Local Hamiltonians in Quantum Computation Funding: Slovak Research and Development Agency, contract No. APVV-0673-07, European Project QAP 2004-IST-FETPI-15848, 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 Thanks: S. Mozes, P. Wocjan, O. Regev, P. Love, S. Lloyd, A. Landahl, A. Hassidim, S. Irani, D. Gottesman, S. Bravyi, ...
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. - PowerPoint PPT Presentation
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Local Hamiltonians inQuantum Computation
Funding: Slovak Research and Development Agency, contract No. APVV-0673-07, European Project QAP 2004-IST-FETPI-15848,
What could we do with them if we had them?
How hard is it to find their properties?
Daniel NagajSlovak Academy of SciencesBratislava, Slovakia
Thanks: S. Mozes, P. Wocjan, O. Regev, P. Love, S. Lloyd, A. Landahl, A. Hassidim, S. Irani, D. Gottesman, S. Bravyi, ...
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/eigenvalue gaps– time independence, translational invariance
4) Constructing Clocks: Linear Time• A combination: domain wall + qutrit
pulse
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-local2-qubit gates:
3-local
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
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]
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]
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
4) Railroad Switch
• One train, two tracks
4) Railroad Switch
• One train, two tracks
4) Railroad Switch
• One train, two tracks
4) Railroad Switch
• One train, two tracks
4) Railroad Switch
• One train, two tracks
4) Railroad Switch
• One train, two tracks
transitions: 3gates: 3
4) Railroad Switch
• One train, two tracks
• The computational subspace: a line again!
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)
• Using a qubit/qutrit railroad switch clock
– the computational subspace
– the dynamics: a quantum walk on a necklace
4) Universality of Quantum (3,2)-SAT
• 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
4) Classical vs. Quantum Problems• 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
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”?
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
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
6) Conclusions & Open Questions
• Hamiltonian Quantum Computers: universal without AQC– Feynman’s Hamiltonian, quantum walk– a computational subspace – where’s the real power of AQC?
• Complexity?– Quantum-3-SAT? Q-2-SAT on a line with low
qudits?
• New (geometric) clocks? – Translational invariance? Simpler geometry?
7) Local Hamiltonians in 1D
• geometric clock, Q-2-SAT in 1D [Aharonov et al.]