Quantum Computationand the Bloch Sphere
Fred WellstoodJoint Quantum Institute
andCenter for Superconductivity Research
Department of PhysicsUniversity of Maryland, College Park, MD
(March 24, 2008)
In principle, a computer can be built that uses quantum mechanics to perform useful calculations.
A quantum computer would be built from quantum bits or "qubits",individual quantum system with two basis states, |0> and |1>
The qubits are coupled together and logic operations are performed by manipulating the quantum state of the entire system. Example: NOT on single qubit:
|0> |1> |1> |0>
α|0> +β|1> α|1> +β|0>
Example: Phase gate on single qubit:|0> |0> |1> eiφ|1>
α|0> +β|1> α|1> +eiφβ|0>
Quantum Mechanics and Quantum Computing
operations need to work on superposition states!
The Principle of SuperpositionSuppose |0> and |1> are two allowed quantum states of a system, then the system can exist in any linear superposition of these states
where α and β are complex numbers>+>= 1|0| βαψ
But we don’t see such states in everyday objects "Schrodinger's cat paradox" (Schrodinger, 1935)
if true in macroscopic objects
+
live dead
"macroscopic quantum superposition"
Quantum Mechanics and Quantum Computing
>+>= 1|0| βαψSuperposition State
- state must be normalized to unity
122 =+ βα
- probability amplitudes α and β can be complex numbers
- then define )cos(θα = )sin(θβ φie=
1)(sin)(cos
)sin()cos(22
2222
=+=
+=+
θθ
θθβα φie
- an overall phase factor has no effect, so we can choose α to be real
>+>=>+>= 1|)sin(0|)cos(1|0| θθβαψ φie
- so we can always write a superposition state in the form:
|0>
|1>
z
y
x
θ
φ
Superposition States as Points on the Bloch Sphere
sphere with radius R=1 …..this is the “Bloch Sphere”
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ 1|
2sin0|
2cos θθ φie
|0>
z
y
x
>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
0|
1|20sin0|
20cos
1|2
sin0|2
cos
φ
φ θθ
i
i
e
e
0=θ
Example: θ = 0
Superposition States as Points on the Unit Sphere
|0>
z
y
x
>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
1|
1|2
sin0|2
cos
1|2
sin0|2
cos
0 ππ
θθ φ
i
i
e
e
πθ =
Example: θ = π, φ = 0
|1>
Superposition States as Points on the Unit Sphere
|0>
z
y
x
2/πθ =
Example: θ = π/2, φ = 0
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
0
>+>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
ππ
θθ φ
i
i
e
e
21|0| >+>
|1>
Superposition States as Points on the Unit Sphere
Superposition States as Points on the Unit Sphere
|0>
z
y
x
2/πθ =
Example: θ = π/2, φ = π/2
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
2
>+>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
i
e
e
i
i
ππ
θθ
π
φ
21|0| >+>
|1>
2/πφ =
21|0| >+> i
To be useful for computation, you need operations that control the state of one qubit based on the state of another.
Controlled NOT or CNOT:Two-qubit operation that flips the second qubit state based on the first qubit state
input state outputstate
|0,0> |0,0>|0,1> |0,1>|1,0> |1,1>|1,1> |1,0>
Example, performing a CNOT operation on α|1,1> + β|0,1> + γ |1,0>
yields:α|1,0> + β|0,1> + γ |1,1>
if true in macroscopic objects
Quantum Entanglement (Schrodinger, 1935) Multiple quantum systems can exist in entangled super-positions of states in which the state of an individual system has no well-defined physical meaning
+
and(dead, live) (live, dead)
Superposition and entanglement are unobservable in ordinary "macroscopic" objects due to interactions with other degrees of freedom and the surrounding world (dissipation and decoherence) … how macroscopic is too macroscopic?
baba >>+>>= 0|1|1|0| βαψ
Quantum Mechanics and Quantum Computing
nA classical computer with an n-bit memory can access 2 states. For example, with n=2 bits the 22 = 4 states are 00, 01, 10 and 11.
