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Is entanglement “robust”?
Lior Eldar
MIT / CTP
Joint work with Aram Harrow
Highlights
 We prove a conjecture of Freedman and Hastings
[‘12] that there exist quantum systems of “robust
entanglement”- called NLTS.
 Tightly connected to the quantum PCP conjecture:
 one of the major open problems in quantum
complexity theory.
 Connected to the conjecture on quantum locallytestable codes (qLTC).
 Implies that the folklore that quantum entanglement
is “fragile” is an artifact of considering spatiallylocalized systems.
Quantum PCP
Complexity perspective
Local Hamiltonians
 n qubits
 Sum of few-body terms, WLOG projectors.
 Fractional energy of a Hamiltonian:
In complexity theory
 Local Hamiltonian problem (LHP):
 Decide: is minΨE(Ψ)=0 or minΨE(Ψ)>1/poly.
 QMA - the quantum analog of MA: quantum
prover, quantum verifier, bounded error.
 Theorem [Kitaev ‘98]: LHP is QMA-complete.
Classical constraint
satisfaction problems (CSP)
 CSP = classical version of LH. All local
terms are diagonal in standard basis.
 Like 3-SAT formulas.
 PCP theorem [AS,ALMSS,Dinur]: NP-hard
even to distinguish minimal energy 0 from
minimal energy > 0.1
Quantum PCP: hardness of LHP
 Approximate-LHP: Given LH, decide whether there
exists a state t such that E(t) = 0 or whether E(t)>1/3 for
all states t.
 Approximate LHP is NP-hard, and in QMA.
 Quantum PCP conjecture [AALV ‘09] : ApproximateLHP is QMA-hard.
NLTS
Entanglement perspective
Quantify entanglement
|0>
U
U
|0>
U
|0>
U
U
|0>
U
|0>
U
Complexity of a
state = minimal
depth circuit
“Robustly-quantum” systems 2:
NLTS
 [Freedman, Hastings ‘13]: No low-energy trivial
states.
 Definition: “trivial states” - depth = O(1).
 NLTS Conjecture: there is a constant c>0, and an
infinite family of LH’s in which all “trivial states”
have fractional energy at least c.
qPCP NLTS
 If NLTS is false  NP to every Hamiltonian
 Suppose Ѱ = U |000..0>.
qPCP
H
U
U+
NLTS
Main Result
Theorem [E., Harrow ‘15]:
There exists an explicit infinite family of 7-local
commuting Pauli Hamiltonians, and a constant c such
that:
 If a circuit U generates a quantum state whose energy
is at most c  U has depth at least Ω(log(# qubits)).
• Proof techniques: products of hyper-graphs, locallytestable codes, Hamiltonian (graph)-powering, degreereduction, uncertainty principle
Perspective on NLTS
Example of highly-entangled states:
quantum code-states
 Fact: |Ψ1>, |Ψ2> are quantum code-states, M local measurements
then <Ψ1|M|Ψ1> = <Ψ2|M|Ψ2>
 Corollary: quantum code-states require large circuit depth.
 Proof:
 Let U be a depth-d circuit : |Ψ⟩ = U|00..0⟩
 Define LH : H = ΣiU|0⟩⟨0|iU+
 H distinguishes Ψ from any orthogonal code-state but is 2d-local
  contradiction.
  no codestate can be locally generated
  Ω(log n) circuit lower-bound.
Most systems have fragile
entanglement
 Many LHs have highly entangled ground states.
 But have tensor product states that are “almost” ground
states.
 Approximating the ground energy is not “crucially”
quantum.
 NLTS means: find (a family of ) LH’s for which the highentanglement property is “robust”.
Ground-state
Why are known LH’s not
robust ? Topology !
 Known systems are embedded on low-dimensional grids
 Approximate by cutting out boxes B:
 Fractional energy of
is
 In a sense - question is unfair!
 Mystery: recent results show that expanding topology
per-se is insufficient for NLTS! [BH’13, AE’13, H’12].
The construction
Our goal:
1. Define a property of quantum states
that arise as ground states of LHs.
2. Show circuit lower bounds for this
property.
3. Show it is “robust” against constantfraction energy of the parent LH.
Lower Bounds for
Quantum Circuits
What is this property?
Low vertex expansion
 Canonical image to remember: Moses parting
the red sea !
 Essentially: two large-measure sets, separated
by large distance, require divine intervention
(quantum circuits of logarithmic depth)
Vertex expansion: analog of
Cheeger’s constant for distributions
 G = hypercube {0,1}n with edges between vertices of dist ≤ m
 V = {0,1}n
 E = { (x,y) : dist(x,y) ≤ m}
 For each S, ∂(S): – the set of strings of S adjacent to Sc, union
the set of strings in Sc adjacent to S.
 The m-th vertex expansion:
Examples of vertex
expansion
 High expansion:
 A single string (no S, with p(S)<1/2)
 Uniform distribution on the hypercube (m at least
√n)
 Low expansion:
 The cat state
 Quantum code states
 Uniform superpositions over classical codes.
Bounded-depth quantum circuits
induce high-expansion dist.
 Claim: Let U be a quantum circuit of depth d, and
 Consider its induced distribution on the first n qubits.
 Its expansion is at least 1/2 for m > 21.5d√n.
Classical case:
Harper’s theorem
 Claim:
 Classical circuit C of depth d (bounded fan-in / fanout), receives n uniform random bits, generates
distribution D on n bits. Then D has no large
separation.
