Long-time simulations for fixed input states on quantum hardware

Quantum computers offer the potential for exponentially more efficient simulation of quantum mechanical systems compared to classical computers. However, current noisy intermediate-scale quantum (NISQ) devices are limited by short coherence times and small qubit counts, hindering the applicability of algorithms designed for fault-tolerant quantum computers such as Trotterization, qubitization, and Taylor series methods. These methods require circuit depths exceeding the capabilities of current hardware. Variational quantum algorithms, which optimize a cost function measured on a quantum computer, offer a promising alternative for NISQ simulations. Early approaches used iterative methods where the state was learned step-by-step. More recent advancements include generalizations of the variational quantum eigensolver for low-lying energy subspaces and quantum-assisted methods that perform all measurements at the algorithm's start. This work refines the Variational Fast Forwarding (VFF) algorithm, enabling long-time simulations using fixed-depth circuits. The original VFF requires a full diagonalization of the short-time evolution operator, which is computationally expensive. The proposed fsVFF algorithm focuses on diagonalizing the energy subspace spanned by the initial state, significantly reducing resource needs, especially when simulating a specific fixed initial state. This approach reduces the qubit count and simplifies the ansatz, leading to improved feasibility on NISQ devices. The authors leverage Quantum No Free Lunch theorems to analyze the required training data and prove the noise resilience of the fsVFF cost function.

The paper reviews existing variational quantum algorithms for quantum simulation, highlighting limitations of methods like Trotterization and qubitization for NISQ devices due to their high circuit depth requirements. It discusses previous variational approaches, such as iterative methods based on action principles and generalizations of the variational quantum eigensolver. The authors also mention recent quantum-assisted methods that avoid classical-quantum feedback loops. The introduction of the original Variational Fast Forwarding (VFF) algorithm is reviewed, along with its limitations regarding the full diagonalization requirement over the entire Hilbert space, which necessitates high resource consumption. This sets the stage for the introduction of fsVFF as a more resource-efficient alternative.

The fsVFF algorithm is presented as a refinement of the VFF algorithm, tailored for fast-forwarding a fixed initial state. The core difference lies in diagonalizing the short-time evolution operator only within the subspace spanned by the initial state and its evolved states. This significantly reduces the qubit requirement (from 2n to n qubits) and allows for simpler ansätze. The algorithm comprises several steps: 1. **Trotter Expansion:** Approximating the short-time evolution operator using a Trotter expansion (e.g., first-order or higher-order Trotter-Suzuki decomposition). 2. **Determining n_eig:** Calculating the number of energy eigenstates with non-zero overlap with the initial state (n_eig). This is done by constructing and evaluating the determinant of the Gramian matrix of state overlaps, obtained through the Hadamard test. An alternative approach involves iterative training with an increasing number of states and observing the convergence of an observable of interest. 3. **Cost Function:** Defining a cost function, C_fsVFF, to measure the overlap between the time-evolved state under the Trotterized unitary and the time-evolved state under the variational ansatz. Minimizing this cost leads to the optimal diagonalization. An alternative randomized training method uses randomly selected states from a continuum, avoiding the explicit calculation of n_eig. 4. **Variational Diagonalization:** Variationally searching for a diagonal compilation of the short-time evolution operator U, which accurately represents its action on the initial state and its evolved states using a classical optimizer (e.g., gradient descent) and measuring gradients using parameter shift rules. The diagonalization ansatz has the form V(α, Δt) = W(θ)D(γ, Δt)W(θ)†, where W(θ) represents the eigenvectors and D(γ, Δt) represents the eigenvalues. 5. **Fast Forwarding:** Using the optimized diagonalized unitary to simulate the system's evolution for a longer time T = NΔt by simply scaling the parameters of the diagonal unitary. The authors provide detailed gradient formulae for the cost function optimization. The algorithm also includes a method for determining the energy eigenvectors and eigenvalues from the obtained diagonalization, either through a simple sampling method or by using fsVFF to reduce the depth of quantum phase estimation (QPE) or quantum eigenvalue estimation (QEE).

