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Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

Tayyab Ali

TL;DR

The paper addresses solving nonlinear differential equations on quantum hardware by using Carleman embedding to map to an infinite-dimensional linear system, truncated at order $N$. It advances a data-access framework that uses Sigma basis (LCNU) and unitary completion to load and implement the truncated system on quantum circuits, achieving near-linear growth in decomposition terms relative to nonzero entries. It provides circuit constructions for the tensor-product components and analyzes barren plateaus in variational solvers, proposing local-cost-function strategies to preserve trainability. Overall, this framework advances practical quantum simulations of nonlinear dynamics by tackling data loading, decomposition, and circuit realization, supported by a public codebase.

Abstract

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order $N$, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component $\mathcal{H}_{j}$ of the decomposition.

Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

TL;DR

The paper addresses solving nonlinear differential equations on quantum hardware by using Carleman embedding to map to an infinite-dimensional linear system, truncated at order . It advances a data-access framework that uses Sigma basis (LCNU) and unitary completion to load and implement the truncated system on quantum circuits, achieving near-linear growth in decomposition terms relative to nonzero entries. It provides circuit constructions for the tensor-product components and analyzes barren plateaus in variational solvers, proposing local-cost-function strategies to preserve trainability. Overall, this framework advances practical quantum simulations of nonlinear dynamics by tackling data loading, decomposition, and circuit realization, supported by a public codebase.

Abstract

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order , on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component of the decomposition.
Paper Structure (9 sections, 3 theorems, 68 equations, 10 figures)

This paper contains 9 sections, 3 theorems, 68 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Carleman Linearization
  3. Nonlinear Vector-Valued Ordinary Differential Equations
  4. Towards Efficient Data Access Model
  5. Linear Combination of Unitaries (LCU)
  6. Linear Combination of Non-Unitaries (LCNU)
  7. Unitary Completion and Circuit Construction for $\mathcal{H}_{j}$
  8. Curse of Dimensionality
  9. Conclusion

Key Result

Theorem 3.3

The unitary completion for $\mathcal{H}_{j} = \otimes_{p}\sigma_{p}$, $\sigma_{p} \in \mathbb{S}$ is given by $\Bar{\mathcal{H}_{j}} = \otimes_{p}\Bar{\sigma_{p}}$, such that $\Bar{\sigma_{p}} = \sigma_{x}$ for $\{\sigma_{+}, \sigma_{-}\}$, and $\Bar{\sigma_{p}} = I$ for the rest of the $\mathbb{S}$

Figures (10)

  • Figure 1: Top: Systematic convergence of Carleman approximation for quadratic model of Bernoulli's equation with $P(x) = 2x$, $Q(x)=2x^{3}$, and $y(0)=1$. The solution is plotted for $x \geq 0$. Bottom Left: Progressive error reduction at increasing values of $N$. Bottom Right: At increasing order of $N$, matrix dimension scales quadratically, however, matrix remains sparse containing only $2N-1$ number of non-zero terms.(source code: https://github.com/ali-tayyab/Carleman-Embedding)
  • Figure 2: Circuit for controlled-$\mathbb{P}_{i}$ to implement the Hamiltonian described in Eqn.().
  • Figure 3: Difference between the number of terms in Sigma and Pauli basis decomposition for the quadratic model of Bernoulli's equation at increasing truncation order.(source code: https://github.com/ali-tayyab/Carleman-Embedding)
  • Figure 4: Circuit construction for $U_{j}$ corresponding to $\mathcal{H}_{j} = \sigma_{-} \otimes \sigma_{+}\sigma_{-} \otimes I$
  • Figure 5: Two $C^{n} X$ gates are combined into one $C^{n-1} X$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 3.5
  • Theorem 3.6
  • proof
  • Example 3.7