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![Page 1: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/1.jpg)
Quantum Information Theory as a Proof Technique
Fernando G.S.L. BrandãoETH Zürich
With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner (MIT), D. Steurer (Cornell), J. Yard (Station Q), Y. Zhou (CMU)
Caltech, February 2013
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Simulating quantum is hard
More than 25% of DoE supercomputer power is devoted
to simulating quantum physics
Can we get a better handle on this simulation problem?
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Simulating quantum is hard, secrets are hard to conceal
More than 25% of DoE supercomputer power is devoted
to simulating quantum physics
Can we get a better handle on this simulation problem?
All current cryptography is based on unproven hardness
assumptions
Can we have better security guarantees for our secrets?
![Page 4: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/4.jpg)
Quantum Information Science…
… gives a path for solving both problems. But it’s a long
journey
QIS is at the crossover of computer science,
mathematics and physics
Physics CS
Math
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The Two Holy Grails of QIS
Quantum Computation:
Use of well-controlled quantum systems for performing
computation
Exponential speed-ups over classical computing
E.g. Factoring (RSA)Simulating quantum systems
Quantum Cryptography:
Use of well-controlled quantum systems for secret key
distribution
Unconditional security based solely on the correctness of
quantum mechanics
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The Two Holy Grails of QIS
Quantum Computation:
Use of well controlled quantum systems for performing
computations.
Exponential speed-ups over classical computing
Quantum Cryptography:
Use of well-controlled quantum systems for secret key
distribution
Unconditional security based solely on the correctness of
quantum mechanics
State-of-the-art: 5 qubits computer
Can prove with high probability that 15 = 3 x 5
![Page 7: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/7.jpg)
The Two Holy Grails of QIS
Quantum Computation:
Use of well controlled quantum systems for performing
computations.
Exponential speed-ups over classical computing
Quantum Cryptography:
Use of well controlled quantum systems for secret key
distribution
Unconditional security based solely on the correctness of
quantum mechanics
State-of-the-art: 5 qubits computer
Can prove with high probability that 15 = 3 x 5
State-of-the-art: 100 km
Still many technological challenges
![Page 8: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/8.jpg)
What If……one can never build a quantum computer?
Answer 1: Then we ought to know why: new physical principle that makesquantum computers impossible?
Answer 2: QIS is interesting and useful independent of building a quantum computer
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What If……one can never build a quantum computer?
Answer 1: Then we ought to know why: new physical principle that makes quantum computers impossible?
Answer 2: QIS is interesting and useful independent of building a quantum computer
This talk
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Outline
• Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture
• Conclusions
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Outline
• Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture
• Conclusions
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Problem 1: For M in H(Cd) (d x d matrix) compute
Very Easy!
Problem 2: For M in H(Cd Cl), compute
Quadratic vs Biquadratic Optimization
Next:Best known algorithm (and best hardness result) using
ideas from Quantum Information Theory
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Problem 1: For M in H(Cd) (d x d matrix) compute
Very Easy!
Problem 2: For M in H(Cd Cl), compute
Quadratic vs Biquadratic Optimization
Next:Best known algorithm (and best hardness result) using
ideas from Quantum Information Theory
![Page 14: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/14.jpg)
Quantum Mechanics
Pure State: norm-one vector in Cd:
Mixed State: positive semidefinite matrix of unit trace:
Quantum Measurement: To any experiment with d outcomes we associate d PSD matrices {Mk} such that ΣkMk=I Born’s rule:
Dirac notation reminder:
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Quantum Entanglement
Pure States:
If , it’s separable
otherwise, it’s entangled.
Mixed States:
If , it’s separable
otherwise, it’s entangled.
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A Physical Definition of Entanglement
LOCC: Local quantum Operations and Classical Communication
Separable states can be created by LOCC:
Entangled states cannot be created by LOCC: non-classical correlations
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The Separability Problem • Given
is it entangled?
• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP
SEP Dε
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The Separability Problem • Given
is it entangled?
• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP
• Dual Problem: Optimization over separable states
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The Separability Problem • Given
is it entangled?
• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP
• Dual Problem: Optimization over separable states
• Relevance: Entanglement is a resource in quantum cryptography, quantum communication, etc…
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Norms on Quantum States EDIT
How to quantify the distance in Weak-Membership?
1. Euclidean Norm (Hilbert-Schmidt): ||X||2 = tr(XTX)1/2
2. Trace Norm: ||X||1 = tr((XTX)1/2)
||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))
3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : {M, I - M} in 1-LOCC
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Norms on Quantum States EDIT
How to quantify the distance in Weak-Membership?
1. Euclidean Norm (Hilbert-Schmidt): ||X||2 = tr(XTX)1/2
2. Trace Norm: ||X||1 = tr((XTX)1/2)
||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))
3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : {M, I - M} in 1-LOCC
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Norms on Quantum States EDIT
How to quantify the distance in Weak-Membership?
1. Euclidean Norm (Hilbert-Schmidt): ||X||2 = tr(XTX)1/2
2. Trace Norm: ||X||1 = tr((XTX)1/2)
||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))
3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : M in 1-LOCC
M in 1-LOCC
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Previous Work
When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable
Beautiful theory behind it (PPT, entanglement witnesses, etc)
Horribly expensive algorithms
State-of-the-art: 2O(|A|log|B|) time complexity for either ||*||2 or ||*||1
Same for estimating hSEP (no better than exaustive search!)
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Hardness Results
When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable
(Gurvits ’02, Gharibian ‘08) NP-hard with ε=1/poly(|A||B|) for ||*||1 or ||*||2
(Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|)) time algorithm with ε=Ω(1) and any ν+μ>0 for ||*||1, unless ETH fails
ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time(Impagliazzo&Paruti ’99)
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Quasipolynomial-time Algorithm
(B., Christandl, Yard ‘11) There is a exp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*||1-LOCC)
2 norm natural from geometrical point of view
1-LOCC norm natural from operational point of view (distant lab paradigm)
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Quasipolynomial-time Algorithm
(B., Christandl, Yard ‘11) There is a exp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*||1-LOCC)
Corollary 1: Solving WSEP(||*||2, ε) is not NP-hard for ε = 1/polylog(|A||B|), unless ETH fails
Contrast with:
(Gurvits ’02, Gharibian ‘08) Solving WSEP(||*||2, ε) NP-hard for ε = 1/poly(|A||B|)
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Quasipolynomial-time Algorithm
(B., Christandl, Yard ‘11) There is a exp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*||1-LOCC)
Corollary 2: For M in 1-LOCC, can compute hSEP(M) within additive error ε in time exp(O(ε-2log|A|log|B|))
Contrast with:
(Harrow, Montanaro ’10) No exp(O(log1-ν|A|log1-μ|B|)) algorithm for hSEP(M) with ε=Ω(1), for separable M
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Algorithm: SoS Hierarchy
Polynomial optimization over hypersphere
Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)
- Round k takes time dim(M)O(k)
- Converge to hSEP(M) when k -> ∞
SoS is the strongest SDP hierarchy known for polynomial optimization (connections with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)
We’ll derive SoS hierarchy by a quantum argument
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Algorithm: SoS Hierarchy
Polynomial optimization over hypersphere
Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)
- Round k SDP has size dim(M)O(k)
- Converge to hSEP(M) when k -> ∞
![Page 30: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/30.jpg)
Algorithm: SoS Hierarchy
Polynomial optimization over hypersphere
Sum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)
- Round k SDP has size dim(M)O(k)
- Converge to hSEP(M) when k -> ∞
SoS is the strongest SDP hierarchy known for polynomial optimization (connections with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)
We’ll derive SoS hierarchy by a quantum argument
![Page 31: Quantum Information Theory as a Proof Technique Fernando G.S.L. Brandão ETH Zürich With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649ca45503460f949642ed/html5/thumbnails/31.jpg)
Classical Correlations are Shareable
Given separable state
Consider the symmetric extension
Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all j in [k], tr\ Bj (ρAB1…Bk) = ρAB
A
B1B2B3B4
Bk
…
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Entanglement is Monogamous(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)
ρAB separable iff ρAB is k-extendible for all k
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SoS as optimization over k-extensions
(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to optimization over k-extendible states(plus PPT (positive partial transpose) test):
(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)
give alternative proof that hierarchy converges
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
gives a proof of equivalence
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SoS as optimization over k-extensions
(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to optimization over k-extendible states(plus PPT (positive partial transpose) test):
(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)
give alternative proof that hierarchy converges(before the hierarchy was even defined :-)
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
gives a proof of equivalence
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How close to separable are k-extendible states?
(Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound
If ρAB is k-extendible
But there are k-extendible states ρAB s.t.
Improved de Finetti Bound
If ρAB is k-extendible:
for 1-LOCC or 2 normk=2ln(2)ε-2log|A| rounds of SoS solves the problemwith a SDP of size |A||B|k = exp(O(ε-2log|A| log|B|))
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How close to separable are k-extendible states?
(Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound
If ρAB is k-extendible
But there are k-extendible states ρAB s.t.
Improved de Finetti Bound (B, Christandl, Yard ‘11)
If ρAB is k-extendible:
for 1-LOCC or 2 norm
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How close to separable are k-extendible states?
Improved de Finetti Bound (B, Christandl, Yard ’11)
If ρAB is k-extendible:
for 1-LOCC or 2 norm
k=2ln(2)ε-2log|A| rounds of SoS solves WSEP(ε) with a SDP of size
|A||B|k = exp(O(ε-2log|A| log|B|))SEP Dε
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Proving…Improved de Finetti Bound (B, Christandl, Yard ’11)
If ρAB is k-extendible:for 1-LOCC or 2 norm
Proof is information-theoretic
Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))
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Proving…
Proof is information-theoretic
Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))
Let ρAB1…Bk be k-extension of ρAB
2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)
For some l<k: I(A:Bl|B1…Bl-1) < 2 log|A|/k
(chain rule)
Improved de Finetti Bound (B, Christandl, Yard ’11)
If ρAB is k-extendible:for 1-LOCC or 2 norm
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Proving…
Proof is information-theoretic
Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))
Let ρAB1…Bk be k-extension of ρAB
2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)
For some l<k: I(A:Bl|B1…Bl-1) < 2 log|A|/k
(chain rule)
What does it imply?
Improved de Finetti Bound (B, Christandl, Yard ’11)
If ρAB is k-extendible:for 1-LOCC or 2 norm
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Quantum Information?Nature isn't classical, dammit,
and if you want to make a simulation of Nature, you'd
better make it quantum mechanical, and by golly it's a wonderful problem, because it
doesn't look so easy.
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Quantum Information?Nature isn't classical, dammit,
and if you want to make a simulation of Nature, you'd
better make it quantum mechanical, and by golly it's a wonderful problem, because it
doesn't look so easy.
Bad news• Only definition I(A:B|C)=H(AC)
+H(BC)-H(ABC)-H(C)
• Can’t condition on quantum information.
