8. quantum noise and quantum operations - · pdf filequantum noise and quantum operations...
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8. Quantum noise and quantum operations
Ideal quantum computation
QUANTUM
COMPUTER
Real quantum computation
QUANTUM
COMPUTER
ENVIRONMENT
Mixed output
state
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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
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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
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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).
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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)
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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
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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.
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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
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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.
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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.
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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.
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Depolarization: With probability the qubit is put in the completely
mixed state
Operator elements
Depolarization for . .
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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
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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!