Building ideal quantum computer from Josephson junction arrays

1
Building ideal quantum computerfrom Josephson
junction arrays
Gianni Blatter, Alban Fauchere, Benoit Doucot,
Mikhail Feigelman, Vadim Geshkenbein, Dmitry
Ivanov, Mathias Troyer, Julien Vidal.
Rutgers University, New Jersey, USA
Landau Institute, Moscow, Russia
Jussieu, Paris, France
Theoretische Physik, ETH Zürich, Switzerland
2
Plan

• Physical requirements on realistic quantum
  computer. What is QC as a physical device? Need
  for a topological protection.
• Simple model of a topological protection dimer
  model on triangular lattice
• Alternative point of view discrete gauge theory
  on a lattice.
• Realistic Josephson junction array topological
  superconductor.
• Model of ideal array
• Ground state and excitations
• Topological protection
• Effects of non-ideality
• Simple physical properties
• Topological insulator
• Conclusions

3
Network model of quantum computing
(David Deutsch, 1985)
initial state

• each qubit can be prepared in some known state,
• each qubit can be measured in a basis,
• the qubits can be manipulated through quantum
  gates
• the qubits are protected from decoherence

final state
4
Advantage of quantum computers
Turing machine
Classical computer
1
0
1
1
1
0
1
1
0
1
1
1
(R, W, M)
Few registers Program
Quantum computer
If problem is solved in Polynomial time on one
machine it is also solved in polynomial time on
another.
Physical proof
Hilbert space - N dimensional, basic states
The complexity of the classical problem grows as
5
Alternative modes of operation
one-parameter (q) qubit
two-parameter (q,Q) qubit
mixing
ac-microwave induced transitions (NMR scheme)
Q
trivial idle state
phase shift
decoupled states
coupled states
fast non-adiabatic switching, amplitude shift
phase shift
Q
NMR scheme
Other qubits in the same device are excited with
probability and a precise addressing
requires long times
With Nqu qubits the distance between resonances
is dD D / Nqu. The transition time of the k-th
qubit is related to the ac-signal V via
6
What is QC as a physical system?
In order to perform useful computation and allow
error correction
Off diagonal matrix elements are small
Memory state depends on the history.
Glass
All this does not distinguish quantum computer
from classical one or a physical glass.
Diagonal matrix elements
Glass
Macroscopic measurements do not distinguish
states!
Quantum Noumenon
I. Kant thing-in-itself
7
Quantum dimer model (triangular lattice)
Constraint each site belongs to one and only one
dimer
Any process consistent with constraint conserves
parity of dimers crossing the line connecting the
boundaries.
Only parity distinguishes two states.
The Hamiltonian Hd mixes even and odd sectors of
the Hilbertspace H and we remain with 2 sectors
8
Quantum dimer liquid (triangular lattice)
Monte Carlo simulations (Moessner Sondhi) on
362-lattice down to T 0.03 t
dimer liquid
staggered
columnar
1
2/3
0
Exact diagonalization on
excitations
energy gap D

• The dimer liquid
• has a finite energy gap
• has no edge states
• is stable under local
• perturbations

9
Discrete gauge theory on a lattice
Constraint (even number
of spins down)
Compare with dimer constraint one spin down.
Elementary process compatible with constraint
Choose topologically non-trivial contour
Fourrier transform states
N2
Constraint i,j sites of
dual lattice
Gauge transformation

Gauge invariancei condition Gauss law
Gauge theory Hamiltonian
10
Phase Dimers Josephson network
Opening Need to make topologically nontrivial
array Need K openings to form K qubits
Superconducting wires with Josephson junctions
H
Each dice is frustrated by flux
Two degenerate states
Josephson circuits ensuring constraints at the
boundary
Even number of clockwise dices at each hexagon
11
Mathematical model
Spin representation for clock (counter clock)
phase on each dice Spins are situated on
lattice bonds
Constraints are defined on the triangular lattice
sites
a
Dynamics
b
(simplest consistent with constraint)
Equivalent to change of the phase of the center
island by
Quasiclassical calculation
12
Solution
1.
2. Hamiltonian without constraint has trivial
ground state (all spins pointing right)
3. Fix the constraint applying projection
operators
4. Hamiltonian also commutes with open strings of