The paper demonstrates the effectiveness of fsVFF through several experiments and simulations: 1. **Gramian Determinant Calculation:** The authors experimentally verify their method for determining n_eig on Honeywell's quantum computer, showing that the determinant of the Gramian matrix accurately indicates the dimension of the subspace. 2. **Hardware Implementation (2-qubit XY chain):** Long-time simulations of the 2-qubit XY spin chain were performed on IBM and Rigetti's quantum computers. The fsVFF algorithm significantly outperformed the iterated Trotter method, maintaining a fidelity above 0.9 for over 600 time steps (compared to 4 time steps for the Trotter method), demonstrating a fast-forwarding ratio of approximately 156-159. The noise resilience of the cost function is evident in the difference between the noisy cost obtained from the quantum computers and the noise-free cost calculated classically. 3. **Numerical Simulations (4-qubit XY model and 8-qubit Fermi-Hubbard model):** Numerical simulations of a 4-qubit XY model and an 8-qubit Fermi-Hubbard model were conducted using noise models based on IBM and trapped-ion architectures. These simulations showed fast-forwarding ratios of 87.5 and 174, respectively, further highlighting the algorithm's potential for larger systems. An adaptive ansatz generation technique was used for these simulations. 4. **Randomized Training:** The authors demonstrate the success of the randomized training approach, achieving low cost values for 5 and 6 qubit XY Hamiltonians with high n_eig, using only a small number of training states per cost function evaluation. 5. **Ansatz Comparison (VFF vs. fsVFF):** A comparison between VFF and fsVFF for simulating the same Hamiltonian shows that fsVFF, with a much shallower ansatz, achieves a significantly lower cost and successful fast forwarding, while VFF fails to find a good diagonalization even with a deeper ansatz. 6. **Observable Convergence:** The authors demonstrate that using fewer than the theoretically predicted number of training states can be sufficient for accurate simulations, as long as the amplitudes of the neglected eigenstates are small. 7. **Energy Estimation:** The effectiveness of fsVFF in reducing the circuit depth of QPE and QEE algorithms for energy estimation is shown. fsVFF enhanced QPE outperforms standard QPE on real and simulated noisy quantum hardware, showing a reduced variation distance from the target probability distribution. Similarly, fsVFF enhanced QEE provides accurate eigenvalue estimations.

The results demonstrate the substantial improvement in the simulation of quantum dynamics on NISQ devices that is afforded by the fsVFF algorithm. The algorithm's focus on a fixed initial state enables the use of significantly simpler ansätze and reduced resource requirements compared to the original VFF algorithm. The achieved fast-forwarding ratios across various systems and hardware platforms strongly support the algorithm's scalability and potential for application to larger and more complex quantum simulations as hardware improves. The noise resilience of the cost function is a crucial factor in the success of the algorithm on NISQ devices. The findings highlight the trade-off between the universality and resource requirements of quantum algorithms, suggesting that sacrificing some generality can significantly improve the practicality of NISQ simulations. Future directions might include exploring the use of error mitigation techniques to further improve performance with larger n_eig and developing problem-inspired ansätze tailored to specific systems.

The fsVFF algorithm represents a significant advancement in the field of NISQ quantum simulation. By focusing on simulating a fixed initial state, fsVFF achieves substantial reductions in resource requirements, allowing for significantly longer and higher-fidelity simulations than previously possible using standard methods. The experimental and numerical results demonstrate the algorithm's effectiveness and scalability across different platforms and quantum systems. Future research could focus on extending fsVFF to address larger systems, incorporating error mitigation techniques, and developing more efficient ansätze based on system symmetries. The exploration of fixed-observable variations of the algorithm holds promising avenues for further improving the feasibility of high-fidelity long-time simulations.

While fsVFF significantly enhances long-time simulations, limitations exist. The algorithm's performance is dependent on the dimension of the subspace spanned by the initial state and its time evolution (n_eig). For systems with large n_eig, the computational resources required for training and implementation might still be significant. The accuracy of the simulations is also limited by the Trotter approximation used for the short-time evolution. While higher-order Trotter approximations can reduce this error, they come at the cost of increased circuit complexity. Furthermore, the choice of ansatz significantly impacts the optimization process and the overall performance. Finding suitable ansätze can be a challenging task, particularly for complex quantum systems.