactions and local fields, together with writing, computation, readout, and minimal stray coupling.The design must also minimize decoherence from uncontrolled interactions with the environment.
Advantages: Bottom-up materials chemistry offers reproducibility because quantum mechanics produces identical molecules or spin complexes when impurities and defects are eliminated.This is presented as an advantage over top-down nanofabrication approaches.
Advantages: Molecularly small qubits involve fewer degrees of freedom and defects, which the review associates with reduced decoherence and shorter timescales.The size advantage applies to both qubits and other architectural components.
Advantages: Insulating molecules and spin complexes avoid moving electrons, enabling a spintronics architecture involving only spin degrees of freedom.The review identifies moving charges or electrons as a source of decoherence.
Disadvantages: Current molecular and ionic spin moments are too small for individual control or readout except in very small numbers, despite algorithms requiring individual qubit access.This is the main size-related disadvantage of chemistry-based spin architectures.
Disadvantages: Dipolar interactions can be potentially fatal to a QIPS, so architectures must suppress them while using chemistry and varied geometries to address both major disadvantages.The review frames architecture selection as the principal response to size and dipolar-interaction constraints.

A. Chemical approaches

Chemistry-based spin qubits seek to suppress decoherence through insulating molecular or ionic building blocks, controlled interactions, reproducibility, and carefully balanced energy scales.

Chemists can improve qubit reproducibility and decoherence control at both nuclear and electronic scales.
Nuclear Chemistry: Reducing nuclear-spin decoherence generally requires even-numbered elements and often costly isotopic purification, constraining chemical choices.
Moving electrons causes intrinsically high decoherence, making insulating single-molecule magnets and rare-earth ions attractive qubit candidates.
Electronic Chemistry: A coherence window requires θD ≫ ∆o ≫ {ωk}, balancing phonon and nuclear-spin decoherence while favoring large anisotropy and weak hyperfine couplings.
Electronic Chemistry: Rare-earth molecules and polyoxometalates expand design options through suitable crystal fields, chemical stability, varied topology, and high negative charge.
Electronic Chemistry: Qubit molecules should minimize defects, structural isomerism, free rotation, and near-degenerate solvent or counterion positions to reduce disorder.
Electronic Chemistry: Inter-qubit interactions must be precisely controlled while dipolar and spin-phonon interactions are suppressed, potentially through antiferromagnetic ordering.
Electronic Chemistry: Exchange interactions are typically stronger in transition-metal systems, around O(100K), than in rare-earth systems, around O(1K).

B. Geometry, Architecture, and Fabrication

Geometry, architecture, substrates, and fabrication strongly affect addressability and decoherence, motivating sparse, antiferromagnetic, remotely coupled, and chemically tailored designs.

Geometry and Architecture: Molecular qubits are difficult to address individually because they are tiny, while probes are large and can interact with only a few qubits.
Geometry and Architecture: Nanometre-scale probing can destructively disturb qubit states, although STM and MFM tips may interact weakly despite their larger surrounding instruments.
Geometry and Architecture: Remote nanometre probes connected by one-dimensional spin chains or nanotubes could couple into qubit arrays without relying on moving electrons.
Geometry and Architecture: Large antiferromagnetically coupled spin arrays could provide larger qubits, nearest-neighbour access, and reduced dipolar errors when each qubit has no net spin.
Geometry and Architecture: Sparse architectures may improve probe access and reduce dipolar errors, but dendrimer supports can introduce decoherence through steric-hindrance-induced disorder.
Geometry and Architecture: Whole-system addressing and molecular cellular automata reduce addressability demands but require long computations and therefore very long decoherence times.
Fabrication: Polyoxometalates offer varied topologies and sizes, with options to encapsulate magnetic ions or host delocalized electrons.
Fabrication: Bimetallic honeycomb oxalate layers can form nuclear-spin-poor antiferromagnetic two-dimensional lattices with two distinct sites, though a compensating cationic layer is required.

V. CONCLUDING REMARKS

The review identifies decoherence—especially from nuclear spins and dipolar interactions—as the central obstacle, while proposing chemistry-based insulating spin qubits and design strategies to address it.

Suppressing decoherence, particularly from nuclear spins and dipolar interactions, remains the main challenge in building a QIPS.
Insulating atomic- or molecular-scale spin qubits offer reduced decoherence and potentially high reproducibility with appropriate care.
The authors anticipate that experimental control of decoherence and multiply entangled spin-qubit systems could enable realistic large-scale architectures, with chemistry playing a major role.