New integral of motion for contours
connecting topologically different (e.g. inner
and outer) boundaries
Choice gives
topologically different degenerate ground states
5.
13
Matrix elements of operators
Take two ground states
For any local operator transition amplitude
Proof 1.Project operators onto the Hilbert
space of states that preserve the constraints,
this operation leaves operators local
2. Local operators preserving the constraint can
be expressed as a product of
which commute with
For such projected operators
Each contains a closed string of
while contains open string. Each has
to enter twice to give non-zero. Thus
14
Topological excitations
Flips spins along contour that has no ends
inside the system
Flips spins along the contour that ends at R
The sign of kinetic energy is different on the
last triangle
Gap for single particle excitations.
Creation excitation at one boundary and dragging
it to another changes the ground state by
Physical meaning change the phase difference
between boundaries or parity of charge on the
inner boundary.
R
Phase and charge are dual
Excitation carries elementary charge 1 (2e)
15
Excitations beyond simple model
Violate the constraint at one hexagon only.
Phase changes by across the line
Flip one dice change phase difference across
by
Conclusion elementary excitation violating
the constraint is half vortex with energy
Need also to flip dices in other hexagons so
their constraints are preserved
To change the topological class (parity) of the
state one needs to create vortex and move it
around the opening
16
The effect of perturbations and corrections

• Sources
• Flux through each rhombus differs from ideal
  local degeneracy is lifted
• Contacts are not exactly
  equal - phase difference across is

• Results
• Appearance of virtual excitations in the ground
  state wave function.
• Kinetic energy of the excitations - motion
  (random -gt localization).
• Perturbation theory in V/r

Non-zero contribution to matrix elements appears
in Lth order of perturbation theory
17
Phase Diagram
Usual superconductor
Insulator
Topological order

• Experimental signatures of the new phase
• Charge 4e superconductor with a gap to charge 2e
  (half flux quantization)
• Macroscopic decoherent times of odd/even states
  echo experiments.

18
Alternative one qubit configuration (no hole)
B
Non trivial topological contour
A
Two boundaries topologically equivalent to
inner/outer boundaries of general (multi-hole)
array
Physical properties attach weak contacts to A
and B boundaries. Measure flux quantization,
charge transfer, phase difference.
19
Topological Insulator
Effective weak link

Two state rhombi
If r sets the largest energy scale in the
problem all low energy states satisfy gauge
invariance condition
Dual to Topological superconductor where
20
Conclusions

• Constructed Josephson junction arrays with
  (multiply) degenerate ground state
• All relaxation times of these low energy states
  are exponentially long
• Topological superconductor is 4e superconductor
  with additional discrete degrees of freedom,
  topological order is the parity of the charge of
  the inner boundary
• Topological insulator liquid of vortices with a
  gap to half vortices.
• Allows exact (adiabatic) simple computing
  operations

Unanswered questions 1. Phase diagram
intermediate phase and phase boundary 2. Arrays
that allow to perform all necessary operations
exactly
21
Quantum Computing

• Classical computer
• the information is stored in
• classical bits, values 0,1
• usual operations NOT, AND, OR
• general purpose device

• Quantum computer
• the information is stored in
• quantum bits (qubits)
• unitary operations, single- and two-
• qubit operations (XOR)
• powerful in calculating specific tasks

Charge, Phase Built in error correction
22
Public KeyEncryption (RSA)
Putting cloak and dagger out of business?
To find one needs to factor N.
Example. If (prime)
Note where period of
If
Fermat theorem has period but has period
Proof. Sets and coincide, thus
23
Quantum computer cracks RSA encryption scheme
Back to business...
Shors algorithm
Need to find period of
Computer
24
Physical implementations
Quantum optics, NMR-schemes Good decoupling
precision
Solid state implementations Good scalability
variability

• trapped atoms (Cirac Zoller)
• photons in QED cavities (Monroe ea, Turchette
  ea)
• molecular NMR (Gershenfeld Chuang)
• 31P in silicon (Kane)

• spins on quantum dots (Loss DiVincenzo)
• 31P in silicon (Kane)
• Josephson junctions, charge (Schön ea, Averin)
• phase (Bocko ea,
  Mooij ea)

All hardware implementations of quantum
computers have to deal with the conflicting
requirements of controllability while minimizing
the coupling to the environment in order to
avoid decoherence.
Have to deal with individual atoms, photons,
spins, Problems with control,
interconnections, measurements.
Have to deal with many degrees of
freedom. Problems with decoherence.