ancilla-assisted quantum information processing

Ancilla-Assisted Quantum Information Processing

Indian Institute of Science Education and Research, Pune

T. S. Mahesh

Acknowledgements

Abhishek Shukla

Swathi Hegde

Hemant Katiyar

Koteswara Rao

Manvendra Sharma

Ravi Shankar

Prof. Anil Kumar

Dr. Vikram Athalye

Prof. Usha Devi

Prof. A. K. Rajagopal

PhD students

MS students

Collaborators

system

ancilla

Ancillary staff: Provide necessary support to the primary activities

or operation of an organization, system, etc.

Dictionary meaning:

ancilla

system

1. Spin-Systems and NMR

2. Measurementsa. Extracting expectation valuesb. Extracting probabilitiesc. Noninvasive measurementsd. Ancilla Assisted State-Tomographye. Ancilla Assisted Process-Tomography

3. Quantum Simulationsa. Particle in a potentialb. Introducing quantum noise

4. Phase Encoding (Quantum Sensors)a. Diffusion in liquidsb. Mapping-out electromagnetic fields

5. Summary

Outline

Nuclear Spin and Magnetic Resonance

Spin ½ (qubit)

Chloroform

B0

EM energy(Radio waves)

0

1 1H


B0

EM energy(Radio waves)

0

1

NMR Signal x Tr[ x ]

Net transverse magnetization

x

Procedure:

Prepare x

t


Ancilla assisted measurement:

1H13C

Prepare

Prepare|+

A1 A2

x

System qubit

Ancillaqubit

x = A1 A2 Am

Am

O. Moussa et al, PRL,104, 160501 (2010)

Prepare

Prepare|+

A

x

System qubit

Ancillaqubit

x = A

Unitary observable

Example: Evaluating Leggett-Garg inequality

t = 0 t 2t

x

↗

x

↗

x

↗

x

↗

x

↗

x

↗time

x(0)x(t) = C12

x(t)x(2t) = C23

x(0)x(2t) = C13

0

0

0

Hamiltonian : H = ½ z

Macrorealistic: K3 = C12 + C23 C13 1

For spin ½ : K3 = 2cos(t) cos(2t) (-3 K3 -1.5)

Athalye, S. S. Roy, TSM, PRL-2011

t

1H13C

A. J. Leggett and A. Garg, PRL-1985

Johannes Kofler, PhD Thesis, 2004

Example: Evaluating Leggett-Garg inequality

1H13C

Athalye, S. S. Roy, TSM, PRL-2011

t = 0 t 2t

x

↗

x

↗

x

↗

x

↗

x

↗

x

↗time

x(0)x(t) = C12

x(t)x(2t) = C23

x(0)x(2t) = C13

0

0

0

Hamiltonian : H = ½ z

Macrorealistic: K3 = C12 + C23 C13 1

For spin ½ : K3 = 2cos(t) cos(2t) (-3 K3 -1.5)

A. J. Leggett and A. Garg, PRL-1985

Johannes Kofler, PhD Thesis, 2004

Extracting probabilities (in computational basis)

crusher

incoherence

convert

measure

Arbitrary 1q density matrix

Diagonal density matrix

Single quantum density matrix

xPrepare tU

U(dephasing channel)

Extracting joint probabilities

t t+tSystem qubit

q(t) q(t+ t)

p( q(t),q(t+ t) ) ?

U(t)x

System qubit

Ancillaqubit

Prepare

Prepare |0

x

U(t)

Suppose Q be an observable, with eigenvalues q = 0 or 1

Extracting joint probabilities: Noninvasive method (Negative Result)Suppose Q be an observable, with eigenvalues q = 0 or 1

t t+tSystem qubit

q(t) q(t+ t)

p( q(t),q(t+ t) ) ?

U(t)x

System qubit

Ancillaqubit

Prepare

Prepare |0

x

U(t)

U(t)x

System qubit

Ancillaqubit

Prepare

Prepare |0

x

U(t)

Discord q = 1---------------------p(0,0) & p(0,1)

Discord q = 0---------------------P(1,0) & p(1,1)

p(q1,q2) p(q1,q3)

time

Q1 Q2 Q3

t2 t3t1

Hemant, Abhishek, Koteswar, TSM, PRA-2013

Extracting joint probabilities

CHsystem

ancilla

Entropic Leggett-Garg Inequality

InformationDeficit:

timeQ1 Q2 Q3

t2 t3 . . .

. . .

t1

System state: 1/2

Dynamical observable : Sz(t) = Ut Sz Ut†

Time Evolution: Ut = exp(iSxt)


CHsystem

ancilla

A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, PRA-2013

Reason for LGI violation:

Classical Probability Theory:

P’(q1,q2) = P(q1,q2,q3)q3

P’(q1,q3) = P(q1,q2,q3)q2

P’(q2,q3) = P(q1,q2,q3)q1

P(q1,q2)

P(q1,q3)

P(q2,q3)

Marginals Grand

Quantum systems do not obey this rule !!

