View source for Chapter 8 - Noise in Quantum Systems

===Introduction===

Noise is the greatest obstacle to building a scalable and reliable quantum computing device.  Furthermore, all realistic quantum systems are noisy.  Therefore, the objective of experimentalists trying to build quantum computing devices is the eliminate as much noise as possible.  In this chapter, the objective will be to understand how to describe noise.

In [[Chapter 3 - Physics of Quantum Information|Chapter 3]], the Schrodinger equation was discussed as the way in which to describe quantum <nowiki>systems'</nowiki> evolution.  The evolution described by [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|Schrodinger's Equation]] is a the evolution of a system which has been isolated from everything else, i.e., it describes a [[Index#C|closed system]].  However, as just stated, realistic systems are noisy and this noise is often due to unwanted interactions with the environment.  There are other noises, for example a gating operation which necessarily has a finite precision.  Such noise can also be described by the representations of open quantum system evolution that are provided in this chapter.  So this chapter is about noise in general.  

===SMR Representation or Operator-Sum Representation===

The operator-sum representation is a method for representing open system evolution.  It now goes by other names such as Kraus representation, or Kraus decomposiion.  However, it originated with [[Bibliography#SMR|Sudarshan, Mathews, and Rau]] in 1961 (SMR) and was later taken up by [[Bibliography#Kraus:83|Kraus]] and others.  Kraus's name is now attached to it due to a set of lecture notes published in the early 1970's.  In this section, it will likely be clear that the description in all its generality was very well (and simply) described by [[Bibliography#SMR|SMR]] and that is the line of argument which will be followed.  

Let us consider a mapping from one density operator to another with no other restrictions.  This can be written as a linear map.
{{Equation | <math>
\rho^\prime = A \rho,
\,\!</math>|8.1}}
or more explicitly
{{Equation | <math>
\rho^\prime_{r^\prime s^\prime} = A_{r^\prime s^\prime,rs} \rho_{rs}.
\,\!</math>|8.2}}
One way to think of this is a linear mapping from one vector to another by a matrix.  (<math>\rho\,\!</math> can be viewed as a vector.  Simply rearrange the elements of the matrix into a column vector.  <math>A \,\!</math> is then a matrix.)  It is apparent that this is a very general mapping.  One that might be considered more general would be to add a constant term.  However, this is, in fact, not more general as it can be absorbed into the definition of <math>A\,\!</math>.  Now, we recall that the density matrix is required to be, Hermitian, positive semi-definite, and have trace one.  Respectively, we write
{{Equation | <math>
\rho = \rho^\dagger,
\,\!</math>|8.3}}
{{Equation | <math>
\rho \geq 0,
\,\!</math>|8.4}}
and
{{Equation | <math>
\mbox{Tr}\rho = 1.
\,\!</math>|8.5}}
The first condition ensures real eigenvalues, the second and third ensure a valid probability interpretation of the density matrix.  
One can show that, given the properties Eqs. [[#eq8.3|(8.3)]], [[#eq8.4|(8.4)]], and [[#eq8.5|(8.5)]], the mapping <math>A\,\!</math> (which can be considered a matrix) has the following properties:
{{Equation | <math>
A_{sr,s^\prime r^\prime} = (A_{rs,r^\prime s^\prime})^*,
\,\!</math>|8.6}}
{{Equation | <math>
x_r^*x_sA_{sr,s^\prime r^\prime}y_{s^\prime}y_{r^\prime} \geq 0,
\,\!</math>|8.7}}
and
{{Equation | <math>
A_{rr,s^\prime r^\prime} = \delta_{s^\prime r^\prime}.
\,\!</math>|8.8}}
We could also introduce a new matrix <math>B\,\!</math> which is related to 
<math>A\,\!</math> by relabeling, by 
{{Equation | <math>
B_{rr^\prime,s s^\prime}\equiv A_{sr,s^\prime r^\prime},
\,\!</math>|8.9}}
with the following properties:
{{Equation | <math>
B_{rr^\prime,s s^\prime}=(B_{ss^\prime,r r^\prime})^*,
\,\!</math>|8.10}}
{{Equation | <math>
z^*_{rr^\prime}B_{rr^\prime,s s^\prime}z_{ss^\prime} \geq 0,
\,\!</math>|8.11}}
and
{{Equation | <math>
B_{rr^\prime,r s^\prime} = \delta_{r^\prime s^\prime}.
\,\!</math>|8.12}}
Now the important point to note is that <math>B\,\!</math> can be considered 
a Hermitian matrix, and as such, it is diagonalizable.  Letting 
<math>\eta(\alpha)\,\!</math> be its eigenvalues and <math>C_{(\alpha)}\,\!</math> the 
corresponding eigenvectors, we see that the mapping 
[[#eq8.2|(8.2)]], can be written as 
{{Equation | <math>
\rho^\prime = \sum_\alpha 
               \eta(\alpha)C_{(\alpha)}^{}\rho C_{(\alpha)}^\dagger.
\,\!</math>|8.13}}

