Download Complexity of one-dimensional spin chains

One-Dimensional Spin Chains
Are Hard
Daniel Gottesman
D. Aharonov, D. Gottesman, J. Kempe, arXiv:0705.4077
S. Irani, arXiv:0705.4067
How Hard is One Dimension?
• Can we simulate one-dimensional quantum
systems on a classical computer?
• Can we find specific properties of onedimensional systems on a classical computer?
What about on a quantum computer?
• Can one-dimensional systems be used to build a
quantum computer?
Can We Simulate 1-D
Classically?
• Simulating the full dynamics is universal for quantum
comptuation: Shepherd, Franz, and Werner (quantph/0512058) construct a universal 1-dimensional quantum
cellular automaton with 12 states per site.
• Finding specific properties, like the ground state energy,
one might expect to be easier, because after all there is just
one number to know, and indeed, finding ground state
properties can be done in many specific cases.
• However, finding ground state properties can actually be
harder than simulating dynamics, since the 1D system
might not be able to find its own ground state!
QMA (Quantum Merlin-Arthur)
QMA is the quantum analogue of NP. If a problem is in
QMA and an instance has the answer “yes”, there exists a
quantum state (a “witness”) and a circuit to check it that gives
the answer “yes” with high probability.
E.g.: k-local Hamiltonian. We are given a Hamiltonian H
consisting of ≤k-qubit interactions, with the promise that the
ground state energy of H is either below E or above E + .
Which is it?
Witness: Many copies of the ground state.
Checking circuit: Apply eiHt conditioned on an ancilla,
and measure the phase induced in the ancilla.
QMA-Completeness
We can reduce a problem L to a problem L’ if, for each
instance of the problem L, we can find an instance of L’with
the same answer (yes/no).
A problem L is complete for QMA if it is in QMA and any
other problem in QMA can be reduced to L. Complete
problems are the hardest problems in QMA -- if you can
solve a QMA-complete problem, you can solve any QMA
problem.
Kitaev defined QMA and showed that 5-local Hamiltonian
(finding the ground state energy of a Hamiltonian with 5body terms) is QMA complete.
Proving QMA-completeness
To show that 5-local Hamiltonian is QMA-complete, we
wish to take an instance x of an arbitrary QMA problem
and find an appropriate Hamiltonian H such that the
ground state of H is 0 if the answer to x is “yes,” and is 
if the answer to x is “no.”
yes/no
witness
ancilla
U1
0
0
0
0
U2
1
U3
2
Checking circuit for the problem x
answer
3
Kitaev’s Solution
Kitaev proves 5-local Hamiltonian is hard by constructing a
Hamiltonian whose ground state encapsulates the history of
the checking circuit.
 = ∑tt
Circuit’s state
at time t
clock
This ground state is enforced by choosing terms in the
Hamiltonian that increment the clock register by 1 and give
an energy penalty unless the state has t = Ut t-1.
Improved by a sequence of results to 2-local Hamiltonians in
2 dimensions (i.e. nearest neighbor interactions only).
1-Dimensional Hamiltonians
What about 2-local Hamiltonians in a 1-dimensional system?
(I.e., nearest-neighbor interactions on a line of spins.)
The analogous classical problem is easy. We have conditions
relating variables yi and yi+1 (i = 1, ..., n). This is equivalent
to a constant number C of problems with n/2 variables: Add
conditions yn/2 = 0, yn/2 = 1, etc. until we have tried all
possibilities for yn/2 and yn/2+1. We keep cutting the problem
in half, giving us a total of Clog n = poly(n) constant-size
problems to solve.
But the quantum problem is still QMA-complete! (At least
using 12-state spins.)
The Difficulty
In a nearest neighbor Hamiltonian, we cannot access the clock
easily. It must be distributed, and each spin’s Hamiltonian can
depend on its location, but not on the time.
Prior work: L. Carroll, Alice in Wonderland
“Well, I’d hardly finished the first
verse,” said the Hatter, “when the
Queen jumped up and bawled out,
‘He’s murdering the time! Off with his
head!’ ... And ever since that,” the
Hatter went on in a mournful tone, “he
won’t do a thing I ask! It’s always six
o’clock now.”
The Solution?
“I want a clean cup,” interrupted the Hatter: “let’s all move one
place on.”
He moved on as he spoke, and
the Dormouse followed him:
the March Hare moved into the
Dormouse’s place, and Alice
rather unwillingly took the
place of the March Hare. The
Hatter was the only one who
got any advantage from the
change: and Alice was a good
deal worse off than before, as
the March Hare had just upset
the milk-jug into his plate.
A Better Solution
Of course, the Hatter is mad, so it is not surprising his
protocol does not work.
Hatter’s protocol: Drink tea - move 1 - repeat
Correct protocol: Drink tea - move n - repeat
First, we modify the checking circuit to be one-dimensional
with a number of extra properties. Later we will write down a
Hamiltonian whose ground state is the modified circuit’s
history.
We divide the spins into adjacent blocks of size n (the number
of qubits in the original checking circuit). We perform gates in
one block, then move all qubits to the next block.
The 12 States of a Site
Active States
Inactive States
Left-moving flag
(moves qubits R)
Unborn (not yet active,
to right of active site)
Turning flag
Dead (finished, to left
of active site)
Right-moving flag
(2 states)
Waiting L (2 states, to
left of active site)
Gate flag (2 states,
moves right)
Waiting R (2 states,
to right of active site)
A 1D Quantum Computer
gate gate
1. Gate flag moves R, does gates, and turns when it hits unborn.
2. Turn flag becomes L flag, which moves L, shifting qubits right
one space as it goes, until it hits dead.
3. L flag turns, becomes R flag, moves R without moving or
altering qubits, and turns when it hits unborn.
4. Repeat steps 2 and 3 until L flag hits dead across a block
boundary. Then it turns to a gate flag, and we return to 1.
Useful Properties
• Exactly one site is active at any given time.
• The active site has exactly one possible move forward in
time and one backwards in time.
• The actions taken only depend on adjacent pairs of spins and
location, not on external time.
• Spins to the left and right of the active site are distinctly
marked.
• Most conditions can be checked locally except, e.g., the
requirement that there are n qubit sites.
• States that cannot be locally checked evolve to something
illegal (gate flag must start and end at block boundary).
From Circuit to Hamiltonian
Add terms to the Hamiltonian to enforce the required
structure and evolution:
Constraints: Add an energy penalty if state does not have the
required form. E.g., penalize these configurations:
Transitions: Penalize states which don’t superpose t and
t+1. For instance, penalize states with only one term of:

