Quantum Noise: The Basics

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Quantum Noise The Basics

• Jacob D. Biamonte biamonte_at_ieee.org

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Section I, background

• How complicated is a single Quantum Bit?
• Measurement
• Measurement operators
• Measurement of a state vector using projective
  measurement
• Density Matrix and the Trace
• Ensembles of quantum states, basic definitions
  and importance
• Measurement of a density state

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Section II, Noise

• Classical noise---The good old days?
• Stochastic processes
• Quantum operations
• The theory of quantum operations
• Unitary Evolution
• Measurement
• Closed vs. Open quantum systems
• Decoherence free subspaces, is it even possible?
• Operator-sum representation
• The Partial Trace
• Quantum operation axioms
• Illustrations of Types of Quantum Noise
•     Some bad channels
• -        Bit Flip
• -        Phase Flip
• -        Bit-phase Flip
• -        Depolarizing Channel
• -        Amplitude Damping
• -        Phase Damping

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Section III

• Distance measures for Probability Distributions
• Distance measures for Quantum States
• Fidelity
• Trace Distance

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How complicated is a single Quantum Bit?
q
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Here is how it works
q
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Measurement operators act on the state space of a
quantum system
Measurement operators act on the state space of a
quantum system Initial state
Operate on the state space with an operator that
preservers unitary evolution
Define a collection of measurement operators for
our state space
Act on the state space of our system with
measurement operators
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Measurement of a state vector using projective
measurement
Operate on the state space with an operator that
preservers unitary evolution
Define observables
Act on the state space of our system with
observables (The average value of measurement
outcome after lots of measurements)
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The Density Matrix and the TraceEnsembles of
quantum states, basic definitions and importance

• Quantum states can be expressed as a density
  matrix
• A system with n quantum states has n entries
  across the diagonal of the density matrix. The
  nth entry of the diagonal corresponds to the
  probability of the system being measured in the
  nth quantum state.
• The off diagonal correlations are zeroed out by
  decoherence, i.e. The system is slowly measured
  and quantum information stops, think about a
  pendulums swing due to friction.
• Unitary operations on a density matrix are
  expressed as
• In other words the diagonal is left as weights
  corresponding to the current states projection
  onto the computational basis after acted on by
  the unitary operator U, much like an inner
  product.
• Trace of a matrix (sum of the diagonal elements)
  
• Unitary operators are trace preserving. The
  trace of a pure state is 1, all information about
  the system is known.
• Operators Commute under the action of the trace
  
• Partial Trace (defined by
  linearity)
• If you want to know about the nth state in a
  system, you can trace over the other states.

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Measurement of a density state
Initial state
Operate on the state space with an operator that
preservers unitary evolution (H gate first bit)
Now act on system with CNOT gate
We still define collections of measurement
operators to act on the state space of our system
The probability that a result m occurs is given
by the equation
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Classical Noise The good old days?
1- p
0
0
p
After some time a bit can flip with a certain
probability, p
p
1
1
1- p
We can write this out using a matrix, the final
states q0 and q1 are found by acting on the
initial states p0 and p1.

• Noise in classical systems can be described using
  the theory of stochastic processes.
• A process chain that acts independently---each
  with a possible error---forms what is known as a
  Markov Process.
• The equation to the left represents a single
  stage process.
• The outputs are directly related to the inputs
  under the action of E the evolution matrix.
• The vector of output probabilities q must be a
  valid distribution---(it must sum to
  one)---meaning that the evolution matrix E must
  be what is known as a positive operator.
• The initial probability p distribution must also
  be a valid distribution.
• The columns of an evolution matrix E must also
  sum to one, known as completeness.

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Quantum Operations---Unitary Evolution

• Similar to what has been mentioned about
  classical states, Quantum states transform as
• An initial state is mapped to a final state under
  a trace preserving action. Think about the
  initial state as a vector of probabilities and
  the final state as a new vector of probabilities
  related to the initial state under the action of
  the operator, like this

The map squiggle E is a quantum operation.
An initial state
Quantum operations allow lots of freedom when
describing changes to a quantum system. Unitary
Transformations
Evolution of the system acted on by an operator
Mention again that we can express this as
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Measurement, another quantum operation
Of course, measurement operators must satisfy
completeness, in our case we have
State of a system before measurement
For
The state of the system is after measurement is
In other words, the off diagonal correlations are
zeroed out. See Exercise 8.2 NC
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As quantum circuit designers, the success of our
work depends on getting as close as we can to
Closed Quantum Systems

• Output of the system is determined by a unitary
  transformation on the input state
• If quantum information science was a land of
  cubicles I would leave the issues of coherent
  evolution to the person in the next cube and when
  he complained about my designs I would resort to
  headphones and continue with my circuit diagrams.
  
• Most work done in quantum computation from a
  computer science perspective does not deal with
  the inherently non-ideal nature of quantum
  systems.
• The goal of quantum test is to use a computer to
  help humans make smarter choices in what states
  should be prepared and what measurements should
  be preformed.
• It is hard to communicate ideas from these
  fields, but ideas form both fields are needed to
  make the theory of quantum test a reality.

