STEP-BY-STEP GUIDE TO NONLINEAR OPTIMIZATION WITH QUANTUM COMPUTERS

JIAQI LENG (UC BERKELEY), YUXIANG PENG (PURDUE)

LEI FAN (UH), XIAODI WU (UMD)

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SESSION 1 | AGENDA

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1. Quantum Hamiltonian Descent (QHD)
• Derivation & Theory
• Implementation on Quantum Hardware
2. Software Library: QHDOPT
3. Step-by-Step Guide to QHDOPT
• Example code
• Applications in energy systems

A SURVEY ON AUDIENCE BACKGROUND IN QC (TBA)

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INTRODUCTION TO �QUANTUM HAMILTONIAN DESCENT

Presenter: Jiaqi Leng (University of California, Berkeley)

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Nonlinear Optimization

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Nesterov’s Accelerated GD (NAG)

NAG & Hamiltonian Dynamics

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Nesterov’s Accelerated GD (NAG)

NAG & Hamiltonian Dynamics

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Hamiltonian Mechanics: From Classical to Quantum

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Physical Interpretations

A Quantum Heavy Ball Method

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QHD FOR CONVEX PROBLEMS

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LARGE QUANTUM SPEEDUPS (PART 1)

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LARGE QUANTUM SPEEDUPS (PART 2)

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QHD ON QUANTUM HARDWARE

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QHD ON QUANTUM HARDWARE

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• T-gate count: >= 1B (before QEC)
• Observation: gate-based implementation requires large, fault-tolerant quantum computers that are not ready yet…
• Can we implement QHD without using gates?

T-Gate Count Estimation of Executing QHD on Universal Quantum Computers (Gate Models)

QHD ON QUANTUM HARDWARE

Analog Quantum Computers

• Analog quantum computing: using one Hamiltonian to simulate another one.
• Pros: Easier to fabricate and scale up.
• Cons: specific-purpose computers, error-prone (no error correction).
• Available devices: D-Wave (5k+qbts), QuEra (200+qbts), Pasqal (200+qbts), etc.

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QHD ON ANALOG QUANTUM COMPUTERS

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HAMILTONIAN EMBEDDING OF QHD

QHDOPT: �A SOFTWARE FOR QUANTUM OPTIMIZATION

Presenter: Yuxiang Peng (Purdue University)

STATE OF QUANTUM4OPTIMIZATION

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Combinatorial

problems

Continuous

problems

Long-term (digital)

Near-term (analog)

Grover’s search [1996]

Quantum annealing [1998]

Quantum approximate optimization algorithm (QAOA) [2014]

Quantum interior point methods [2023]

Quantum SDP solvers [2019]

Quantum central path method [2023]

Quantum Hamiltonian descent [2023]:

EMPIRICAL QHD FOR YOU!

• What are required for you to leverage QHD in real use cases?

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None of these is easy!

(Even for quantum experts!)

4. Familiar with quantum devices and their analysis

1. Understand quantum mechanics

2. Know the design of QHD for specific instances

3. Know quantum programming

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You

Problem instance

QHDOPT

Python interface

Various quantum devices

Minimum

QHD is for Everyone!

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univariate

bivariate

What Can QHDOPT Solve?

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SymPy Interface

QP Interface

QHDOPT: User Interface

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Through a new paradigm of quantum computing

Implementation of QHDOPT

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QHD evolution

Kinetic

Potential

obj. func.

BUT

Most quantum devices are spatially discrete

Spatial discretization of QHD evolution

Implementation of QHDOPT

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Most quantum devices have local interactions and controls

BUT, again

Example:

unary encoding

Implementation of QHDOPT

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Quantum platforms have

different programming interfaces

SimuQ [PYLW, 24]

On D-Wave, IonQ, and QuTiP (classical simulator)

Programmed in SimuQ’s Hamiltonian Modeling Language

Compiled by SimuQ’s compiler

Provider’s compilation to machine instructions

SimuQ Compilation

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NLOPT problem

Continuous-space

Hamiltonian Evo.

Discrete-space Hamiltonian Evo.

