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]