Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Itay HenInformation Sciences Institute, USC

NIPS Quantum Machine Learning Workshop

December 12, 2015

Fidelity-Optimized Quantum State Estimation

Joint work with Amir Kalev, UNM

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Disclaimer

disclaimer: no Quantum Machine Learning per se here,

Disclaimer: 1) a renunciation of any claim to or connection with;2) disavowal; 3) a statement made to save one’s own ass.

but… machine learning for quantum systems

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

The Problem

an oven is emitting identical copies of the same unknown quantum state in a steady flow.objective: find out what this

state iswhat measurements should we perform?

oven measurementapparatus . .. . .. . ..

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

The Problem

what is the optimal sequence of measurements that would yield the best

estimate with the smallest error and in the least amount of

measurements?

oven measurementapparatus . .. . .. . ..

an oven is emitting identical copies of the same unknown quantum state in a steady flow.

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

some probability theory the protocol some results conclusions and applications for actual quantum machine learning

Outline

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Probability theory

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

given an emitted state , what is the probability of getting the outcome ?

Some probability theory

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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 ?

Some probability theory

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

now, given the sequence of outcomes , what is the probability that the emitted state is ? for that, we have Bayes’ law:

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

Dec 12, 2015NIPS Quantum Machine Learning Workshop

now, given the sequence of outcomes , what is the probability that the emitted state is ? for that, we have Bayes’ law:

Some probability theory

probability of gettingthe 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

Dec 12, 2015NIPS Quantum Machine Learning Workshop

we thus end up with:

where:

Some probability theory

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

The protocol

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

equipped with the above probability measure, we can give a general “learning” protocol for optimized adaptive tomography:

The protocol

1. perform measurement in

a randomly-chosen basis

2. based on record of measurement

outcomes thus far, find most-likely

state

4. compute optimal basis for next measurement

5. execute measurement

in optimal basis 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?

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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 , we get:

Most likely state

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

now, we can rewrite

as:

put differently:

Most likely state

where

and

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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.

Determining next basis of measurement

we would like to maximize the fidelity of the emitted state with our best guess after the measurement

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

how do we do that? let’s say that the chosen basis of

measurement in the k-th step is:

Determining next basis of measurement

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 on all outcomes “so far” with the real state.

but, we have already calculated that, it’s simply

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

of course, we do not know which of the outcomes

we’ll get.

Determining next basis of measurement

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

Dec 12, 2015NIPS Quantum Machine Learning Workshop

but what is:

exactly?

Determining next basis of measurement

it’s:

putting it all together, we find that the optimal basis is simply:

where

and

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Some Results

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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?

Next basis of measurement: an example

oven measurement

apparatus

. .. . .. . ..

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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.

Next basis of measurement: an example

oven measurement

apparatus

. .. . .. . ..

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Meaning of results what does the requirement of maximizing

where

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 cannot possibly guess what the outcome is!

putting it all together, we arrive at:

and

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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.

Meaning of results

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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!

Meaning of results

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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?

The qubit case: first few measurements

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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…

The qubit case: first few measurements

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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 measurementschosen in optimal way.• fully-optimized.

no surprise, learning methods are superior.

Numerical results: the qubit case

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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.

Numerical results: the qudit case (d=4)

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Conclusions and what’s next

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

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”. in some situations, learning curve can be optimized, provided that we can utilize all the gathered information to maximize our knowledge about desired quantities.

Conclusions

Itay Hen

Dec 12, 2015NIPS Quantum Machine Learning Workshop

Itay HenInformation Sciences Institute, USC

NIPS Quantum Machine Learning Workshop

December 12, 2015

Fidelity Optimized Quantum State Estimation

Joint work with Amir Kalev, UNM

Thank You!