A quantum computer can access superposition statesand entangled states. With n qubits, this gives of order states.2
2n
Example: for n=1 qubit we can have:
21 0 +
=xψ
11 =ψ0=oψ
21 i 0 −
=− yψ2
1 i 0 +=yψ
Example: for n= 2 qubits we can have 36 product states such as:
21 0 −
=−xψ
⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
21 0
21 0 i
xyψ
1111 =ψ00=ooψ
12
1 i 01 ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=yψ
211 00
1
+=eψ
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=− 2
1 00xoψ
011 =oψ101 =oψ
plus entangled states (can’t be written as product) such as:
211 00
3
ie
+=ψ
211 00
2
−=eψ 2
11 i 004
−=eψ
Key Question: can a useful quantum computer be built in practice?
Answer: Definitely maybe.
Main Experimental Challenge: All systems experience noise and interact with other quantum systems (the outside world), and this eventually destroys the delicate quantum superposition states. This is called decoherence.
Decoherence is best understood using density matrix.
Here we will just try to understand how you can manipulate the quantum state of a multi-qubit system to perform operations.
The extra states can be used to tackle some very difficult tasks: - use Shor's algorithm to factor large numbers quickly and
break RSA encrypted messages, - simulating other quantum systems, - efficiently searching large data-bases (Grover’s Algorithm)?
|0>
z
y
x
>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
0|
1|20sin0|
20cos
1|2
sin0|2
cos
φ
φ θθ
i
i
e
e
0=θ
Example: θ = 0
Single qubit control operations as rotations on the Bloch sphere
|0>
z
y
x
>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
1|
1|2
sin0|2
cos
1|2
sin0|2
cos
0 ππ
θθ φ
i
i
e
e
πθ =
Example: θ = π, φ = 0
|1>
starting from |0>rotate about the y-axis by π….. πy-pulse….or … NOT since such a rotation would also change |1> to |0>
Single qubit control operations as rotations on the Bloch sphere
|0>
z
y
x
2/πθ =
Example: θ = π/2, φ = 0
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
0
>+>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
ππ
θθ φ
i
i
e
e
21|0| >+>
|1>
starting from |0>rotate about the y-axis by π/2… π/2-pulse….or ...since two such rotations would produce NOT
NOT
Single qubit control operations as rotations on the Bloch sphere
|0>
z
y
x
2/πθ =
Example: θ = π/2, φ = π/2
21|0|
1|4
sin0|4
cos
1|2
sin0|2
cos
2
>+>=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛=Ψ
i
e
e
i
i
ππ
θθ
π
φ
21|0| >+>
|1>
2/πφ =
starting from rotate about the z-axis by π/2. This is πZ/2 or “π/2 phase gate”since it will increase phase term for any state by π/2”
21|0| >+> i
( ) 21|0| >+>
Single qubit control operations as rotations on the Bloch sphere
- Consider a 2-level system with energy splitting ΔE. - Cool system to temperature T << ΔE/kb and it will relax to |0>.- Apply power (a perturbation) continuously at frequency f = ΔE/h.
0
1
ΔE=hf
Apply power for short time --> Small amplitude to be in 1
0
1
ΔE=hf
Keep applying power --> eventually
system pumped entirely into 1
(NOT gate or π-pulse)
0
1
ΔE=hf
Keep applying power --> system pumped
back down to 0
System cycles back and forth between 0 and 1 deterministically at well-defined rate (Rabi frequency Ω) set by power and tuning. Stopping power at appropriate time can produce NOT or
Basic Idea for Driving a System - Rabi Oscillations
NOT
Two-level System Dynamics
State of a system described by wavefunction Ψ that satisfies time-dependent Schrodinger’s Equation
ψψ Ht
i =∂
∂h
For a two-level system with Hamiltonian Ho that is being driven at frequency ω with a perturbing energy H’, we can write H in matrix form as:
⎞⎛⎞⎛⎞⎛ )cos()cos(00 tVEtVE oo ωω⎟⎟⎠
⎜⎜⎝
=⎟⎟⎠
⎜⎜⎝
+⎟⎟⎠
⎜⎜⎝
=+=11 )cos(0)cos(*0
'EtVtVE
HHH o ωω
where: Eo = energy of ground state, E1 = energy of excited stateV cos(wt) = <0|H’|1>
and where:
⎟⎟⎠
⎞⎜⎜⎝
⎛>=
01
0| ⎟⎟⎠
⎞⎜⎜⎝
⎛>=
10
1| ⎟⎟⎠
⎞⎜⎜⎝
⎛>=Ψ
βα
|
Two-level System Dynamics
Plug into Schrodinger’s Equation:
ψψ Ht
i =∂
∂h ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂∂∂
βα
ωω
β
α
1)cos()cos(
EtVtVE
ti
ti
o
h
h
Write as two coupled equations:
βαωβ
βωαα
1)cos(
)cos(
EtVt
i
tVEt
i o
+=∂∂
+=∂∂
h
h
Fairly nasty…guess solution of form: (this will always work!)
