Gate-based quantum simulation of Gaussian bosonic circuits on exponentially many modes

Gate-based Quantum Simulation of Gaussian Bosonic Circuits on Exponentially Many Modes

The presented paper introduces a novel framework for the simulation of Gaussian Bosonic (GB) circuits through gate-based quantum computing, specifically targeting the simulation of GB circuits on states over (2ⁿ⁾ modes using an ((n+1))-qubit quantum computer. The authors' methodology involves encoding the initial bosonic state's expectation values over quadrature operators and their covariance matrix into a qubit-state. The quantum circuit then mimics the symplectic propagators induced by the GB gates.

Key Contributions

1. Simulation Framework: The authors propose a new approach to simulate the action of GB circuits on exponentially many bosonic modes using a quantum circuit. By encoding the expectation values of quadrature operators of the initial bosonic state and its covariance matrix into a quantum state, they translate the problem from the infinite-dimensional bosonic Hilbert space to the discrete qubit space.
2. Dictionary for Gate Mapping: The paper introduces a comprehensive dictionary mapping between GB gates and qubit gates. This mapping allows the effective simulation of GB circuits' symplectic propagators on a gate-based quantum computer. Specifically, it shows how particle-preserving GB gates correspond to unitary operations in the qubit picture, while non-particle-preserving gates translate to imaginary-time evolutions.
3. BQP-Complete Problem: A significant theoretical contribution is the demonstration that the quantum evolution of particle-preserving GB circuits, acting on Gaussian states over exponentially many modes, leads to BQP-completeness. This result implies that such GB circuit simulations are as powerful as universal quantum computations.
4. Numerical Simulations: The framework has been validated through numerical simulations of an interferometer acting on approximately 8 billion modes, illustrating the practical applicability and potential of this approach.

Theoretical Underpinnings

The study leverages complexity theory classes, particularly the Bounded-Error Quantum Polynomial (BQP) class. A problem falls into BQP if a quantum computer can solve it in polynomial time with a small probability of error. By presenting a set of tasks within GB circuits that adhere to these constraints, the authors effectively demonstrate the BQP-completeness for specific GB evolutions.

Detailed Methodology

1. Initial State Encoding: The quadrature operators' expectation values of the starting bosonic state are encoded into a qubit-state. Technically, the initial qubit state represents the position and momentum expectation values, and their covariance matrix, of the initial bosonic state.
2. Evolution Through Qubit Circuits: The evolution of the encoded state is simulated using qubit circuits that reflect the symplectic transformations of the corresponding GB circuits. This involves detailed construction and operation of qubit gates that mimic the behavior of bosonic gates:

• Phase Gates: Mapped to (R_y) rotations on the symplectic qubit, conditioned on the state of the register qubits.
• Beamsplitters: Correspond to controlled (R_y) rotations that act trivially on the symplectic qubit but affect the register qubits.
• Squeezing Gates: Require imaginary-time evolution which is implemented via linear combinations of unitaries, with post-selection required for successful gate completion.

Efficient Simulation Conditions: The framework is efficient under specific conditions, such as having a polynomial-time preparation of input qubit-states and requiring the quantum circuit to involve only polynomially many gates.

Implications and Future Directions

The methodology bridges a significant gap between bosonic systems’ continuous variable representations and the discrete qubit computations of quantum computers. The theoretical assertion of BQP-completeness suggests this approach could fundamentally expand our understanding and capabilities in quantum simulations.

Practical Implications:

• Quantum Optics and Photonics: The formalism provides a new lens through which to approach traditional problems in quantum optics, potentially enabling more complex simulations of photonic systems.
• Quantum Simulations: It offers a feasible pathway to simulate large-scale, complex quantum systems using gate-based quantum computers, significantly enhancing the simulation capacity.

Theoretical Implications:

• Complexity Theory: By establishing the equivalence in computational power between certain GB circuits and universal quantum computers, this research adds depth to our understanding of quantum computational complexity.
• Quantum Information Science: The framework underlines potential approaches to tackle non-particle-preserving systems’ simulation, which could extend beyond the Gaussian states domain.

Future Developments: Future research could extend this work to non-Gaussian state simulations and explore efficient implementations of higher-order moment simulations. Another interesting avenue would be integrating this framework with emerging quantum computing architectures to enhance the practical scalability and execution of these simulations.

Conclusion

This paper provides a detailed, methodologically sound approach to gate-based quantum simulation of GB circuits on exponentially many modes. By formalizing a dictionary between GB and qubit gates and demonstrating BQP-completeness, the authors open new avenues for efficient quantum simulations, thus making a significant addition to the landscape of quantum information research.