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z0M7*KH4d2$⺠4RUWV52$5eWhOe8D@2$MHpEg2$^h=\[633j0AA f33f3@*<^ ʚ;bxl8ʚ;g4EdEd0ppp@<4dddd$w0X4Results(f
Cycles and the Infinite Chain:
Family of translationallyinvariant Hamiltonians {Hn} for a cycle of nt 21dimensional particles
Spectral gap is 1/poly(n)
For any state in the ground space of Hn, and any m, there exist regions of size of m whose entanglement entropy is W (min{m,n}).
Entanglement bounds for a constant fraction of regions of size m.
Bounds hold in the limit as t tends towards infinity.
NbB63.@L
B6b*ZO~$1D Area Law$f
Upper bound on the entanglement entropy for any ground state of a 1D Hamiltonian H, independent of region size but exponentially dependent on 1/D, where D is the spectral gap of H. [Hastings 07]
Gottesman and Hastings: is the dependence on 1/D tight?
Family of 1D Hamiltonians with unique ground state with regions whose entanglement entropy is W(poly(1/D)).
Previously, best known such lower bound was W(log(1/D))
arXiv:0901.1108
Construction present here gives a similar result.
PPPP2PP/^01 ViHamiltonian Construction Basics (f6Type I terms (illegal pairs)
ab><ab
Energy penalty for: & .xxxabxxxxx& .
Type II terms (transition rules)
(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f6Type I terms (illegal pairs)
ab><ab
Energy penalty for: & .xxxabxxxxx& .
Type II terms (transition rules)
(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f`oOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KapConstruction Overview(f
2PConstruction Overview(f
^PPP3Target Ground State(f8Target Ground State(fTarget Ground State(f9Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f:Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f;Target Ground State(fTarget Ground State(fTarget Ground State(fFirst Round(f,The Hamiltonian& so farfHtrans = sum of terms from transition rules as applied to all neighboring pairs of particles.
Hlegal = sum of terms from illegal pairs
H = Htrans + Hlegal
(^G
Z
% $$3$3$3(HY*
WellFormed StatesfA state in the standard basis is wellformed if it is of the form:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
& or any substring of these forms
Can be checked by local checks
RDZ&ZZ;ZZBZZZZ!3" $3$3$3(G
SWellFormed StatesfCan not enforce by local checks that a legal state has a control state.
E.g., eeee& & eee must be a legal state.
If the state is bracketed ( a < on left end and a > on the right end), then a legal state must be exactly on of the following three forms:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
JHZ+ZZ&ZZ;ZZZZZH* $$W 3 3 3$P
Properties of WellFormed States!!(f.
Htrans + Hlegal is closed over the subspace spanned by wellformed states.
(All additional terms will be diagonal in standard basis).
For each wellformed state, at most one transition rule applies in the forward direction and at most one transition rule applies in the reverse direction.
"O;
<;$3$3$3(,State Graph(f2
Nodes set of all state in standard basis.
Edges  directed edge from state A to state B if B can be obtained by applying one transition rule to A.
Wellformed states are disconnected from the rest of the graph.
State graph restricted to wellformed states form disjoint paths.&e3 dq Hamiltonians Restricted to Paths!!(f
+The Hamiltonian restricted to a single path,,(f&
If Hlegal has a nonzero entry, the minimum eigenvalue of H restricted to the subspace spanned by states in the path is W(1/l3) which is W(1/n6). [Kitaev 02]
If Hlegal is all zero, the state which is the uniform superposition of states in the path has zero energy.
`PPPPPPPPP pb333 P
#
^ w
Bad States(fWill check for these states by showing that they evolve (via forward or backward application of the transition rules) to illegal states.Zfs(How To Check for Bad States: An Example))(f
ZZZ333 +(How To Check for Bad States: An Example))(f
ZZZ333 ,(How To Check for Bad States: An Example))(f
ZZZ333 (How To Check for Bad States: An Example))(f
ZZZ333 .(How To Check for Bad States: An Example))(f
ZZZ333 /
Need to Show:(fDA path that starts with
corresponds to a zero energy state for H.
