[Paper Review] Towards quantum simulation of spin systems using continuous variable quantum devices

Razieh Annabestani, Brajesh Gupt+1 more|arXiv (Cornell University)|Sep 20, 2020

Quantum Computing Algorithms and Architecture40 references4 citations

TL;DR

This paper proposes mapping spin Ising Hamiltonians to continuous variable (CV) quantum devices via the Jordan-Schwinger transformation, enabling time evolution simulation using CV gate decompositions. It demonstrates that Gaussian Boson Sampling devices can estimate ground state energies by measuring Hafnians of covariance matrices, offering a path to hybrid classical-quantum algorithms like the CV variational quantum eigensolver.

ABSTRACT

We study Bosonic representation of spin Ising model with the application of simulating two level systems using continuous variable quantum processors. We decompose the time evolution of spin systems into a sequence of continuous variable logical gates and analyze their structure. We provide an estimation of quantum circuit scaling with the size of the spin lattice system. Furthermore, we discuss the possibility of using a Gaussian Boson sampling device to estimate the ground state energy of Ising Hamiltonian. The result has potential application in developing hybrid classical-quantum algorithms such as continuous variable version of variational quantum eigensolver.

Motivation & Objective

• To enable quantum simulation of spin systems on continuous variable quantum processors by mapping spin Hamiltonians to bosonic modes.
• To decompose the time evolution operator of the Ising model into universal CV quantum gates for resource estimation.
• To explore the feasibility of using near-term Gaussian Boson Sampling devices to estimate the ground state energy of Ising Hamiltonians.
• To establish a connection between the Hafnian of a covariance matrix and the expectation values of transverse Ising model terms in CV systems.
• To develop a framework for hybrid classical-quantum algorithms, such as a continuous variable version of the variational quantum eigensolver.

Proposed method

• Mapping the transverse Ising Hamiltonian to a bosonic representation using the Jordan-Schwinger transformation, resulting in quadratic and quartic interaction terms in CV modes.
• Decomposing the time evolution operator into a sequence of continuous variable logical gates using a universal CV gate set.
• Analyzing multi-mode terms in the Hamiltonian to derive compact expressions for gate components and enabling efficient circuit synthesis.
• Using the Wick's theorem and Hafnian formalism to relate the expectation values of quartic terms to the Hafnian of the covariance matrix.
• Engineering a Gaussian Boson Sampling circuit with doubled optical modes to map the target covariance matrix Σ to a valid physical covariance matrix σ_A, enabling Haf(Σ) estimation via post-selected single-photon detection.
• Implementing zero-mean Gaussian initial states via displacement operations and post-selecting on the (1,1,…,1) photon detection pattern to extract Hafnian values.

Experimental results

Research questions

• RQ1 Can spin systems described by the Ising Hamiltonian be efficiently mapped to continuous variable quantum processors using bosonic representations?
• RQ2 How can the time evolution operator of the Ising model be decomposed into a sequence of universal continuous variable quantum gates for simulation?
• RQ3 Can Gaussian Boson Sampling devices estimate the ground state energy of an Ising Hamiltonian by measuring the Hafnian of the system's covariance matrix?
• RQ4 What are the resource requirements for simulating a spin lattice of size N using this CV-based approach, particularly in terms of optical modes and gate operations?
• RQ5 Is it possible to extract the Hafnian of a target covariance matrix Σ from a physical Gaussian state in a GBS device, despite non-linear mapping constraints?

Key findings

• The Jordan-Schwinger transformation successfully maps the transverse Ising Hamiltonian into a bosonic Hamiltonian with quadratic and quartic interaction terms in continuous variable modes.
• The time evolution operator of the Ising model is decomposable into a sequence of continuous variable quantum gates, enabling full simulation on a CV quantum processor.
• The expectation value of quartic terms in the transverse Hamiltonian is proportional to the Hafnian of the covariance matrix, as per Wick's theorem.
• By doubling the number of optical modes, a valid physical covariance matrix σ_A can be constructed such that Haf(Σ) = √Haf(A), enabling estimation of the target Hafnian.
• Post-selection on the (1,1,…,1) photon detection pattern in a Gaussian Boson Sampling device allows measurement of Haf(Σ), though it requires single-photon detectors and is probabilistic.
• The method enables a scalable framework for estimating ground state energies of Ising-type Hamiltonians on near-term CV quantum devices, supporting hybrid classical-quantum algorithms like the CV variational quantum eigensolver.