1. Computational Complexity of Quantum Systems
Sandy Irani
Computer Science Department
University of California, Irvine
Visiting Institute for Quantum Information, Caltech

2. Complexity of Quantum Systems
Can the evolution of a quantum system be simulated efficiently on a classical computer?
How hard is to approximate interesting properties of a quantum system (for example, ground state energy)?
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.

3. One-dimensional Quantum Systems
Density Matrix Renormalization Group (DMRG) method has been successful at computing the ground state and time evolution of a quantum system. [White][Vidal][Verstraete and Cirac][Schollwock]
1-D quantum systems can perform universal quantum computation with nearest neighbor interactions.
1-D quantum cellular automaton can perform universal quantum computation.
[Watrous][Shepherd, Franz and Werner]

4. Complexity of Quantum Systems
Adiabatic Evolution [Farhi, Goldstone, Gutman and Sipser]
Can a quantum computation be embedded in the ground state of a quantum system?
QMA-completeness
Can we prove that computing properties of a quantum system are hard with respect to a complexity class?
Entanglement
Bounded entanglement accounts for the success of some numerical methods. Can we formally bound the entanglement for classes of quantum systems?

5. Complexity Classes
Resources required to solve a problem grow with the size of the input
Example: time required to sort n numbers will grow with n.
Question: How does the time grow with n?
How to measure time complexity?
Time on a particular platform (language, compiler, processor, etc.)
Number of instructions executed
Steps on a Turing Machine
Gates in a circuit
In the case of QC, the set of all 2-qubit gates is a continuous set.

6. Polynomial Time Class P:
The set of problems that can be solved on a “reasonable” (classical) model of computation [Turing Machine, PC, circuit, etc.] such that the complexity is bounded by a polynomial as the size of the input increases.
(e.g. sorting n numbers in time n log n )
For simplicity, we often refer to decision problems (yes, no answers).
Decision problem <-> Language
– set of “yes” instances
In the case of QC, the set of all 2-qubit gates is a continuous set.

7. The Class NP
A language L is in the class NP, if there is a deterministic polynomial time algorithm V and polynomial p such that
If x is in L, there exists y (witness, certificate) such that |y|<p(|x|) and
V(x,y)=“accept”
If x is not in L, for all y such that |y|<p (|x|)
V(x,y)=“reject”
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.

8. Quantum Complexity ClassesNP P

9. Boolean 3-Satisfiability Problem
Input: boolean formula expressed as the disjunction of clauses each of which contains three literals:
Output: “Yes” or “No” depending on whether there is a truth assignment of the variables which causes each clause to evaluate to true.
Clause 1
Clause 2
Clause m
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.

10. Complete Problems Hardest problems in a complexity class.
Any problem in the class can be reduced to a complete problem.
3-SAT
Generic
Problem
In NP
“Yes”
instances
Satisfiable
Formulae
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.
“No”
instances
Unsatisfiable
Formulae

11. Original Proof that 3SAT is NP-complete
Polynomial time verifier V
(circuit, Turing machine,
Random-Access-Machine…)
For a problem in NP +
Problem input
3-SAT Formula
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.
There is a satisfying
assignment for the
3-SAT formula
There is a witness
that causes V to
accept
IFF
Encode the dynamic process of a computation (verifier) into a
static boolean formula.
[Cook, Levin 1971]

12. Quantum Complexity Classes
NP-complete
Problems NP P

13. What about quantum complexity?
A problem P is in BPP if there exists a universal family of polynomial size circuits Cn (n denotes the length of the input) such that for each x of length n:
If P(x)=“yes”, Prob[Cn(x)=1] > 2/3
If P(x)=“no”, Prob[Cn(x)=0] > 2/3 r Cn
0/1
QC by its very nature is probabilistic, so we need first to define its counterpart in classical computation.
By repeating, can always boost these probabilities so that they are e and 1-e.
Talk about universal.
I chose to define BPP in terms of circuits but any reasonable classical defn of an algorithm would suffice. x

