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Local Hamiltonians in Quantum Computation

Local Hamiltonians in Quantum Computation. What could we do with them if we had them ? How hard is it to find their properties?. Daniel Nagaj Slovak Academy of Sciences Bratislava , Slovakia.

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Local Hamiltonians in Quantum Computation

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  1. Local Hamiltonians inQuantum Computation What could we do with them if we had them?How hard is it to find their properties? Daniel Nagaj Slovak Academy of SciencesBratislava, Slovakia Funding:Slovak Research and Development Agency, contract No. APVV-0673-07, European Project QAP 2004-IST-FETPI-15848, Thanks: S. Mozes, P. Wocjan, O. Regev, P. Love, S. Lloyd, A. Landahl, A. Hassidim, S. Irani, D. Gottesman, S. Bravyi, ...

  2. 1) Local Hamiltonians • Two questions about local Hamiltonians • continuous-time quantum computing BQP universality • interesting (ground) state propertiesQMA-complete problems • Stronger results: • small locality, simple geometry • small energy × time cost • large promise/eigenvaluegaps • time independence, translational invariance

  3. 1) Outline • Computation & circuits • NP-completeness of Satisfiability • Feynman, reversible computation • Hamiltonian quantum computers • Two Hamiltonian problems • Local Hamiltonian [Kitaev] • Quantum k-SAT [Bravyi] • A clock workshop • clocks for QMA results • clocks for BQP universality • Adiabatic quantum computing

  4. 2) The Class NP • Questions (yes/no), whose answers are easy to check • FactoringDoes 114991 havea factor smaller than 60? • Graph isomorphismAre these two graphs isomorphic? • SatisfiabilityIs there a bit string avoidingall the bad assignments? disallowed substrings

  5. 2) The Class NP • Questions (yes/no), whose answers are easy to check • Merlin tries to convince Arthur a yes case: there existsa witness, on which C outputs yes a no case: for allinputs, C outputs no

  6. 2) NP-complete problems • Knowing how to solve one NP-hard problem would let us solve all NP problems • Could this circuit ever output 1?Does this verifier circuit have a witness? • 3-SAT is NP-complete (NP-hard, also in NP)[Cook,Levin] 3-local conditions

  7. 2) The Class QMA • questions (yes/no), whose answers are easy to checkon a quantum computer • Merlin tries to convince Arthur a yes case: there exists a witness, on which C outputs yeswith high probability (p  a) a no case: on any input, V outputs yes only with a small probability (p  b)

  8. 3) Reversible Computing & Quantum Circuits • How to implement a reversible computation in a physical system? [Feynman] • The Schrődinger equation • unitary time evolution • physical Hamiltonians: local • Quantum circuits • also reversible

  9. 3) Feynman’s Hamiltonian Computer

  10. 3) Hamiltonian Quantum Computation • a pointer particle(clock register) • the workspace(work register) • Feynman’s Hamiltonian computer • The Hamiltonian • A quantum walk on a “line”

  11. 3) Hamiltonian Quantum Computation • a pointer particle(clock register) • the workspace(work register) • Feynman’s Hamiltonian computer • The Hamiltonian • A quantum walkon a “line” • The output

  12. 3) Boosting the Success Probability

  13. 3) The Local Hamiltonian Problem work register after t gates • The history state • a state encoding the progress of a quantum computation • encodes also the result of • A ground state • a Hamiltonian with energy penalties for • non-history states (bad computation) • states with computations yielding `no’ • if a circuit can output `yes’, a `good’ history state exists • the ground state of H then has low energy

  14. 3) The Local Hamiltonian Problem • The history state • a state encoding the progress of a quantum computation • Kitaev’s (k-)Local Hamiltonian computation (history)

  15. 3) The Local Hamiltonian Problem • The history state • a state encoding the progress of a quantum computation • Kitaev’s (k-)Local Hamiltonian initialization final answer

