Download Reversible universal quantum computation within translation

Reversible universal quantum computation within translation invariant systems
K. G. H. Vollbrecht and J. I. Cirac
Max-Planck-Institut für Quantenoptik, Garching
Phys. Rev. A 73, 012324 (2006)
The Pointer
Abstract
We show how to perform reversible universal quantum computation on a translationally invariant pure state, using
only global operations based on next-neighbor interactions. We do not need not to break the translational symmetry of
the state at any time during the computation. Since the proposed scheme fulfills the locality condition of a quantum
cellular automata, we present a reversible quantum cellular automaton capable of universal quantum computation.
Quantum Computation Scheme I
The main idea is to break the symmetry by a structure called the pointer. The
pointer consists out of two neighboring atoms, one in level 2 and one in level 3.
Starting point: all atoms prepared in state 0.
0
Once given a pointer it can be moved back an forward by the following
sequence:
2
3
0
1
0
0
0
0
1
0
0
1
0
Pointer
Motivation:
Neutral atoms confined in (quasi) periodic optical potentials and manipulated by
lasers provide us with one of the most promising avenues to implement a
quantum computer or to perform quantum simulations. At the moment, quantum
computation with atoms in optical lattices is hindered by several major obstacles.
One is the lack of addressability.
Use 2 internal levels
Lack of addressability
|a〉
2
3
x
x
2
3
0
0
Once the symmetry is broken it is easy to construct a usual quantum
computation scheme. The pointer can be used to address single sites and to run
the computation without having the possibility of addressing single atoms.
30 Ù 31
1. Step:
2
Local operations
no addressability at all
00000000000000
11111111111111
translation + reflection symmetric operations
only next-neighbor interactions
In the classical situation this
translation + reflection symmetric state at
restrictions
forces
the
computation to be completely
trivial.
any time during the computation
reversible: measurements only at the end of computation
state stays pure at any time
S
YE
Is Quantum Computation still possible ?
3
The main part of the presented protocol will work with only two types of
simple operations explained in the following.
We assume having many atoms in a row but the following restrictions:
!
How one can encode several qubits ?
0
2
0
3
1
4
2
1
0
2
0
3
0
H
H
H
H
H
H
H
H
H
H
H
H
H
We apply the same Hamiltonian on every site. For example
1
H
C-Not
1. Step:
2. Step:
3. Step:
4. Step:
5. Step:
6. Step:
1
H~ 1 2 + 2 1
The pointer can be used to address single
atoms and to do arbitrary local operations.
This is done by moving the pointer next to a
atom and applying a nearest neighbor
interaction that only acts on atoms next to a 3
level.
Local operations: e.g.
Operations
Consider worst case
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
Initialization
0
0
1. Step:
10 Ù12
2. Step:
20 Ù23
3
2
1
2
3
Problem:
Some of the quantum computers are to close together such that they interact
with each other resulting in a corrupted pair of quantum computers.
2
- all atoms interact with nearest neighbors at the same time.
0
Step 3:
Run ensemble quantum computation scheme.
Quantum Computation Model
- the laser used for the logic operations has to focus close to (or even beyond) the diffraction limit
in order to address a single atom.
0
Step 2:
By a initialization sequence every 1 is turned into
two pointers. This creates two quantum computers of
type A and B at a time.
Two-qubits-gate
Next-neighbor interaction
} qubit
|b〉
0
Moving Pointer
1. Step: 03Ù04
2. Step: 42 Ù40
3. Step: 03 Ù04
4. Step: 14 Ù13
5. Step: 42 Ù41
6. Step: 13 Ù14
7. Step:
2 Ù3
Las
er
0
Step 1:
Introduce some noise, such that some of the states randomly turn into 1.
Random state with probabilities ε for 1 and 1-ε for 0 .
By a sequence of operations a C-NOT can be
generated between the two atoms that are left
and right of the pointer.