14. What about quantum complexity?
A problem P is in BPP if there exists a universal family of polynomial size circuits Cn (n denotes the length of the input) such that for each x of length n:
If P(x)=“yes”, Prob[Cn(x)=1] > 2/3
If P(x)=“no”, Prob[Cn(x)=0] > 2/3 r Cn
0/1
QC by its very nature is probabilistic, so we need first to define its counterpart in classical computation.
By repeating, can always boost these probabilities so that they are e and 1-e.
Talk about universal.
I chose to define BPP in terms of circuits but any reasonable classical defn of an algorithm would suffice. x
1-1/e
1-1/e

15. Quantum Complexity ClassesNP P
BPP

16. Quantum Circuits U1 U6 U5 U2 U4 U7 U3 U8 M

17. Universal Set of Gates Two qubit gates suffice:
Any arbitrary unitary operation can be expressed exactly as a product of unitary operations, each of which acts non-trivially on only two qubits.
There is a finite universal set of gates:
There exists a finite set of gates such that any unitary operation can be approximated to arbitrary accuracy by a quantum circuit involving only those gates.
In the case of QC, the set of all 2-qubit gates is a continuous set.

18. BQP
A problem L is in BQP if there exists a universal family of polynomial size quantum circuits Cn (n denotes the length of the input) such that for each x of length n:
If P(x)=“yes”, Prob[Cn(x)=1] > 2/3
If P(x)=“no”, Prob[Cn(x)=0] > 2/3
[Deutsch][Yao][Bernstein,Vazirani]
Answer bit Cn

19. Quantum Complexity ClassesNP P
BQP
BPP

20. The Class Merlin-Arthur
A language L is in the class MA, if there is a randomized polynomial time algorithm C and polynomial p such that
If x is in L, there exists y such that |y|<p(|x|) and
Prob[C(x,y)=1] > 2/3
If x is not in L, for all y such that |y|<p (|x|)
Prob[C(x,y)=0] > 2/3
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.

21. The Class Merlin-Arthur
A language L is in the class MA, if there is a randomized polynomial time algorithm C and polynomial p such that
If x is in L, there exists y such that |y|<p(|x|) and
Prob[C(x,y)=1] > 2/3
If x is not in L, for all y such that |y|<p (|x|)
Prob[C(x,y)=0] > 2/3
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.
1-1/e
1-1/e

22. Quantum Complexity ClassesMA NP P
BQP
BPP

23. QMA
A language L is in the class QMA, if there is a polynomial sized quantum circuit C and polynomial p such that
If x is in L, there exists , a quantum state on at most p(|x|) qubits
Prob[C(x, )=1] > 2/3
If x is not in L, for all such that is a quantum state on p(|x|) qubits,
Prob[C(x, )=0] > 2/3
[Kitaev]
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.

24. QMA
A language L is in the class QMA, if there is a polynomial sized quantum circuit C and polynomial p such that
If x is in L, there exists , a quantum state on at most p(|x|) qubits
Prob[C(x, )=1] > 2/3
If x is not in L, for all such that is a quantum state on p(|x|) qubits,
Prob[C(x, )=0] > 2/3
[Kitaev]
1-1/e
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.
1-1/e

25. Quantum Complexity Classes
PSPACE
QMA MA NP P
BQP
BPP

26. Quantum Complexity Classes
Where does
the problem of computing
ground energy fit into
this picture?
PSPACE
QMA MA NP P
BQP
BPP

27. Quantum Complexity Classes
Where does
the problem of computing
ground energy fit into
this picture?
PSPACE
QMA-complete
QMA MA NP P
BQP
BPP

28. Restrictions on the Hamiltonian
A Hamiltonian H is a d-state k-local Hamiltonian if it acts on a system of d-state particles and can be written as a sum of terms, each of which acts non-trivially on a set of k particles.
H is an r-dimensional d-state Hamiltonian if it is a d-state 2-local Hamiltonian and each term interacts non-trivially on only nearest neighbor particles arranged in an r-dimensional grid.
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.

