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Complexity of Quadratic Bosonic Hamiltonian Simulation: $\mathsf{BQP}$-Completeness and $\mathsf{PostBQP}$-Hardness

Lilith Zschetzsche, Refik Mansuroğlu, Norbert Schuch

Abstract

The computational complexity of simulating the dynamics of physical quantum systems is a central question at the interface of quantum physics and computer science. In this work, we address this question for the simulation of exponentially large bosonic Hamiltonians with quadratic interactions. We present two results: First, we introduce a broad class of quadratic bosonic problems for which we prove that they are $\mathsf{BQP}$-complete. Importantly, this class includes two known $\mathsf{BQP}$-complete problems as special cases: Classical oscillator networks and continuous-time quantum walks. Second, we show that extending the aforementioned class to even more general quadratic Hamiltonians results in a $\mathsf{PostBQP}$-hard problem. This reveals a sharp transition in the complexity of simulating large quantum systems on a quantum computer, as well as in the difference in complexity between their simulation on classical and quantum computers.

Complexity of Quadratic Bosonic Hamiltonian Simulation: $\mathsf{BQP}$-Completeness and $\mathsf{PostBQP}$-Hardness

Abstract

The computational complexity of simulating the dynamics of physical quantum systems is a central question at the interface of quantum physics and computer science. In this work, we address this question for the simulation of exponentially large bosonic Hamiltonians with quadratic interactions. We present two results: First, we introduce a broad class of quadratic bosonic problems for which we prove that they are -complete. Importantly, this class includes two known -complete problems as special cases: Classical oscillator networks and continuous-time quantum walks. Second, we show that extending the aforementioned class to even more general quadratic Hamiltonians results in a -hard problem. This reveals a sharp transition in the complexity of simulating large quantum systems on a quantum computer, as well as in the difference in complexity between their simulation on classical and quantum computers.

Paper Structure

This paper contains 13 sections, 3 theorems, 53 equations, 1 figure.

Table of Contents

  1. Post-Processing of Measurement Data
  2. -hardness of generic Gaussian Hamiltonians (Theorem \ref{['thm:PostBQP']})
  3. PostBQP-completeness Construction
  4. Formal Solution to the Bosonic Problem
  5. Separation of Quantum Circuit Logic and Dynamics on the Clock Register
  6. Dynamics of the clock register: Tight binding model
  7. Time Evolution and Read-Out
  8. Solution of the Tight-Binding Model on the Clock Register
  9. Alternative PostBQP-hard families of bosonic Hamiltonians
  10. Coordinate Transformations
  11. Most General Quadratic Bosonic Hamiltonian
  12. Positivity of coupling matrices $A$ and $C$
  13. Symplectic Transformations and Generalized Hopping

Key Result

Theorem 1

Problem prob:QO is -complete.

Figures (1)

  • Figure 1: Venn diagram depicting inclusion of different classes of Hamiltonians in the class of inertially coupled bosons, cf. Eq. \ref{['eq:QO']}. We define the class of inertially coupled bosons as a subset of quadratic bosonic Hamiltonians and prove that their simulation can be formulated as a -complete decision problem. This unifies previous proofs for -completeness of classical and quantum oscillators as well as quantum walks and general hopping Hamiltonians extending the class of inertially coupled bosons. Shades of blue indicate -completeness and Hamiltonians in red define a -hard problem, in general.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof