1. QEC conditions
Energy barriers
A quantum error correction scheme must
satisfy the condition
Where Ψi are code words and Cab is a
Hermitian matrix. For our code words, we
used Coherent Spin States (CSS). Typically a
CSS is expressed in the form
While this condition cannot be strictly
satisfiedwith a CSS, it can be approximately
satisfied.Furthermore the fit of this
approximation can be arbitrarily improved by
increasing the number of atoms in the
coherent state.
Supposing that the orthogonally of the code
words holds true once the number of qubits
surpasses some threshold N0 It follows that
any qubits in excess of this contribute to the
distance of the correction scheme.
The defining characteristicof self correcting
memories is the scaling of the coherence
time with system size. An argument must be
made from either energeticsor dynamics
that adding additional qubits is economical.
[figure]
(probability heat map on the bloch sphere beta = 5)
We have shown that the energy requisite to
move a certain distance on the bloch sphere
within a certain time frame is given by E =
(θn/t)2.Thus an arbitrary confinementof
errors can be achieved with additional
qubits. Knowing the energy barrier between
codewords lets us estimate the probability of
transition between them.
Simulation
We model the environment as a series of
harmonic oscillators to derive the master
equation for the system:
We have simulated the time evolution of the
density matrix for the case of 2 – 8 qubits
and can thus calculate the expectation value
of any observable. Of particular interest is
the Husimi Q function and the variance in Sx,
Sy and Sz.
Quantum Error Correction in systems of
ultra-cold atoms
Louis Tessler1 and Tim Byrnes1,2
Contact: IIAOPSW@gmail.com
Introduction
Using cold atoms for quantum computing
has come into vogue in recent years. Cold
atom ensembles exhibit quantum behavior
while also being somewhat macroscopic.
Combined, these properties imbue them
with superior coherence characteristics.
Previously our group has shown
entanglementand universal gates are
possible in this context.Inour paper we show
that the an ensemble with a Hamiltonian H =
-S2 has error correcting properties.This key
piece will paves the way for practical
quantum computing based on cold atom
technology.
Results
8 qubits T=0
T=2 Γ
T=10 Γ
Conclusions
We have shown that H = -S2 has desirable
properties for error correction.We have
shown consistencywith the established
framework of quantum error correction and
further demonstratedthe desired properties
numerically. In future work we will expand
the correction scheme to the case of more
than one ensemble – qubit.
1. New York University, Shanghai
2. National Instituteof Informatics,Tokyo
Gottesman, Daniel PhD thesis
Brown et al, arXiv:1411.6643v1
NSFC,
NTT Basic Research Laboratories,
Shanghai Research Challenge Fund.
Pictured above is the simulated time evolution of
a state with a single error compare this with the Q
function for a state without error.
Cold atoms
Recent experimental work by Polzik has
shown that ensembles of ultra-coldatoms
can be used for quantum information related
tasks like teleportation.Furthermore it is
possible to interface ensembles with other
quantum technologiesby. Thus it is worth
exploring the implications of a quantum
computer based entirely on such ensembles.
T. Byrnes, et al., Optics Communications (2014)
Polzik, E.S, doi:10.1038/nphys2631
No go Theorems
Brown et al, arXiv:1411.6643v1
In general we can
side step the no-go
results by violating
the assumption of
locality.
Furthermore the
no-go theorems do
not rule out self
correction in 4 or
higher dimensions.
The relevant question is how many
neighbors each qubit has (the effective
dimensionality)not the actual number of
spatial dimensions. Thus 4d is not a serious
problem