Download Implementations of Quantum Information

Implementations of
Quantum Information I
Barry C. Sanders
IQIS, University of Calgary, www.iqis.org
CQCT, Macquarie University, Sydney, Australia, www.qcaustralia.org
Montreal August 2005
Implementations
One task in physics is to implement quantum information
processing, i.e. realize quantum communication and
quantum information processing.
There are many challenges because of imperfections in
systems and decoherence, but there are promising
techniques such as quantum error correction to
overcome these problems.
Many candidates for physical quantum information
processing, and we study some of these here.
I. Introduction
 General qubit state:
 General multiqubit state:
 Density matrix:
One Universal Set of Gates
Identity and Not gates:
Hadamard gate:
Phase gates:
Controlled phase gate (equivalent to ^X=CNOT under local unitaries:
Goals
 Encode qubits in physical system.
 Process these qubits.
 Universal set of gates for quantum computation.
 Qubit-specific readout.
 Store qubits.
 Minimize decoherence.
 Correct errors.
Problems
 Qubit may in Hilbert space larger than two
dimensions: truncate!
 Coupling to environment ==> decoherence.
 Imperfect gates so they don’t effect precisely
the desired transformation
 Preparation and readout
DiVincenzo Criteria
1.
2.
3.
4.
5.
Scalability.
Ability to initialize.
Long decoherence times.
Universal set of quantum gates.
Qubit measurement capability.
Additional Criteria
6. Interconvertibility between physical qubits.
7. Faithfully transmit flying qubits.
Investigate proposals
 Trapped ions
 Nuclear spins
 Spin-based quantum dot qubit
 Photons
Our General Approach
 Identify the physical qubits in a given system.
 Determine the Hamiltonian(s) governing
dynamics of the qubits:
 The Hamiltonian generates the unitary
evolution operator, which performs
processing:
The Hamiltonian
 The Hamiltonian operator is a function of
operators concerning degrees of freedom of
the system.
 If quantum information is encoded in
positions x1 and x2 of two particles, then
with … representing other relevant operators.
Real systems are highly complicated, and
creating an effective model is high art!
Harmonic Oscillator
 A simple harmonic oscillator in one dimension
is described by
 The particle has mass m and angular frequency
w, which is independent of amplitude.
 Number operator
has spectrum 0,1,2,….
 Eigenstates corresponding to number of quanta
are {|n}, e.g. photons (for light) or phonons (for
vibrations).
Phonons
 Number of quanta are increased or decreased
by creation or annihilation operators:
 The position operator can be represented by
 The conjugate momentum operator can be
represented by
The Environment
 The Hamiltonian generates unitary evolution,
which corresponds to dynamics in a closed
system, but the system must be open for
preparation and readout.
 The openness is the coupling of the system to
the environment; e.g. a puck sliding on ice is
slowed by frictional coupling to ice and air
resistance so ice and air are part of the puck’s
environment.
Growing the Hamiltonian
 The Hamiltonian for the entire model must
include system and environment.
 If the environment has dynamical degrees of
freedom {ci}, these are included in the
Hamiltonian; extend previous Hamiltonian:
 The system+environment state
evolves
according to unitary evolution generated by
this bigger Hamiltonian.
The Reduced State
 The state of the system+environment is not
useful to us; we just want to know the state of
the system.
 We discard all information about the
environment by tracing the density matrix for
system+environment over environment
degrees of freedom:
 The state of the system is, in general mixed.
 Decoherence-free subspaces and quantum
error correction are designed to protect purity.
Summary
 Goals are to encode quantum information in a
physical system and realize quantum gates and single
qubit measurements, perhaps with subsequent
dynamics dependent on these measurement results.
 Dynamics determined by Hamiltonian, which
generates the evolution operator describing the gates
and circuits.
 Systems are necessarily coupled to the environment,
and decoherence-free subspaces and quantum error
correction are designed to protect against
environment-induced degradation.
II. Trapped ions
Trapped Ions
 The trapped ion system is an early and
promising medium for realizing quantum
information processing.
 Ions are charged atoms, and electric fields are
used to confine or move these ions in a lattice.