• I(A:B|C)ρ ≈ 0 doesn’t imply ρ is approximately separablein 1-norm (Ibinson, Linden, Winter ’08)
Good news• I(A:B|C) still defined
• Chain rule, etc. still hold
• I(A:B|C)ρ=0 implies ρ is separable
(Hayden, Jozsa, Petz, Winter‘03)
information theory
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Chain rule:2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)
Then
Proving the Bound
Thm (B, Christandl, Yard ‘11) For ||*||1-LOCC or ||*||2
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Conditional Mutual Information Bound
• Coding Theory Strong subadditivity of von Neumann entropy as state redistribution rate (Devetak, Yard ‘06)
• Large Deviation Theory Hypothesis testing for entanglement (B., Plenio ‘08)
( )
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hSEP equivalent to
1. Injective norm of 3-index tensors
2. Minimum output entropy quantum channel
3. Optimal acceptance probability in QMA(2)
4. 2->4 norm of projectors:
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Unique Games Conjecture(Unique Games Conjecture, Khot ‘02) For every ε>0 it’s NP-hard to tell for a system of equations xi + xj = c mod k
YES) More than 1-ε fraction constraints satisfiableNO) Less than ε fraction satisfiable
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Unique Games Conjecture(Unique Games Conjecture, Khot ‘02) For every ε>0 it’s NP-hard to tell for a system of equations xi + xj = c mod k
YES) More than 1-ε fraction constraints satisfiableNO) Less than ε fraction satisfiable
(Raghavedra ‘08) UGC implies 2-level SoS gives the best approximation algorithm for all Constraints Satisfaction Problems (max-cut, vertex cover, …)
Major barrier in current knowledge of algorithms for combinatorial problems
(Arora, Barak, Steurer ‘10) exp(nO(ε)) time algorithm for UG
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Small Set Expansion Conjecture(Small Set Expansion Conjecture, Raghavendra, Steurer ‘10) For every ε, δ>0 it’s NP-hard to tell for a graph G = (V, E) whether
YES) Φ(M) < ε for a region M of size ≈ δ|V|,
NO) Φ(M) > 1-ε for all regions M of size ≈ δ|V|,
Expansion:
M
V
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Small Set Expansion Conjecture
(Raghavendra, Steurer ‘10) Small Set Expansion ≈ Unique Games(Barak, B, Harrow, Kelner, Steurer, Zhou ‘12) Rough estimate 2->4 norm of projector onto top eigenspace of graphs ≈ Small Set Expansion
Expansion:
M
V
(Small Set Expansion Conjecture, Raghavendra, Steurer ‘10) For every ε, δ>0 it’s NP-hard to tell for a graph G = (V, E) whether
YES) Φ(M) < ε for a region M of size ≈ δ|V|,
NO) Φ(M) > 1-ε for all regions M of size ≈ δ|V|,
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time under ETH
5. Improvement (hardness specific P) would imply exp(O(log2(n))) time lower bound to Unique Games under ETH
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time under ETH
5. Improvement (hardness specific P) would imply exp(O(log2(n))) time lower bound to Unique Games under ETH
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time under ETH
5. Improvement (hardness specific P) would imply exp(O(log2(n))) time lower bound to Unique Games under ETH
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show one cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time (under ETH)
5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound to Unique Games (under ETH)
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show one cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time (under ETH)
5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound on Unique Games (under ETH)
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)
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Quantum Bound on SoS Implies
1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.
2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)
3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time
4. Other quantum arguments show one cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time (under ETH)
5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound on Unique Games (under ETH)
(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)Both improvements can be casted as open problems in quantum information theory
Maybe to solve UGC we should learn more about QIT?
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Outline
• Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture
• Conclusions
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Quantum Mechanics, again
Dynamics in quantum mechanics is given by a
Hamiltonian H:
Equilibrium properties are also determined by H
Thermal state:
Groundstate:
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Constraint Satisfaction Problems vs Local Hamiltonians
k-arity CSP:
Variables {x1, …, xn}, alphabet Σ
Constraints:
Assignment:
Unsat :=
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Constraint Satisfaction Problems vs Local Hamiltonians
k-arity CSP:
Variables {x1, …, xn}, alphabet Σ
Constraints:
Assignment:
Unsat :=
k-local Hamiltonian H:
n qudits in
Constraints:
qUnsat := E0 : min eigenvalue
H1
qudit
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C. vs Q. Optimal AssignmentsFinding optimal assignment of CSPs can be hard
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C. vs Q. Optimal AssignmentsFinding optimal assignment of CSPs can be hard
Finding optimal assignment of quantum CSPs can be even harder
(BCS Hamiltonian groundstate, Laughlin states for FQHE,…)
Main difference: Optimal Assignment can be a highly entangled state (unit vector in )
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Mean-Field……consists in approximating groundstate by a product state
is a CSP
Successful heuristic in Quantum Chemistry Condensed matter
Folklore: Mean-Field good when Many-particle interactions Low entanglement in state
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Mean-Field……consists in approximating groundstate by a product state
is a CSP
Successful heuristic in Quantum Chemistry Condensed matter
Folklore: Mean-Field good when Many-particle interactions Low entanglement in state
Can we make folklore more rigorous?