A path that does not start with
contains a state that has an illegal pair
OR does not have states that are bracketed. (Start with < and end with > ).<!EInitializing Qubits(f
Hinit =  ><  Penalty for state U
Ensures that ground state corresponds to a path whose initial state has qubits set to
* , V , 333 ,pEnforcing Bracketed States(fHbracket = I   ><    >< 
H = n( Htrans + Hlegal ) + Hinit + Hbracket
n( Htrans + Hlegal ) term ensures there are no bracket terms in the middle.
Hbracket term gives an energy benefit for having brackets at the end
>\Z ZZZZ , , 9f<ff
f 333 /;G2Entropy of Entanglement(f
Entropy of Entanglement(f'Entropy of Entanglement(fFinite Cycle of size tn(ffChange Hlegal so that the pair is allowed.
Wellformed states look like:
A sequence from a to a is a segment.
H is closed on the set of wellformed states for a fixed set of segments.
*ZZZZZZ
u M$ $ $$3$3$3(Finite Cycle Cont. (ffDH = p(n)(Hlegal + Htrans +Hinit) + Hsize
For p(n) large enough, using the Projection Lemma of KempeKitaevRegev, we can assume that the ground state of H is composed of tensor projects of ground states for finite chains.
Ground state for finite chain of length l is
Ground state for H will have form:
J8
$$$$$$$$
$ $ $$3$3$3(t.Hsize,f
fHsize= (1/n)I
  >< 
+ (n1)/Tn[ ><  +  ><  +  ><  ]
Tl is the number of standard basis states in the support of the ground state for a segment of length l.
if and only if l=n
Otherwise
&Q! $ 4ff 3 3 3$$["Ground States for the Finite Cycle0#"(f(f<
n orthogonal ground states, each a translation one site over.
For any region size a constant fraction of the regions of that size have high entanglement.
Superposition of all n states is translationally invariant and for every region size, all regions of that size have high entanglement.
ZZZ#ZZ ZZZZZZ#ff 3 3 3$n4
Open Problems(f
Improve gap lower bound on entropy as a function of 1/D.
For a given region size m, can we achieve high entropy for all regions of size m?
Unique ground state for finite cycle and infinite chain?
Can we achieve high entanglement entropy for all region sizes simultaneously for a translationallyinvariant Hamiltonian?
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For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
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For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
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Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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Introduction
IntroductionPrevious WorkTranslationInvarianceResultsResults1D Area Law Hamiltonian Construction Basics Hamiltonian Construction Basics Hamiltonian Construction BasicsOne Dimensional HamiltoniansOne Dimensional HamiltoniansOne Dimensional HamiltoniansOne Dimensional HamiltoniansConstruction OverviewConstruction OverviewTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateTarget Ground StateFirst RoundThe Hamiltonianso farWellFormed StatesWellFormed States!Properties of WellFormed StatesState Graph!Hamiltonians Restricted to Paths,The Hamiltonian restricted to a single pathBad States)How To Check for Bad States: An Example)How To Check for Bad States: An Example)How To Check for Bad States: An Example)How To Check for Bad States: An Example)How To Check for Bad States: An ExampleNeed to Show:Initializing QubitsEnforcing Bracketed StatesEntropy of EntanglementEntropy of EntanglementEntropy of EntanglementFinite Cycle of size tnFinite Cycle Cont. Hsize#Ground States for the Finite CycleOpen ProblemsFonts UsedDesign TemplateEmbedded OLE Servers
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Energy penalty for: & .xxxabxxxxx& .
Type II terms (transition rules)
(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f6Type I terms (illegal pairs)
ab><ab
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(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f`oOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
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[AGIK07]Z@ZJZRZZZKZ33JR333KapConstruction Overview(f
2PConstruction Overview(f
^PPP3Target Ground State(f8Target Ground State(fTarget Ground State(f9Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f:Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f;Target Ground State(fTarget Ground State(fTarget Ground State(fFirst Round(f,The Hamiltonian& so farfHtrans = sum of terms from transition rules as applied to all neighboring pairs of particles.