 Quantum information is encoded in the
electron energy level.
 Coupling is obtained via collective motion,
which is quantized (with the quanta called
photons).
Ion
 Charged atom - number of electrons is greater
than or less than number of protons.
 Concerned with outermost electron orbiting
“shielded” nucleus.
 Angular momentum J is a vector sum of spin s
and orbital angular momentum L: J=s+L.
 Spectroscopic notation:
 For L, use S for L=0, P for L=1, D for L=2, …
Atom-Photon interactions
Photon absorption from
ground to excited state.
Stimulated emission from
ground to excited state.
Emitted photon is a copy of the “trigger” photon.
Spontaneous emission from
ground to excited state.
Emitted photon is random in direction and phase.
For S-P transitions, rate is once per nanosecond
and, for S-D, rate is once per second: S-D is better.
Cirac-Zoller (1995) proposal: N ions in a linear trap, each interacting
with a separate laser beam. Ions are confined by harmonic potentials
in each of x, y, z directions with x frequency much less than for y,z.
Excitation of alkali ion dipole-forbidden
transition
Driving the Ion
 Each laser beam acts on one ion located at the node of
the laser field standing wave.
 There are two excited states, with transition to q=0 or
q=1 determined by laser polarization.
 Ions share a collective centre-of-mass motion with
energy restricted to zero or one phonon.
e1
e0
g
Single-Qubit Rotation
 Laser field is tuned to the |g—|e0 transition
with the laser polarization set to q=0.
 The Rabi frequency W is the strength of the
atom-field interaction; f is the laser phase.
 The Hamiltonian for the nth ion is
 For evolution time t=kp/W,
Interaction Hamiltonian
 Laser acts on nth ion and is detuned by centreof-mass motion angular frequency nx.
 Change of electronic energy is accompanied
by creation or annihilation of one phonon.
 For q the angle between laser beam and x-axis,
and k = 2p/l for l the laser field wavelength.
Unitary Evolution
 Apply the laser beam for time
 Evolution is given by
 This transformation doesn’t change
 Effects transformation
Two-Qubit Gate: ^Z
Start
Um1,0(0)
Un2,1(0)
Um1,0(0)
|gm|gn|0
|gm|gn|0
|gm|gn|0
|gm|gn|0
|gm|e0n|0
|gm|e0n|0
|gm|e0n|0
|gm|e0n|0
|e0m|gn|0
-i|gm|gn|1
i|gm|gn|1
|e0m|gn|0
|e0m|e0n|0
-i|gm|e0n|1
-i|gm|e0n|1 -|e0m|e0n|0
Universal Quantum Computation
 The Cirac-Zoller scheme enables universal
quantum computation for ions by combining
single-qubit rotations with the two-qubit ^Z.
 Requires atoms to be cold.
 Sorensen and Molmer introduced a
bichromatic off-resonant driving schemes that
allows ions to be ‘warm’: two-photon
processes interfere to minimize sensitivity of
ions to vibrational state.
XX
Independent of Phonon Number
 Two-photon transition via intermediate states
 Two-photon transition rate for |ggn —|een:
 Same transition rate for |egn —|gen.
 Two-photon transitions interfere to remove n
from the effective two-photon Rabi frequency.
Architecture for a Large-Scale
Computer
 Kielpinski-Monroe-Wineland, Nature 2002.
 Static ion traps may be limited to a few dozen
ions so KMW suggested multiple quantum
registers with quantum communication
between these registers.
 Trapped ions are stored in quantum memory
registers, with ions shuttled between registers.
 Ions are transported quickly and then recooled
by sympathetic cooling (via cooling process on
a different species of ion in the
neighbourhood.)
Electrode
Segments
Interaction
Region
Memory
region
Decoherence-Free Subspace
 Ion transport presents another problem: qubit
evolves according to |g+|e|g+eia|e.
 The parameter a is random and is due to
varying magnetic field strength along the ion’s
path, resulting in random fluctuations in the
energy separation of|g and |e.
 Reduce this decoherence by encoding the
logical qubit as |0=|ge and |1=|eg.
 The logical qubits are invariant (up to
unimportant phase) under stochastic magnetic
fields.
