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Hamiltonian simulation with explicit formulas for Digital-Analog Quantum Computing

Mikel Garcia-de-Andoin, Thorge Müller, Gonzalo Camacho

TL;DR

The paper tackles the challenge of compiling digital-analog quantum circuits for simulating arbitrary two-body Hamiltonians by introducing a constructive, polynomial-time protocol. It builds a positive semidefinite coupling matrix $\mathbf{B}$ from the target and source Hamiltonians and performs an eigendecomposition to express $\mathbf{B}$ as a sum of rank-1 terms, subsequently re-assembling these into a DAQC circuit using local unitaries on a ZZ source Hamiltonian. The result is a circuit with at most $12N^2$ digital-analog blocks and a total analog time bounded by $t_A \le \mathrm{tr}(\mathbf{B})$, with numerical evidence showing favorable, near-constant scaling of $t_A$ with system size for typical problem ensembles. This approach removes the need for expensive preprocessing optimizations, enabling scalable quantum simulation across larger qubit systems and potentially extending to broader Hamiltonians and architectures.

Abstract

Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.

Hamiltonian simulation with explicit formulas for Digital-Analog Quantum Computing

TL;DR

The paper tackles the challenge of compiling digital-analog quantum circuits for simulating arbitrary two-body Hamiltonians by introducing a constructive, polynomial-time protocol. It builds a positive semidefinite coupling matrix from the target and source Hamiltonians and performs an eigendecomposition to express as a sum of rank-1 terms, subsequently re-assembling these into a DAQC circuit using local unitaries on a ZZ source Hamiltonian. The result is a circuit with at most digital-analog blocks and a total analog time bounded by , with numerical evidence showing favorable, near-constant scaling of with system size for typical problem ensembles. This approach removes the need for expensive preprocessing optimizations, enabling scalable quantum simulation across larger qubit systems and potentially extending to broader Hamiltonians and architectures.

Abstract

Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.

Paper Structure

This paper contains 8 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Digital Analog Quantum Computing
  3. Constructive protocol for Hamiltonian simulation
  4. Numerical calculations
  5. Conclusions
  6. SQG parametrization
  7. Proof of main result
  8. From the decomposition of $\mathbf{B}$ to the DAQC circuit

Figures (2)

  • Figure 1: Problem Hamiltonian decomposition and DAQC circuit sketch. The colored blocks corresponds to analog blocks with evolution time $t_k$. The full circuit that simulates the evolution under the problem Hamiltonian for a time $T$, $TH_\text{P}=\sum_k t_kH_\text{P}^{(k)}$, is composed of up to $12N^2$ digital-analog blocks. The particular values of the rotation angles of the SQGs and $t_k$ are given in Eq. \ref{['eq:fulldecomposition']}.
  • Figure 2: $\log t_\text{A}$ for random problems of size $N$. The problem matrices ${\textbf{B}}$ are generated with uniform values in the range $[-1,1]$ and then normalized with the elementwise infinite norm. The solid line shows the average obtained over $10^4$ runs. The dashhed line shows the upper bound calculated as $t_\text{A}\leq 3N\lvert\tilde{\lambda}_\text{min}\rvert$. The colored regions represent the area between the maximum and minimum values obtained.