 Proof:
 In D - identify two subsets S,T with measure >1/3.
 Consider the pre-images under C: C-1(S), C-1(T).
 They are at most O(√n) apart. (Harper’s theorem).
 So S,T are at distance 2d√n.
Toy proof: CAT state
 CAT state  has low expansion.
 Let U be a circuit of depth d that generates CAT : |CAT>
= U |00…0>
 Then H = U Σi|0><0|i U+ distinguishes |CAT> from
 |cat> : |00…0> - |11…1> (because <00..0| U+|cat> =
0)
 H is 2d local.
cat metric explanation
• Local Hamiltonians examine “pairs” |x><y|
• Cannot affect pairs (x,y) if d(x,y) > locality !
•  d(U) > log(n).
Quantum circuits also limited by Harper
 Start with Ψ = U |00…0>, depth(U) = d.
 Ψ’s expansion minimized at set A.
 Write Ψ = |A>+|Ac>.
 Consider the state Ψ’ = |A>-|Ac>.
A
Ac
Distinguishing Hamiltonian
 Put m = √n2O(d)
 Consider H = U Σi|0><0|i U+.
 Take the √n-degree Chebyshev polynomial of H, C(H).
 C(H) is m–local.
 By assumption: <Ψ’, Ψ> is small.
  Ψ, Ψ’ have eigenvalues 1, and <1-1/poly in H.
  Ψ, Ψ’ have eigenvalues 1, <0.7 in C(H).
 Claim: this is sufficient to conclude that the distribution
of Ψhas high expansion. Why?
Recall CAT example:
Local Hamiltonians: examine “pairs” of strings in density matrix.
Cannot affect pairs whose distance exceeds locality !
Large energy gap  high
expansion
Ac
AT
∂(A) – only place they differ  ∂(A)
must have large (relative) mass
high expansion
Locally-Testable Codes
are classical NLTS
Goal of this section
 So far: low expansion  high circuit depth
 Next goal: Hamiltonian where any low-energy state
has low expansion.
Local testability
 Turns out that “local testability” is a “turnkey” property.
 Definition: Locally-testable codes
 A subspace C of GF(2)n.
 A local tester: set of check terms {Ci} such that for
every word w :
Example: Hadamard code
 Encode x in {0,1}n as the function
c(w) = <w,x>
Express as a truth table of length 2n.
 Rate is log(n).
 It is locally testable [BLR] !
 Sample a,b at random
 Accept if and only if c(a) + c(b) = c(a+b).
 Reads only 3 bits from c
 LTC with ρ ≥ 1/3.
LTC = Classical NLTS
 We would like a locally defined system that preserves
low expansion in the presence of noise.
 “toy” example: distributions on LTCs
 Uniform distribution on a code – low expansion
 Noisy uniform distribution on a code – could have
high expansion.
 Noisy uniform distribution on an LTC – low
expansion !
How do we make this
property quantum ?
 A hypergraph product [Tillich-Zémor ‘09]:
 Takes in two classical codes C
 Produces a quantum code Q = C xTZ C.
 We show:
 If C is LTC  Q has a “residual” property of local
testability.
 Informally, means clustering.
The hyper-graph product code
[Tillich-Zémor ‘09]
 Parity-check code C, bi-partite graph G(A,B).
 Consider the transpose of C, CT – G(B,A).
 Generate two new linear codes:
 Bits are A x A, B x B
 Checks are X checks: B ⊗ I +I ⊗ A, Z checks: I ⊗
B+A ⊗ I]
AxA
BxB
Last tool: uncertainty
principle
 Suppose you have a quantum code with “residual”
local testability.
 Noisy words cluster around the original code.
 But what if they all cluster around the same
codeword ?
 One cluster  no low-expansion !!
 What is the answer ? The uncertainty principle ?
 There is a high-degree of uncertainty  at least two
clusters.
Putting it all together
 Take the Hadamard code C
 Reduce its degree as a bi-partite graph: C  C’.
 Apply the hypergraph-product to C’: derive a
quantum CSS code from a classical code.
 Argue: the local constraints of the code are an NLTS
local Hamiltonian.
Constructed LH is NLTS!
 Fix a low-energy quantum state.
 It superposes nontrivially on at least two clusters.
 These clusters correspond to different code-words
of an LTC with large distance.
 Distribution is low expansion
 So any U approximating state has d(U) > log(n).
Clustering around affine spaces !
 Live “footage” from F2n
_
Summary of NLTS-LH
 Hamiltonian corresponds to quantum CSS code.
 n qubits, O(n) local terms, each is 7-local.
 Evades previous no-go’s:
 Not expanding enough for [AE’13]
 Not 2-local so doesn’t contradict [BH’13].
 Low girth so [H’12] doesn’t apply.
 Rate is only log(n).
 Distance is √n.
Take-home message
Results
 Entanglement is not “inherently fragile”.
 Local Hamiltonians can “expel” trivial
states from the low side of the spectrum.
 They need to have an expanding topology.
 Expansion per-se is not enough !
 You need an extra structure. What is it ?
 Local testability !
Future directions
 Find NLTS Hamiltonians that actually do
something useful.
 To break the log(n) lower-bound, one needs to
abandon “light-cone” arguments and encode
computational problems…
 If you can make it QMA-hard – it’s the qPCP
conjecture.
 Try to find qLTCs – even with moderate
locality.