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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,...

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1) Local Hamiltonians Two questions about local Hamiltonians –continuous-time quantum computing BQP universality –interesting (ground) state properties QMA-complete problems Stronger results: –small locality, simple geometry –small energy × time cost –large promise/eigenvalue gaps –time independence, translational invariance

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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

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Questions (yes/no), whose answers are easy to check Factoring Does 114991 have a factor smaller than 60? Graph isomorphism Are these two graphs isomorphic? Satisfiability Is there a bit string avoiding all the bad assignments? 2) The Class NP disallowed substrings

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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

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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

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questions (yes/no), whose answers are easy to check on 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

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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

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3) Feynmans Hamiltonian Computer

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3) Hamiltonian Quantum Computation Feynmans Hamiltonian computer The Hamiltonian A quantum walk on a line a pointer particle (clock register) the workspace (work register)

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3) Hamiltonian Quantum Computation Feynmans Hamiltonian computer The Hamiltonian A quantum walk on a line The output a pointer particle (clock register) the workspace (work register)

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3) Boosting the Success Probability

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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

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3) The Local Hamiltonian Problem The history state –a state encoding the progress of a quantum computation Kitaevs (k-)Local Hamiltonian computation (history)

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3) The Local Hamiltonian Problem The history state –a state encoding the progress of a quantum computation Kitaevs (k-)Local Hamiltonian final answer initialization

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The history state –a state encoding the progress of a quantum computation Kitaevs (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

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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

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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

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MAX-k-SAT –NP-complete for k2 MAX-2-sat k-SAT –easy for k=2 –NP-complete for k3 3-SAT –with dits (3,2)-SAT is NP-complete simple in 1D for all dits 3) Classical vs. Quantum Problems

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MAX-k-SAT –NP-complete for k2 MAX-2-sat k-SAT –easy for k=2 –NP-complete for k3 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 k2 2-local Ham, even in 2D Quantum-k-SAT –easy for k=2 –QMA 1 -complete for k4 k=4, using 3-local projectors –universal: Quantum-3-SAT –with qudits QMA 1 -complete: Q-(5,3)-SAT universal: Q-(3,2)-SAT QMA 1 -c.: Q-(11,11)-SAT in 1D

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

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4) Constructing Clocks: Linear Time Domain wall clock

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

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Domain wall clock –used by Kitaev (5-local Hamiltonian is QMA 1 - 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-local 2-qubit gates: 5-local

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

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4) Constructing Clocks: Linear Time Domain wall clock with 4D particles (4D = made from 2 qubits) Quantum 4-SAT is QMA 1 -complete [Bravyi] (4,2,2)=(2,2,2,2) transitions: 4-local 2-qubit gates: 4-local

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4) Constructing Clocks: Linear Time Pulse clock –Feynmans pointer particle idea

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Pulse clock –Feynmans pointer particle idea –needs initialization the dead state problem: bad for Quantum k-SAT` 4) Constructing Clocks: Linear Time transitions: 2-local 2-qubit gates: 4-local

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4) Constructing Clocks: Linear Time Pulse clock –Feynmans pointer particle idea –needs initialization Qutrit pulse transitions: 2-local 2-qubit gates: 4-local

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4) Constructing Clocks: Linear Time Pulse clock –Feynmans 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

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

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4) Constructing Clocks: Linear Time A combination: domain wall + qutrit pulse Quantum (3,2,2)-SAT is QMA 1 -complete Q-4-SAT from 3-local projectors: QMA 1 - 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

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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

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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]

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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]

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4) Constructing Clocks: Beyond the Line Quantum (5,3)-SAT is QMA 1 -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

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4) Railroad Switch One train, two tracks

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4) Railroad Switch One train, two tracks

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4) Railroad Switch One train, two tracks

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4) Railroad Switch One train, two tracks

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4) Railroad Switch One train, two tracks

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

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

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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)

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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

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MAX-k-SAT –NP-complete for k2 MAX-2-sat k-SAT –easy for k=2 –NP-complete for k3 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 k2 2-local Ham, even in 2D Quantum-k-SAT –easy for k=2 –QMA 1 -complete for k4 k=4, using 3-local projectors –universal: Quantum-3-SAT –with qudits QMA 1 -complete: Q-(5,3)-SAT universal: Q-(3,2)-SAT QMA 1 -c.: Q-(11,11)-SAT in 1D

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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?

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5) Efficient Simulation of Quantum Circuits Use a Hamiltonian Computer –[AvDKLLR]: AQC is universal 3-local, L 17 –[Mizel,Lidar]: AQC is universal 4-loc,al L 4 –use a better one... 3-local, L 7 –go fast! [Lloyd] 3-local, L 2 log 2 L

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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: L 2 log 2 L

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6) Conclusions & Open Questions Hamiltonian Quantum Computers: universal without AQC –Feynmans Hamiltonian, quantum walk –a computational subspace –wheres 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?

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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]

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