Hlegal = sum of terms from illegal pairs
H = Htrans + Hlegal
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WellFormed StatesfA state in the standard basis is wellformed if it is of the form:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
& or any substring of these forms
Can be checked by local checks
RDZ&ZZ;ZZBZZZZ!3" $3$3$3(G
SWellFormed StatesfCan not enforce by local checks that a legal state has a control state.
E.g., eeee& & eee must be a legal state.
If the state is bracketed ( a < on left end and a > on the right end), then a legal state must be exactly on of the following three forms:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
JHZ+ZZ&ZZ;ZZZZZH* $$W 3 3 3$P
Properties of WellFormed States!!(f.
Htrans + Hlegal is closed over the subspace spanned by wellformed states.
(All additional terms will be diagonal in standard basis).
For each wellformed state, at most one transition rule applies in the forward direction and at most one transition rule applies in the reverse direction.
"O;
<;$3$3$3(,State Graph(f2
Nodes set of all state in standard basis.
Edges  directed edge from state A to state B if B can be obtained by applying one transition rule to A.
Wellformed states are disconnected from the rest of the graph.
State graph restricted to wellformed states form disjoint paths.&e3 dq Hamiltonians Restricted to Paths!!(f
+The Hamiltonian restricted to a single path,,(f&
If Hlegal has a nonzero entry, the minimum eigenvalue of H restricted to the subspace spanned by states in the path is W(1/l3) which is W(1/n6). [Kitaev 02]
If Hlegal is all zero, the state which is the uniform superposition of states in the path has zero energy.
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corresponds to a zero energy state for H.
A path that does not start with
contains a state that has an illegal pair
OR does not have states that are bracketed. (Start with < and end with > ).<!EInitializing Qubits(f
Hinit =  ><  Penalty for state U
Ensures that ground state corresponds to a path whose initial state has qubits set to
* , V , 333 ,pEnforcing Bracketed States(fHbracket = I   ><    >< 
H = 3( Htrans + Hlegal ) + Hinit + Hbracket
3( Htrans + Hlegal ) term ensures there are no bracket terms in the middle.
Hbracket term gives an energy benefit for having brackets at the end
>\Z ZZZZ , , 9f<ff
f 333 /;G2Entropy of Entanglement(f
Entropy of Entanglement(f'Entropy of Entanglement(fFinite Cycle of size tn(ffChange Hlegal so that the pair is allowed.
Wellformed states look like:
A sequence from a to a is a segment.
H is closed on the set of wellformed states for a fixed set of segments.
*ZZZZZZ
u M$ $ $$3$3$3(Finite Cycle Cont. (ffDH = p(n)(Hlegal + Htrans +Hinit) + Hsize
For p(n) large enough, using the Projection Lemma of KempeKitaevRegev, we can assume that the ground state of H is composed of tensor projects of ground states for finite chains.
Ground state for finite chain of length l is
Ground state for H will have form:
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fHsize= (1/n)I
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if and only if l=n
Otherwise
&Q! $ 4ff 3 3 3$$["Ground States for the Finite Cycle0#"(f(f<
n orthogonal ground states, each a translation one site over.
For any region size a constant fraction of the regions of that size have high entanglement.
Superposition of all n states is translationally invariant and for every region size, all regions of that size have high entanglement.
ZZZ#ZZ ZZZZZZ#ff 3 3 3$n4
Open Problems(f
Improve gap lower bound on entropy as a function of 1/D.
For a given region size m, can we achieve high entropy for all regions of size m?
Unique ground state for finite cycle and infinite chain?
Can we achieve high entanglement entropy for all region sizes simultaneously for a translationallyinvariant Hamiltonian?
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In principle, if we know the Hamiltonian of a quantum system, we know its dynamics completely.
Figuring out the Hamlitonian to describe a particular quantum system can be a very difficult problem much of 20th c. physics has been dedicated to solving this problem.
The time evolution of a system is easy to solve once you know the energy eigenstates of a system. Also, understanding the state of the system in its lowest energy state is a fundamental entity to learn about a system. Therefore, a great deal of work (and computational power) has been spent on determining the eigenstates and corresponding eigenvalues of a system.
For our purposes, the Hamiltonian will be a 2^n x 2^n matirx although in all the cases that are interesting for us, it will have a compact represenation.