04Ù02
42 Ù43
20 Ù21
42 Ù43
20 Ù21
04 Ù02
2
3
To read out a single atom the pointer is moved
next to it and transferred it conditioned on its
state into level 4. The number of atoms in level
4 can be counted.
0
1Ù2
Measurement
This Hamiltonian acts only at the green marked neighbored pairs. If the
interaction time is suitably chosen the levels 1 and 2 are exchanged.
Here 1 Ù 2 denotes that every atoms in level 1 is changed to level 2 and vice
versa.
1. Step:
2. Step:
2
3
3
0
0
0
0
0
2
0
The number of corrupted quantum computers
can be controlled by the probability ε. When ε is
chosen sufficiently small the relative number of
corrupted quantum computers tends to zero.
3
0
Problem:
choosing ε small
2
3
The probability ε can be
chosen in such a way, that
the number of quantum
computers stays constant,
while the resources scales
only quadratically.
!
0
# quantum computers
Solution:
choose ε ~ 1/n2
choose #atoms ~ n2
# quantum computers
const
Final Quantum Computation Scheme
Starting point: all atoms prepared in state 0.
0
4
30 Ù 34
Count #4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Step 1’:
We apply a unitary rotation such that every
atom is in a superposition of 0 and 1.
ε 1 + 1− ε 0
Apply |0〉
How one can manipulate different qubits without addressability ?
Next-Neighbor operations
How one can readout qubits without addressability ?
0
Rules of the game
We consider atoms having five internal levels: |0〉 …|4〉. 0
0
0
0
0
0
0
0
0
0
0
3
2
0
0
0
0
0
0
0
0
0
0
3
H
12
H
0
H
3
H
12
H
H ~ 10 20 + 20 10 + 01 02 + 02 01
0
0
We apply only global operations, e.g., operations that affect all atoms in
the same way. In particular, we allow only reflection + translation
symmetric next-neighbor interactions.
0
0
H
0
H
21
H
0
H
3
H
0
H
We apply the same nearest neighbor Hamiltonian on every site. For example
Initially all atoms are in |0〉
0
1
2
H
0
0
Reversible: Measurements are done only at the end of the computation.
φ
Ensemble Quantum Computation
10 Ù 20
This Hamiltonian acts only at the green marked neighbored pairs. Since the
chosen Hamiltonians commute, their actions can be calculated one after
another. Here 10 Ù 20 denotes that the every pair 10 has to change to 20 and
vice versa. Due to the reflection symmetry also 01 changes to 02 and vice
versa.
These operations are enough to permit a classical computation. For quantum
computation we need a third part of operation, that do not allow for such a
simple notation. The third type of operation uses again Hamiltonians of the
above form but with arbitrary interaction times.
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
=
A single pointer can not be created without addressability. Therefore we
have to change the scheme to a ensemble quantum computation scheme.
0
3
2
1
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
1
2
m
1− ε
+ 1− ε
m−1
ε
+ 1− ε
m−2
ε
3
Instead of running one quantum computer it is possible to run several quantum
computers in parallel, like it is known for NMR quantum computation. In our
scheme we will need to types of above described basic quantum computes:
2
|0
|0
|0
〉
〉
0 〉
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
+….
Initialization
Step 2’:
We apply the initialization and run an
ensemble quantum computation scheme.
1. Step:
10 Ù12
2. Step:
20 Ù23
Basic quantum computer:
Version A
2
3
0
Version B
0
0
0
0
0
0
0
0
0
3
2
The second version B is just a reflection symmetric version of A. If a quantum
computation is applied, all quantum computers are doing the same calculation
in parallel. At the end of the computation a measurement can be applied as
described above. The number of atoms in level 4 is counted.
1− ε
m
+ 1− ε
m−1
ε
+ 1− ε
m−2
ε
2
|0
|0
|0
〉
〉
3 〉
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
1
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
1
2
3
0
0
0
0
0
0
0
0
3
2
1
2
The superposition of randomly distributed quantum computers give the same
measurement results as in above quantum computation scheme I.