If all of the <math>\eta(\alpha)\,\!</math> are positive, then a factor of <math>\sqrt{\eta(\alpha)}\,\!</math> can be absorbed into the  <math>C_{(\alpha)}\,\!</math>  and the map can be written as 
{{Equation | <math>
\rho^\prime = \sum_\alpha C_{(\alpha)}^{}\rho C_{(\alpha)}^\dagger.
\,\!</math>|8.14}}
This is what is sometimes called the Operator-Sum representation or Kraus decomposition, originally given in  [[Bibliography#SMR|SMR]].

===Modelling Open System Evolution===

Noise in a quantum system can arise in two different ways.  The first is through imperfect controls; a unitary transformation is not implemented exactly as one would like due to experimental limitations.  The second is through an unwanted interaction with another system, usually called a bath or environment.  As will be shown later, both of these can be modelled using completely positive maps if the assumption is made that the system and bath are initially uncorrelated.  The definition, concept, and assumption of complete positivity will be discussed in a later section.

====Initially Uncorrelated System and Bath====

The dynamics of an open quantum system coupled to a bath is formally obtained from the evolution <math>U=\exp(-iHt)\,\!</math> under the combined system-bath Hamiltonian 
{{Equation | <math>
H=H_{S}\otimes I + I\otimes H_{B} + \sum_{\gamma }S_{\gamma }\otimes B_{\gamma },
\,\!</math>|8.15}}
where <math>H_{S}\,\!</math> is the Hamiltonian for the system alone, <math>H_{B}\,\!</math> is the
Hamiltonian for the bath alone, and the <math>S_{\gamma }\,\!</math> and <math>B_{\gamma }\,\!</math> are
operators on the system and the bath respectively.  This last term couples the system to the bath so that the system is no longer considered a closed quantum system and the bath and system do not evolve independently.  If we assume that the system and the bath are decoupled at the beginning of the experiment.  We may write, at <math>t = 0\,\!</math>,
<center> <math>
\rho(t=0) = \rho_S \otimes \rho_B.
\,\!</math></center>
We then act with a unitary operator on the closed system (our system plus the bath) and trace (or average) over the bath using a [[Index#P|partial trace]] to get
{{Equation | <math>
\rho_S(t)=\mathrm{Tr}_{B}[{U}\left( \rho_S (0)\otimes \rho_{B}(0)\right) 
          {U}^{\dagger }],
\,\!</math>|8.16}}
where <math>\rho_S (0)\,\!</math> is the initial density matrix of the (open) system, and <math>\rho_{B}(0)\,\!</math> is the initial density matrix of the bath.  If we take the bath to be in a state <math>\rho_B = \sum_i \mu_i|i\rangle\langle i|\,\!</math>, then we can write Eq. [[#eq8.16|(8.16)]] as 
{{Equation|<math>\begin{align}
\rho_S^\prime(t) &= \sum_{i,j} \langle j|(U_{tot}(t)
\rho_S(0)\otimes \mu_i|i\rangle\langle i| U_{tot}^\dagger (t)|j\rangle 
  \\
              &= \sum_{i,j}\mu_i \langle j|U_{tot}(t) |i\rangle
 \rho_S(0)\langle i| U_{tot}^\dagger(t) |j\rangle.
\end{align}
</math>|8.17}} 
Now we may define a new index <math>\alpha \equiv (i,j)\,\!</math> and also we may define
{{Equation | <math>
A_\alpha(t) \equiv \sqrt{\mu_i}\;\langle j|U_{tot}(t) |i\rangle,
\,\!</math>|8.18}}
and in an analogous way <math>A_\alpha^\dagger\,\!</math> so that the [[Index#O|operator-sum representation]] is given by 
{{Equation | <math>
\rho_S^\prime (t)= \sum_\alpha A_\alpha(t) \rho_S A_\alpha^\dagger(t). 
\,\!</math>|8.19}}
Note the relation to Eq.[[#eq8.14|(8.14)]].  In some places the definition [[#eq8.18|(8.18)]] will be used without the factor <math>\sqrt{\mu_i}\,\!</math>.  This will be of little consequence, but should be noted. 

The equation [[#eq8.19|(8.19)]] is used very frequently for quantum operations and is known by several names: the operator 
sum decomposition and the Kraus decomposition.  However, when it was first derived, 
it was called the eigenvalue decomposition [[Bibliography#SMR|SMR]] and 
the map was called, and rather appropriately, a dynamical map.  
The <math>A\,\!</math>s are often called [[Index#K|Kraus operators]].  