A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, PRA-2013

Extracting GRAND probabilities: Suppose Q be an observable, with eigenvalues q = 0 or 1

0 tSystem qubit

Q(0) q(t)

p(q(0),q(t),,q(nt)) ?

(n-1)t

q((n-1)t)

nt

Q(nt)

xSystem qubit

nancillaqubits

x

U(t) U(t)Prepare

Prepare |0

Prepare |0

Prepare |0

U(t) U(t)

Illegitimate Joint Probability

P(q1,q2,q3)is illegitimate !!

Violation ofEntropic LGI


Quantum State Tomography

Tomography:

Quantum State TomographyComplete characterization of complex density matrix

- Requires a series of measurements all starting from same initial condition

= +

Obtained bymeasuring

z

Obtained bymeasuring x and y

9 different experimentscarried out

3-unknowns

15-unknowns

Measure: x(1) |00|,

x(1) |11|,

|00| x(2),

|11| x(2),

After rotations:II, XI, YI, IX, IY, XX, XY, YX, YY

Complexsignal ofTwo-qubits

Quantum State Tomography: Scaling

n-qubit system:

n 2nNumber of experiments ~

Observables per experiment

22n

Number unknowns in the density matrix

= n2n

n-qubits

number of experiments

2 23

4

7

11

19

2n x 2n density matrix

System qubits

ancilla qubits

|00…0

System qubits

|00…0

ancilla qubits

Ucomp

System qubits

ancilla qubits

Utomo

x

Ancilla Assisted Quantum State Tomography:

(n+a)-qubit system:

n 2(n+a)Number of experiments ~

Observables per experiment

22n

Number unknowns in the density matrix

= n

2n - a

Nieuwenhuizen & coworkers, PRL-2004

Ancilla Assisted Quantum State Tomography: Scaling

(a)(n)

n2n - a

Abhishek, Koteswar, TSM, PRA-2013

Ancilla Assisted Quantum State Tomography:

Fidelity: 0.95

3-system qubits, 2-ancilla qubits


Ancilla Assisted Quantum State Tomography: Noisy Measurements


Quantum Process Tomography:- Characterizes the process (unitary or nonunitary)

Standard method:

1

1

1

1

matrix

tomo

tomo

tomo

tomo

b1

b2

b3

b4

() = mn EmEn†

mn

Ancilla Assisted Process Tomography:- Characterizes the process (unitary or nonunitary)

Using a single ancilla qubit

11

11

matrix

(on system)tomo

() = mn EmEn†

mn

Altepeter et al, PRL-2003

Single-Shot Process Tomography:- Characterizes the process (unitary or nonunitary)

Using two ancilla qubits

11

11

matrix

process

(on system)

x

() = mn EmEn†

mn

Schrodinger equation: iħ (d/dt) |(t) = H |(0)

|(t) = exp(-iHt)|(0)

H = T + V

KineticP2/2m

Potential

Do not commute

exp(-i H dt) exp(-i V/2 dt) . exp(-i T dt) . exp(-i V/2 dt)

Trotter approximation:

Quantum Simulation: Particle in a potential (1D)

(with spin-1/2 nuclei)

|111 |110 |101 |100 |011 |010 |001 |000

x

exp(-i H dt) exp(-i V/2 dt) . exp(-i T dt) . exp(-i V/2 dt)

Circuit for Diagonal Unitary

Trotter form:

Quantum Simulation: Particle in a potential (1D)

exp(-i H dt) exp(-i V/2 dt) .Uiqft. exp(-i T’ dt) . Uqft . exp(-i V/2 dt)

position

Ancilla Assited Quantum Simulation:

Initial state

Final state(after

Simulation)

Ravi Shankar, Swathi Hegde, TSM, PLA-2013

Ancilla Assited Quantum Simulation: Ravi Shankar, Swathi Hegde, TSM, PLA-2013

Experiments Theory

chloroform

1H (system)

13C (ancilla: environment)

System

Ancilla

Time

System

Ancilla

Time

kicks

Cory & coworkersPRA, 2003

Simulating quantum noise:

chloroform

1H (system)

13C (environment)

Simulating quantum noise:

Has applications in optimizing dynamical decoupling sequences

Swathi & TSM (on-going work)

Measuring diffusion

B0

|0+|1 |0+ei|1

Price, Concepts in NMR-1997

31P

Trimethylphosphite(300 K, DMSO, fixed conc.)

Measuring diffusion

Abhishek, Manvendra, TSM, CPL-2013

B0

|0…0+|1…1 |0…0+ein|1…1

Measuring diffusion: NOON states

31P


PreparingNOON states

Converting tosingle-quantum

states


10-qubits

31P


Measuring diffusion: NOON states


Mapping RF Intensity with NOON states:


31P

Summary:

Ancilla qubits play an important role in practical quantum processors

Provide efficient ways to measure expectation values and joint probabilities

Assist in Quantum State Tomography and Quantum Process Tomography

Assist in direct read-out of probabilities in quantum simulation

Can induce controlled quantum noise on the system qubits

Can participate in preparing large NOON states – have applications in quantum sensors

ancilla-assisted quantum information processing

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