Perturbative gadgets with constant bounded interactions
This presentation is the property of its rightful owner.
Sponsored Links
1 / 30

Perturbative gadgets with constant-bounded interactions PowerPoint PPT Presentation


  • 93 Views
  • Uploaded on
  • Presentation posted in: General

Perturbative gadgets with constant-bounded interactions. Sergey Bravyi 1 David DiVincenzo 1 Daniel Loss 2 Barbara Terhal 1. 1 IBM Watson Research Center 2 University of Basel. Classical and Quantum Information Theory Santa Fe, March 27, 2008. arXiv:0803.2686. Outline:. Physics.

Download Presentation

Perturbative gadgets with constant-bounded interactions

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Perturbative gadgets with constant bounded interactions

Perturbative gadgets with constant-bounded interactions

Sergey Bravyi1David DiVincenzo1Daniel Loss2Barbara Terhal1

1 IBM Watson Research Center

2 University of Basel

Classical and Quantum Information TheorySanta Fe, March 27, 2008

arXiv:0803.2686


Perturbative gadgets with constant bounded interactions

Outline:

Physics

Perturbative gadgets

High-energy fundamentaltheory, full Hamiltonian H(simple)

High-energy simulator Hamiltonian H(simple)

Rigorous, but unphysical scaling of interactions

Non-rigorous territory

Effective low-energy Hamiltonian Heff (complex)

Low-energy target Hamiltonian Htarget(complex)

Goal: develop a rigorous formalism for constructinga simulator Hamiltonian within a physical range of parameters


Perturbative gadgets with constant bounded interactions

Motivation:

1. Htarget is chosen for some interesting ground-state properties

Toric code model

Quantum loop models

Briegel-Raussendorf cluster state

2. Htarget is chosen for some computational hardness properties

Quantum NP-hard Hamiltonians

Adiabatic quantum computation


Perturbative gadgets with constant bounded interactions

What is realistic simulator Hamiltonian ?

  • Only two-qubit interactions

  • Norm of the interactions is bounded by a constant (independent of the system size)

  • Each qubit can interact with a constant number of otherqubits (bounded degree)

  • Nearest-neighbor interactions on a regular lattice(desirable but not necessary)

Main question: given a target Hamiltonian, what groundstate properties can be reproduced by a realistic

simulator Hamiltonian?


Perturbative gadgets with constant bounded interactions

Main question: given a target Hamiltonian, what groundstate properties can be reproduced by a realistic

simulator Hamiltonian?

Wish list:

Today’s talk

  • Ground-state energy; small ``extensive’’ error

  • Expectation values of extensive observables (e.g. average magnetization) small extensive error

  • Expectation values of local observables

  • Spectral gap

  • Topological Quantum Order

Gapped Hamiltonians ?


Perturbative gadgets with constant bounded interactions

Some Terminology:


Perturbative gadgets with constant bounded interactions

Example: 2D Heisenberg model

2-local

Pauli degree =12+1=13

Interaction strength


Perturbative gadgets with constant bounded interactions

Main result

*Can be improved to Pauli degree = 3


Perturbative gadgets with constant bounded interactions

Main improvement: interaction strength of the simulator is reduced from J poly(n) to O(J)

Shortcoming: can not go beyond a small extensive error

Idea of perturbation gadget

[Kempe, Kitaev, Regev 05] 3-local to 2-local

[Oliveira, Terhal 05] k-local to 2-local on 2D lattice

[Bravyi, DiVincenzo, Oliveira, Terhal 06] k-local to 2-local for stoquastic Hamiltonians

[Biamonte, Love 07] simulator with XZ,X,Z only

[Schuch, Verstraete 07] simulator with Heisenberg interactions

[Jordan, Farhi 08] k-th order perturbative gadgets


Perturbative gadgets with constant bounded interactions

If the only purpose of the simulator H is to reproduce the ground state

energy, why don’t we “simulate” Htarget simply by computing its ground

state energy with a small extensive error ?

1. We hope that H reproduces more than just the ground state energy (for example, expectation values of extensive observables)

2. Computing the ground state energy of Htarget with a small extensive error is NP-hard problem even for classical Hamiltonians

[see Vijay Vazirani, “Approximation Algorithms”, Chapter 29]

Hardness of Approximation = PCP theorem


Perturbative gadgets with constant bounded interactions

Our result:


Perturbative gadgets with constant bounded interactions

Hamiltonians on a regular lattice


Perturbative gadgets with constant bounded interactions

The Simulation Theorem: plan of the proof

  • Add ancillary high-energy “mediator” qubits to the“logical” qubits acted on by Htarget

  • Choose appropriate couplings between the mediatorand the logical qubits

  • Construct a unitary operator generating an effectivelow-energy Hamiltonian acting on the system qubits

  • Apply Lieb-Robinson type arguments to bound the error

Follows old ideas

new


Perturbative gadgets with constant bounded interactions

unpertubed Hamiltonian

perturbation


Perturbative gadgets with constant bounded interactions

Toy model: why interesting ? Perturbation gadgets


Perturbative gadgets with constant bounded interactions

2nd order Lemma: the lower bound


Perturbative gadgets with constant bounded interactions

2nd order Lemma: the upper bound


Perturbative gadgets with constant bounded interactions

Local block-diagonalization: Schrieffer-Wolff transformation


Perturbative gadgets with constant bounded interactions

Basic properties:


Perturbative gadgets with constant bounded interactions

Global block-diagonalization


Perturbative gadgets with constant bounded interactions

Generalization: combining SW-formalism with the coupled cluster method

Coupled cluster method [F. Coester 1958]: heuristic simulation algorithm formany-body quantum systems. One of the most powerful techniques in the

modern quantum chemistry. Main idea: use variational states

Where C is so called creation operator

It is expected that ground states of realistic Hamiltonians can be approximated by taking into account only subsets Γof small size (C is a local operator)


Perturbative gadgets with constant bounded interactions

Generalization: combining SW-formalism with the coupled cluster method


Perturbative gadgets with constant bounded interactions

Generalization: combining SW-formalism with the coupled cluster method


Perturbative gadgets with constant bounded interactions

Main question: given a target Hamiltonian, what groundstate properties can be reproduced by a realistic

simulator Hamiltonian? Remains largely open…

Wish list:

Today’s talk

  • Ground-state energy; small ``extensive’’ error

  • Expectation values of extensive observables;small extensive error

  • Expectation values of local observables

  • Spectral gap

  • Topological Quantum Order

Gapped Hamiltonians ?


  • Login