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Design nearly optimal quantum algorithm for linear differential equations via Lindbladians

Zhong-Xia Shang, Naixu Guo, Dong An, Qi Zhao

TL;DR

A new quantum algorithm for solving ODEs by harnessing open quantum systems is proposed, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices.

Abstract

Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called non-diagonal density matrix encoding, which leverages the inherent non-unitary dynamics of Lindbladians to encode general linear ODEs into the non-diagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.

Design nearly optimal quantum algorithm for linear differential equations via Lindbladians

TL;DR

A new quantum algorithm for solving ODEs by harnessing open quantum systems is proposed, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices.

Abstract

Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called non-diagonal density matrix encoding, which leverages the inherent non-unitary dynamics of Lindbladians to encode general linear ODEs into the non-diagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.

Paper Structure

This paper contains 15 sections, 19 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Preliminaries
  2. Symbol conventions
  3. Quantum linear algebra
  4. Quantum algorithms for open system simulation
  5. Amplitude amplification and amplitude estimation
  6. Solving linear ODEs via Lindbladians
  7. Non-diagonal density matrix encoding
  8. Encoding ODEs into Lindbladians
  9. Measuring properties
  10. Extracting $|\mu_T\rangle$
  11. Algorithm and complexity
  12. Time-independent case
  13. Time-dependent case
  14. Lower bound discussion
  15. Relations between Lindbladians and non-Hermitian physics

Key Result

Theorem 1

For semi-dissipative and homogeneous linear ODEs, assume we are given access to $\alpha_V$-block encoding $U_{V(t)}$ of $V(t)$ with $\alpha_V\geq \max_t \|V(t)\|$ and quantum state preparation unitary $U_{\mu_0}$: $\ket{0}_n\rightarrow |\mu_0\rangle$, and suppose that the smallest nonzero eigenvalue

Figures (1)

  • Figure 1: High-level illustration of the algorithm. The dynamics of a linear ODE is embedded into the intrinsic non-unitary dynamics of a Lindbladian and the solution to the ODE is encoded into a non-diagonal block of the density matrix evolved according to the Lindbladian.

Theorems & Definitions (30)

  • Definition 1: Non-diagonal density matrix encoding (NDME)
  • Theorem 1: Quantum ODE solver via Lindbladian, direct access model
  • Theorem 2: Quantum ODE solver via Lindbladian, square root access model
  • Definition 2: Block encoding chakraborty2019powergilyen2019quantum
  • Lemma 1: Multiplication of block encodings chakraborty2019powergilyen2019quantum
  • Lemma 2: Polynomial of a block encoding gilyen2019quantum
  • Lemma 3: Polynomial approximations of positive power functions gilyenthesisgilyén2022improvedquantumalgorithmsfidelity
  • Lemma 4: Polynomial approximations of square root function
  • proof
  • Lemma 5: Quantum algorithm for time-independent Lindbladian li2023opensystem
  • ...and 20 more