U
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Real quantum systems are Open

• Output of the system is determined by a unitary
  transformation on the principal system and the
  environment.
• Notice that the final state, e(r) might not be
  related by a unitary transformation to the
  initial state, r. In fact

U
In words we trace over the environment, leaving
our system in a product of the this trace and its
no longer unitary state.
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Assume U is a CNOT gate and the principle system
is the control bit, this is an open quantum
system, an interaction between an environment and
a qubit.
Ucnot
(1)
(2)
(3)
(4)
Notice now the difference from Equations (2) and
(3), do you see how the off diagonal entries
change after the action of the CNOT gate occurs?
This slide should show you how irritating
environments can be to experimentalists.
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The Ring of Fire
Operator-Sum Representation
Physically Motivated Axioms
System Coupled to Environment
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The Partial Trace
Lets pretend we have our very own Hilbert space,
its a big place but not so big that we cant add
another
Suppose we want to trace over A in the basis
Given,
and
Thus, we have shown that a operator can take a
form
Thus for any gate we may write
and this is not necessarily a unitary operator,
but dont be silly the stuck at model is still
stupid!
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Operator-sum Representation

• A representation in terms of operators on the
  principal systems Hilbert space alone.
• where
  is an operator on the principal state space,
• U is the part of U that acts on the system
  alone.

This is known as the operator sum representation
The point is that you want to find operators that
represent the environment the system is attached
to.
Example
Suppose that the environment applies a CNOT gate.
Assume that the environment E starts in state
Operator sum
This is not the best example
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Trace Preservation

• In this model, the operation elements must
  satisfy the completeness relation
• Since this relationship is true for all r it
  follows that

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Trace Preservation

• This equation is satisfied by quantum operations
  which are trace-preserving.
• When extra information about what occurred in the
  process is obtained by measurement, the quantum
  operation can be non-trace-preserving, that is

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Axioms of Quantum Operations

• We define a quantum operation e as a map from the
  set of density operators of the input space Q1 to
  the set for Q2 with the following three
  properties
• A1 is the
  probability that the process e occurs when r is
  the initial state. Thus,
  .
• Note that, with this definition, the correctly
  normalized final state is

You see the error is present in the state vector
or not, but it is only observed with some
probability, this can be related directly to
classical noise
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Axioms of Quantum Operations

• A2 e is a convex-linear map on the set of
  density matrices, that is, for probabilities
  pi,
• A3 e is a completely positive map. e(A) is
  positive for any positive operator A in Q1.
  Furthermore, this must hold for applying the map
  to any combined system RQ1.

The point is that probability is not created or
destroyed, unless you cross the street while
reading a book on quantum computing.
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The Axioms and Operator-sum

• Theorem 8.1 The map e satisfies axioms A1, A2,
  and A3 if and only if
• For some set of operators Ei which map the
  input Hilbert space to the output Hilbert space,
  and
• Proof (Nielsen/Chuang, pages 368-370)

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Bit Flip Channel
A bit flip channel flips bits, could you have
guessed? For a single bit it has an operator sum
representation with operation elements as follows
q
Example
Consider
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Bit Flip Channel

• Bloch sphere representation, the state on the
  x-axis are left alone while the y-z plane is
  uniformly contracted

q
y
z
x
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Phase Flip Channel

• Corresponds to a measurement in the 0gt, 1gt
  basis, with the result of the measurement unknown
• Operation Elements

q
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Phase Flip Channel

• The z axis is left alone, and the x and y
  components are uniformly contracted

y
x
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Bit-phase Flip

• A combination of bit flip and phase flip
• Operation Elements

q
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Bit-phase Flip

• Bloch vector is projected along y-axis, x and z
  components of the Bloch vector are contracted

y
x
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Depolarizing Channel

• Qubit is replaced with a completely mixed state
  I/2 with probability p, it is left untouched with
  probability 1-p
• The state of the quantum system after the noise
  is

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Depolarizing Channel

• The Bloch sphere contracts uniformly

y
x
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Amplitude Damping

• Noise introduced by energy dissipation from the
  quantum system
• Emitting a photon
• The quantum operation

can be thought of as the probability of losing a
photon E1 changes 1gt into 0gt - i.e. losing
energy E0 leaves 0gt alone, but changes amplitude
of 1gt
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Amplitude Damping

• Bloch sphere Representation
• The entire sphere shrinks toward the north pole,
  0gt

y
x
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Phase Damping

• Describes the loss of quantum information without
  the loss of energy
• Electronic states perturbed by interacting with
  different charges
• Relative phase between energy eigenstates is lost
• Random phase kick, which causes non diagonal
  elements to exponentially decay to 0
• Operation elements
• l probability that photon scattered without
  losing energy

Phase Damping
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Operation Elements for Important Single Bit
operations
Amplitude Dampening
Depolarizing channel
Phase Damping
Phase Flip
Bit Flip
Bit-Phase Flip
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Distance measures for Probability Distributions

• We need to compare the similarity of two
  probability distributions
• Two measures are widely used trace distance and
  fidelity
• Trace distance also called L1 distance or
  Kalmogorov distance
• Trace Distance of two probability distributions
  px and qx
• The probability of an error in a channel is equal
  to the trace distance of the probability
  distribution before it enters the channel and the
  probability distribution after it leaves the
  channel

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Distance measures for Probability Distributions

• Fidelity of two probability distributions
• When the distributions are equal, the fidelity is
  1

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Distance measures for Quantum States

• How close are two quantum states?
• The trace distance of two quantum states r and s
• If r and s commute, then the quantum trace
  distance between r and s is equal to the
  classical trace distance between their
  eigenvalues
• The trace distance between two single qubit
  states is half the ordinary Euclidian distance
  between them on the Bloch sphere

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Fidelity of Two Quantum States

• When r and s commute (diagonal in the same basis)
• The fidelity of a pure state and an
  arbitrary state r
• That is, the square root of the overlap