notes about the bloch sphere - umd physics sphere n… · a . quantum gate. is just a unitary...
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Fred Wellstood Joint Quantum Institute
and Center for Nanophysics and Advanced Materials Department of Physics University of Maryland
College Park Maryland
ASC short course September 4 2016 wellsquidumdedu
Notes About the Bloch Sphere
gt
+gt
=Ψ 1|
2sin0|
2cos θθ φie
2
Any pure quantum state of a qubit can be visualized as a point on a unit sphere This is the ldquoBloch Sphererdquo
Felix Bloch Nobel Prize in 1952 for NMR
III Quantum states and gates
(ref 55)
|0gt
z
y
x
gt=
gt
+gt
=
gt
+gt
=Ψ
0|
1|20sin0|
20cos
1|2
sin0|2
cos
φ
φ θθ
i
i
e
e
0=θ
Example θ = 0
3
III Quantum states and gates
|0gt
z
y
x
gt=
gt
+gt
=
gt
+gt
=Ψ
1|
1|2
sin0|2
cos
1|2
sin0|2
cos
φ
φ
φ
ππ
θθ
i
i
i
e
e
e
πθ =
Example θ = π
|1gt 4
III Quantum states and gates
|0gt
z
y
x
2πθ =
Example θ = π2 φ = 0
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
0
gt+gt=
gt
+gt
=
gt
+gt
=Ψ
ππ
θθ φ
i
i
e
e
21|0| gt+gt
|1gt 5
III Quantum states and gates
|0gt
z
y
x
2πθ =
Example θ = π2 φ = π2
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
2
gt+gt=
gt
+gt
=
gt
+gt
=Ψ
i
e
e
i
i
ππ
θθ
π
φ
21|0| gt+gt
|1gt
2πφ =
21|0| gt+gt i
6
III Quantum states and gates
A quantum gate is just a unitary linear operation U on the states of the qubits For one qubit a gate operation can be visualized as a rotation on the Bloch sphere
gt
+gt
=Ψ 1|
2sin0|
2cos θθ φie
Starting from |0gt any state can be reached by first rotating about y (or x) by angle θ and then about z by angle φ
These ldquotwordquo operations form a universal gate set for a single qubithellip a 1-qubit quantum computer 7
III Quantum gates
- Consider qubit with energy splitting ∆E - Apply power (a perturbation) at frequency f = ∆Eh
0
1
∆E=hf
Start in |0gt and apply power for short time --gt develop small amplitude to be in |1gt
Implementing Quantum Gates
8
Rotating states on the Bloch sphere using a ldquoRabi Oscillationrdquo
0
1
∆E=hf
Start in |0gt and apply power for short time --gt develop small amplitude to be in |1gt
0
1
∆E=hf
Keep applying power --gt eventually system pumped entirely into 1 (NOT gate or π-pulse)
Implementing Quantum Gates
9
- Consider qubit with energy splitting ∆E - Apply power (a perturbation) at frequency f = ∆Eh
Rotating states on the Bloch sphere using a ldquoRabi Oscillationrdquo
0
1
∆E=hf
Start in |0gt and apply power for short time --gt develop small amplitude to be in |1gt
0
1
∆E=hf
Keep applying power --gt eventually system pumped entirely into 1 (NOT gate or π-pulse)
0
1
∆E=hf
Keep applying power --gt system pumped back down to 0 (stimulated emission)
Implementing Quantum Gates
10
- Consider qubit with energy splitting ∆E - Apply power (a perturbation) at frequency f = ∆Eh
Rotating states on the Bloch sphere using a ldquoRabi Oscillationrdquo
Rabi oscillations with zero detuning
-015-01
-0050
00501
015
-500 0 500 1000 1500 2000
V
t (micros)
mic
row
ave
driv
e
11
saturation 50
State of system cycles back and forth between |0gt and |1gt at well-defined rate (Rabi frequency) set by drive power Stop at appropriate time to make a NOT or NOT
0
02
04
06
08
1
-100 0 100 200 300 400 500
P 1
t (micros)
I I Rabi - 1944 Nobel Prize
for NMR
(ref 56)
π2-pulse
Spin Echo
-015-01
-0050
00501
015
-500 0 500 1000 1500 2000t (ns)
Vπ-pulse
π2-pulse π2-pulse
τ τ
mic
row
ave
driv
e
12
at fo1