29. Local Hamiltonian Problem
d-STATE k-LOCAL HAMILTONIAN
Let H be a d-state k-local Hamiltonian. Then (H, E0,D) is in d-STATE k-LOCAL HAMILTONIAN if the ground state E0 of H is at most E. The system must satisfy the promise that either E0 is at most E or at least E + D.
D will be 1/poly in the size of the system
Theorem: 2-State 5-Local HAMILTONIAN is QMA-complete [Kitaev]
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.

30. 5-local Hamiltonian Given a quantum verifier C( )
Find 5-local Hamiltonian H such that
If there is a such that C accepts with high probability, then the ground state of H has energy at most E.
If for every probability C accepts is low, then every eigenstate of H has energy at least E+D.
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.

31. Classical/Qauntum Turing machine computation (Verfier)
Set of variables with possible settings
Instance of k-Sat (set of constraints on the variables)
Variable settings
Cost of the assignment
Quantum circuit (Verfier)
Set of particles each with a finite set of states
Hamiltonian H – sum of terms with energy constraints
Quantum state
Energy of the state

32. 5-local Hamiltonian
Suppose we have a quantum verifier circuit that is composed of T gates operating on n qubits. (Both witness and ancillary qubits).
Will have a system composed of n computation qubits and T+1 clock quibits.
Let denote a possible quantum witness.
Let denote the state of the n quibits after the first t gates have been applied.
The constraints of the 5-local Hamiltonian will ensure that any low energy state must have the form:
Then one final constraint that places an energy penalty on a state for which the outcome of the verifier is 0.
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.

33. 2-quibit system
Encode constraints onto an eigenstate that corresponds to an eigenvector of 0.
Forbid
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.

34. 2-quibit system
Enforce that amplitude of is the same as the amplitude of
Any 0 eigenvalue will be of form
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.

35. 2-quibit system Any 0 eigenstate of this Hamiltonian Must be
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.

36. 5-local Hamiltonian
Suppose that gate t operates on qubits 1 and 2 with unitary operator U
Want a 5-qubit terms that ensures for every state of the form:
The following state has equal amplitude: t
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.
t+1

37. 5-local Hamiltonian
Suppose that gate t operates on qubits 1 and 2 with unitary operator U
Want a 5-qubit terms that ensures for every state of the form:
The following state has equal amplitude:
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.

38. 5-local Hamiltonian I -U I -U+ 100xx 110xx 100xx 110xx
Applying the gate U to quibits 1 and 2 in parallel for every other possible setting of the other qubits.
The value of the other qubits does not change. I
110xx
-U+

39. Dynamic vs. Static Qauntum Computation
Quantum circuit: dynamic
Different gates applied at different points in time.
Qubits physically situated on a line not limiting.
Ground state of a Hamiltonian: static
Ground state must satisfy constraints imposed by Hamiltonian.
Complexity of ground state may be affected by
Locations of particles
Types of terms allowed in Hamiltonian

40. QMA-Complete Problems
5-Local Hamiltonian is QMA-complete [Kitaev]
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.

41. QMA-Complete Problems
5-Local Hamiltonian is QMA-complete [Kitaev]
2-Local Hamiltonian is QMA-complete [Kempe, Kitaev,Regev]
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.

42. QMA-Complete Problems
5-Local Hamiltonian is QMA-complete [Kitaev]
2-Local Hamiltonian is QMA-complete [Kempe, Kitaev,Regev]
2-Dim 2-State Hamiltonian is QMA-complete [Oliveira, Terhal]
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.

43. QMA-Complete Problems
5-Local Hamiltonian is QMA-complete [Kitaev]
2-Local Hamiltonian is QMA-complete [Kempe, Kitaev,Regev]
2-Dim 2-State Hamiltonian is QMA-complete [Oliveira, Terhal]
1-Dim 12-state Hamiltonian is QMA-complete.[Aharonov, Gottesman, Irani,Kempe]
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.

44. The Classical Analog 1D MAX-2-SAT with d-state variables is in P:
- Divide line in half.
- For each d2 possible values for variables at boundary,
recursively solve n/2 variable sub-problem on each half.
- Select solution that satisfies most constraints.
Must solve 2d2 sub-problems of size n/2
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.