We should note that, in order for the density matrix to keep its trace equal to one, i.e., for the map to be trace-preserving, 
<center><math>
\sum_\alpha A_\alpha^\dagger A_\alpha = I_S.  
\,\!</math></center>

It can be shown that this agrees with the most general quantum evolution consistent with the condition of complete positivity, known as the operator sum representation (OSR) [[Bibliography#SMR|SMR]], [[Bibliography#Kraus:83|Kraus]], [[Bibliography#Schumacher:96a|Schumacher]]: 
{{Equation|<math>\begin{align}
 \mathcal{E}_{t}(\rho(0)) &\equiv  \rho (t)   \\
  &= \!\sum_{\mu \nu }\!A_{\mu \nu }(t)\rho (0)A_{\mu \nu }^{\dagger}(t) 
\end{align}
</math>|8.20}} 

====Fixed-Basis Operations====

Another expression for the time-dependent density operator is the following: 
{{Equation|<math>\begin{align}
 \mathcal{E}_{t}(\rho(0)) =  \sum_{\alpha ,\beta }\!\chi _{\alpha \beta }(t)K_{\alpha }\rho(0)
         K_{\beta }^{\dagger}.
\end{align}
</math>|8.21}} 
The operators <math>A_{\mu \nu }(t)=\sqrt{\lambda _{\nu }}\langle
\mu |U(t)|\nu \rangle \,\!</math> where <math>U=\exp (-iHt/\hbar )\,\!</math> and the initial bath
density matrix is written as <math>\rho _{B}(0)=\sum_{\nu }\lambda _{\nu }|\nu
\rangle \langle \nu |\,\!</math> [[Bibliography#Lidar:CP01|Lidar, et al.]]. They satisfy the normalization
condition: <math>\sum_{\mu }A_{\mu \nu }^{\dagger }A_{\mu \nu } = I_{S}\,\!</math>.  The matrix <math>\chi _{\alpha \beta
}(t)=\sum_{i}b_{i,\alpha }b_{i,\beta }^{\ast }\,\!</math> is a time-dependent,
hermitian coefficient matrix defined by a transformation of the Kraus
operators to a fixed operator basis <math>K_{\alpha }\,\!</math>: <math>A_{\mu \nu
}(t)=\sum_{\alpha }b_{\mu \nu ;\alpha }(t)K_{\alpha }\,\!</math>. 

We can express our <math>A_\alpha\,\!</math> in terms of a fixed basis of operators.  
The <math>A_\alpha\,\!</math> are functions of time (since <math>U_{tot}\,\!</math> is) 
and are thus ''not'' fixed operators.  We can expand these <math>A\,\!</math>'s in terms of a complete fixed 
chosen basis <math>\{K_m\}\,\!</math>: 
{{Equation | <math>
A_\alpha = \sum_m b_{\alpha,m}(t) K_m,
\,\!</math>|8.22}}
so that the time-dependence is in the <math>b\,\!</math>'s.  Now we would like to see how Eq. [[#eq8.21|(8.21)]] transforms under a change of basis, that is, when the <math>K\,\!</math>'s are different.  We will do this to first order in time.  We should note that the very general form of the Hamiltonian (Eq. [[#eq8.15|(8.15)]]) tells us that each term of the Hamiltonian could be made Hermitian depending on the order and grouping in the terms of the sum.  We will not see at any point where this choice will make a significant difference in our analysis since most of the bases we use are arbitrary.  Thus, to first order (and without the factor <math>\sqrt{\mu_i}\,\!</math>)
{{Equation | <math>
A_\alpha \approx \delta_{ij} + 
(-i\Delta t)\sum_\gamma S_\gamma \langle i | B_\gamma |j\rangle.
\,\!</math>|8.23}}
Since the <math>S_\gamma\,\!</math> are Hermitian, they can be expanded in a 
complete basis for the algebra of the <math>d\times d\,\!</math> 
unitary matrices, plus 
the identity if the system is <math>d\,\!</math>-dimensional.  Therefore, under a 
unitary transformation of the operators <math>K\,\!</math> or for that matter <math>A\,\!</math>, 
the operators transform as (see the note above equ.(\ref{bsnKs}))
{{Equation | <math>
K^\prime_m = UK_m U^\dagger = \sum_n U a_{mn} \lambda_n U^\dagger,
\,\!</math>|8.24}}
where the set <math>\{\lambda_n\}\,\!</math> are a complete set of basis elements 
for the algebra of <math>d\,\!</math>-dimensional unitary matrices, with <math>n\,\!</math> ranging 
from zero to <math>d^2-1\,\!</math> with <math>\lambda_0\equiv I\,\!</math>.  This transformation 
can be written as 
{{Equation|<math>\begin{align}
A^\prime_\alpha &= \sum_m b_{\alpha m} K^\prime_m   \\
                &= \sum_{m,n,p} b_{\alpha m} a_{mn} R_{np} \lambda_p,
\end{align}
\,\!</math>|8.25}} 
where 
{{Equation | <math>
U\lambda_n U^\dagger = \sum_p R_{np} \lambda_p   
\,\!</math>|8.26}}
defines an action by the adjoint representation of the group.  
Alternatively, we can redefine the <math>b_{\alpha m}\,\!</math> such that 
{{Equation | <math>
\tilde{b}_{\alpha n} =  b_{\alpha m} a_{mn},
\,\!</math>|8.27}}
so that our fixed basis stays fixed and we look at the transformation as an active, rather than a passive one.  (That is, we transform the object instead of the basis.)

