Quantum Computation and the Bloch Sphere - UMD … ph… · Quantum Computation and the Bloch Sphere Fred Wellstood Joint Quantum Institute and Center for Superconductivity Research

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"


>+>= 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>


|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>



|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


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>


|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


|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| >+>


- 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 )

Quantum Computation and the Bloch Sphere - UMD … ph… · Quantum Computation and the Bloch Sphere Fred Wellstood Joint Quantum Institute and Center for Superconductivity Research

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