Quantum Spin Adapted Models Tackle SU(2) Symmetry
Spin-Adapted
Spin adapted representations increase simulation efficiency in quantum computing by directly using quantum systems' innate symmetries, especially the challenging non-Abelian SU(2) total-spin symmetry. The exponential growth of Hilbert space size with particle number is a fundamental difficulty in quantum system simulation. Despite classical algorithms' success in reducing computing costs using non-Abelian symmetries, quantum algorithms have struggled to include them.
Problems with Conventional Methods Total spin eigenstates may require an exponentially high number of coefficients to represent them in terms of the eigenbasis, the common computational basis. Most quantum computing methods use simpler Abelian symmetries. Previous research on non-Abelian symmetries focused on maintaining symmetry conservation with non-spin-adapted bases, which can be computationally difficult and may not avoid hardware noise or Trotterization from breaking symmetry. Explicit unitary basis modifications, like the quantum Schur transformation, are theoretically possible but impractical on most quantum hardware because they employ qubits, whose length and local dimensions rise with system scale.
New spin-adapted structure Researchers have developed a novel formalism for building quantum algorithms directly in a total-spin operator eigenbasis to circumvent these issues. This concept uses the symmetric group approach (SGA) to generate spin-adapted quantum Hamiltonians and associated unitarizes.
A truncation technique for total-spin eigenstates' internal degrees of freedom is innovative in this context. This method defines a hierarchy of spin-adapted subspace encodings onto quantum registers with increasing precision. In a site antiferromagnetic Heisenberg model, intermediate total spin values can be shortened to a maximum value, creating ever-larger subspaces that accurately describe the model's low-energy behaviour.
The step encoding, the other popular spin eigenstate encoding approach, is less suitable for quantum computing applications than height encoding, which retains quantum operator locality. However, step encoding often produces very non-local projector operators. The height encoding and height variable truncation provide sparse and local qubit Hamiltonians, making them excellent for quantum simulations on hardware with constrained connectivity.
Key benefits and advantages:
This approach generates a hierarchy of spin-adapted Hamiltonians for the antiferromagnetic Heisenberg model whose ground-state energy and wave function rapidly converge to their exact equivalents, even with very small truncation thresholds. Ground state energy estimates for a 16-site Heisenberg chain converged quickly. Hardware suitability: Truncated Hamiltonians can be encoded into sparse and local qubit Hamiltonians, making them suitable for quantum simulations on hardware with limited connectivity. Avoids Impractical Changes: This method achieves results comparable to those of complex unitary basis modifications like the quantum Schur transformation without explicitly implementing them on most qubit-based quantum devices. For smaller truncated subspaces, the approach may reduce qubit count and improve noise resilience. Using the lowest non-trivial truncation subspace, a ground state approximation for a 16-site chain only needs 3 qubits for the singlet state (down from 9) or 7 for the triplet state (down from 14 for the barStrunc le 3/2 subspace). This reduced qubit count could increase noise robustness in hardware quantum simulations of low-energy states. Reducing Sampling Overhead: Quantum algorithms can be formulated in total-spin eigenbases, so the generated wave functions can be sampled there instead of the projected spin eigenbasis. Significant compression of the ground-state wave function has been shown, which should reduce the sampling overhead needed for observable quantum state estimation. See also National Quantum Computing Centre Gets NPL Ion Trap.
Application and Demo:
The researchers solved the one-dimensional antiferromagnetic Heisenberg Hamiltonian using their method. A resource-efficient adiabatic state-preparation technique was used to create shallow quantum-circuit approximations for the ground states of the Heisenberg Hamiltonian in various symmetry sectors. This includes singlet and triplet variations. Even with slight truncation values, numerical simulations showed that these approximation time-evolution unitaries approximate low-energy beginning states' temporal dynamics. The adiabatic schedules using simple linear ramp functions have very high final instantaneous fidelities.
Future Outlook:
Future research will use real quantum hardware to extend this protocol to non-integrable spin Hamiltonians and electronic structure Hamiltonians to test its scalability. Using spin-adapted operator-specific scalable error mitigation approaches, the researchers seek to enable utility-scale quantum simulations. Compressing the ground-state wave function in the spin-adapted basis should speed up quantum simulations by reducing sampling overhead during observable estimation. More compact wave functions could improve methods like sample-based quantum diagonalisation (SQD).
Finally, our novel strategy directly embeds physical systems into spin-adapted representations for effective quantum simulations. Resource economy, noise robustness, and reduced sampling overhead are advantages for existing and near-term quantum devices with this method.
