Fidelity-Optimized
Quantum State Estimation
Itay Hen
Information Sciences Institute, USC
NIPS Quantum Machine Learning
Workshop
December 12, 2015
Joint work with Amir Kalev, UNM
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Disclaimer
disclaimer: no Quantum Machine Learning per se here,
but… machine learning for quantum systems
Disclaimer: 1) a renunciation of any claim to or connection with;
2) disavowal; 3) a statement made to save one’s own ass.
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The Problem
oven
measurement
apparatus
an oven is emitting identical copies of the same
unknown quantum state in a steady flow.
objective: find out what this state is
what measurements should we
perform?
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The Problem
oven
measurement
apparatus
an oven is emitting identical copies of the same
unknown quantum state in a steady flow.
what is the optimal sequence of measurements
that would yield the best estimate with the
smallest
error and in the least amount of measurements?
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Outline
Itay Hen

some probability theory

the protocol

some results

conclusions and applications for
actual quantum machine learning
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Probability theory
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Some probability theory
given an emitted state
getting
the outcome
?

Itay Hen
, what is the probability of
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Some probability theory
given an emitted state , what is the probability of
getting
the outcome
after a single measurement?
 what is the probability of getting the sequence of
outcomes
?

Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Some probability theory
given an emitted state , what is the probability of
getting
the outcome
after a single measurement?
 what is the probability of getting the sequence of
outcomes
?


now, given the sequence of outcomes
, what is the
probability that the emitted state is ? for that, we
have Bayes’ law:
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Some probability theory
now, given the sequence of outcomes
, what is the
probability that the emitted state is ? for that, we
have Bayes’ law:

probability of getting
the state given the
sequence of outcomes
probability of getting
the sequence of
outcomes given the
state
the a priori
probability
of the state
the probability
of obtaining the
sequence of
outcomes

let us assume for simplicity (we don’t have to) that we have no
knowledge about oven, i.e., that
.

moreover, we have
Itay Hen
NIPS Quantum Machine Learning Workshop
.
Dec 12, 2015
Some probability theory

we thus end up with:

where:
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The protocol
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The protocol

equipped with the above probability measure, we can give
a
general
“learning” protocol for optimized adaptive
1. perform
2. based on record of
measurement in a
tomography:
randomly-chosen
basis
measurement
outcomes thus far,
find most-likely state
5. execute measurement
in optimal basis
4. compute optimal basis
for next measurement
3. exit if convergence
criterion has been
reached, otherwise:
remaining questions:
how do we calculate most-likely state / best guess?
how do we determine the optimal measurement
basis?
Dec 12, 2015
Itay
Hen
NIPS Quantum Machine Learning Workshop

Most likely state

given a list of outcomes from all measurements thus far.
what should be our best guess in the k-th step for the
emitted state ?
this actually depends on how we define “best”. let’s say
we’d like
to maximize the fidelity of our guess with the real thing.


obviously, we don’t know what
is, but we know the
probability of occurrence for each state, so we can guess:

plugging in what we already have for
Itay Hen
NIPS Quantum Machine Learning Workshop
, we get:
Dec 12, 2015
Most likely state

now, we can rewrite

as:

put differently:
where
and
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Determining next basis of measurement
given a list of prior measurement outcomes,
how shall we determine the next basis of
measurements?


is there a simple clear answer?
we have to carefully state what we would like
accomplished.

we would like to maximize the
fidelity of the emitted state with
our best guess after the
measurement
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Determining next basis of measurement

how do we do that?
let’s say that the chosen basis of measurement in the
k-th
step is:
 let’s assume that after the measurement is carried
out, the
obtained outcome is
with
.

we would like to maximize the fidelity of our best
guess
based
all state.
outcomes “so far”
with
theonreal
but, we have already calculated that, it’s simply

Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Determining next basis of measurement

of course, we do not know which of the outcomes
we’ll get.
we must therefore average over all possible
outcomes,
namely:

here,
is the probability of obtaining the
n-th
outcome given what we know so far about the emitted
state:

Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Determining next basis of measurement

but what is:
exactly?

it’s:
putting it all together, we find that the optimal basis is
simply:

where
and
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Some Results
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Next basis of measurement: an example

example: an oven is emitting copies of a qubit.

let’s say the following outcomes have been obtained:
•
4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.

in which direction should the next measurement be
performed?

first, what’s the “most likely” state?
oven
Itay Hen
measurement
apparatus
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Next basis of measurement: an example

example: an oven is emitting copies of a qubit.

let’s say the following outcomes have been obtained:
•
4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.

in which direction should the next measurement be
performed?

clearly, the best guess is up-z.
oven
Itay Hen
measurement
apparatus
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Meaning of results
what does the requirement of maximizing

where
and
mean exactly?
it tells us that we should find a basis of
measurements
such that all outcomes are equally probable!

it tells us to perform a measurement in a basis that
we
Dec 12, 2015
Itay
Hen
NIPS
Quantum what
Machine Learning
Workshop
cannot
possibly
guess
the outcome
is!

Meaning of results

•
going back to the example, outcomes are:
4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.

in which direction should the next measurement be
performed?

if we perform a measurement in the
z-direction, we have a pretty good guess
of what of the outcome is going to be.

this is not the case in the x and y
directions. but, there’s more
certainty in the x direction.
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Meaning of results

•
going back to the example, outcomes are:
4 up-z, 3 up-x and 3 down-x, 2 up-y and 2 down-y.

in which direction should the next measurement be
performed?

if we perform a measurement in the
z-direction, we have a pretty good guess
of what of the outcome is going to be.

this is not the case in the x and y
directions. but, there’s more
certainty in the x direction.

we should therefore measure in the y direction!
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The qubit case: first few
measurements
consider the first few iterations of the protocol in the qubit case


an oven is emitting qubits one by one…

protocol dictates that we perform the first measurement in some
random direction. let’s call the outcome up-z.

what does the protocol say about next basis of measurement?
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
The qubit case: first few
measurements
consider the first few iterations of the protocol in the qubit case


an oven is emitting qubits one by one…

protocol dictates that we perform the first measurement in some
random direction. let’s call the outcome up-z.

what does the protocol say about next basis of measurement?
should be in a basis “orthogonal to z”. measurement direction
should be
on equator of Bloch sphere. let’s call the outcome up-x.

what about the next measurement basis? orthogonal to z and x,
namely
y.


next one is more complicated…
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Numerical results: the qubit case

repeatedly performing a numerical experiment thousands of
times, we calculated the mean infidelity (with respect to the
true state) as a function of number of measurements.
compared several methods:
• random-basis
measurements.
• repeated measurements
in the x, y, z directions.
• x, y, z measurements
chosen in optimal way.
• fully-optimized.


no surprise, learning
methods are superior.
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Numerical results: the qudit case
(d=4)
another example: qudit with d=4. again, mean
infidelity as a
function of number of measurements.

here, we’re assuming that the available
measurements are
only “local Pauli”, i.e.,
xx, xy, xz,…,zz.


comparing a random
sequence of local Pauli
measurements with an
optimized sequence.
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Conclusions and what’s
next
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015
Conclusions

optimized adaptive tomography helps!

easily extended to emitted mixed states, generalized
measurements, etc.
what’s next? we have seen that the first few
optimizations
of the measurement bases yield “orthogonal” or,
mutually
unbiased, bases. this procedure can therefore be
used to
generate sets of MUBs (or, so we believe).

can be carried over to machine learning protocols,
e.g.,
Wiebe et al’s “Quantum
Hamiltonian Learning”. inDec 12, 2015
Itay Hen
NIPS Quantum Machine Learning Workshop

Thank You!
Fidelity Optimized
Quantum State Estimation
Itay Hen
Information Sciences Institute, USC
NIPS Quantum Machine Learning
Workshop
December 12, 2015
Joint work with Amir Kalev, UNM
Itay Hen
NIPS Quantum Machine Learning Workshop
Dec 12, 2015