  16. 3) The Local Hamiltonian Problem • The history state • a state encoding the progress of a quantum computation • Kitaev’s (k-)Local Hamiltonian • is the ground state energy of H less than a or more than b? • 5-local Hamiltonian: QMA-complete

  17. 3) The Local Hamiltonian Problem • Local Hamiltonian [Kitaev] • an analogue of classical MAX-k-SAT • is the ground state energy of the wholeHless than a or more than b? • Quantum k-SAT [Bravyi] • an analogue of classical k-SAT • Hamiltonian: a sum of projectors.Can they all be satisfied? • How to prove they are hard? • encode any q. computation U into the ground state of some H • knowing the ground state energy of H meansknowing whether U can ever output `yes’

  18. 3) Encoding a Quantum Computation • Stronger results? • interactions: a few particles with low dimensionality • a simple geometry of interactions • locally checkable encoding, initialization and output • unique transitions ... large eigenvalue gaps • possible transitions out of the computational subspace... requires large energy penalties • possibly a quantum PCP theorem one day? • look for a unique solution: Quantum k-SAT

  19. 3) Classical vs. Quantum Problems • MAX-k-SAT • NP-complete for k≥2 • MAX-2-sat • k-SAT • easy for k=2 • NP-complete for k≥3 • 3-SAT • with dits • (3,2)-SAT is NP-complete • simple in 1D for all dits

  20. 3) Classical vs. Quantum Problems • MAX-k-SAT • NP-complete for k≥2 • MAX-2-sat • k-SAT • easy for k=2 • NP-complete for k≥3 • 3-SAT • with dits • (3,2)-SAT is NP-complete • simple in 1D for all dits • k-local Hamiltonian • QMA-complete for k≥2 • 2-local Ham, even in 2D • Quantum-k-SAT • easy for k=2 • QMA1-complete for k≥4 • k=4, using 3-local projectors • universal: Quantum-3-SAT • with qudits • QMA1-complete: Q-(5,3)-SAT • universal: Q-(3,2)-SAT • QMA1-c.: Q-(11,11)-SAT in 1D

  21. 4) Constructing Clocks • two registers(clock/work) • requirements: locality • check the encoding • transitions • initialization & readout • time progression • linear/nonlinear • geometricclock

  22. 4) Constructing Clocks: Linear Time • Domain wallclock

  23. 4) Constructing Clocks: Linear Time transitions: 3-local 2-qubit gates: 5-local • Domain wallclock • used by Kitaev (5-local Hamiltonian) • easy to check initialization, output, single active site

  24. 4) Constructing Clocks: Linear Time transitions: 3-local 2-qubit gates: 5-local • Domain wallclock • used by Kitaev (5-local Hamiltonian is QMA1-complete) • easy to check initialization, output, single active site • 3-local Hamiltonian [Kempe & Regev] • suppressing bad transitions: projection lemma • 2-local Hamiltonian [Kempe, Kitaev, Regev, Oliveira & Terhal] • effective 3-local interactions: gadgets, even in 2D

  25. 4) Constructing Clocks: Linear Time • Domain wall clock with 4D particles(4D = made from 2 qubits)

  26. 4) Constructing Clocks: Linear Time • Domain wall clock with 4D particles (4D = made from 2 qubits) • Quantum 4-SAT is QMA1-complete [Bravyi] (4,2,2)=(2,2,2,2) transitions: 4-local 2-qubit gates: 4-local

  27. 4) Constructing Clocks: Linear Time • Pulse clock • Feynman’s pointer particle idea

  28. 4) Constructing Clocks: Linear Time • Pulse clock • Feynman’s pointer particle idea • needs initialization • the dead state problem: bad for Quantum k-SAT` transitions: 2-local 2-qubit gates: 4-local

  29. 4) Constructing Clocks: Linear Time • Pulse clock • Feynman’s pointer particle idea • needs initialization • Qutrit pulse transitions: 2-local 2-qubit gates: 4-local

  30. 4) Constructing Clocks: Linear Time • Pulse clock • Feynman’s pointer particle idea • needs initialization • Qutrit pulse • uses qutrits • needs initialization transitions: 2-local 2-qubit gates: 4-local transitions: 2-local 2-qubit gates: 3-local

