[Paper Review] Long-time simulations with high fidelity on quantum hardware

Joe Gibbs, Kaitlin Gili+6 more|arXiv (Cornell University)|Feb 8, 2021

Quantum Computing Algorithms and ArchitectureComputer Science56 references35 citations

TL;DR

The paper introduces fixed state Variational Fast Forwarding (fsVFF), a NISQ-friendly quantum algorithm that achieves long-time, high-fidelity simulations by diagonalizing the evolution operator on the subspace spanned by a fixed initial state, enabling >600 time steps with fidelity above 0.9 on current hardware.

ABSTRACT

Moderate-size quantum computers are now publicly accessible over the cloud, opening the exciting possibility of performing dynamical simulations of quantum systems. However, while rapidly improving, these devices have short coherence times, limiting the depth of algorithms that may be successfully implemented. Here we demonstrate that, despite these limitations, it is possible to implement long-time, high fidelity simulations on current hardware. Specifically, we simulate an XY-model spin chain on the Rigetti and IBM quantum computers, maintaining a fidelity of at least 0.9 for over 600 time steps. This is a factor of 150 longer than is possible using the iterated Trotter method. Our simulations are performed using a new algorithm that we call the fixed state Variational Fast Forwarding (fsVFF) algorithm. This algorithm decreases the circuit depth and width required for a quantum simulation by finding an approximate diagonalization of a short time evolution unitary. Crucially, fsVFF only requires finding a diagonalization on the subspace spanned by the initial state, rather than on the total Hilbert space as with previous methods, substantially reducing the required resources. We further demonstrate the viability of fsVFF through large numerical implementations of the algorithm, as well as an analysis of its noise resilience and the scaling of simulation errors.

Motivation & Objective

• Motivate and enable long-time quantum simulations on noisy intermediate-scale quantum devices.
• Develop a resource-efficient method to fast-forward quantum dynamics for a fixed initial state.
• Reduce circuit depth and width requirements compared to full space diagonalization.
• Provide theoretical guarantees on faithfulness and noise resilience of fsVFF.
• Demonstrate hardware and numerical evidence of extended high-fidelity simulation.
• Analyze error scaling and potential eigenstate/eigenvalue insights from fsVFF outputs.

Proposed method

• Replace full unitary diagonalization with a fixed-state diagonalization of the short-time evolution operator U(Δt).
• Use a cost function C_fsVFF that measures overlap between U^k|ψ0> and the trained diagonalized form for k up to n_eig (training states).
• Represent V(α,Δt)=W(θ)D(γ,Δt)W(θ)† with W and D as trainable circuits capturing eigenvectors and eigenvalues on the initial-subspace.
• Estimate n_eig via Krylov subspace logic using Gramian determinants Det(G(k)) with Hadamard tests to determine the subspace dimension.
• Train parameters θ and γ by classical optimization using gradients measured on quantum hardware.
• Fast-forward by applying W(θ_opt)D(γ_opt,NΔt)W(θ_opt)† to propagate the initial state for N steps.

Experimental results

Research questions

• RQ1 Can a fixed-state diagonalization of the short-time evolution operator faithfully reproduce dynamics for all subsequent times within the initial-state subspace?
• RQ2 What is the resource overhead (qubits, circuit depth, parameters) when restricting diagonalization to the initial-state subspace compared to full-space VFF?
• RQ3 How does fsVFF perform in terms of fidelity and noise resilience on current NISQ hardware?
• RQ4 What are the practical limits (n_eig, Δt, ansatz design) for achieving long-time simulations on superconducting and photonic quantum devices?
• RQ5 Can fsVFF provide useful spectral information (eigenvalues/eigenstates) from subspace diagonalization with limited resources?

Key findings

• fsVFF achieves fidelity greater than 0.9 for over 600 time steps on Rigetti and IBM hardware for a 2-qubit XY spin chain.
• Iterated Trotter fails to maintain fidelity beyond a few steps, while fsVFF maintains high fidelity over hundreds of steps, demonstrating a large fast-forwarding ratio.
• FsVFF requires only half the number of qubits compared to full VFF, enabling larger-scale simulations on current hardware.
• The trained fsVFF cost remains significantly lower on hardware than the noisy cost, showing noise resilience of the cost function.
• Numerical simulations show fsVFF effectiveness in 4-qubit XY and 8-qubit Fermi-Hubbard models, supporting NISQ viability.
• The approach enables potential eigenstate and eigenvalue inference within the subspace via time-series methods or reduced-depth QPE.