Qubits as Parafermions

Abstract

Qubits are neither fermions nor bosons. A Fock space description of qubits leads to a mapping from qubits to parafermions: particles with a hybrid boson-fermion quantum statistics. We study this mapping in detail, and use it to provide a classification of the algebras of operators acting on qubits. These algebras in turn classify the universality of different classes of physically relevant qubit-qubit interaction Hamiltonians. The mapping is further used to elucidate the connections between qubits, bosons, and fermions. These connections allow us to share universality results between the different particle types. Finally, we use the mapping to study the quantum computational power of certain anisotropic exchange Hamiltonians. In particular, we prove that the XY model with nearest-neighbor interactions only is not computationally universal. We also generalize previous results about universal quantum computation with encoded qubits to codes with higher rates.

17 pages, no figures. v3: This version to appear in J. Math. Phys., special issue on quantum computation

References in corpus (11)

• Quantum Entanglement in Fermionic Lattices
• Classical simulation of noninteracting-fermion quantum circuits
• Control of Spin Dynamics of Excitons in Nanodots for Quantum Operations
• Indirect Interaction of Solid-State Qubits via Two-Dimensional Electron Gas
• Cancellation of Spin-Orbit Effects in Quantum Gates Based on the Exchange Coupling in Quantum Dots
• Polynomial-Time Simulation of Pairing Models on a Quantum Computer
• Analytical and numerical study of hardcore bosons in two dimensions
• Anisotropic Spin Exchange in Pulsed Quantum Gates
• Constructing Qubits in Physical Systems
• Fermionic Linear Optics and Matchgates
• Power of Anisotropic Exchange Interactions: Universality and Efficient Codes for Quantum Computing

Cited by in corpus (28)

• Quantum tensor product structures are observable-induced
• Emerging quantum computing algorithms for quantum chemistry
• Resource-efficient digital quantum simulation of $d$-level systems for photonic, vibrational, and spin-$s$ Hamiltonians
• Qubit-excitation-based adaptive variational quantum eigensolver
• Electron-Phonon Systems on a Universal Quantum Computer
• Efficient quantum circuits for quantum computational chemistry
• Digital quantum computation of fermion-boson interacting systems
• Efficient generation of graph states for quantum computation
• Symmetry Principles in Quantum Systems Theory
• Universal Quantum Logic from Zeeman and Anisotropic Exchange Interactions
• Encoded Recoupling and Decoupling: An Alternative to Quantum Error Correcting Codes, Applied to Trapped Ion Quantum Computation
• Universal Existence of Exact Quantum State Transmissions in Interacting Media
• Extending matchgates into universal quantum computation
• Quantum Computing in the Presence of Detected Spontaneous Emission
• Universal Fault-Tolerant Quantum Computation in the Presence of Spontaneous Emission and Collective Dephasing
• Canonical entanglement for two indistinguishable particle
• Simulating para-Fermi oscillators
• Cooper-Pair Condensates with Non-Classical Long-Range Order on Quantum Devices
• Avoiding leakage and errors caused by unwanted transitions in Lambda systems
• Exact dynamics of interacting qubits in a thermal environment: Results beyond the weak coupling limit
• Real-time simulation of flux qubits used for quantum annealing
• Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states
• Quantum computation from fermionic anyons on a 1D lattice
• Efficient measurement schemes for bosonic systems
• A representation of complex rational numbers in quantum mechanics
• Quantum Simulation of Molecules without Fermionic Encoding of the Wave Function
• Generalized Grassmann variables for quantum kit (k-level) systems and Barut-Girardello coherent states for su(r+1) algebras
• Many-Fermion Simulation from the Contracted Quantum Eigensolver without Fermionic Encoding of the Wave Function