Quantum Gates for Discrete Subgroups Achieve 216-Element Fidelity with Reduced T-Gate Costs

The quest for practical quantum computation hinges on identifying minimal sets of quantum gates capable of performing any desired calculation, and recent work addresses this challenge within a specific mathematical framework. Sebastian Osorio Perez from the University of Maryland, Edison M. Murairi and Henry Lamm from Fermi National Accelerator Laboratory, along with Erik J. Gustafson, demonstrate a primitive gate set for quantum computations based on a discrete subgroup of, a mathematical group with 216 elements. This achievement simplifies the required quantum operations, as the team provides qubit decompositions for essential gates including inversion, multiplication, trace calculation, and a group Fourier transform. Crucially, the resulting gate costs for complex calculations, such as modelling shear viscosity, are significantly lower than those estimated for other contemporary approaches, representing a substantial step towards more efficient quantum algorithms.

This achievement establishes qubit decompositions for essential gates including inversion, group multiplication, the trace, and the group Fourier transform, paving the way for implementing quantum algorithms within this group structure. Calculations estimate that simulating shear viscosity would require a number of T gates comparable to, or lower than, those needed by other current approaches.

Quantum Simulation of Lattice Gauge Hamiltonians

Research in quantum simulation of lattice gauge theories is rapidly advancing, with scientists exploring how to use quantum computers to model the behavior of quantum fields, particularly those described by lattice gauge theory, a key tool in particle physics. A central focus is formulating these theories in a Hamiltonian form, crucial for quantum simulation because quantum computers naturally operate on Hamiltonian systems. Maintaining gauge symmetry within this formulation presents a significant challenge, requiring innovative approaches to preserve the essential physics of the theory. Researchers are developing quantum algorithms designed to calculate quantities of interest in lattice QCD and QED, including particle masses and correlation functions.

Efficiently preparing the initial quantum state and performing measurements are critical bottlenecks, leading to investigations of variational methods and optimization strategies. Current quantum computers are noisy, so scientists are also developing techniques to mitigate these errors and improve simulation accuracy. A notorious problem in lattice QCD, the sign problem, is also being addressed with potential quantum solutions. Tensor networks, like Projected Entangled Pair States, are being explored as both classical and potential quantum approaches to simulating these theories. Recent trends focus on developing near-optimal state preparation techniques to minimize quantum resource requirements.

Researchers are also constructing Ginsparg-Wilson Hamiltonians, designed to have improved chiral symmetry properties, making them more suitable for simulating fermions. Variational Quantum Algorithms, like the Variational Quantum Eigensolver, are popular choices for tackling these simulations on near-term quantum computers. Quantum Imaginary Time Evolution is used to find the ground state of a Hamiltonian, and scientists are exploring how to simulate the time evolution of lattice gauge theories on quantum computers. A major focus is on reducing the number of qubits and quantum gates required for simulations through more efficient algorithms and state preparation techniques.

Hybrid quantum-classical approaches, combining the strengths of both classical and quantum computation, are also gaining prominence. Research extends to exploring different gauge groups beyond the standard SU(3), and drawing connections between lattice gauge theory and condensed matter systems. Improved chiral symmetry is a key goal, alongside robust error mitigation strategies for obtaining accurate results on noisy quantum computers.

SU(3) Computation Cost Dominated by Fourier Transform

Scientists have achieved a primitive gate set for computations utilizing a discrete subgroup of SU(3), specifically the 216-element group. This work establishes qubit decompositions for essential gates including inversion, group multiplication, the trace, and the group Fourier transform. The team measured the number of T gates required for a fiducial calculation of shear viscosity, finding a need for approximately 7. 1x 10 12 T gates when using the HI Hamiltonian and 3. 5x 10 12 T gates for the HKS Hamiltonian.

Experiments revealed that the cost of the group Fourier transform dominates simulations, accounting for over 99% of the computation regardless of the Hamiltonian used. The team determined that using the Σ(72x 3) group reduces gate costs by a factor of 10 35 compared to other methods. Furthermore, the total synthesis error for the HKS Hamiltonian and HI Hamiltonian follows a predictable pattern based on simulation parameters. Data shows that the number of T gates necessary for the primitive gates follows a consistent ordering from cheapest to most expensive. Extrapolations suggest that other gate sets will likely require significantly more T gates than the Σ(72x 3) group. These results, competitive with recent work by other teams, demonstrate a significant reduction in computational cost compared to previous estimates, achieving a T gate density of roughly 1 per Σ(72x 3) register per clock cycle.

SU(3) Quantum Computation via Discrete Subgroup Mapping

Scientists have detailed the construction of a foundational set of gates for digital quantum computation utilizing a discrete subgroup, specifically Σ(72 × 3), of SU(3). The researchers successfully devised qubit decompositions for the necessary primitive gates, including inversion, group multiplication, the trace, and the group Fourier transform. This establishes a pathway for implementing quantum algorithms within this group structure. The resulting gate set demonstrates a favorable resource cost. The achievement lies in mapping the complex mathematical structure of Σ(72 × 3) onto a qubit-based quantum computer.

This involved defining how elements within the group can be represented and manipulated using qubits, and identifying the minimal set of gates required for universal quantum computation within this framework. The team developed a specific encoding scheme, utilizing nine qubits and three qutrits, to represent group elements and their corresponding quantum states. This encoding allows for efficient implementation of the necessary gates and facilitates quantum simulations. Researchers acknowledge that the resource estimates presented are dependent on the specific simulation being performed and may vary for different quantum algorithms. Future research directions include exploring the practical implementation of these gates on actual quantum hardware and investigating the potential for further optimization of the encoding scheme. Additionally, the team suggests that this approach could be extended to other discrete subgroups of SU(3) and potentially to other symmetry groups, broadening the scope of quantum algorithms that can be efficiently implemented.