QUANTUM TOMOGRAPHY WITH AN APPLICATION TO A CNOT GATE

OUTLINE

Quantum State Tomography Finite Dimensional Infinite Dimensional (Homodyne)

Quantum Process Tomography (SQPT) Application to a CNOT gate Related topics

QUANTUM STATE TOMOGRAPHY

QST “is the process of reconstructing the quantum state (density matrix) for a source of quantum systems by measurements on the system coming from the source.”

The source is assumed to prepare states consistently

QUANTUM STATE TOMOGRAPHY

Simply put:

Do this a lot

FINITE DIMENSIONAL SPACE

Typically easier to work with Know a priori how many coefficients to

expect The value of n is known

FINITE DIMENSIONAL SPACE

Easily approached via linear inversion Ei is a particular measurement outcome

projector S and T are linear operators

.

FINITE DIMENSIONAL SPACE

Use measured probabilities and invert to obtain density matrix Sometimes leads to nonphysical density

matrix!

.

MAXIMUM LIKELIHOOD ESTIMATION

“the likelihood of a set of a parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values”

The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state

QST FOR ONE QUBIT

Example from class: 1 qubit

Repeatedly measure sigma x

FINITE DIMENSIONAL SPACE

FOUND r1!

INFINITE DIMENSIONAL SPACE

The value of n is unknown!

Make multiple homodyne measurements Obtain Wigner function

Find density matrix

HOMODYNE MEASUREMENTS

Analogous to constructing 3d image from multiple 2d slices

Goal is to determine the marginal distribution of all quadratures

QUANTUM PROCESS TOMOGRAPHY

In QPT, “known quantum states are used to probe a quantum process to find out how the process can be described”

QUANTUM PROCESS TOMOGRAPHY

In essence:

QUANTUM PROCESS TOMOGRAPHY

In practice:

QPT

J.L. O’Brien: “The idea of QPT is to determine a completely positive map ε, which represents the process acting on an arbitrary input state ρ”

Am are a basis for operators acting on ρ

QPT

Choose set of operators: Use input states:

QPT

Form linear combination

Do QST to determine each

Write them as a linear combination of basis states

QPT

Solve for lambda Now write

And solve for beta (complex)

QPT

Combine to get

Which follows that for each k:

QPT

Define kappa as the generalized inverse of beta

And show that satisfies

QPT FOR A SINGLE QUBIT

OPERATORS BASIS

QPT FOR A SINGLE QUBIT

Use input states

Now QST on output

QPT FOR A SINGLE QUBIT

Use QST to determine

QPT FOR A SINGLE QUBIT

Results correspond to

Now beta and lambda can be determined, but due to the particular basis choice and the Pauli matrices:

QPT FOR A SINGLE QUBIT

Finally arriving to:

APPLICATION TO CNOT

J.L. O’Brien et al used photons and a measurement-induced Kerr-like non-linearity to create a CNOT gate

CNOT

QPT IN PRACTICE

Φa are input states Ψb are measurement analyzer setting cab is the number of coincidence detections

RESULTS

Average gate fidelity: 0.90 Average purity: 0.83 Entangling Capability: 0.73

RELATED TOPICS

Ancilla-Assisted Process Tomography (AAPT) d2 separable inputs can be replaced by a suitable

single input state from a d2-dimensional Hilbert space

Entanglement-Assisted Process Tomography (EAPT) Need another copy of system

Tangle

SOURCES

“Quantum Process Tomography of a Controlled-NOT Gate” http://quantum.info/andrew/publications/

2004/qpt.pdf Quantum Computation and Quantum

Information Michael A. Nielsen & Isaac L. Chuang

Wikipedia