2-local Hamiltonian Evo.

Quantum device

QHD

Spatial discretization

Hamiltonian embedding

Analog compilation

NLOPT

refined solution

Discrete-space measurement res.

Quantum device measurement res.

Ham. Emb.

Decoding

Classical

Refinement

QHDOPT Post-Processing (Refinement)

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Time-to-solution (s)

Beats Ipopt on low-dim complicated bivariate problems

How Does QHDOPT Perform?

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Beats Ipopt, BARON on 50-dim QP-like problems

Wall-clock time

QHDOPT on D-Wave devices

Ipopt, BARON on a laptop

Timeout limit: 20min

50-dim non-convex

quadratic programming

Beats Ipopt,

matches BARON

on 50-dim QP problems

Current capability

How Does QHDOPT Scale?

QHDOPT for Nonconvex QP

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• 75-dimensional nonconvex QP problems: 600 digital qubits, ~5000 physical qubits on D-Wave (due to limited qubit connectivity)
• DW-QHD is better than the rest, including DW-QAA, and classical GD, interior points, and some local search heuristic.

Q & A

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USE QHDOPT: A STEP-BY-STEP GUIDE�

Presenter: Lei Fan (University of Houston)

QHD EXAMPLE

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Target Problem

Preprocessing:

Raw Output

QHD EXAMPLE

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Step 1: Modeling with SymPy or QP format

Step 2: Select the solver of QHD

Step 3: Obtain and post-process output

PENALTY METHOD + IPOPT

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Add slack variables

• Reformulate the problem into a box-constrained problem.
• Design the penalty factors.
• Use QHD to solve the box-constrained problem
• Use IPOPT to refine the solution

APPLICATION IN ENERGY SYSTEMS

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Electricity to Hydrogen Production System

Mingze Li, Siyuan Wang, Lei Fan, Zhu Han, “Integrated Quantum Hamiltonian Descent with Interior Point Method for Optimal Schedule of Hybrid Electricity-to-Hydrogen System” IEEE KEPC 2025.

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APPLICATION IN ENERGY SYSTEMS

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Hydrogen Production Efficiency Curve

APPLICATION IN ENERGY SYSTEMS

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Simulation Results

• Pure-IPOPT: solve the problem with a single random initial point.
• IPOPT 1k: solve the problem with 1k random initial point.
• QHD: QHD solver
• QHD-IPOPT: QHD generate initial points for IPOPT

• Good initial points produced by QHDOPT, when the instance size is manageable by quantum machine.
• As the instance size increases, the current quantum machine cannot provide a good solution due to the size of the quantum machine.

WHAT KIND OF APPLICATIONS CAN YOU ENVISION?

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REFERENCES

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1. Jiaqi Leng, Ethan Hickman, Joseph Li, and Xiaodi Wu. “Quantum Hamiltonian Descent”, arXiv:2303.01471 (2023).
2. Jiaqi Leng, Yufan Zheng, Zhiyuan Jia, Lei Fan, Chaoyue Zhao, Yuxiang Peng, and Xiaodi Wu. “Quantum Hamiltonian Descent for Non-smooth Optimization”, under review.
3. Samuel Kushnir, Jiaqi Leng, Yuxiang Peng , and Lei Fan, Xiaodi Wu. “QHDOPT: A software for nonlinear optimization with quantum Hamiltonian decent”,  INFORMS Journal on Computing, 2024
4. Jiaqi Leng, Yufan Zheng, and Xiaodi Wu. “A Quantum-Classical Performance Separation in Nonconvex Optimization”, arXiv:2311.00811 (2023).
5. Mingze Li, Siyuan Wang, Lei Fan, Zhu Han, “Integrated Quantum Hamiltonian Descent with Interior Point Method for Optimal Schedule of Hybrid Electricity-to-Hydrogen System”, IEEE KEPC 2025.
6. Li, Mingze, Lei Fan, and Zhu Han. "Quantum Hamiltonian Descent based Augmented Lagrangian Method for Constrained Nonconvex Nonlinear Optimization." IEEE QCE 2025

THANKS FOR YOUR ATTENTION!

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