45. Why the difference?
Ground state encodes the entire computation of the quantum verifier.
The ground state is a superposition of the computation at each point in time.
Can encode an extra dimension in the ground state: time.
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.

46. Translational Invariance
1D construction – sum of terms for each neighboring pair of particles. Terms are position-dependent.
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.

47. Translational Invariance
1D construction – sum of terms for each neighboring pair of particles. Terms are position-dependent.
Can the 1D construction be made translationally-invariant?
Translationally invariant modification that can be used for 1D universal adiabatic computation.
[Nagaj-Wocjan, Janzin-Wocjan-Zhang]
Degenerate
1D Local Hamiltonian is QMA-complete, even what all two-particle terms are the same. [Kay]
Requires position-dependent 1-particle terms.
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.

48. Translation-Invariance
Can we make the 1D construction translationally-invariant with a Hamiltonian which has a non-degenerate 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.
Bounded entanglement accounts for the success of some numerical methods [DMRG, MERA, PEPS…]
Are there classes of systems which are guaranteed to have ground states with low entanglement?
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.

49. Entropy of Entanglement
Density matrix r for the ground state of a system of n particles.
Let A be a contiguous subset (region) of the particles and B its complement
rA = trB( r ).
Entropy of entanglement for region A:
SA = S(rA) = tr(rA log rA ).
How does entanglement entropy scale with the size of the region or the spectral gap in the worst case?
For 1D systems, previously known bounds, ground state entanglment entropy scales logarithmically with the region size.
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.

50. Results Finite 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,n-m}).
[Irani, arXiv: ]

51. Results Cycles and the Infinite Chain:
Family of translationally-invariant Hamiltonians {Hn} for a cycle of nt 21-dimensional 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.
[Irani, arXiv: ]

52. 1D Area Law
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))
[Gottesman, Hatsings arXiv: ]
Construction present here gives a similar result.

53. Hamiltonian Construction Basics
Type I terms (illegal pairs)
|ab><ab|
Energy penalty for: ….xxxabxxxxx….
Type II terms (transition rules)
½(|ab><ab| + |cd><cd| - |ab><cd| - |cd><ab|)
In the case of QC, the set of all 2-qubit gates is a continuous set.
…xxxxabxxxx…
…xxxxcdxxxx…

54. Hamiltonian Construction Basics
Type I terms (illegal pairs)
|ab><ab|
Energy penalty for: ….xxxabxxxxx….
Type II terms (transition rules)
½(|ab><ab| + |cd><cd| - |ab><cd| - |cd><ab|)
In the case of QC, the set of all 2-qubit gates is a continuous set.
…xxxxabxxxx…
…xxxxcdxxxx…
ab -> cd

55. Hamiltonian Construction Basics
Each contains no
illegal pairs.
In the case of QC, the set of all 2-qubit gates is a continuous set.
Ground State =
[Kitaev02]

56. Construction Overview
Set of Transition Rules:
Target ground state is a sequence of states (each one transitions to the next via a transition rule). Sequence corresponds to a process that creates entanglement.
Set of Illegal Pairs:
Ensure non-degeneracy.
Energy penalty for any state which is not part of the target ground state.
In the case of QC, the set of all 2-qubit gates is a continuous set.

57. One Dimensional Hamiltonians
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] w f f W
In the case of QC, the set of all 2-qubit gates is a continuous set.

58. One Dimensional Hamiltonians
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] w f f W
In the case of QC, the set of all 2-qubit gates is a continuous set. … e e w f W E …

59. One Dimensional Hamiltonians
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] w f f W
In the case of QC, the set of all 2-qubit gates is a continuous set. … e e w f W E …

60. One Dimensional Hamiltonians
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] w f f W
In the case of QC, the set of all 2-qubit gates is a continuous set. … e e w f W E … … e e f W W E …

61. Construction Overview
Circles: Single states w W
<
> J f h
Diamonds: Two-dimensional subsystems e+ E1 U0 u g I i
In the case of QC, the set of all 2-qubit gates is a continuous set.
Control states:
Standard basis: specify state type for each site
And then 0 or 1 for each 2D subsystem J f h g I i

