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Variational Quantum Framework for Partial Differential Equation Constrained Optimization

Amit Surana, Abeynaya Gnanasekaran

TL;DR

It is demonstrated that by using an alternative tensor product decomposition which better exploits the sparsity and structure of linear systems arising from PDE discretizations, one can substantially overcome the key computational bottleneck in VQLS arising from commonly employed Pauli basis for the linear combination of unitary (LCU) decomposition step within VQLS.

Abstract

We present a novel variational quantum framework for linear partial differential equation (PDE) constrained optimization problems. Such problems arise in many scientific and engineering domains. For instance, in aerodynamics, the PDE constraints are the conservation laws such as momentum, mass and energy balance, the design variables are vehicle shape parameters and material properties, and the objective could be to minimize the effect of transient heat loads on the vehicle or to maximize the lift-to-drag ratio. The proposed framework utilizes the variational quantum linear system (VQLS) algorithm and a black box optimizer as its two main building blocks. VQLS is used to solve the linear system, arising from the discretization of the PDE constraints for given design parameters, and evaluate the design cost/objective function. The black box optimizer is used to select next set of parameter values based on this evaluated cost, leading to nested bi-level optimization structure within a hybrid classical-quantum setting. We present detailed computational error and complexity analysis to highlight the potential advantages of our proposed framework over classical techniques. We implement our framework using the PennyLane library, apply it to a heat transfer optimization problem, and present simulation results using Bayesian optimization as the black box optimizer.

Variational Quantum Framework for Partial Differential Equation Constrained Optimization

TL;DR

It is demonstrated that by using an alternative tensor product decomposition which better exploits the sparsity and structure of linear systems arising from PDE discretizations, one can substantially overcome the key computational bottleneck in VQLS arising from commonly employed Pauli basis for the linear combination of unitary (LCU) decomposition step within VQLS.

Abstract

We present a novel variational quantum framework for linear partial differential equation (PDE) constrained optimization problems. Such problems arise in many scientific and engineering domains. For instance, in aerodynamics, the PDE constraints are the conservation laws such as momentum, mass and energy balance, the design variables are vehicle shape parameters and material properties, and the objective could be to minimize the effect of transient heat loads on the vehicle or to maximize the lift-to-drag ratio. The proposed framework utilizes the variational quantum linear system (VQLS) algorithm and a black box optimizer as its two main building blocks. VQLS is used to solve the linear system, arising from the discretization of the PDE constraints for given design parameters, and evaluate the design cost/objective function. The black box optimizer is used to select next set of parameter values based on this evaluated cost, leading to nested bi-level optimization structure within a hybrid classical-quantum setting. We present detailed computational error and complexity analysis to highlight the potential advantages of our proposed framework over classical techniques. We implement our framework using the PennyLane library, apply it to a heat transfer optimization problem, and present simulation results using Bayesian optimization as the black box optimizer.
Paper Structure (45 sections, 14 theorems, 147 equations, 7 figures, 3 algorithms)

This paper contains 45 sections, 14 theorems, 147 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. PDE Constrained Optimization
  4. Heat Transfer Optimization Problem Formulation
  5. Explicit Time Discretization
  6. Implicit Time Discretization
  7. Optimization Problem
  8. Methods
  9. BVQPCO Framework
  10. Variational Quantum Linear Solver
  11. LCU Decomposition
  12. State Preparation
  13. Ansatz Selection
  14. Cost Function
  15. Classical Optimizers
  16. ...and 30 more sections

Key Result

Lemma 1

For the linear system (eq:linsys), following bounds hold for the different VQLS cost functions where, $\kappa$ and $N=2^n$ are the condition number and size of $\mathbf{A}$ in (eq:linsys) respectively, and $\epsilon$ is the error tolerance with $\rho$ being the trace norm, $|\psi\rangle$ being the exact solution of eq:linsysq, and $|\psi(\mathbf{\theta}_*)\rangle$ being the approximate VQLS solu

Figures (7)

  • Figure 1: Design of coating layers for a flat plate subject to a time-dependent flux.
  • Figure 2: Top: Schematic showing transformation of a PDE constrained optimization problem into a variational quantum form. Bottom: Flow diagram of the BVQPCO framework. VQLS uses an inner level optimization to solve the linear system constraints, arising from the discretization of the underlying PDEs, for given design parameters, and evaluate the quantities related to design cost/objective function. A black box optimizer, e.g., BO is used for the outer level optimization to select next set of parameters values based on the evaluated design cost.
  • Figure 3: Quantum circuit to prepare state $\Tilde{\mathbf{b}}$ in \ref{['eq:linsysexp']} with $T_0(x)= 2(\frac{\Delta t}{k\Delta x}q^1 ), q^i = q^1 = 50 \forall i, k=1.0, \Delta t=0.25, \Delta x=0.2$ (a) 3 qubits, $n_x = 2, n_t=4$, and (b) 4 qubits $n_x = 4, n_t=4$. (c) Mottonen state preparation using built-in PennyLane implementation for 4 qubits, $n_x = 4, n_t=4$ case.
  • Figure 4: Real-valued ansatz (modified circuit 9) on 4 qubits comprising of Hadamard gate (H), SWAP gates and rotation about y-axis gates (RY).
  • Figure 5: Schematic showing different stages in going from the PDE to the VQLS solution, associated variables and the error in approximations. The case of explicit/forward Euler scheme is shown as an example. Similar flow applies for the case of implicit/backward Euler scheme.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • Remark 2
  • ...and 13 more