Julia Kempe CNRS & LRI, Univ. of Paris-Sud Alexei Kitaev Caltech Oded Regev Tel-Aviv University

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

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

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

U |’ = U | | Quantum Computation • A qubit is described by a unit vector in two-dimensional space: • | = c0|0 + c1|1 such that |c0|2+|c1|2=1 • |0 and |1 are simply two orthogonal vectors • An n-qubit system is described by a unit vector in a 2n dimensional space: • |C{0,1}nsuch that|| |||2=1 • An operation on an n-qubit system is described by a unitary matrix: • U C2n2nsuch that UU†=I(i.e., unitary)

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

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

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

Local Hamiltonian Problem Kitaev’s quantum Cook-Levin Theorem (’99): Local Hamiltonian is QMA-complete. • Def. k-local Hamiltonian problem: • Input: k-local Hamiltonian , , Hi 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 “witness = ground state”

Intuition: Formula: , Hamiltonians: H2 local Hamiltonians H1 Satisfying assignment is groundstate of • Energy-penalty 1 for each unsatisfied constraint. • x1x2 … xn| H |x1x2 … xn  = #unsatisfied constraints Local Hamiltonian Problem Penalties for: x1x2x3 = 010 x3x4x5 = 100 …

MAX2SAT is NP-complete • 2-local Hamiltonian is NP-hard Results Classical Quantum • 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] • MAX3SAT is NP-complete 2-local Hamiltonian ?? New result: 2-local Hamiltonian is QMA-complete • 1-local Hamiltonian is in P

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

V Vx Classical Cook-Levin Theorem • Thm: SAT is NP-complete • Proof: First, given a verifier V, encode the input into V x1 x2 … y1 y2 … 0 0 … y1 y2 … 0 0 … input x 1 1 witness y witness y ancilla 0 ancilla 0

propagationclauses output clause Classical Cook-Levin Theorem • Thm: SAT is NP-complete • Proof: Next, create a tableau of variables and 3 kinds of clauses. z01 ancilla clauses z02 z03 z04 ancilla z0N z1N z2N zTN time = 0 1 2 3 4 … T

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

output clause 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 z01 ancilla clauses z02 z03 propagationclauses   z04 ancilla qubits z0N zTN |0|1 |2 … |T

output clause Quantum Cook-Levin Theorem • Instead of tensoring the columns, we put them in superposition • So the witness is a sum over history |0|1|2 … |T z01 ancilla clauses z02 z03 propagationclauses z04 ancilla qubits z0N zTN | |0|0+|1|1+…+|T|T

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

t T-t Reducing Locality 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: Sclock - space of legal time-states is preserved (invariant) ▪ Thm [KempeRegev]: 3-local Hamiltonian is QMA-complt. |tt|  |1010|t,t+1 |tt-1|  |110100|t-1,t,t+1

… Spectrum: H H’ = H + V Energy gap: ||H||>>||V|| 0 groundspace S Three-qubit gadget 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?

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

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

Heff=P1P2P3 3-local H=P1P2P3 3-local 2 2 3 3 P2XB P3XC ZZ B C ZZ ZZ A P1XA 1 1 Terms in H’=H+V are 2-local Fine-tune the energy gap = -3 Three-qubit gadget

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

P2XB |100  |110 Ex.: P1XA P3XC |000 |111 Three-qubit gadget 2 3 P2XB P3XC B C S={|001,|010,|100, |110,|101,|011} =-3 Energy gap: A P1XA S={|000, |111} 0 1 V++ S  S V-+ V+- Third order: S S Theorem:

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

Adiabatic theorem: g(t) gap between ground- and first excitedstate If then the final state arbitrarily close to groundstate of HP. Idea of Adiabatic Computation* • Start in the groundstate of a Hamiltonian H0 (easy to prepare) • Encode problem as a Hamiltonian HP (groundstate gives solution) • Adiabatically (slowly!) evolve from H0 to HP *E. Farhi, J. Goldstone, S. Gutmann, M. Sipser:“Quantum Computation by Adiabatic Evolution”, q-p/’00

U Adiabatic simulation*: • Hfinal • groundstate = • Kitaev’s “history state” • Hinitial • groundstate • |0…0 |0 T’=poly(T): The gap between groundstate and first excited state is 1/poly(T) at all times. H(t) = (1-t/T’)Hinitial +t/T’ Hfinal Adiabatic Computation simulates quantum computation Standard quantum circuit: |0…0 |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", FOCS’04, p.42-51

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

-2ZA -1P1XA -1P2XA H=P1P2 Heff=P1P2 “Proxy Interaction”: (with A. Landahl) -1Z1ZA -1X2XB -2YAYB Heff=Z1X2 Other applications of the gadget(work in progress) “Interaction at a distance”: H=Z1X2 only XX,YY,ZZ available Useful for Hamiltonian-based quantum architectures

Julia Kempe CNRS & LRI, Univ. of Paris-Sud Alexei Kitaev Caltech Oded Regev Tel-Aviv University