62. Construction Overview
< e+ e+ e+ u+ u+ u+ g+ E+ E+ w W W W
>
EPR Pairs
In the case of QC, the set of all 2-qubit gates is a continuous set.
Entangled states e+ E1
Unentangled states U0 u
Waiting states w W

63. Target Ground State e+ U+ U+ E+
< h W W
>

64. Target Ground State e+ U+ U+ E+
< h W W
>

65. Target Ground State < > < > e+ U+ U+ E+ h W W e+ e+ g U+

66. Target Ground State < > < > e+ U+ U+ E+ h W W e+ e+ g U+

67. Target Ground State < > < > < > e+ U+ U+ E+ h W W e+

68. Target Ground State < > < > < > < > e+ U+ U+h W W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
>

69. Target Ground State < > < > < > < > < >h W W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
>

70. Target Ground State < > < > < > < > < >h W W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
>

71. Target Ground State < > < > < > < > < >h W W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
>

72. Target Ground State < > < > < > < > < >h W W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ u+ E+ E+
< w f
>

73. Target Ground State < > < > < > < > < >h W W
>
< f W
> e+ e+ g U+ E+
< W W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ u+ E+ E+
< w f
>

74. Target Ground State < > < > < > < > < >h W W
>
< f W
> e+ e+ g U+ E+
< W W
> e+ e+ U+ E+ E+
< f W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ u+ E+ E+
< w f
>

75. Target Ground State < > < > < > < > < >h W W
>
< f W
> e+ e+ g U+ E+
< W W
> e+ e+ U+ E+ E+
< f W
> e+ e+ u+ g E+
< W W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ u+ E+ E+
< w f
>

76. Target Ground State < > < > < > < > < >h W W
>
< f W
> e+ e+ g U+ E+
< W W
> e+ e+ U+ E+ E+
< f W
> e+ e+ u+ g E+
< W W
> e+ e+ U+ E+ E+
< h W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ u+ E+ E+
< w f
>

77. Target Ground State < > < > < > < > < >h W W
>
< f W
> e+ e+ g U+ E+
< W W
> e+ e+ U+ E+ E+
< f W
> e+ e+ u+ g E+
< W W
> e+ e+ U+ E+ E+
< h W
> e+ e+ u+ g E+
< w W
> e+ e+ u+ g E+ …
< w w
> e+ e+ u+ i E+
< w w
> e+ e+ e+ E+ E+ E+
< h
> e+ e+ u+ E+ E+
< w f
>

78. The Hamiltonian…so far
H = Htrans + Hlegal
Htrans = sum of terms from transition rules as applied to all neighboring pairs of particles.
Hlegal = sum of terms from illegal pairs
Will need to select these terms so that any standard basis state outside the target ground state has an energy penalty.
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.

79. Well-Formed States
Will assume for now that the ground state is a superposition of bracketed standard basis states:
the leftmost particle will be in state and the rightmost particle will be in state
Will late add a term to give an energy penalty to any state that is not bracketed.
<
>
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. …
<
>

80. Well-Formed States
A state in the standard basis is well-formed if it is of the form:
Can be checked by local checks…i.e. illegal pairs:
[anystate] [Uppercase][LowerCase],…. … … … … e e
< h U U E+ E+
> W W … … … … e e u u i E+ E+
>
< w w … … … … … e e u u n E+ E+
>
< w w W W
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. … … … … … e e u u n U U E+ E+
>
< W W u e e w
<

81. Bad States
< e e e e u g E w w
>
< e u g E E E E w w
>
< e e u u u u g E w
>
< e1 e u u g E0 w w w
>
Will check for these states by showing that they evolve (via forward or backward application of the transition rules) to illegal states.

82. Properties of Well-Formed States
Htrans + Hlegal is closed over the subspace spanned by well-formed states.
(All additional terms will be diagonal in standard basis).
For each well-formed state, at most one transition rule applies in the forward direction and at most one transition rule applies in the reverse direction.
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.

83. State Graph 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.
Well-formed states are disconnected from the rest of the graph.
State graph restricted to well-formed states form disjoint paths.
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.

