Perturbative gadgets with constant-bounded interactions

Sergey Bravyi1

David DiVincenzo1

Daniel Loss2 Barbara Terhal1

Classical and Quantum Information TheorySanta Fe, March 27, 2008

1 IBM Watson Research Center2 University of Basel

arXiv:0803.2686

High-energy fundamentaltheory, full Hamiltonian H (simple)

Effective low-energy Hamiltonian Heff

(complex)

High-energy simulator Hamiltonian H (simple)

Low-energy target Hamiltonian Htarget

(complex)

Outline:Physics Perturbative gadgets

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

Rigorous, but unphysical scaling of interactions

Rigorous, but unphysical scaling of interactions

Non-rigorous territory

Non-rigorous territory

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 ?

1. Only two-qubit interactions

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

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

4. 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:

1. Ground-state energy; small ``extensive’’ error

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

3. Expectation values of local observables

4. Spectral gap

5. Topological Quantum Order

Gap

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Ham

ilton

ians

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Today’s talk

Some Terminology:

Example: 2D Heisenberg model

2-local

Pauli degree =12+1=13

Interaction strength

Main result

*Can be improved to Pauli degree = 3

[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

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

[Schuch, Verstraete 07] simulator with Heisenberg interactions

Idea of perturbation gadget

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

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 groundstate 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

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

2. Choose appropriate couplings between the mediatorand the logical qubits

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

4. Apply Lieb-Robinson type arguments to bound the error

new

Fol

low

s ol

d id

eas

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 themodern 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 Γ of 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:

1. Ground-state energy; small ``extensive’’ error

2. Expectation values of extensive observables;small extensive error

3. Expectation values of local observables

4. Spectral gap

5. Topological Quantum Order

Gap

ped

Ham

ilton

ians

?

Today’s talk