First and Second Quantized Digital Quantum Simulations Show Factor of 2 Resource Reduction for Bosonic Systems

Bosonic systems underpin many areas of physics, from superconductivity to cosmology, yet simulating their behaviour presents a significant computational challenge. Mathias Mikkelsen and Hubert Okadome Valencia, both from QunaSys Inc., alongside their colleagues, now demonstrate a detailed comparison of different approaches to digitally simulating these systems using quantum computers. Their work investigates how efficiently various computational ‘mappings’, methods for representing bosonic particles as quantum bits, perform when calculating key properties of these systems, such as the reduced density matrix and the behaviour of standard bosonic Hamiltonians. The team reveals that first quantized mappings, particularly a unary approach, offer substantial advantages in gate efficiency, and importantly, can achieve comparable or even superior performance to second quantized methods in terms of both qubit and gate requirements for realistic scenarios, paving the way for more practical quantum simulations of complex bosonic systems.

Scientists investigated how these mappings perform when calculating the reduced density matrix, a key component in describing the behaviour of many-body systems, and for simulating standard bosonic Hamiltonians. The team demonstrates that first quantized mappings require fewer computational resources to represent certain properties of the reduced density matrix compared to second quantized approaches, achieving a reduction in resources by a factor of two. While the number of computational steps increases more rapidly with system size for one particular first quantized mapping, a thorough numerical analysis reveals it still outperforms several second quantized methods for realistic scenarios. Importantly, the binary first quantized mapping achieves comparable efficiency to the most efficient unary first quantized mapping when simulating the Bose-Hubbard model and harmonic traps, suggesting it offers a balance between qubit and gate efficiency for practical applications. The authors acknowledge that the binary first quantized mapping provides upper bounds on resource requirements, and the actual values may be lower, while the unary mappings offer precise calculations.

Researchers also explored the impact of truncating the local Hilbert space within the second quantized mappings to reduce computational cost. They found that significant truncation is required to achieve comparable resource usage, potentially introducing inaccuracies, particularly for systems lacking strong on-site repulsion. Future work could focus on refining these mappings and exploring their application to more complex quantum systems, potentially advancing the development of quantum simulations and algorithms.

First and Second Quantization for Molecules

This research investigates methods for representing molecular systems on quantum computers, comparing first and second quantization approaches. Classical computers struggle with the complexity of many-body quantum systems like molecules, motivating the search for efficient quantum simulation techniques. First quantization directly represents particles, such as electrons, using qubits to encode their positions and spins, while second quantization focuses on operators that describe the addition or removal of particles from specific orbitals. This second approach is often more natural for describing fermionic systems and can lead to more efficient quantum circuits. The primary goal of this work is to determine which approach leads to more efficient quantum circuits, minimizing the number of gates and execution time for simulating molecular properties.

Scientists explored the use of reduced density matrices, mathematical objects that capture essential quantum correlations, and how to represent them using both first and second quantization. They employed the Bogoliubov Hamiltonian Mapping, a technique within second quantization that transforms the original Hamiltonian into a form suitable for quantum computation. They also utilized operator decomposition techniques to break down complex operators into simpler gates and leveraged the Qiskit software framework to optimize circuits. The number of CNOT gates serves as a key metric for circuit complexity, with fewer gates indicating a shorter and more feasible quantum circuit.

The research consistently demonstrates that second quantization, particularly using the Bogoliubov Hamiltonian Mapping, leads to more efficient quantum circuits than first quantization, especially for larger systems. This efficiency stems from the ability to represent particle correlations more compactly. The Bogoliubov Hamiltonian Mapping proves particularly effective in reducing the number of CNOT gates required for simulating molecular properties. While the one-particle reduced density matrix is often easier to implement, it may not fully capture all important quantum effects. The two-particle reduced density matrix provides a more complete description but requires more complex circuits. The Qiskit optimizer can further reduce the number of CNOT gates, although the extent of the reduction varies depending on the method and system size. For certain cases, the circuits are already quite optimized, and Qiskit optimization provides minimal further improvement.

These findings have significant implications for the field of quantum chemistry, providing guidance on how to design more efficient quantum algorithms for simulating molecular properties. The development of more efficient algorithms is crucial for making quantum computing a practical tool for solving real-world problems in chemistry and materials science, particularly in the near term when quantum computers have limited resources. This research highlights the importance of choosing the right representation and optimization techniques to minimize the complexity of quantum circuits and helps researchers estimate the resources required to simulate specific molecular systems on a quantum computer.