Superposition of two single-qubit states with
partial prior knowledge using NMR
Shruti Dogra
Indian Institute of Technology Madras, Chennai
In collaboration with: George Thomas and Sibasish Ghosh
Institute of Mathematical Sciences, Chennai, India
and Dieter Suter
Technische Universität Dortmund, D-44221 Dortmund, Germany
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Superposition of unknown quantum states
There is a no-go theorem stating that there exists no protocol to
obtain a desired superposition of two unknown pure quantum
states, |ψ1 i and |ψ2 i to get1 :
|Ψi = a|ψ1 i + b|ψ2 i.
(1)
However, with partial prior knowledge, |hχ|ψ1 i| and |hχ|ψ2 i|, one
can obtain superposed state of the form,
hχ|ψ1 i
Nψ
hχ|ψ2 i
|ψ1 i + b
|ψ2 i ,
a
|Ψ0 i =
N
|hχ|ψ2 i|
|hχ|ψ1 i|
where |χi is a known referential state.
Such a protocol is proposed recently by Oszmaniec et al (Phys.
Rev. Lett. 116, 110403 (2016)).
1 Phys. Rev. Lett. 116, 110403 (2016).
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
The problem
Input states: |ψ1 i and |ψ2 i. Let |χi be a reference state, whose
respective overlaps hχ|ψ1 i and hχ|ψ2 i are known.
z |χi
|ψ1 i
φ
y
x
|ψ2 i
We presented an experimentally feasible protocol to superpose n
pure states of a qudit and experimentally performed the
superposition of two pure single-qubit states on a two-qubit NMR
quantum information processor2 .
Calculated success probabilities for different prior information.
2 S. Dogra, G. Thomas, S. Ghosh, and D. Suter, arXiv:1702.02418 [quant-ph] (2017).
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme
(A) Encoding
Consider a system of two coupled spins 1/2 with the Hamiltonian
H = −ΩA AZ − ΩX XZ + JAZ XZ
4
2
3
|01i
1
Shruti Dogra
|11i
|10i
|00i
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme
(A) Encoding
Consider a system of two coupled spins 1/2 with the Hamiltonian
H = −ΩA AZ − ΩX XZ + JAZ XZ
4
2
3
|01i
1
Shruti Dogra
4
|11i
2
|10i
3
|01i
1
|00i
|11i
|10i
|00i
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme
(A) Encoding
Consider a system of two coupled spins 1/2 with the Hamiltonian
H = −ΩA AZ − ΩX XZ + JAZ XZ
4
2
4
|11i
3
|01i
1
2
|10i
3
|01i
1
|00i
|11i
|10i
|00i
Input states: ψ1 = c00 |0i + c01 |1i and ψ2 = c10 |0i + c11 |1i.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme
(A) Encoding
Consider a system of two coupled spins 1/2 with the Hamiltonian
H = −ΩA AZ − ΩX XZ + JAZ XZ
4
2
4
|11i
3
|01i
1
2
|10i
3
|01i
1
|00i
|11i
|10i
|00i
Input states: ψ1 = c00 |0i + c01 |1i and ψ2 = c10 |0i + c11 |1i.
State of two-qubit system,
p
|ηi = a|0i|ψ1 i + b|1i|ψ2 i,
a2 + b2 = 1.
Here first qubit is considered as ancilla, and second qubit is
considered as system-qubit.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme...
(B) Removing the impalpable global phases
Assuming the global phases (eιγ1 and eιγ2 ) with states |ψ1 i and
|ψ2 i respectively,
|η 0 i = eιγ1 a|0i|ψ1 i + eιγ2 b|1i|ψ2 i.
eιγ1 =
hχ|ψ1 i
|hχ|ψ1 i|
and eιγ2 =
hχ|ψ2 i
|hχ|ψ2 i| .
A phase gate, (γ1 − γ2 )1z leads to,
|η 00 i = eι(
Shruti Dogra
γ1 +γ2 )
2
(a|0i|ψ1 i + b|1i|ψ2 i).
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Theoretical scheme...
(B) Removing the impalpable global phases
Assuming the global phases (eιγ1 and eιγ2 ) with states |ψ1 i and
|ψ2 i respectively,
|η 0 i = eιγ1 a|0i|ψ1 i + eιγ2 b|1i|ψ2 i.
eιγ1 =
hχ|ψ1 i
|hχ|ψ1 i|
and eιγ2 =
hχ|ψ2 i
|hχ|ψ2 i| .
A phase gate, (γ1 − γ2 )1z leads to,
|η 00 i = eι(
γ1 +γ2 )
2
(a|0i|ψ1 i + b|1i|ψ2 i).
(C) Superposition
A Hadamard gate on the first qubit leads to,
|η 000 i = |0i ⊗ (a|ψ1 i + b|ψ2 i) + |1i ⊗ (a|ψ1 i − b|ψ2 i).
First qubit measurement in basis {|0i, |1i} gives the desired result,
Nψ (a|ψ1 i + b|ψ2 i) ∝ |Ψi.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental realization
We use a system of two nuclear spins and NMR as a tool to
demonstrate the superposition of two single-qubit states.
NMR quantum computer harnesses the intrinsic magnetic
properties of the nuclear spins, manipulated by specifically tailored
radio frequency pulses (quantum gates) to perform the desired
computational tasks.
The criteria for a quantum computer to work was put forward by
David P. DiVincenzo in 2000. This involves: a working medium,
ample coherence times, a set of implementable operations, ability
to initialize the system in a desired state, and the ability to read
the output.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
NMR Quantum Computation
1
O
H
1
H
1
H
System
13
C
Cl
Quantum Gates
Cl
Cl
2
Initial state preparation
Final readout
19
F
C=C
19
Shruti Dogra
19
F
F
I
1
1−p 1
-2
|1i
p
|0i
+ 12
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
NMR Quantum Computation
( π2 )y
1
1
H=√
1
2
Hadamard
System
Quantum Gates
Initial state preparation
Controlled NOT