tEi
tEi
etB
etAo
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
=
=
h
h
1
)(
)(
β
αPlug into Schrodinger’s Equation
notice that this says that the amplitude β to be found in |1> will change based on amplitude α to be in |0>
tEitEitEi
tEitEi
o
tEi
etBEetAtVetBt
i
etBtVetAEetAt
i
o
oo
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−
+=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂∂
hhh
hhh
h
h
11
1
)()()cos()(
)()cos()()(
1ω
ω
tEitEi
o
tEi
o
tEietBtVetAEetAE
ttAei
ooo ⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−
+=+∂
∂ hhhhh1
)()cos()()()( ω
For the first equation, we find:
Clean things up:tEEi o
etBtVttAi
⎟⎠⎞
⎜⎝⎛ −
−=
∂∂ hh
1
)()cos()( ω
For simplicity, let’s assume we are on resonance ( )
tiettVBttAi ωω −=
∂∂ )cos()()(
h
oEE −= 1ωhexpand this term
( ) ( )titititi
ti
etBVeeetVB
ettVBttAi
ωωωω
ωω
21)(22
)(
)cos()()(
+=+
=
=∂
∂
−
−h
This term is changing very rapidly and is far from resonance at ω…so it can be dropped…. “rotating wave approximation”
)(2
)(
)(2
)(
tAi
VttB
tBi
VttA
h
h
=∂
∂
≅∂
∂
)(2
)( 2
2
2
tAVttA
⎟⎠⎞
⎜⎝⎛−=
∂∂
h
Assuming A(0) = 1, solution is:
( )2cos)( ttA Ω=
h
V=Ω is the Rabi frequency
take another time derivative of the 1st equation and use 2nd to eliminate dB/dt
( )2sin)( titB Ω−=
( )
( )
( ) ( ) >Ω−>Ω=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
Ω−
Ω=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛>=Ψ
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
1|2sin0|2cos
2sin
2cos
)(
)(|
1
11
tEitEi
tEi
tEi
tEi
tEi
etiet
eti
et
etB
etA
o
oo
hh
h
h
h
h
βα
Take out an overall phase factor of ( )htiEo−exp
( ) ( ) >Ω−>Ω>=Ψ⎟⎠⎞
⎜⎝⎛ −
−1|2sin0|2cos|
01 tEEietit h
prob
abilt
y
0
1
t
P0=|α|2
P1=|β|2
Ω/2π Ω/4π0
>⎟⎠⎞
⎜⎝⎛+>⎟
⎠⎞
⎜⎝⎛>=Ψ 1|
2sin0|
2cos| θθ φie
Also notice this is now in the familiar “polar coordinate” form:
where andtΩ=θ2
01 πφ −⎟⎠⎞
⎜⎝⎛ −
= tEEh
( )
( )
( ) ( ) >Ω−>Ω=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
Ω−
Ω=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛>=Ψ
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
1|2sin0|2cos
2sin
2cos
)(
)(|
1
11
tEitEi
tEi
tEi
tEi
tEi
etiet
eti
et
etB
etA
o
oo
hh
h
h
h
h
βα
Take out an overall phase factor of ( )htiEo−exp
( ) ( ) >Ω−>Ω>=Ψ⎟⎠⎞
⎜⎝⎛ −
−1|2sin0|2cos|
01 tEEietit h
Rabi Oscillation on the Bloch Sphere
|0>
|1>
z
y
x( ) 21|0| >+>
( ) 21|0| >+> i
dφ/dt =ω01
tΩ=θ
To make NOT gate, stop driving at t = π/ΩProblem: Show that this will NOT any state!
The behavior of a state on the Bloch sphere is completely analogous to a magnetic moment precessing in a magnetic field oriented along the z-axis.
Rabi Oscillations are completely analogous to nuclear magnetic resonance (NMR). In NMR, a static magnetic field Bz is applied and then resonant rf magnetic fields are applied at frequency f to drive a nuclear spins at resonance (ω= γBz )
