Julia Kempe CNRS & LRI, Univ. of Paris-Sud Alexei Kitaev Caltech Oded Regev Tel-Aviv University FSTTCS, Chennai, December 18 th, 2004 The Complexity of the Local Hamiltonian Problem

Results Result: 2-local Hamiltonian is QMA complete 2-local adiabatic computation is equivalent to standard quantum computation Also implies:

Outline Introduction Quantum computing QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Quantum Computation A qubit is described by a unit vector in two- dimensional space:A qubit is described by a unit vector in two- dimensional space: –|  = c 0 |0  + c 1 |1  such that |c 0 | 2 +|c 1 | 2 =1 –|0  and |1  are simply two orthogonal vectors An n-qubit system is described by a unit vector in a 2 n dimensional space:An n-qubit system is described by a unit vector in a 2 n dimensional space: –|   C {0,1} n such that || |  || 2 =1 An operation on an n-qubit system is described by a unitary matrix:An operation on an n-qubit system is described by a unitary matrix: –U  C 2 n  2 n such that UU † =I (i.e., unitary) |  U |  ’  = U | 

Quantum Computation The model of quantum computation is as strong as classical computationThe model of quantum computation is as strong as classical computation Moreover, there exists a small set of quantum gates that are universalMoreover, there exists a small set of quantum gates that are universal [Deutsch ’95, Barenco et al. ’95, DiVincenzo’95] H |0  |1  U CNOT “Hadamard” Quantum complexity theory is born!Quantum complexity theory is born!

The class NP “yes” instance: x  L exists witness: y 1 (accept) “no” instance: x  L for all y 0 (reject) VV NP – Nondeterministic Polynomial Time Def: L  NP if there is a poly-time verifier V and a polynomial p s.t. Cook-Levin Theorem: SAT is NP-complete

The class QMA “yes” instance: x  L 1 (accept) “no” instance: x  L exists witness |  for all |  0 (reject) prob 1-  0 (reject) prob  1 (accept) prob  prob 1-  UU QMA – Quantum Merlin Arthur Def: L  QMA if there is a poly-time quantum verifier U and a polynomial p s.t.

Local Hamiltonian Problem Def. k-local Hamiltonian problem: Input: k-local Hamiltonian,, H i acts on  k qubits, a<b constants Promise: Smallest eigenvalue of H either  a or  b (b-a const.) Output: 1 if H has eigenvalue  a 0 if all eigenvalues of H  b Kitaev’s quantum Cook-Levin Theorem (’99): Local Hamiltonian is QMA-complete. “witness = ground state”

Local Hamiltonian Problem Intuition: Formula: Penalties for: x 1 x 2 x 3 = 010 x 3 x 4 x 5 = 100 … Satisfying assignment is groundstate of Energy-penalty 1 for each unsatisfied constraint.  x 1 x 2 … x n | H |x 1 x 2 … x n  = #unsatisfied constraints Hamiltonians:, H1H1 H2H2 local Hamiltonians

Classical Quantum MAX3SAT is NP-complete log|x|-local Hamiltonian is QMA-compl. [Kitaev’99] 5-local Hamiltonian is QMA-complete [Kitaev’99] 3-local Hamiltonian is QMA-complete [KempeRegev’02] MAX2SAT is NP-complete 2-local Hamiltonian is NP-hard 2-local Hamiltonian ?? 1-local Hamiltonian is in P Results New result: 2-local Hamiltonian is QMA-complete

Adiabatic Computation Quantum computers can simulate adiabatic computation [Farhi et al. 00] Adiabatic computation can simulate quantum computers [ADKLLR 04] In fact, 3-local adiabatic computation is enough [ADKLLR 04] New result: 2-local adiabatic computation can simulate quantum computers

Outline Introduction Quantum Computation QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Classical Cook-Levin Theorem Thm: SAT is NP-complete Proof: First, given a verifier V, encode the input into V x1x2…y1y2…00…x1x2…y1y2…00… 1 input x witness y ancilla 0 V y1y2…00…y1y2…00… 1 witness y ancilla 0 VxVx

Classical Cook-Levin Theorem propagation clauses output clause z 01 z 02 z 03 z 04 z 0N z 1N z 2N z TN time = … T ancilla clauses ancilla Thm: SAT is NP-complete Proof: Next, create a tableau of variables and 3 kinds of clauses.

