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Quantum Optimization via Gradient-Based Hamiltonian Descent
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Proposes gradient-based Quantum Hamiltonian Descent (QHD), adding gradient terms to the QHD Hamiltonian, with Lyapunov convergence proofs, a gate-complexity analysis, and 2D numerical comparisons against SGDM, NAG, and QHD.
Problem
Quantum Hamiltonian Descent (QHD) uses only function values, which limits its convergence speed and global-search reliability on non-convex continuous optimization problems.
Approach
Extends the QHD Hamiltonian, derived from the Bregman Lagrangian, with gradient-dependent terms (a gradient-shifted momentum and a gradient-norm term) obtained via canonical quantization, and proves continuous-time convergence of the resulting Schrödinger dynamics in function values and gradient norms.
Key results
- Gradient-based QHD is reported to reach a final objective on the Cube-Wave function nearly an order of magnitude lower than QHD and two orders lower than SGDM and NAG.
- Theory gives E[f(X_t)] <= O(t^-2) for convex f with beta=0, and E[||grad f(X_t)||^2] <= O(t^-2) under an extra gradient-norm condition.
- Gate/query complexity analysis gives O(K) zeroth-order queries and Otilde(alpha d h K L) first-order queries, linear in dimension d.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| incorporate gradient information into the Hamiltonian | QHD · convex objective | function value | Helped |
| incorporate gradient information into the Hamiltonian | QHD · convex objective | gradient norm | Helped |
| use standard QHD without gradient information instead of gradient-based QHD | QHD · convex objective | function value | Hurt |
| incorporate gradient information into the Hamiltonian | QHD · Styblinski-Tang | success probability | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Michalewicz | function value | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Cube-Wave | function value | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Rastrigin | function value | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Michalewicz | success probability | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Cube-Wave | success probability | Helped |
| incorporate gradient information into the Hamiltonian | QHD · Rastrigin | success probability | Helped |
Surprising findings
- The Hamiltonian-based dynamics are not claimed to be equivalent to the high-resolution ODE, and the authors use a step size much larger than the commutator-based theory suggests, observing convergence anyway.
Limitations
- Numerical experiments are limited to 2D problems due to exponential overhead of quantum simulation.
- Convergence of the discrete-time algorithm is not established; the step size is treated as an independent parameter.
- Convergence theorems require convexity or an extra gradient-norm condition on f, while the non-convex experiments lack theory.
Experimental setup
Datasets: Styblinski-Tang function Michalewicz function Cube-Wave function Rastrigin function convex test function f(x,y)=(x+y)^4/256+(x-y)^4/128
Benchmarks: Styblinski-Tang Michalewicz Cube-Wave Rastrigin convex quartic test function
Metrics: expected function value expected squared gradient norm success probability
Baselines: SGDM NAG QHD
Builds on: Quantum Hamiltonian Descent Bregman Lagrangian high-resolution ODE Nesterov accelerated gradient
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