Simulating Hamiltonian dynamics in a programmable photonic quantum processor using linear combinations of unitary operations

Technical Summary: Simulating Hamiltonian Dynamics in a Programmable Photonic Quantum Processor Using Linear Combinations of Unitary Operations

Problem Statement
Efficiently simulating the dynamic evolution of physical and molecular systems is a fundamental application of quantum computing. While the Lie-Trotter-Suzuki approximation (Trotterization) serves as a basic algorithm for quantum dynamic simulation, its accuracy often falls short of theoretical predictions, requiring high-order formulas that increase circuit depth and operational costs. To address this, multi-product algorithms—linear combinations of multiple Trotterizations—have been proposed to achieve high-order precision with lower circuit depth. However, implementing these algorithms via Linear Combinations of Unitaries (LCU) faces significant experimental challenges. Specifically, standard LCU implementations suffer from low success probabilities due to the entanglement between ancillary and data registers, which generates unwanted linear combination terms. This trade-off between achieving high simulation precision and maintaining a viable success probability has limited the practical application of multi-product algorithms.

Methodology
The authors propose and experimentally demonstrate a modified multi-product algorithm assisted by Oblivious Amplitude Amplification (OAA).

1. Algorithmic Modification:

   • The core approach involves linearly combining multiple low-order Trotterizations ($S_1$) with specific coefficients ($c_q$) to cancel low-order error terms, thereby improving the approximation to higher orders (e.g., from $O(t^3)$ to $O(t^5)$ or higher).
   • To overcome the low success probability inherent in standard LCU circuits (where the probability drops as $1/(\sum |c_q|)^2$), the authors integrate OAA. This technique amplifies the success probability to near-unity, similar to the mechanism in Grover's algorithm, without requiring the algorithm to be unitary in the traditional sense.
   • Theoretical analysis shows that by choosing iteration numbers $L(q)$ such that $\sum |c_q| \approx 1.969$ (yielding a base success probability of $\approx 25\%$), a single step of OAA ($N=1$) can amplify the success probability to nearly 100% while maintaining the error scaling of the multi-product method.

2. Experimental Platform:

   • The algorithm was implemented on a programmable four-qubit silicon-photonic quantum simulator.
   • The chip integrates 451 optical components fabricated on a Silicon-on-Insulator platform using CMOS processes.
   • The system utilizes a mapping where quantum dit (qudit) states are translated into multiple qubit states, allowing for high-fidelity entangling operations.
   • The hardware executes a sequence of programmable controlled-controlled-unitary (CCU) gates. The LCU circuit is realized by preparing an ancillary register in $|00\rangle$, applying controlled-unitary operations corresponding to the Trotter steps, and performing projective measurements on the ancillary register.
   • The OAA circuit is implemented by repeating the sequence $U_{OAA} = (-WRW^\dagger R)^N$, where $W$ is the LCU circuit and $R$ is an amplitude flip operator.

Key Contributions

• Algorithmic Innovation: The proposal of a modified multi-product algorithm that combines multi-product Trotterization with OAA to simultaneously achieve high simulation precision and nearly deterministic success probability.
• Hardware Milestone: The demonstration of the first multi-qubit controlled-unitary gate in integrated photonics.
• First-of-its-Kind Simulation: The first implementation of Hamiltonian dynamics simulation specifically within LCU circuits on a programmable photonic chip.
• Experimental Validation: A complete experimental benchmarking of the algorithm against standard Trotterization and the original multi-product algorithm.

Results
The authors simulated the dynamics of a coupled electron and nuclear spin system governed by a Rabi-type Hamiltonian.

• Precision: Theoretical and experimental results showed that the modified multi-product algorithm achieved significantly higher precision than both conventional Trotterization and the original multi-product algorithm for evolution times $t \leq 30$. The simulation error was defined as the discrepancy between the exact state and the output state.
• Success Probability: Without OAA, the modified multi-product algorithm exhibited a success probability of approximately 26.33% for $t \leq 30$. With the implementation of OAA ($N=1$), the success probability was amplified to 99.79%, approaching a deterministic outcome.
• Fidelity: The experimental data showed strong agreement with theoretical predictions, yielding mean classical statistical fidelities of $F_c = 0.996 \pm 0.005$ (with OAA) and $F_c = 0.996 \pm 0.004$ (without OAA).
• Limitations: For evolution times $t > 30$, deviations from the exact solution were observed due to high-order errors and non-unitary effects, and the high success probability could not be fully retained.

Significance and Claims
The paper claims to certify the multi-product algorithm in a small-scale quantum simulator based on linear combinations of operations. The work demonstrates that by combining multi-product algorithms with OAA, it is possible to relax the success probability constraints that typically hinder LCU-based simulations. The authors state that this approach promises practical implementations for quantum dynamics simulations. Furthermore, they note that since the linearly combined operations in LCU do not need to be strictly unitary, the method holds potential for simulating the dynamics of non-Hermitian systems. The successful integration of these algorithms on a reprogrammable silicon-photonic chip suggests a pathway toward scaling up quantum hardware to efficiently simulate complex physical and molecular systems.