Final readout
CN OT12
Shruti Dogra
1
 0
=
 0
0
1
−1
( π2 )2x
1
2J
0
1
0
0
0
0
0
1
( π2 )2−y

0
0 

1 
0
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
NMR Quantum Computation
System
Pseudopure state |ψi
ρ = 1−
n In×n + |ψihψ|
Quantum Gates
Thermal equilibrium
Initial state preparation
4
Final readout
Eigenvalues in a two-qubit system:
|00i: 21 (ω1 + ω2 + 12 J12 )
|01i: 21 (ω1 − ω2 − 12 J12 )
|10i: 21 (−ω1 + ω2 − 12 J12 )
|11i: 21 (−ω1 − ω2 + 12 J12 )
Transitions:
Qubit-1: ω1 ± 12 J12
Qubit-2: ω2 ± 21 J12
Shruti Dogra
2
|11i
3
|01i
1
|10i
|00i
Pseudo-pure state
4
2
|11i
3
|01i
1
|10i
|00i
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
NMR Quantum Computation
Tomography

a11
 a
12 − ιb12

ρ = 
a13 − ιb13
a14 − ιb13
a12 + ιb12
a22
a23 − ιb23
a24 − ιb24
a13 + ιb13
a23 + ιb23
a33
a34 − ιb34
a14 + ιb14
a24 + ιb24
a34 + ιb34
a44




Fidelity:
System
Quantum Gates
Initial state preparation
Final readout
F =p
T r(ρ†theory ρexpt )
p
(T r(ρ†theory ρtheory )) (T r(ρ†expt ρexpt ))
where ρtheory and ρexpt denote the
theoretical and experimental density
matrices respectively.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental system
A two-qubit system:
1
13
H
C
(A), (C): Pseudopure state NMR spectra with a small detection
angle
(B), (D): Thermal equilibrium NMR spectra
1
(A)
13
H
(C)
(B)
(D)
8.2
Shruti Dogra
C
8.0
7.8 ppm
80
79
78
77 ppm
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
NMR pulse sequence
Encoding
2δ
1
H
13
ȳˆ θ0
θ0 θ1
θ1 ẑ ȳˆ
l̂0
ˆ
l0′ ˆ
l1
l̂1′
Measure
C
(i)
ˆl0
ˆl0
0
ˆl1
ˆl0
1
∆ 90
1
2J
1
2J
(ii)(iii)(iv)
3π
= cos( 3π
2 + φ0 )x̂ + sin( 2 + φ0 )ŷ,
= cos(φ0 )x̂ + sin(φ0 )ŷ,
= cos(π + φ1 )x̂ + sin(π + φ1 )ŷ, and
= cos( π2 + φ1 )x̂ + sin( π2 + φ1 )ŷ.
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental design
Various different pairs of states are
encoded.
A two-qubit tomography at the end
of step (iii).
A two-qubit tomography at the end
of step (iv).
A single-qubit partial quantum
state tomography mimicing the
single-qubit measurement.
Single-qubit measurement:
subspace spanned by |00i, |01i.
Normalization: Gz ( π2 )1y
Shruti Dogra
Encoding
2δ
1
H
13
∆ 90
ȳˆ θ0
θ0 θ1
θ1 ẑ ȳˆ
l̂0
ˆ
l0′ ˆ
l1
l̂1′
Measure
C
(i)
1
2J
1
2J
(ii)(iii)(iv)
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Measurement