+

Boundary Conditions
“Then you keep moving round, I suppose?” said Alice.
“Exactly so,” said the Hatter: “as the things get used up.”
“But what happens when you come to the beginning
again?” Alice ventured to ask.
“Suppose we change the subject,” the March Hare
interrupted, yawning. “I’m getting tired of this. I vote the
young lady tells us a story.”
Luckily, in our case, boundary conditions are easy. In the first
block, the gate flag projects ancilla qubits on 0, and in the last
block, the gate flag projects the answer qubit on 1. Any state
that does not satisfy the circuit will have an energy penalty.
Ground State of the Hamiltonian
There are three subspaces of the Hilbert space:
• States which have the wrong structure: Everything in this
subspace has high energy because of the constraints in the
Hamiltonian.
• States which have the wrong structure, but cannot be
locally checked (e.g., have m qubits instead of n): These
states must violate a transition rule after at most O(m2)
transitions, so have a (polynomially small) positive energy.
• States which have the right structure and n qubits: The
transition rules and boundary conditions select only a
correct history state as the ground state of the Hamiltonian.
Conclusion and Summary
• There exist 1-dimensional spin systems where finding
the ground state energy is QMA-complete. The same
construction also shows an adiabatic quantum computer
can be built using 1-dimensional systems.
• As a consequence, there exist 1-D systems that cannot
find their own ground state in a reasonable time.
• Can this be done with less that 12 states per site? In
particular, can it be done with qubits?
• Are random 1-D Hamiltonians QMA-complete? What
about other properties (e.g., correlation functions)?
• Can one-dimensional systems exhibit other sorts of
interesting quantum phenomena?
Please wake your neighbor