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Exponential Quantum Speedup for Simulation-Based Optimization Applications

Jonas Stein, Lukas Müller, Leonhard Hölscher, Georgios Chnitidis, Jezer Jojo, Afrah Farea, Mustafa Serdar Çelebi, David Bucher, Jonathan Wulf, David Fischer, Philipp Altmann, Claudia Linnhoff-Popien, Sebastian Feld

TL;DR

This article proves that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component, and presents two practically relevant use cases that fall within this subgroup of QuSO problems.

Abstract

The simulation of many industrially relevant physical processes can be executed up to exponentially faster using quantum algorithms. However, this speedup can only be leveraged if the data input and output of the simulation can be implemented efficiently. While we show that recent advancements for optimal state preparation can effectively solve the problem of data input at a moderate cost of ancillary qubits in many cases, the output problem can provably not be solved efficiently in general. By acknowledging that many simulation problems arise only as a subproblem of a larger optimization problem in many practical applications however, we identify and define a class of practically relevant problems that does not suffer from the output problem: Quantum Simulation-based Optimization (QuSO). QuSO represents optimization problems whose objective function and/or constraints depend on summary statistic information on the result of a simulation, i.e., information that can be efficiently extracted from a quantum state vector. In this article, we focus on the LinQuSO subclass of QuSO, which is characterized by the linearity of the simulation problem, i.e., the simulation problem can be formulated as a system of linear equations. By cleverly combining the quantum singular value transformation (QSVT) with the quantum approximate optimization algorithm (QAOA), we prove that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component. Finally, we present two practically relevant use cases that fall within this subgroup of QuSO problems.

Exponential Quantum Speedup for Simulation-Based Optimization Applications

TL;DR

This article proves that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component, and presents two practically relevant use cases that fall within this subgroup of QuSO problems.

Abstract

The simulation of many industrially relevant physical processes can be executed up to exponentially faster using quantum algorithms. However, this speedup can only be leveraged if the data input and output of the simulation can be implemented efficiently. While we show that recent advancements for optimal state preparation can effectively solve the problem of data input at a moderate cost of ancillary qubits in many cases, the output problem can provably not be solved efficiently in general. By acknowledging that many simulation problems arise only as a subproblem of a larger optimization problem in many practical applications however, we identify and define a class of practically relevant problems that does not suffer from the output problem: Quantum Simulation-based Optimization (QuSO). QuSO represents optimization problems whose objective function and/or constraints depend on summary statistic information on the result of a simulation, i.e., information that can be efficiently extracted from a quantum state vector. In this article, we focus on the LinQuSO subclass of QuSO, which is characterized by the linearity of the simulation problem, i.e., the simulation problem can be formulated as a system of linear equations. By cleverly combining the quantum singular value transformation (QSVT) with the quantum approximate optimization algorithm (QAOA), we prove that a large subgroup of LinQuSO problems can be solved with up to exponential quantum speedups with regards to their simulation component. Finally, we present two practically relevant use cases that fall within this subgroup of QuSO problems.
Paper Structure (19 sections, 44 theorems, 41 equations, 12 figures, 1 table)

This paper contains 19 sections, 44 theorems, 41 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Simulation-based Optimization
  4. Quantum Optimization
  5. Data input
  6. Quantum Linear Algebra Subroutines
  7. Amplitude Arithmetic
  8. Data readout
  9. Methodology
  10. Data input
  11. Amplitude Arithmetic
  12. Assembling QSim
  13. Examples
  14. Unit Commitment
  15. Topology Optimization
  16. ...and 4 more sections

Key Result

Theorem 1

Given an objective function $f:\lbrace 0,1\rbrace^n\rightarrow\mathbb{R}$, the quantum circuit defined by $U\left(\beta, \gamma\right)$ in eq:QAOA yields $\mathop{\mathrm{arg\,min}}\limits_x f(x)$ for $p\rightarrow \infty$, and $\beta_i \coloneqq 1-i/p$, and $\gamma_i \coloneqq i/p$. where $U_M(\beta_i) \coloneqq e^{-i\beta_i H_{M}}$, $U_C(\gamma_i) \coloneqq e^{-i\gamma_i H_{C}}$, $H_C\coloneqq\

Figures (12)

  • Figure 1: Quantum circuit implementation of a $(k,l)$-UCU as defined in \ref{['def:UCU']}. Thesymbol is used as shorthand to iterate over all possible control combinations on the applied wire.
  • Figure 2: Circuit of a $(\sqrt{s_r s_c}, n+3, \varepsilon_1+\varepsilon_2)$-block-encoding of a matrix $A$ given corresponding sparse-access oracles $O_r$, $O_c$, and $O_A$ as defined in \ref{['lem:OBE']}. $D_s$ is defined as the map $\ket{0}^{\otimes q}\mapsto \frac{1}{\sqrt{s}}\sum_{k=1}^s \ket{k}$ (with $2\leq s \leq 2^q$ -- for an implementation requiring no ancillary qubits and a depth of $\mathcal{O}(\log_2 s)$ see Ref. Shukla2024. Purely for notational simplicity, AQE is visualized by a uniformly controlled $R_y$ rotation (cf. \ref{['lem:AQE']}).
  • Figure 3: Quantum circuits implementing an $(\alpha,a+1,\varepsilon)$-block-encoding of $P(A)$ via QSVT for arbitrary given $A\in \mathbb{C}^{L\times R}$ and even or odd polynomials $P\in \mathbb{C}\left[x\right]$. The blue boxes show implementations of projector-controlled phase shift operators $\tilde{\Pi}_{\phi_k}$ and $\Pi_{\phi_k}$ using a clean ancillary qubit from the top wire.
  • Figure 4: Quantum circuit implementing a $(1,n+2,0)$-block-encoding of the diagonal matrix $\textnormal{diag}(\textnormal{Re}(\ket{\psi}))$ as defined in \ref{['thm:diagonal-blockencoding']}. Implementations for the operators $G_p$ and $W_p$ are displayed in \ref{['fig:diagonal-blockencoding-Gp']} an \ref{['fig:diagonal-blockencoding-Wp']} in the Appendix.
  • Figure 5: Quantum circuit implementing $\mathcal{Q}$ from \ref{['lem:exp-val']}.
  • ...and 7 more figures

Theorems & Definitions (101)

  • Definition 1: MINLP
  • Definition 2: Summary Statistic Information
  • Definition 3: Quantum simulation-based optimization
  • Definition 4: Linear Quantum simulation-based optimization
  • Theorem 1: Quantum Approximate Optimization Algorithm farhi2014quantumsack2021
  • Lemma 1: Grover Mixer Bärtschi2020GroverMixers
  • Definition 5: Uniformly Controlled Unitary Yuan2023optimalcontrolled
  • Lemma 2: Implementing UCGs 10044235
  • Lemma 3: Quantum State Preparation 10044235
  • Theorem 2: Controlled Quantum State Preparation Yuan2023optimalcontrolled
  • ...and 91 more