a11
 a12 − ιb12

ρ=
a13 − ιb13
a14 − ιb13
a12 + ιb12
a22
a23 − ιb23
a24 − ιb24
(i) Direct readout, and (ii) Gz ( π2 )2y
Shruti Dogra
a13 + ιb13
a23 + ιb23
a33
a34 − ιb34

a14 + ιb14
a24 + ιb24 

a34 + ιb34 
a44
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Example
Encoding
2δ
H
13
ȳˆ θ0
θ0 θ1
θ1 ẑ ȳˆ
l̂0
ˆ
l0′ ˆ
l1
l̂1′
Measure
C
1
2J
(i)
(ii)
(ii)(iii)(iv)
(E) ρexp , F = 0.997
(iv)
(D) ρexp , F = 0.994
|0i
|1i
(C) ρexp , F = 0.996
Shruti Dogra
Imaginary
|0i
|1i
Real
Imaginary
Real
|00i
|01i
|10i
|11i
|10i
|11i
|00i
|01i
|10i
|11i
|00i
|01i
|00i
|01i
|10i
|11i
|00i
|01i
|10i
|11i
|00i
|01i
|10i
|11i
|0i
|1i
|0i
|1i
|0i
|1i
|0i
|1i
Real
Real
|10i
|11i
Imaginary
|00i
|01i
Real
(B) |ψ2 ihψ2 |
|00i
|01i
|10i
|11i
|0i
|1i
|0i
|1i
|0i
|1i
|0i
|1i
(A) |ψ1 ihψ1 |
1
2J
|0i
|1i
|0i
|1i
+ e |1i)
Imaginary
Imaginary
(F) ρth = |ΨihΨ|
|0i
|1i
|0i
|1i
a=b
1
ιπ
2
|0i
|1i
√1 (|0i
2
|0i
|1i
|ψ1 i = |0i
∆ 90
Real
Imaginary
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental outcomes
Input states:
|ψ1 i = |0i
|ψ2 i = √12 (|0i + eιφ2 |1i)
|Ψi = a|ψ1 i + b|ψ2 i
a=b
(D1)
θ0 = φ 0 = 0 0
θ1 = 450 , φ1 = 1800
(D2)
(C1)
θ0 = φ 0 = 0 0
θ1 = 450 , φ1 = 900
(C2)
(B1)
θ0 = φ 0 = 0 0
θ1 = 450 , φ1 = 450
(B2)
(A1)
θ0 = φ 0 = 0 0
θ1 = 450 , φ1 = 00
(A2)
79
φ2
Fc
1
0
0.993
2
π
4
0.990
3
π
2
0.996
4
π
0.994
Shruti Dogra
ρth = |ΨihΨ|
0.73
0.3
0.12
0.3
0.73
0.21 − 0.21i
0.12
0.21 + 0.21i
0.73
0. − 0.3i
0.12
0. + 0.3i
0.73
−0.3
−0.3
0.12
78
ppm
79
78
ρexp + 1 I2×2
4
0.77
0.28
0.10
0.28
0.76
0.19 − 0.23i
0.08
0.19 + 0.23i
0.76
0. − 0.3i
0.
+
0.3i
0.07
0.76
−0.29
−0.29
0.08
ppm
Ff inal
0.996
0.995
0.997
0.997
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental outcomes....different weights
√
|ψ1 i = 12 (|0i + 3|1i)
√
|ψ2 i = 12 ( 3|0i + |1i)
Table : Here are the various experimental outcomes. |Ψi = a|ψ1 i + b|ψ2 i . As
per column 4, values of 2δ = 900 , 53.130 , and 36.870 correspond to the
weights a = b, a = 2b and a = 3b respectively. Fc is the fidelity with which
input states are encoded and and Ff inal is the fidelity of final superposed
state.
a:b
2δ
Fc
1:1
900
0.993
2:1
53.130
0.988
3:1
36.870
0.984
Shruti Dogra
ρexp + 14 I2×2
ρth = |ΨihΨ|
0.47
0.47
0.47
0.47
0.35
0.42
0.42
0.5
0.28
0.37
0.37
0.48
Ff inal
0.47
0.46
0.46
0.51
0.37
0.42
0.42
0.52
0.31
0.37
0.37
0.5
0.998
0.999
0.999
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Experimental outcomes...with assumed global phases
|ψ1 i = eιγ1 ( 12 |0i +
γ1
γ2
√
3
2 |1i)
Fc
0
0
0.993
0
2π
3
0.991
ρth
eιγ1
2
(|0i + e
= |ΨihΨ| =
γ1
γ2
0.47
0.47
0.47
0.47
ιπ √
4
0
0
0.988
0
2π
3
0.989
3
2 |0i
+ 12 |1i)
0.47
0.46
0.46
0.46
eιγ2
2
√
0.46
0.51 0.46
0.46
( 3|0i + e
0.12 − 0.36i
0.31
0.44
0.21 + 0.33i
0.46
0.18 + 0.29i
Ff inal
ρexp + 14 I2×2
Shruti Dogra
0.47
0.47 0.47
0.47
0.47
0.12 + 0.36i
Fc
3|1i), |ψ2 i =
√
ρexp + 14 I2×2
ρth = |ΨihΨ|
|ψ1 i =
and |ψ2 i = eιγ2 (
ι2π
3
0.998
0.999
|1i)
Ff inal
0.21 − 0.33i
0.35
0.18 − 0.29i
0.29
0.974
0.981
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Superposition of n-qudits
d-dimensional states: |Ψ1 id , |Ψ2 id , . . . |Ψn id
a1 , a2 , . . . an be the desired weights
Consider d-dimensional referential state |χid .
Known overlaps, |hχ|Ψj id |2 = cj , where j ∈ {1, 2, . . . , n}
We begin with the initial state,
1 0
(a |0in + a02 |1in + · · · + a0n |n − 1in ) ⊗ |Ψ1 id ⊗ · · · ⊗ |Ψn id
N 1
qP
n
0
02
√Qn ak
N=
j=1 aj and ak =
c
(j6=k,j=1)
Shruti Dogra
j
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Superposition of n-qudits
1
1
1
1
Controlled-swap operations3 , CS2,3
CS2,4
. . . CS2,n
(a01 |0in ⊗ |Ψ1 id ⊗ |Ψ2 id ⊗ |Ψ3 id ⊗ · · · ⊗ |Ψn id
1/N
+a02 |1in ⊗ |Ψ2 id ⊗ |Ψ1 id ⊗ |Ψ3 id ⊗ · · · ⊗ |Ψn id
+.........
2
+a0n |n − 1in ⊗ |Ψn id ⊗ |Ψ3 id ⊗ · · · ⊗ |Ψ1 id ).
Nn
Projective measurements, In×n ⊗ Id×d ⊗ k=3 (|χid hχ|d )k
 


