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Perturbative gadgets with constant-bounded interactions

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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

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

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

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?

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 ?

Some Terminology:

Example: 2D Heisenberg model

2-local

Pauli degree =12+1=13

Interaction strength

Main result

*Can be improved to Pauli degree = 3

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

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

Our result:

Hamiltonians on a regular lattice

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

unpertubed Hamiltonian

perturbation

Toy model: why interesting ? Perturbation gadgets

2nd order Lemma: the lower bound

2nd order Lemma: the upper bound

Local block-diagonalization: Schrieffer-Wolff transformation

Basic properties:

Global block-diagonalization

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)

Generalization: combining SW-formalism with the coupled cluster method

Generalization: combining SW-formalism with the coupled cluster method

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 ?