Quantum Information Theory: Present Status and Future Directions Julia Kempe CNRS & LRI, Univ. de Paris-Sud, Orsay, France Newton Institute, Cambridge, August 24 th, 2004 The Complexity of Local Hamiltonians

Joint work with Oded Regev and Alexei Kitaev Result: 2-local Hamiltonian is QMA complete J. K., Alexei Kitaev and Oded Regev, quant-ph/ local adiabatic computation is equivalent to standard quantum computation Also implies:

Outline A Bit of History QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

A Bit of (ancient) History Complexity Theory: classify “easy” and “hard”

A Bit of (ancient) History NP – Nondeterministic Polynomial Time: Def. L  NP if there is a poly-time verifier V and a polynomial p s.t. V “yes” instance: x  L witness: y 1 (accept) V “no” instance: x  L for all “witnesses” y 0 (reject)

Example: SAT Formula:  SAT iff there is a satisfying assignment for x 1,…,x n (i.e. all clauses true simultaneously) false 1 - true 0 = 1, 1 = 0 V=  (y) “yes” instance:   SAT witness: y=011000… 1 (true, accept) “no” instance:   SAT for all “witnesses” y=010110… 0 (false, reject) V=  (y)

NP complete A language is NP complete if it is in NP and as hard as any other problem in NP. Cook-Levin Theorem: SAT is NP-complete  L  SAT y=011000… y 11  V  x 0  SAT y=010110… 0  L y x  NPNP-complete

NP complete Cook-Levin Theorem: 3SAT is NP-complete 3SAT: 3 variables per clause 3 variables 2SAT is in P (there is a poly time algorithm). MAX2SAT is NP-complete MAX2SAT: Input: Formula with 2 variables per clause, number m Output: 1 (accept) if there is an assignment that violates  m clauses 0 (reject) all assignments violate >m clauses

QMA V “yes” instance: x  L yes 1 (accept) V “no” instance: x  L no witness: |  for all “witnesses” |  0 (reject) prob 1-  0 (reject) prob  1 (accept) prob  prob 1-  QMA – Quantum Merlin Artur = BQNP = “Quantum NP” Def. L  QMA if there is a poly-time quantum verifier V and a polynomial p s.t.

More recent (quantum) History QMA – Quantum Merlin Artur = BQNP Def. L  QMA if there is a poly-time quantum verifier V and a polynomial p s.th. First studied in [Knill’96] and [Kitaev’99] – called it BQNP “ QMA” coined by [Watrous’00] – also: group-nonmembership  QMA Kitaev’s quantum Cook-Levin Theorem (’99): Local Hamiltonian is QMA-complete.

Local Hamiltonians 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

Local Hamiltonians 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

NP and QMA NP-completeness: QMA-completeness? x1x2…y1y2…00…x1x2…y1y2…00…   1 x y 0 Verifier V: input witness ancilla  …

NP and QMA NP-completeness: QMA-completeness? y1y2…00…y1y2…00…    1 … y 0 Verifier V x : … … 3-clauses check: propagation output z 01 z 02 z 03 z 04 z 0N z 1N z 2N z TN t = … T |  |0  … CC H |1  … CC H   |  0  |  1  …  |  T  ? Verifier U x : ancilla qubits witness ancilla input ancilla No local way to check!

NP and QMA NP-completeness: QMA-completeness? y1y2…00…y1y2…00…    1 … y 0 Verifier V x : 3-clauses check: propagation output |  |0  … CC H |1  … CC H ? Verifier U x : ancilla qubits witness ancilla input |  |0  = |  0  |  1  … |  T   |0|0 |1|1 |2|2 |T|T |   |  0  |0  + |  1  |1  +…+ |  T  |T  witness = sum over history

NP-completeness: QMA-completeness: 3SAT is NP-complete 2SAT is in P log|x|-local Hamiltonian is QMA-compl. [Kitaev’99] 5-local Hamiltonian is QMA-compl. [Kitaev’99] 3-local Hamiltonian is QMA-compl. [KempeRegev’02] but: MAX2SAT is NP-complete 2-local Hamiltonian is NP-hard 2-local Hamiltonian???? 1-local Hamiltonian is in P More recent (quantum) History Is 2-local Hamiltonian QMA-complete??

Outline A Bit of History QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Kitaev’s log-local Construction Local Hamiltonians check: H= J in H in + J prop H prop + H out || |1  Verifier U x : witness = sum over history m N-m T T=poly(N) input propagation output Computation qubits Time register {|0 , |1 ,…, |T  }

Kitaev’s log-local Construction H= J in H in + J prop H prop + H out Verifier: U x =U T U T-1 …U 1 To show: If U x accepts with prob.  1-   on input | , 0 , then H has eigenvalue  . If U x accepts with prob.   on all | , 0 , then all eigenvalues of H  ½-  .