n
n
n−1
O
1 X   Y hχ|Ψj id 
ak
|k − 1in |Ψk id 
|χid
√
N
cj
m=1
k=1
3
(j6=k,j=1)
1
(a1 |0in |Ψ1 id + a2 |1in |ψ2 id + · · · + an |n − 1in |ψn id )
N
Fourier transformation
!
n−1
n X
1 X
j(k−1)
√
|jin ⊗
f
ak |Ψk id
N n j=0
k=1
4
Projective measurement: |0in h0|n ⊗ Id×d
3 Phys. Rev. Lett. 116, 110403 (2016).
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
Superposition of n-qudits
Final state: |Ψi =
N√
Ψ
N n
4
Pn
k=1 ak
Q
hχ|Ψj id
n
√
(j6=k,j=1)
cj
Success probability ,
|Ψk id
Qn
2
2
NΨ
j=1 cj NΨ
P = 2 = Pn
.
2
N n
j=1 aj cj n
4 S. Dogra, G. Thomas, S. Ghosh, and D. Suter, arXiv:1702.02418 [quant-ph] (2017).
Shruti Dogra
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p
1
O
H
Thank you!
1
H
1
H
13
C
Cl
19
Cl
Cl
2
19
F
F
C=C
19
Shruti Dogra
F
I
1
1−p 1
-2
|1i
p
|0i
+ 12
Indian Institute of Technology Madras, Chennai
Superposition of two single-qubit states with partial p