Barriers in Hamiltonian Complexity

Umesh V. Vazirani U.C. Berkeley

QuantumComplexity

Theory

Condensed MatterTheory

Hamiltonian Complexity

Classical Quantum

Constraint SatisfactionProblem

Local Hamiltonian

Solution Ground State

Condensed Matter Theory

• Describing ground states of local Hamiltonians,and understanding their properties.

• Problem: n qubit state described by 2n complexnumbers.

• Ground states of realistic systems have concisedescriptions.

• Reason: Limited entanglement.

Area Laws

Assuming area laws, beautiful sequence of results showing how to simulate quantum systems efficiently using tensor networks, MERAs and PEPs. [Vidal; Verstraete & Cirac, ...]

Quantum Complexity Theory

f (x1,L , xn )=c1 ∧L ∧cm H =H1 +L + Hm

Quantum Classical

Local Hamiltonian k-SAT

NP-hard to findassignment min # UNSAT clauses

[Kitaev] QMA-hard to find ground

state

Quantum Complexity Theory

f (x1,L , xn )=c1 ∧L ∧cm H =H1 +L + Hm

Quantum Classical

Local Hamiltonian k-SAT

NP-hard to findassignment min # UNSAT clauses

[Kitaev] QMA-hard to find ground

state

M[Gottesman, Irani 09] Ground states of translationally invariant 1-D

Hamiltonians hard unless BQEXP = QMAEXP

Quantum Complexity Theory

f (x1,L , xn )=c1 ∧L ∧cm H =H1 +L + Hm

Quantum Classical

Local Hamiltonian k-SAT

NP-hard to findassignment min # UNSAT clauses

?? QMA-hard to find any low

energy state?

PCP Theorem

Classical

approx

Two Major Challenges in Hamiltonian Complexity

• Prove or disprove a quantum PCP theorem.

• Prove the area law for 2-D and 3-D gapped local Hamiltonians.

Local Hamiltonians

• n qubit system

• Hamiltonian: H = 2n x 2n hermitian matrix.

• Energy operator: eigenstates of H are states with definite energy. Energy = eigenvalue.

• k-local if each term acts non-trivially on k qubits.

• Each term assigns energy penalty to state.

• Interested in structure and eigenvalue (energy) associatedwith lowest eigenstate (ground state).

H =H1 +L + Hm

3SAT as a local Hamiltonian Problem

f (x1,L , xn )=c1 ∧L ∧cm

• n bits ---> n qubits • Clause ci = x1 v x2 v x3 corresponds to 8x8 Hamiltonian matrix acting on first 3 qubits:

• Satisfying assignment is eigenvector with evalue 0.• All truth assignments are eigenvectors with eigenvalue = # unsat clauses.

hi =

10

00

00

00

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

[Kitaev] Given a local hamiltonian H = H1 + ... + Hm

it is QMA-hard to determine the minimum eigenvalue (ground state energy) of H to within 1/poly(n)

PCP Theorem: Polynomial time procedure to convert 3-SAT formula f into g:• f satisfiable implies g satisfiable• f unsatisfiable implies g < 1-c satisfiable.

[Kitaev] Given a local hamiltonian H = H1 + ... + Hm it is QMA-hard to determine the minimum eigenvalue (ground state energy) of H to within 1/poly(n).

PCP Theorem: Polynomial time procedure to convert 3-SAT formula f into g:• f satisfiable implies g satisfiable• f unsatisfiable implies g < 1-c satisfiable.

Quantum PCP: Given a local hamiltonian H = H1 + ... + Hm

is it QMA-hard to determine the minimum eigenvalue (ground state energy) of 1/m H to within c for some constant c?

i.e. Is there a quantum poly time alg that converts any local hamiltonian H into H’ such that |H’| = O(1) and if H has promise gap 1/poly(n) then H’ has promise gap constant c.

Quantum PCP Formulation

• Well balanced question: No strong intuition to call it a quantum PCP conjecture. • [Aharonov, Arad, Landau, V 2008] Proof of quantum gap amplification using the detectability lemma. Dinur’s proof uses GA + degree reduction + alphabet reduction.

• Can define quantum PCP in terms of proof checking. i.e. is there a quantum state that can be checked by accessing only constant number of qubits. The two definitions are equivalent.

Area LawFor gapped local Hamiltonians H = H1 + ... + Hm , ground state has low entanglement. Gapped = e1 - e0 > c.

[Hastings 2007] Proof of area law for 1-D systems. [Aharonov, Arad, Landau, V 2010] Simplified proof for frustration-free 1-D systems, using detectability lemma.

Area LawFor gapped local Hamiltonians H = H1 + ... + Hm , ground state has low entanglement. Gapped = e1 - e0 > c.

How to quantify entanglement?

Quantifying EntanglementA B

ψ = ci∑ ai ⊗ bi

Schmidt decomposition: {|ai>}, {|bi>} orthonormal sets

Entanglement rank = number of non-zero terms

Entanglement rank = 1 iff product state.

Entanglement entropy = classical entropy of {ci2}

Detectability Lemma

• H = H1 + ... + Hm Assume Hi = I - Pi where Pi is a projection matrix. Assume gap = e1 - e0.

• Frustration-free: Assume ground energy = 0. i.e. ground state satisfies all m constraints.

• The normalized operator G = (I - 1/m H) fixes the ground state, but shrinks all other evectors by a factor of (1 - gap/m). So if H is gapped, i.e. gap = constant, then shrinkage ~ 1/m.

• Can a local operator achieve constant shrinkage?

Overall Idea of Proof (of 1D area law)

• Step 1: Show that there is a product state |a> x |b> which has constant overlap (inner product) with the ground state.

• Step 2: Show that this implies that the ground state has constant entanglement.

Overall Idea of Proof (of 1D area law)• Step 1: Show that there is a product state |a> x |b> which has constant overlap (inner product) with the ground state.

• Step 2: Show that this implies that the ground state has constant entanglement. To prove step 2, repeatedly apply a transformation to |a> x |b> that moves it closer to the ground state without increasing its entanglement entropy much.

The detectability lemma gives exactly such a transformation.

Overlap r implies entropy = O(1/e log 1/re log D)

To prove Step 1: Assume for contradiction that the maximum overlap between ground state and a product state is at most 2-l for some large constant l.

Consider the product state above corresponding to the ground state.

Since it has small overlap with the ground state, there is a measurement that can distinguish the two with probability at least 1 - 2-l.

Use the detectability lemma to show that such a measurement can be done locally (on O(l) qudits).

Conclude that the entanglement across the boundary is proportional to l.

ρA ρB

The Numbers• Measurement on 2l qudits distinguishes product state from ground state with probability 1 - exp(-el), where e = gap.

• This implies entanglement entropy of el across this boundary.

• Now by monogamy of entanglement:

el

el/2 el/2

• e l log l < l log D => l < exp(1/e log D) M

• So overlap > exp(-el), with l < exp(1/e log D)

• Overlap r implies entropy = O(1/e log 1/re log D)

• So entanglement entropy = O(1/e log D exp(1/e log D))

Conclusions

• Proving area law in more than 1-D and quantum PCP theorem are two major challenges.

• To prove 2-D case sufficient to consider frustration-free Hamiltonians. i.e. detectability lemma applies.

• Would be interesting to know if area law breaks down for any interaction graph.