[Paper Review] Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation

Benjamin Commeau, M. Cerezo+4 more|arXiv (Cornell University)|Sep 5, 2020

Quantum Computing Algorithms and ArchitectureComputer Science42 references50 citations

TL;DR

Introduces Variational Hamiltonian Diagonalization (VHD), a hybrid quantum-classical method to approximately diagonalize a Hamiltonian, enabling fast-forwarded quantum dynamics with fixed-depth circuits and no Trotter error. Demonstrates operational意味 and trainability, with numerical validation on XY models.

ABSTRACT

Dynamical quantum simulation may be one of the first applications to see quantum advantage. However, the circuit depth of standard Trotterization methods can rapidly exceed the coherence time of noisy quantum computers. This has led to recent proposals for variational approaches to dynamical simulation. In this work, we aim to make variational dynamical simulation even more practical and near-term. We propose a new algorithm called Variational Hamiltonian Diagonalization (VHD), which approximately transforms a given Hamiltonian into a diagonal form that can be easily exponentiated. VHD allows for fast forwarding, i.e., simulation beyond the coherence time of the quantum computer with a fixed-depth quantum circuit. It also removes Trotterization error and allows simulation of the entire Hilbert space. We prove an operational meaning for the VHD cost function in terms of the average simulation fidelity. Moreover, we prove that the VHD cost function does not exhibit a shallow-depth barren plateau, i.e., its gradient does not vanish exponentially. Our proof relies on locality of the Hamiltonian, and hence we connect locality to trainability. Our numerical simulations verify that VHD can be used for fast-forwarding dynamics.

Motivation & Objective

• Motivate near-term quantum dynamical simulation by reducing circuit depth and eliminating Trotter error.
• Develop a variational scheme to diagonalize an entire Hamiltonian so that time evolution can be simulated efficiently.
• Provide an operational cost function with a meaningful fidelity bound and prove absence of shallow barren plateaus under locality assumptions.
• Demonstrate through numerical experiments that VHD enables fast-forwarded dynamics beyond coherence times."
• Show how pre-training with Variational Fast Forwarding (VFF) can further reduce resource requirements.

Proposed method

• Propose an ansatz H̃(θ,γ)=W(θ)D(γ)W†(θ) that diagonalizes H approximately.
• Use a hardware-efficient, layered circuit for W(θ) and a local, tractable D(γ) consisting of Z^k terms.
• Define a cost C_VHD(θ,γ)=||H−H̃(θ,γ)||_HS^2/d and a normalized version to guide training.
• Evaluate cost terms c_pqk(θ)=Tr(σ^pq WZ^k W†)/d via Hadamard-test circuits and compute gradients.
• Iteratively optimize (θ,γ) in a hybrid quantum-classical loop until a termination condition based on a desired fidelity is met.
• Optionally pre-train with Variational Fast Forwarding (VFF) to supply a good initialization for θ and γ.

Experimental results

Research questions

• RQ1 Can a variationally diagonalized Hamiltonian provide a fixed-depth circuit for accurate long-time quantum dynamics?
• RQ2 Does the VHD cost function have an operational meaning in terms of fidelity, and can it guarantee a termination criterion?
• RQ3 Under locality assumptions, does the VHD cost landscape avoid barren plateaus and remain trainable as system size grows?
• RQ4 How effective is VFF pre-training in reducing quantum-resource requirements before executing VHD?
• RQ5 What is the practical performance of VHD in fast-forwarding dynamics for representative models such as the XY Heisenberg chain?

Key findings

• VHD yields a fixed-depth quantum circuit to approximate time evolution e^−iHt up to time T via V(T)=W e^{−iDT} W†.
• The VHD cost provides a bound linked to average simulation fidelity, enabling a meaningful termination condition.
• Under locality conditions (local a-local H and local D), the gradient variance does not vanish exponentially with n, avoiding shallow-depth barren plateaus (Theorem 1).
• Numerical experiments on the 1D XY model show that VHD reduces the cost to below 1e−9 (n=3), 1e−8 (n=4), and 1e−5 (n=5) after optimization.
• Pre-training with VFF helps reduce initial cost and accelerates convergence, but full VHD optimization further eliminates Trotter error by exactly diagonalizing H.
• Fast-forwarded simulations using VHD maintain average infidelity below 10^−3 for times up to about 10^3 for n=3–5 qubits, outperforming VFF alone.