Can we put limitations on use of mean-field and related schemes?
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The Local Hamiltonian Problem and Quantum Complexity Theory
ProblemGiven a local Hamiltonian H, decide if E0(H)=0 or E0(H)>Δ
E0(H) : minimum eigenvalue of H
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The Local Hamiltonian Problem and Quantum Complexity Theory
ProblemGiven a local Hamiltonian H, decide if E0(H)=0 or E0(H)>Δ
E0(H) : minimum eigenvalue of H
Thm (Kitaev ‘99) The local Hamiltonian problem is QMA-complete for Δ = 1/poly(n)
(analogue Cook-Levin thm)
QMA is the quantum analogue of NP, where the proof and the computation are quantum
Input Witness
U1
…. U5U4 U3 U2
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The meaning of it
It’s believed QMA ≠ NP
Thus there is generally no efficient classical description of groundstates of local Hamiltonians
What’s the role of the promise gap Δ on the hardness?
…. But first, what happens for CSP?
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PCP TheoremPCP Theorem (Arora et al ’98, Dinur ‘07): There is a ε > 0 s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm
- NP-hard even for Δ=Ω(m)
- Equivalent to the existence of Probabilistically Checkable Proofs for NP.
- Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n1-ε-factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)
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PCP TheoremPCP Theorem (Arora et al ’98, Dinur ‘07): There is a ε > 0 s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm
- NP-hard even for Δ=Ω(m)
- Equivalent to the existence of Probabilistically Checkable Proofs for NP.
- Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n1-ε-factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)
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PCP TheoremPCP Theorem (Arora et al ’98, Dinur ‘07): There is a ε > 0 s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm
- NP-hard even for Δ=Ω(m)
- Equivalent to the existence of Probabilistically Checkable Proofs for NP.
- Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (n1-ε-factor)) (obs: Unique Game Conjecture is about the existence of strong form of PCP)
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Quantum PCP?The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E0(H)=0 or (ii) E0(H) > εm.
- (Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.
- Equivalent to estimating mean groundenergy to constant accuracy (eo(H) := E0(H)/m)
- And to estimate the energy at constant temperature
- At least NP-hard (by PCP Thm) and in QMA
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Quantum PCP?The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E0(H)=0 or (ii) E0(H) > εm.
- (Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.
- Equivalent to estimating mean groundenergy to constant accuracy (eo(H) := E0(H)/m)
- And to estimate the energy at constant temperature
- At least NP-hard (by PCP Thm) and in QMA
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Quantum PCP?The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E0(H)=0 or (ii) E0(H) > εm.
- (Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.
- Equivalent to estimating mean groundenergy to constant accuracy (eo(H) := E0(H)/m)
- And to estimate the energy at constant temperature
- At least NP-hard (by PCP Thm) and in QMA
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Quantum PCP?The qPCP conjecture: There is ε > 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H with m local terms determine whether (i) E0(H)=0 or (ii) E0(H) > εm.
- (Bravyi, DiVincenzo, Loss, Terhal ‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.
- Equivalent to estimating mean groundenergy to constant accuracy (eo(H) := E0(H)/m)
- And to estimate the energy at constant temperature
- At least NP-hard (by PCP Thm) and in QMA
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Quantum PCP?
NP
QMAqPCP?
?
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Previous Work and Obstructions
(Aharonov, Arad, Landau, Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm (gap amplification)
But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignment
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Previous Work and Obstructions
(Aharonov, Arad, Landau, Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm (gap amplification)
But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignment
(Bravyi, Vyalyi ’03; Arad ’10; Hastings ’12; Freedman, Hastings ’13; Aharonov, Eldar ’13, …) No-go for large class of commuting Hamiltonians and almost commuting Hamiltonians
But: Commuting case might always be in NP
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Approximation in NP(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.
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Approximation in NP(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.
Let {Xi} be a partition of the sites with each Xi having m sites.
X1
X3X2
m < O(log(n))
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Approximation in NP(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.
