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Local Hamiltonians in Quantum Computation

This study investigates the role of local Hamiltonians in quantum computation, addressing pivotal questions about their properties and the complexities entailed in their characterization. We delve into the impact of local Hamiltonians on continuous-time quantum computing, grounding state properties, and QMA-complete problems. By examining small locality and geometry, energy costs, and the promise/eigenvalue gaps, we aim to elucidate what can be achieved with local Hamiltonians and the computational challenges associated with them. The implications for quantum algorithms and complexity theory are significant, offering new insights and potential pathways for research.

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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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