  31. 4) Constructing Clocks: Linear Time • A combination: domain wall + qutrit pulse

  32. 4) Constructing Clocks: Linear Time • A combination: domain wall + qutrit pulse • Quantum (3,2,2)-SAT is QMA1-complete • Q-4-SAT from 3-local projectors: QMA1-complete • a qutrit from a pair of qubits (00,01±10) • a 3-local Hamiltonian (a new construction) • energy separation: b-a = O(L-4) (old result: L-10) transitions: 3-local 2-qubit gates: 3-local

  33. 4) Constructing Clocks: Beyond the Line • Quantum 2-SAT (with qudits) • progress the clock by 2-local interactions • pulse clock: initialization problem • domain wall with qubits : 3-local • solution: use qutrits

  34. 4) Constructing Clocks: Beyond the Line • Quantum 2-SAT (with qudits) • how to apply a 2-qubit gate by interacting with a single work qubit at a time? • Triangle clock [Eldar, Regev]

  35. 4) Constructing Clocks: Beyond the Line • Quantum 2-SAT (with qudits) • how to apply a 2-qubit gate by interacting with a single work qubit at a time? • Triangle clock [Eldar, Regev]

  36. 4) Constructing Clocks: Beyond the Line • Quantum (5,3)-SAT is QMA1-complete [Eldar, Regev] • apply a 2-qubit gate by interacting with a single work qubit at a time • use only 2-local clock transitions • Triangle clock

  37. 4)Railroad Switch • One train, two tracks

  38. 4)Railroad Switch • One train, two tracks

  39. 4)Railroad Switch • One train, two tracks

  40. 4)Railroad Switch • One train, two tracks

  41. 4)Railroad Switch • One train, two tracks

  42. 4)Railroad Switch • One train, two tracks transitions: 3gates: 3

  43. 4)Railroad Switch • One train, two tracks • The computational subspace: a line again!

  44. 4)Universality of Quantum 3-SAT • Using a railroad switch clock • fast, universal quantum computation with a Q-3-SAT Hamiltonian • made from 3-local projectors • resources: • the computational subspace • protected by a gap O(L-1) • not against everything (loss of a pointer)

  45. 4)Universality of Quantum (3,2)-SAT • Using a qubit/qutrit railroad switch clock • the computational subspace • the dynamics: a quantum walk on a necklace

  46. 4) Classical vs. Quantum Problems • MAX-k-SAT • NP-complete for k≥2 • MAX-2-sat • k-SAT • easy for k=2 • NP-complete for k≥3 • 3-SAT • with dits • (3,2)-SAT is NP-complete • simple in 1D for all dits • k-local Hamiltonian • QMA-complete for k≥2 • 2-local Ham, even in 2D • Quantum-k-SAT • easy for k=2 • QMA1-complete for k≥4 • k=4, using 3-local projectors • universal: Quantum-3-SAT • with qudits • QMA1-complete: Q-(5,3)-SAT • universal: Q-(3,2)-SAT • QMA1-c.: Q-(11,11)-SAT in 1D

  47. 5) Adiabatic Quantum Computing • Ground states and optimization problems • a cost function h(z) of an optimization problem • A Hamiltonian Algorithm [FGGS] • use a time-dependent, slowly changing Hamiltonian • Adiabatic Theorem • start in the ground state, end up in the ground state • how slow is “slow”?

  48. 5)Efficient Simulation of Quantum Circuits • Use a Hamiltonian Computer • [AvDKLLR]: AQC is universal3-local, L17 • [Mizel,Lidar]: AQC is universal4-loc,al L4 • use a better one... 3-local, L7 • go fast! [Lloyd]3-local, L2 log2L

  49. 5)Efficient Simulation of Quantum Circuits • Unique transitions • a computational subspace • The Hamiltonian • Dynamics • a quantum walk • no need to go adiabatically • 3-local & fast: L2 log2L

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