Let {Xi} be a partition of the sites with each Xi having m sites. Then there are products states ψi in Xi s.t.
Ei : expectation over Xi
deg(G) : degree of GΦ(Xi) : expansion of Xi
S(Xi) : entropy of groundstate in Xi
X1
X3X2
m < O(log(n))
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Approximation in NP(B., Harrow ‘12) Let H be a 2-local Hamiltonian on qudits with interaction graph G(V, E) and |E| local terms.
Let {Xi} be a partition of the sites with each Xi having m sites. Then there are products states ψi in Xi s.t.
Ei : expectation over Xi
deg(G) : degree of GΦ(Xi) : expansion of Xi
S(Xi) : entropy of groundstate in Xi
X1
X3X2
Approximation in terms of 3 parameters:
1. Average expansion2. Degree interaction graph3. Average entanglement groundstate
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Approximation in terms of degree
No classical analogue:
(PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σβ/deg(G)γ
Parallel repetition: C -> C’ i. deg(G’) = deg(G)k ii. Σ’ = Σk
ii. Unsat(G’) > Unsat(G)(Raz ‘00) even showed Unsat(G’) approaches 1 exponentially fast
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Approximation in terms of degree
No classical analogue:
(PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σβ/deg(G)γ
Contrast: It’s in NP determine whether a Hamiltonian H is s.t E0(H)=0 or E0(H) > αd3/4/deg(G)1/8
Quantum generalizations of PCP and parallel repetition cannot both be true (assuming QMA not in NP)
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Approximation in terms of degree
No classical analogue:
(PCP + parallel repetition) For all α, β, γ > 0 it’s NP-complete to determine whether a CSP C is s.t. Unsat = 0 or Unsat > α Σβ/deg(G)γ
Bound: ΦG < ½ - Ω(1/deg) implies
Highly expanding graphs (ΦG -> 1/2) are not hard instances
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Approximation in terms of degree
1-D
2-D
3-D
∞-D
…shows mean field becomes exact in high dim
Rigorous justification to folklore in condensed matter physics
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Approximation in terms of average entanglement
The problem is in NP if entanglement of groundstate satisfies a subvolume law:
Connection of amount of entanglement in groundstate and computational complexity of the model
X1
X3X2
m < O(log(n))
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New Classical Algorithms for Quantum Hamiltonians
Following same approach we also obtain polynomial time algorithms for approximating the groundstate energy of
1. Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07)2. Dense Hamiltonians, improving on (Gharibian, Kempe ‘10)3. Hamiltonians on graphs with low threshold rank, building on
(Barak, Raghavendra, Steurer ‘10)
In all cases we prove that a product state does a good job and use efficient algorithms for CSPs.
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Proof Idea: Monogamy of Entanglement
Cannot be highly entangled with too many neighbors
Entropy quantifies how entangled can be
Proof uses information-theoretic techniques (chain rule of conditional mutual information, informationally complete POVMs, etc) to make this intuition precise
Inspired by classical information-theoretic ideas for bounding convergence of SoS hierarchy for CSPs (Tan, Raghavendra ‘10, Barak, Raghavendra, Steurer ‘10)
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Quantum Information Method• Condensed Matter Physics: Tensor Network States (Cirac, Hastings, Verstraete, Vidal, …)
• Mathematical Physics: Area Law from Exponential Decay of Correlations (B., Horodecki ‘12)
• Computational Complexity: Lower bounds on LP extensions for Travel Salesman Problem (Fiorini et al ‘11)
• Compressed Sensing: Better low rank matrix recovery methods (Gross et al ‘10)
• Etc, see (Drucker, de Wolf ‘09) for more
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ConclusionsQIT useful to bound SoS hierarchy
- Quasi-polynomial algorithm for deciding entanglement - Connections 2->4 norm, Small Set Expansion - New approach to resolve UGC (improve quantum SoS bound and/or quantum hardness)
QIT useful to bound efficiency of mean-field theory
- Cornering quantum PCP - Poly-time algorithms for planar and dense Hamiltonians
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Thank you!