Note that changing the Hamiltonian by some external means (for example, focusing a laser on an atom) is how one change the Hamiltonian for just the right amount of time so that a desired unitary tranformation is applied to our system. $.pDQ
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What are the minimal set of properties a Hamiltonian must have in order for its ground state to have a high degree of entanglement?
Do symmetries such as translationalinvariance limit entanglement?
Onedimensional systems.
Entropy of entanglement when traced down to a contiguous region in the 1D chain.
,jj333 @
Previous Workf
Local Hamiltonian is QMAcomplete, even for 1D systems.
Adiabatic computation on 1D systems can perform universal quantum computation [AIGK07]
Can these constructions be made translationallyinvariant?
Translationally invariant modification that can be used for 1D universal adiabatic computation.
[NagajWocjan, JanzinWocjanZhang]
Degenerate
1D Local Hamiltonian is QMAcomplete, even what all twoparticle terms are the same. [Kay]
Requires positiondependent 1particle terms.
PP;PaP%!PP[P.PPPP ;_%T.333PTTranslationInvariancef
Can we make the 1D construction translationallyinvariant with a Hamiltonian which has a nondegenerate ground state?
If the system is described by a single Hamiltonian term applied to all pairs of particles (with bounded precision), how do we encode a circuit?
Show high entanglement in the ground state.
For 1D systems, previously known bounds, ground state entanglment entropy scales logarithmically with the region size.
P
P,PPwPPP w333,"=?#Results(f1Finite 1D chain:
Single Hamiltonian term H that operates on two particles of dimension 21, when applied to every neighboring pair in a finite chain of n particles:
Unique ground state
Spectral gap 1/poly(n)
The entropy of entanglement of a region of size m on either end of the chain is W(min{m,nm}).
Z Z!Z ZZZ{ 4Results(f
Cycles and the Infinite Chain:
Family of translationallyinvariant Hamiltonians {Hn} for a cycle of nt 21dimensional particles
Spectral gap is 1/poly(n)
For any state in the ground space of Hn, and any m, there exist regions of size of m whose entanglement entropy is W (min{m,n}).
Entanglement bounds for a constant fraction of regions of size m.
Bounds hold in the limit as t tends towards infinity.
NbB63.@L
B6b*ZO~$1D Area Law$f
Upper bound on the entanglement entropy for any ground state of a 1D Hamiltonian H, independent of region size but exponentially dependent on 1/D, where D is the spectral gap of H. [Hastings 07]
Gottesman and Hastings: is the dependence on 1/D tight?
Family of 1D Hamiltonians with unique ground state with regions whose entanglement entropy is W(poly(1/D)).
Previously, best known such lower bound was W(log(1/D))
arXiv:0901.1108
Construction present here gives a similar result.
PPPP2PP/^01 ViHamiltonian Construction Basics (f6Type I terms (illegal pairs)
ab><ab
Energy penalty for: & .xxxabxxxxx& .
Type II terms (transition rules)
(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f6Type I terms (illegal pairs)
ab><ab
Energy penalty for: & .xxxabxxxxx& .
Type II terms (transition rules)
(ab><ab + cd><cd  ab><cd  cd><ab)
.!3!.
,Hamiltonian Construction Basics (f`oOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KOne Dimensional Hamiltonians(f>Two types of states:
Control states: h g f i I H
Passive States: W w E e U u < >
Transition rules apply to control state and a state to the right or left.
May move control state to the left or right.
For example: >
[AGIK07]Z@ZJZRZZZKZ33JR333KapConstruction Overview(f
2PConstruction Overview(f
^PPP3Target Ground State(f8Target Ground State(fTarget Ground State(f9Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f:Target Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(fTarget Ground State(f;Target Ground State(fTarget Ground State(fTarget Ground State(fFirst Round(f,The Hamiltonian& so farfHtrans = sum of terms from transition rules as applied to all neighboring pairs of particles.
