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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. Measurements
a. Extracting expectation values
b. Extracting probabilities
c. Noninvasive measurements
d. Ancilla Assisted State-Tomography
e. Ancilla Assisted Process-Tomography
3. Quantum Simulations
a. Particle in a potential
b. Introducing quantum noise
4. Phase Encoding (Quantum Sensors)
a. Diffusion in liquids
b. Mapping-out electromagnetic fields
5. Summary
Outline
Nuclear Spin and Magnetic Resonance
Spin ½ (qubit)
Chloroform
B0
EM energy(Radio waves)
0
1 1H
Nuclear Spin and Magnetic Resonance
B0
EM energy(Radio waves)
0
1
NMR Signal x Tr[ x ]
Net transverse magnetization
x
Procedure:
Prepare x
t
Nuclear Spin and Magnetic Resonance
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
qubitq(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
qubitq(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)
Hemant, Abhishek, Koteswar, TSM, PRA-2013
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
Hemant, Abhishek, Koteswar, TSM, PRA-2013
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
Abhishek, Koteswar, TSM, PRA-2013
Ancilla Assisted Quantum State Tomography: Noisy Measurements
Abhishek, Koteswar, TSM, PRA-2013
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
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
Trimethylphosphite(300 K, DMSO, fixed conc.)
PreparingNOON states
Converting tosingle-quantum
states
Abhishek, Manvendra, TSM, CPL-2013
10-qubits
31P
Trimethylphosphite(300 K, DMSO, fixed conc.)
Measuring diffusion: NOON states
Abhishek, Manvendra, TSM, CPL-2013
Mapping RF Intensity with NOON states:
Abhishek, Manvendra, TSM, CPL-2013
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
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