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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

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

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

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

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

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

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

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

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

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

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

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.

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

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)

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

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

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.

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.

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

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

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

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

Quantum computer cracks RSA encryption scheme

Back to business...

Shors algorithm

Need to find period of

Computer

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.