84. Hamiltonians Restricted to Paths
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.

85. The Hamiltonian restricted to a single path
If Hlegal has a non-zero 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.
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.

86. Bad States
< e e e e u g E w w
>
< e u g E E E E w w
>
< e e u u u u g E w
>
< e1 e u u g E0 w w w
>
Will check for these states by showing that they evolve (via forward or backward application of the transition rules) to illegal states.

87. First Round Special control states for the first round force the
< J W W W
> e+ I U+ U+
Special control states for
the first round force the
left end and the right end
to agree that it is the first
round.
Will be used to check that
state is symmetric about
center.
< W W W
> e+ u+ I U+
< W W W
> e+ u+ u+ I
< W W W
> e+ u+ u+ I
< w W W
> e+ u+ u+ I
< w w W
> e+ u+ u+ I
< w w w
> e+ u+ u+ i
< w w w
>

88. How To Check for Bad States: An Example
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.
< e e e e u f E w w
>

89. How To Check for Bad States: An Example
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.
< e e e u u i w w w
>
< e e e e u f E w w
>

90. How To Check for Bad States: An Example
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.
< e e e u u I w w w
>
< e e e u u i w w w
>
< e e e e u f E w w
>

91. How To Check for Bad States: An Example
< e e e I U U W W W
>
...
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.
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92. How To Check for Bad States: An Example
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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.
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93. Need to Show:
A path in the state graph corresponding to the target ground state is a zero energy state for H.
Any standard basis state that is not contained in a path corresponding to the target ground state either:
Contains an illegal pair
Evolves via forward or backwards transitions to a state with an illegal pair.
Is not bracketed.

94. Initializing Qubits Hinit = | >< | Penalty for state U-
Ensures that ground state corresponds to a path whose initial state has qubits set to U- U-
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. … … U+ U+
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> m m

95. Enforcing Bracketed States
Hbracket = 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
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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.

96. Entropy of Entanglement…
O(n2)
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. …

97. Entropy of Entanglement…
O(n2)
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. … A

98. Entropy of Entanglement…
O(n2)
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. … A

99. Finite Cycle of size tn Change Hlegal so that the pair is allowed.
Well-formed states look like:
A sequence from a to a is a segment.
H is closed on the set of well-formed states for a fixed set of segments.
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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.
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100. Finite Cycle Cont. H = p(n)(Hlegal + Htrans +Hinit) + Hsize
For p(n) large enough, using the Projection Lemma of Kempe-Kitaev-Regev, 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:
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.

101. Hsize
Hsize= (1/n)I
- | >< |
+ (n-1)/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
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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.

102. Ground States for the Finite Cycle
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. … … … … …
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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.

103. Open Problems
Improve gap – lower bound on entropy as a function of 1/D.
Area law for 2D?
Better characterization of the complexity of translationally invariant systems?
Interesting classes of Hamiltonians for which finding ground energy is in BQP?
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.

104. First Round Special control states for the first round force the
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> e+ I U+ U+
Special control states for
the first round force the
left end and the right end
to agree that it is the first
round.
Will be used to check that
state is symmetric about
center.
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> e+ u+ I U+
< W W W
> e+ u+ u+ I
< W W W
> e+ u+ u+ I
< w W W
> e+ u+ u+ I
< w w W
> e+ u+ u+ I
< w w w
> e+ u+ u+ i
< w w w
>

105. How To Check for Bad States: An Example
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.
< e e e e u f E w w
>

106. How To Check for Bad States: An Example
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.
< e e e u u i w w w
>
< e e e e u f E w w
>

107. How To Check for Bad States: An Example
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.
< e e e u u I w w w
>
< e e e u u i w w w
>
< e e e e u f E w w
>

108. How To Check for Bad States: An Example
< e e e I U U W W W
>
...
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.
< e e e u u I w w w
>
< e e e u u i w w w
>
< e e e e u f E w w
>

109. How To Check for Bad States: An Example
< e e J U U U W W W
>
< e e e I U U W W W
>
...
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.
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>
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>
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