Post on 25-Feb-2016
description
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
• 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
1) Outline
• Questions (yes/no), whose answers are easy to check
• FactoringDoes 114991 havea factor smaller than 60?
• Graph isomorphismAre these two graphs isomorphic?
• Satisfiability Is there a bit string avoidingall the bad assignments?
2) The Class NP
disallowed substrings
• Questions (yes/no), whose answers are easy to check
• Merlin tries to convince Arthur
a yes case: there exists a witness, on which C outputs yes
a no case: for all inputs, C outputs no
2) The Class NP
• 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]
2) NP-complete problems
3-local conditions
• 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 yes with high probability (p a)
a no case: on any input, V outputs yes
only with a small probability (p b)
2) The Class QMA
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
3) Feynman’s Hamiltonian Computer
3) Hamiltonian Quantum Computation• Feynman’s
Hamiltonian computer
• The Hamiltonian
• A quantum walk on a “line”
a pointer particle(clock register)the workspace(work register)
3) Hamiltonian Quantum Computation• Feynman’s
Hamiltonian computer
• The Hamiltonian
• A quantum walkon a “line”
• The output
a pointer particle(clock register)the workspace(work register)
3) Boosting the Success Probability
• 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
3) The Local Hamiltonian Problem
work register after t gates
3) The Local Hamiltonian Problem• The history state
– a state encoding the progress of a quantum computation
• Kitaev’s (k-)Local Hamiltonian
computation (history)
3) The Local Hamiltonian Problem• The history state
– a state encoding the progress of a quantum computation
• Kitaev’s (k-)Local Hamiltonian
final answerinitialization
• 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
3) The Local Hamiltonian Problem
• Local Hamiltonian [Kitaev]– an analogue of classical MAX-k-SAT– is the ground state energy of the whole H
less 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 means
knowing whether U can ever output `yes’
3) The Local Hamiltonian Problem
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
• 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
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
3) 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
4) Constructing Clocks• two registers
(clock/work)
• requirements: locality– check the encoding– transitions– initialization & readout
• time progression – linear/nonlinear
• geometricclock
4) Constructing Clocks: Linear Time
• Domain wallclock
4) Constructing Clocks: Linear Time
• Domain wallclock
– used by Kitaev (5-local Hamiltonian)– easy to check initialization, output, single
active site
transitions: 3-local2-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
4) Constructing Clocks: Linear Time
transitions: 3-local2-qubit gates:
5-local
4) Constructing Clocks: Linear Time
• Domain wall clock with 4D particles(4D = made from 2 qubits)
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-local2-qubit gates:
4-local
4) Constructing Clocks: Linear Time
• Pulse clock
– Feynman’s pointer particle idea
• Pulse clock
– Feynman’s pointer particle idea– needs initialization
• the dead state problem: bad for Quantum k-SAT`
4) Constructing Clocks: Linear Time
transitions: 2-local2-qubit gates:
4-local
4) Constructing Clocks: Linear Time
• Pulse clock
– Feynman’s pointer particle idea– needs initialization
• Qutrit pulse
transitions: 2-local2-qubit gates:
4-local
4) Constructing Clocks: Linear Time
• Pulse clock
– Feynman’s pointer particle idea– needs initialization
• Qutrit pulse
– uses qutrits– needs initialization
transitions: 2-local2-qubit gates:
4-local
transitions: 2-local2-qubit gates:
3-local
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.]
• diffusion clock [Cirac et al.]
• translationally invariant, HQCA [Nagaj & Wocjan]