Completeness H= J in H in + J prop H prop + H out Verifier: U x =U T U T-1 …U 1 To show: If U x accepts with prob.  1-   on input | , 0 , then H has eigenvalue  . If U x accepts with prob.   on all | , 0 , then all eigenvalues of H  ½-  .  |H in |   =0  |H prop |   =0  |H out |   

5-local Hamiltonians Log-local terms: Idea (Kitaev): unary | t  | 11…100…0  tT-t | t  t |  | 10  10| t,t+1 | t  t-1 |  | 110  100| t-1,t,t+1 Penalise illegal time states: S clock - space of legal time-states is preserved (invariant)

3-local Hamiltonians 5-local terms: | t  t-1 |  | 110  100| t-1,t,t+1 Idea [KR’02]: | 110  100| t-1,t,t+1  | 1  0| t (| 1  0| t )| S clock = | t  t-1 | Give a high energy penalty to illegal time states to effectively prevent transitions outside S clock : H S clock

Outline A Bit of History 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 H’ = H + V Spectrum: H … 0 groundspace S Energy gap:  ||H||>>||V|| What is the effective Hamiltonian in the lower part of the spectrum? Three-qubit gadget

Perturbation Theory H’ = H + V Spectrum: H 0 groundspace S Energy gap:  ||H||>>||V|| SS Case 1: Energy gap  >>> ||V|| S SS V -- - restriction of V to S V ++ - restriction of V to S  What is the effective Hamiltonian in the lower part of the spectrum? Projection Lemma: H eff = V --   (same spectrum)  =O(||V|| 2 /  )

Perturbation Theory H’ = H + V Spectrum: H 0 groundspace S Energy gap:  ||H||>>||V|| Theorem: SS What is the effective Hamiltonian in the lower part of the spectrum? Case 2: Fine tune the energy gap  > ||V|| S SS V -- - restriction of V to S V ++ - restriction of V to S 

Perturbation Theory H 0 groundspace S Energy gap:  Theorem: SS First order Second order Third order The lower spectrum of H’ is close to the spectrum of H eff (under certain conditions). H’ = H + V

Perturbation Theory H 0 groundspace S Energy gap:  Theorem: SS First order: ||V|| 2 <<  The lower eigenvalues (<||V||) of H’ are close to the eigenvalues of H eff (under certain conditions). Projection Lemma H’ = H + V

Three-qubit gadget H=P 1 P 2 P 3 3-local B A C ZZ P1XAP1XA P2XBP2XB P3XCP3XC Terms in H’ are 2-local H eff =P 1 P 2 P 3 3-local

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

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

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

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

Three-qubit gadget B A C H’ = H + V 2 P2XBP2XB 3 P3XCP3XC 1 P1XAP1XA Theorem:

Three-qubit gadget B A C H’ = H + V 2 P2XBP2XB 3 P3XCP3XC 1 P1XAP1XA Theorem:

Three-qubit gadget B A C H’ = H + V 2 P2XBP2XB 3 P3XCP3XC 1 P1XAP1XA Theorem:

Three-qubit gadget B A C H’ = H + V 2 P2XBP2XB 3 P3XCP3XC 1 P1XAP1XA Theorem:  =  -3 0 H 0  -1 V H eff const.

2-local Hamiltonian is QMA-complete start with the QMA-complete 3-local Hamiltonian replace each 3-local term by 3-qubit gadget

Outline A Bit of History QMA Local Hamiltonians Previous Constructions The 3-qubit Gadget Implications Adiabatic computation Other applications of the technique

Implications for Adiabatic Computation Adiabatic computation [Farhi et al.’00]: “track” the groundstate of a slowly varying Hamiltonian Standard quantum circuit: | 0…0  |T|T T gates *D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation", lanl-report quant- ph/ Adiabatic simulation*: H initial groundstate | 0…0  | 0  H final groundstate H(t) = (1-t/T’)H initial +t/T’ H final T’=poly(T): If gap 0 (H(t))- 1 (H(t)) between groundstate and first excited state is 1/poly(T)

Implications for Adiabatic Computation 2-local adiabatic computation is equivalent to standard quantum computation Our result also implies: *D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation", lanl-report quant- ph/ | 0…0  | 0  adiabat H(t) = (1-t/T’) H in + t/T’ H prop Log-local*: Replace with 2-local: H(t) = (1-t/T’)(H in +J clock H clock ) + t/T’(H prop gadget +J clock H clock )

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

References Quantum Complexity : J. Kempe, A. Kitaev, O. Regev: “The Complexity of the local Hamiltonian Problem”, quant-ph/ , to appear in Proc. FSTTCS’04 J. Kempe and O. Regev: "3-Local Hamiltonian is QMA-complete", Quantum Information and Computation, Vol. 3 (3), p (2003), lanl-report quant-ph/ Adiabatic Computation : D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computationis Equivalent to Standard Quantum Computation", lanl-report quant-ph/ , to appear in FOCS’04 *Photo: Oded Regev: Ladybug reading “3-local Hamiltonian” paper

MA V “yes” instance: x  L yes witness: y 1 (accept) V “no” instance: x  L no for all “witnesses” y 0 (reject) MA – Merlin-Artur: Def. L  MA if there is a poly-time verifier V and a polynomial p s.t. 0 (reject) prob 1-  prob  1 (accept) prob  prob 1- 