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Hamiltonian Simulation with Optimal Sample Complexity

Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, Theodore J. Yoder

TL;DR

This work studies the sample-based Hamiltonian simulation problem, where the Hamiltonian is encoded in an unknown density matrix ρ = (H + cI)/Tr(H + cI). It proves that the Lloyd–Mohseni–Rebentrost protocol is asymptotically optimal in both evolution time t and precision δ, requiring only n = O(t^2/δ) copies of ρ to implement e^{-iρ t} on a target state, independently of system dimension. The authors extend the framework to simulate Hermitian polynomials of multiple input states, including linear combinations, commutators, and more general polynomials, with provably optimal sample complexities and clear constructions. They demonstrate practical applications such as commutator-based simulations for orthogonality testing and coherent state addition, and show that the LMR protocol enables a universal quantum computer using only partial swaps and simple single-qubit states, albeit with polynomial overhead. The paper thus positions sample-based Hamiltonian simulation as a powerful, broadly optimal tool with broad algorithmic and computational implications in quantum information processing.

Abstract

We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631--633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.

Hamiltonian Simulation with Optimal Sample Complexity

TL;DR

This work studies the sample-based Hamiltonian simulation problem, where the Hamiltonian is encoded in an unknown density matrix ρ = (H + cI)/Tr(H + cI). It proves that the Lloyd–Mohseni–Rebentrost protocol is asymptotically optimal in both evolution time t and precision δ, requiring only n = O(t^2/δ) copies of ρ to implement e^{-iρ t} on a target state, independently of system dimension. The authors extend the framework to simulate Hermitian polynomials of multiple input states, including linear combinations, commutators, and more general polynomials, with provably optimal sample complexities and clear constructions. They demonstrate practical applications such as commutator-based simulations for orthogonality testing and coherent state addition, and show that the LMR protocol enables a universal quantum computer using only partial swaps and simple single-qubit states, albeit with polynomial overhead. The paper thus positions sample-based Hamiltonian simulation as a powerful, broadly optimal tool with broad algorithmic and computational implications in quantum information processing.

Abstract

We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631--633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.

Paper Structure

This paper contains 20 sections, 16 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. LMR Protocol versus State Tomography
  3. LMR Protocol is Optimal
  4. Pure State Discrimination and the Optimality of LMR for Pure States
  5. Metrological View of Grover's Search
  6. The LMR Protocol is Optimal for Pure States
  7. Generalized LMR for Simulation of Hermitian Polynomials
  8. Simulating Linear Combinations
  9. Simulating the Commutator and Anticommutator
  10. Simulating Hermitian Polynomials in the Input States
  11. Applications of Commutator Simulation
  12. Coherent state addition
  13. Orthogonality Testing
  14. Universality of LMR
  15. Outlook
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\rho \in \mathrm{D}(\mathcal{H}_\mathsf{A})$ and $\sigma \in \mathrm{D}(\mathcal{H}_\mathsf{A} \otimes \mathcal{H}_\mathsf{B})$ be two unknown quantum states and $t \in \mathbb{R}$ (can be either positive or negative). Then there exists a quantum algorithm that transforms $\sigma_{\mathsf{A}\ma as long as the number of copies of $\rho$ is $n = O(t^2 / \delta)$. In other words, this quantum al

Figures (3)

  • Figure 1: The gadget to create a state $\rho'$. Here the controlled-cross-cross gate is a controlled-swap, and the waste bin indicates the partial trace. The $H$-gate is a single-qubit Hadamard gate (not to be confused with the Hamiltonian) and the measurement is in the $Z$-basis.
  • Figure 2: The gadget to create $\rho'^{(r)}$. Here $S_k$ is the permutation of $k$ registers given in \ref{['eq:perm']}, and the waste bins indicate the partial trace. The $H$-gate is a single-qubit Hadamard gate and measurement is in the $Z$-basis. In Ekert2002 they use the same circuit, but use the measurement outcomes to perform spectrum estimation.
  • Figure 3: Connectivity graph for qubits in our model. Each circle represents a qubit. Qubits connected by a solid line can have the Heisenberg interaction applied between them. The qubit $q_*$ can be prepared in the state $|0\rangle$ or $|+\rangle.$

Theorems & Definitions (39)

  • Theorem 1: LMR14
  • proof
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • ...and 29 more