Hlegal = sum of terms from illegal pairs
H = Htrans + Hlegal
(^G
Z
% $$3$3$3(HY*
WellFormed StatesfA state in the standard basis is wellformed if it is of the form:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
& or any substring of these forms
Can be checked by local checks
RDZ&ZZ;ZZBZZZZ!3
!"#$%&'()*+,./012" $3$3$3(G
SWellFormed StatesfCan not enforce by local checks that a legal state has a control state.
E.g., eeee& & eee must be a legal state.
If the state is bracketed ( a < on left end and a > on the right end), then a legal state must be exactly on of the following three forms:
(<+e)(e)*(w+u)*(g+I+f)(W+U)*(E)*(>+e)
(<+e)(e)*(h+J)(W+U)*(E)*(>+e)
(<+e)(e)*(w+u)*(i)(E)*(>+e)
JHZ+ZZ&ZZ;ZZZZZH* $$W 3 3 3$P
Properties of WellFormed States!!(f.
Htrans + Hlegal is closed over the subspace spanned by wellformed states.
(All additional terms will be diagonal in standard basis).
For each wellformed state, at most one transition rule applies in the forward direction and at most one transition rule applies in the reverse direction.
"O;
<;$3$3$3(,State Graph(f2
Nodes set of all state in standard basis.
Edges  directed edge from state A to state B if B can be obtained by applying one transition rule to A.
Wellformed states are disconnected from the rest of the graph.
State graph restricted to wellformed states form disjoint paths.&e3 dq Hamiltonians Restricted to Paths!!(f
+The Hamiltonian restricted to a single path,,(f&
If Hlegal has a nonzero entry, the minimum eigenvalue of H restricted to the subspace spanned by states in the path is W(1/l3) which is W(1/n6). [Kitaev 02]
If Hlegal is all zero, the state which is the uniform superposition of states in the path has zero energy.
`PPPPPPPPP pb333 P
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corresponds to a zero energy state for H.
A path that does not start with
contains a state that has an illegal pair
OR does not have states that are bracketed. (Start with < and end with > ).<!EInitializing Qubits(f
Hinit =  ><  Penalty for state U
Ensures that ground state corresponds to a path whose initial state has qubits set to
* , V , 333 ,pEnforcing Bracketed States(fHbracket = I   ><    >< 
H = 3( Htrans + Hlegal ) + Hinit + Hbracket
3( Htrans + Hlegal ) term ensures there are no bracket terms in the middle.
Hbracket term gives an energy benefit for having brackets at the end
>\Z ZZZZ , , 9f<ff
f 333 /;G2Entropy of Entanglement(f
Entropy of Entanglement(f'Entropy of Entanglement(fFinite Cycle of size tn(ffChange Hlegal so that the pair is allowed.
Wellformed states look like:
A sequence from a to a is a segment.
H is closed on the set of wellformed states for a fixed set of segments.
*ZZZZZZ
u M$ $ $$3$3$3(Finite Cycle Cont. (ffDH = p(n)(Hlegal + Htrans +Hinit) + Hsize
For p(n) large enough, using the Projection Lemma of KempeKitaevRegev, we can assume that the ground state of H is composed of tensor projects of ground states for finite chains.
Ground state for finite chain of length l is
Ground state for H will have form:
J8
$$$$$$$$
$ $ $$3$3$3(t.Hsize,f
fHsize= (1/n)I
  >< 
+ (n1)/Tn[ ><  +  ><  +  ><  ]
Tl is the number of standard basis states in the support of the ground state for a segment of length l.
if and only if l=n
Otherwise
&Q! $ 4ff 3 3 3$$["Ground States for the Finite Cycle0#"(f(f<
n orthogonal ground states, each a translation one site over.
For any region size a constant fraction of the regions of that size have high entanglement.
Superposition of all n states is translationally invariant and for every region size, all regions of that size have high entanglement.
ZZZ#ZZ ZZZZZZ#ff 3 3 3$n4
Open Problems(f
Improve gap lower bound on entropy as a function of 1/D.
For a given region size m, can we achieve high entropy for all regions of size m?
Unique ground state for finite cycle and infinite chain?
Can we achieve high entanglement entropy for all region sizes simultaneously for a translationallyinvariant Hamiltonian?
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Slide Titles9_ajlajl. 2
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