8. Quantum noise and quantum operations

Ideal quantum computation

QUANTUM

COMPUTER

Real quantum computation

QUANTUM

COMPUTER

ENVIRONMENT

Mixed output

state

8.1 Classical noise and Markov processes

Single bit flip on hard drive due to e.g. fluctuating magetic field

Probability to flip

during (long) time

Matrix notation

Initial probabilities , final probabilities

Properties of evolution matrix

Positivity: All elements non-negative real numbers

Completeness: Sum of all elements in a column

8.1 Quantum operations

Notation

Input state , output state . A general quantum operation is

where is a map .

Examples:

Unitary transformation

Measurement

Approaches

OPERATOR SUM

REPRESENTATION

PHYSICALLY MOTI-

VATED AXIOMS

SYSTEM COUPLED

TO ENVIRONMENT

System coupled to environment

ENVIRONMENT

PRINCIPAL

SYSTEM

Ideal, closed system

Non-ideal, open system

Total input state , rotation of system-environment .

Partial trace over environment

For no (principal) system-environment interaction

Derivation: Bit flip (not in book).

Operator sum representation

One can formulate the quantum operation in terms of operators

acting on the principal system only

Let be an orthonormal basis for the environment

and (without loss of generality)

an operator sum representation of . The operation elements satisfy

a completeness relation:

Since this holds for all we must have (trace preserving)

Generalization: To include measurement (not trace preserving) we demand

Interpretation: The quantum operation can be understood as taking

the state and (randomly) replacing it with the state

with probability

Physically motivated axioms

A1: The probability that occurs is

A2: is a convex-linear map on density matrices

A3: is a completely positive map, i.e. is positive for all

Then one can show that the map satisfies A1,A2,A3 if and only if

,

Non-uniqueness

The operator sum representation is not unique

where is a unitary matrix.

8.3 Examples of quantum noise and operations

Bloch sphere representation

From 2.72 (hand in) you know that one can write a single qubit state as

where the Bloch vector

and .

It can be shown that the a trace preserving quantum operation

corresponds to a map

where is a real, symmetric matrix, is a real, orthogonal matrix

with and is a real vector

describes i) deformation, ii) rotation and iii) displacement of

the Bloch sphere

Basic quantum operations

Bit flip: With probability the state of the qubit is flipped

Operator elements

Bit flip for . Left: Set of all pure states. Rigth: States after bit flip operation.

Phase flip: With probability the state ”phase” of the qubit is flipped

Operator elements

Bit flip for . Left: Set of all pure states. Rigth: States after phase flip operation.

Bit-phase flip: With probability there is a bit-flip and a phase flip

Operator elements

Bit flip for . Left: Set of all pure states. Rigth: States after bit-phase flip operation.

Depolarization: With probability the qubit is put in the completely

mixed state

Operator elements

Depolarization for . .

Amplitude damping: The system ”emits energy” and approaches the

ground state

Operator elements

Amplitude damping for . .

For we have

Generalized amplitude damping: The qubit is ”thermalized”, i.e. approaches

Phase damping: The system is ”dephased” in the basis

Operator elements

Using freedom to unitarily transform operator elements we get

with

This is just the phase flip!