Quantum Cook-Levin Theorem Let us try to extend this to the quantum setting |  |0  … |1  ancilla qubits UxUx

Quantum Cook-Levin Theorem Let us try to extend this to the quantum setting The naïve attempt does not work There is no local way to check local consistency propagation clauses output clause z 01 z 02 z 03 z 04 z 0N z TN ancilla clauses ancilla qubits   |  0  |  1  |  2  … |  T 

Quantum Cook-Levin Theorem Instead of tensoring the columns, we put them in superposition So the witness is a sum over history propagation clauses output clause z 01 z 02 z 03 z 04 z 0N z TN ancilla clauses ancilla qubits |   |  0  |0  + |  1  |1  +…+ |  T  |T  |  0  |  1  |  2  … |  T 

Quantum Cook-Levin Theorem H= J in H in + J prop H prop + H out input propagation output Computation qubits Time register {|0 , |1 ,…, |T  } Thm [Kitaev]: Local Hamiltonian is QMA-complete Proof: Expect the witness described before. Construct the following Hamiltonians.

Reducing Locality tT-t | t  t |  |10  10| t,t+1 | t  t-1 |  |110  100| t-1,t,t+1 Notice that we have log-local terms: Thm [Kitaev]: 5-local Hamiltonian is QMA-complete Proof idea: Use unary encoding |t  | 11…100…0  Penalise illegal time states: S clock - space of legal time-states is preserved (invariant) ▪ Thm [KempeRegev] : 3-local Hamiltonian is QMA-complt.

Outline Introduction Quantum Computation QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Idea: use perturbation theory to obtain effective 3- local Hamiltonians from 2-local ones by restricting to subspaces What is the effective Hamiltonian in the lower part of the spectrum? Three-qubit gadget … H’ = H + V Spectrum: H 0 groundspace S Energy gap:  ||H||>>||V||

Perturbation Theory What is the effective Hamiltonian in the lower part of the spectrum? Energy gap  > ||V|| S SS V -- - restriction of V to S V ++ - restriction of V to S  H’ = H + V Spectrum: H 0 groundspace S Energy gap:  ||H||>>||V|| SS

Perturbation Theory H’ = H + V Spectrum: H 0 groundspace S Energy gap:  ||H||>>||V|| Theorem: SS Energy gap  > ||V|| S SS V -- - restriction of V to S V ++ - restriction of V to S  The lower spectrum of H’ is close to the spectrum of H eff (under certain conditions).

Three-qubit gadget H=P 1 P 2 P 3 3-local B A C ZZ P1XAP1XA P2XBP2XB P3XCP3XC Terms in H’=H+V are 2-local H eff =P 1 P 2 P 3 3-local Fine-tune the energy gap  =  -3

Three-qubit gadget B A C ZZ Energy gap:  S={|000 , |111  } S  ={|001 ,|010 ,|100 , |110 ,|101 ,|011  } 0  =  -3

Three-qubit gadget B A C 2 P2XBP2XB 3 P3XCP3XC 1 P1XAP1XA Theorem: Third order: S S S   V -+ V +- V ++ SS Ex.: P1XAP1XA P3XCP3XC |000  |100   |110  |111  P2XBP2XB Energy gap:  S={|000 , |111  } S  ={|001 ,|010 ,|100 , |110 ,|101 ,|011  } 0  =  -3

2-local Hamiltonian is QMA-complete Start with the QMA-complete 3-local Hamiltonian Replace each 3-local term by a 3-qubit gadget

Outline Introduction Quantum Computation QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Start in the groundstate of a Hamiltonian H 0 (easy to prepare) Encode problem as a Hamiltonian H P (groundstate gives solution) Adiabatically (slowly!) evolve from H 0 to H P *E. Farhi, J. Goldstone, S. Gutmann, M. Sipser: “Quantum Computation by Adiabatic Evolution”, q-p/’00 Adiabatic theorem: g(t) gap between ground- and first excited state If then the final state arbitrarily close to groundstate of H P. Idea of Adiabatic Computation*

Adiabatic Computation simulates quantum computation Standard quantum circuit: *D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation", FOCS’04, p Adiabatic simulation*: H initial groundstate | 0…0  | 0  H final groundstate = Kitaev’s “history state” H(t) = (1-t/T’)H initial +t/T’ H final T’=poly(T): The gap between groundstate and first excited state is 1/poly(T) at all times. U |T|T | 0…0  T gates

Adiabatic Computation simulates quantum computation Adiabatic simulation*: H initial groundstate | 0…0  | 0  H final groundstate = Kitaev’s “history state” H(t) = (1-t/T’)H initial +t/T’ H final Result: 2-local adiabatic computation is equivalent to standard quantum computation Use the gadget to replace everything by 2-local terms.

Other applications of the gadget (work in progress) “Interaction at a distance”: H=P 1 P 2 H eff =P 1 P 2  -1 P 1 X A  -1 P 2 X A  -2 Z A “Proxy Interaction”: (with A. Landahl) H=Z 1 X 2 only XX,YY,ZZ available H eff =Z 1 X 2  -2 Y A Y B  -1 Z 1 Z A  -1 X 2 X B Useful for Hamiltonian-based quantum architectures