Summary
The Cirac-Zoller proposal for quantum computation in
an ion trap is one of the first and influential, especially
in the conception of creating two-qubit unitary gates.
The Sorensen-Molmer proposal relaxes constraints on
the temperature of the ions by using an off-resonant
bichromatic driving field.
The Kielpinski-Monroe-Wineland proposal shows how
to surpass the scaling arguments for static ions by
using multiple registers, quantum communication, and
decoherence-free subspaces.
III. Various Proposals
Various proposals
 In ion-trap quantum computation, quantum
information is stored in electronic levels, and
laser fields drive transitions with vibrations
acting as the bus.
 Another early and promising proposal
concerns encoding quantum information onto
nuclei and using magnetic fields for readout.
 Here we discuss this nuclear magnetic
resonance technique plus coupled quantum
dots and also photons.
Nuclear Spin Quantum Computation
 Nuclear spins - DiVincenzo Science 1995, Cory
et al and Gershenfeld & Chuang both 1995.
 Qubit is realized as a nucleus with gm its
magnetic dipole moment, in a static magnetic
field B0 along z-axis and an alternating timedependent magnetic field B1 with angular
frequency w along y axis; the Hamiltonian is
 Couple to specific qubit by tuning w.
Two-Qubit Coupling
 This Hamiltonian generates a unitary
evolution that is a combination of single-qubit
rotation (by tuning w ) and two-qubit
operations.
 Quantum computing has been performed in
liquid state; unfortunately initialization is
difficult as the density matrix is always close
to identity.
 Scalability may be resolved using solid state.
Loss and DiVincenzo QDot QComputer
 A quantum dot confines
an electron in all 3
dimensions: artificial
atom.
 Qubit corresponds to
spin state of quantum
dot’s excess electron.
 Higher energies ignored.
 The T-gate is a magnetic gate that rotates the electron spin:
single-qubit rotations.
 The J-gate is the coupling term effected by inter-dot electron
quantum tunneling.
Effects gate operation:
A-Gates
BAC (10-3) T
J-Gates
B
(2 T)
Barrier
e-
e-
Kane’s Proposal 1998
31P+
31P+
Si Substrate
The J-gate controls e-e collisions,
hence the coupling of 31P’s.
Photons
 The electromagnetic field consists of modes
identified, for example, by frequency, spatial
characteristics, and polarization.
 Each mode is a harmonic oscillator, and the
number state |n corresponds to n photons.
 A qubit can be a superposition of no photon
and one photon in a mode, a|0+b|1 (singlerail qubit), or exaclty one photon in a
superposition of two modes, a|01+b|10
(dual-rail qubit).
Dual-Rail Photonic Qubit
 A beam splitter produces the unitary operation
 The beam splitter rotates the dual-rail qubit.
 Ideal “Kerr nonlinear medium”
 The Kerr interaction provides
the two-qubit operation
sufficient for universal
circuits.
 Unfortunately c is miniscule
or, if enhanced, accompanied
by unacceptably high photon
losses.
Chuang & Yamamoto 1995
Knill-Laflamme-Milburn 2001
 A solution to weak
nonlinearity provided
by GC teleportation
gate, which teleports
input to “processed”
output by feeding
entangled ancillas
produced offline.
Gottesman & Chuang 1999
 KLM developed a circuit to produce Kerr nonlinearity rarely,
but successes are signaled by photodetection events.
 Successful outputs are provided as ancillas for a GC type of
teleporter; by only supplying proper states, the quantum
computation is not hindered by the local success rate of the
nonlinear gate.
One-way computation
 More recently Raussendorf and Briegel introduced one-way
quantum computation with cluster states, based on processing
highly entangled states via successive single qubit
measurements.
 This approach seems promising for realizing quantum
computation with photons.
Raussendorf and Briegel 2001
Summary
Surveyed a few proposals for quantum computation
using ions, nuclei, quantum dots, and photons.
Ignored schemes such as optical cavities,
superconducting qubits, and fullerene.
The theory can be understood once the model
Hamiltonian is determined, and a lot of work is
underway to realize such Hamiltonian dynamics and
understand the environment.