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Perturbative gadgets with constant-bounded interactions

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.

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Perturbative gadgets with constant-bounded interactions

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  1. 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

  2. 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

  3. 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

  4. 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?

  5. 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 ?

  6. Some Terminology:

  7. Example: 2D Heisenberg model 2-local Pauli degree =12+1=13 Interaction strength

  8. Main result *Can be improved to Pauli degree = 3

  9. 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

  10. 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

  11. Our result:

  12. Hamiltonians on a regular lattice

  13. 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

  14. unpertubed Hamiltonian perturbation

  15. Toy model: why interesting ? Perturbation gadgets

  16. 2nd order Lemma: the lower bound

  17. 2nd order Lemma: the upper bound

  18. Local block-diagonalization: Schrieffer-Wolff transformation

  19. Basic properties:

  20. Global block-diagonalization

  21. 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)

  22. Generalization: combining SW-formalism with the coupled cluster method

  23. Generalization: combining SW-formalism with the coupled cluster method

  24. 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 ?

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