It is interesting to note that upon comparison of Eqs. [[#eq8.21|(8.21)]] and [[#eq8.22|(8.22)]] we can identify <math>b\,\!</math>s and <math>K\,\!</math>s as [[Bibliography#Lidar:CP01|Lidar, et al.]]
{{Equation | <math>
K_\gamma = S_\gamma,\;\;\;\;\;\;\; 
b_{\gamma,m} = (-i\Delta t)\langle i | B_\gamma |j\rangle.
\,\!</math>|8.28}}

===Unitary Degree of Freedom in the OSR===

The operator sum decomposition is not unique.  There is a freedom in choosing the operators in Eq. [[#eq8.14|(8.14)]] (or [[#eq8.19|(8.19)]].  It turns out, and this will be shown, that this is equivalent to having the freedom to choose a basis for the bath.  

====Unitary Freedom====

Let us consider an operator-sum decomposition, of the form given in Eq. [[#eq8.19|(8.19)]] which represents some quantum process whereby one quantum system, a bath, interacts with another and is then traced out.  Now consider another given by a linear combination of the operators 
<math>A_\alpha\,\!</math>, 
{{Equation | <math>
D_\alpha = \sum_\beta u_{\beta\alpha}A_\alpha.
\,\!</math>|8.29}}
Now, let us construct the operator-sum representation for the set of operators <math> \{ D_\beta \}\,\!</math>.  This is 
{{Equation | <math>
\sum_\beta D_\beta\rho_S D_\beta^\dagger = \sum_{\alpha,\gamma}
u_{\beta\alpha}A_\alpha \rho_S u^*_{\beta\gamma}A_\gamma^\dagger.
\,\!</math>|8.30}}
Rewriting this as 
{{Equation | <math>
\sum_\beta D_\beta\rho D_\beta^\dagger = \sum_{\alpha,\gamma}
(u_{\gamma\beta})^\dagger u_{\beta\alpha} A_\alpha \rho A_\gamma^\dagger,
\,\!</math>|8.31}}
where <math>(u_{\gamma\beta})^\dagger = u^*_{\beta\gamma}\,\!</math>,
(i.e. the  <math>(u_{\gamma\beta})\,\!</math> are elements of a matrix  <math>U\,\!</math>)
we see that if
<math>(u_{\gamma\beta})^\dagger u_{\beta\alpha}=\delta_{\gamma\alpha}\,\!</math>,
then 
{{Equation | <math>
\sum_\beta D_\beta\rho_S D_\beta^\dagger = \rho_S^\prime.
\,\!</math>|8.32}}
Therefore the two different sets of operators <math> \{ A_\alpha \}\,\!</math> and <math> \{ D_\beta \}\,\!</math> give rise to the same open-system evolution of <math>\rho_S\,\!</math> if they are related by a unitary transformation of the form of <math> U \,\!</math>.  Note that if the two sets do not have the same number of elements, then we may either append zeroes (as is done
in [[Bibliography#NielsenChuang:book|Nielsen and Chuang's book]]) or let <math>U\,\!</math> be a <nowiki>''right unitary matrix''</nowiki>, which means the matrix <math>U^\dagger U = I\,\!</math>, but the matrix need not be square or
have <math>UU^\dagger  = I\,\!</math>.

====Physical Interpretation of the Unitary Freedom====

It is fairly to show that this unitary degree of freedom is associated with a change of the bath basis.  Let us first recall
{{Equation|<math>
A_{ij} = \langle i| U_{tot}^\dagger(t) |j\rangle.
\,\!</math>|8.33}} 
So that when we write <math>u_{\beta\alpha}A_{\alpha}\,\!</math> we interpret this as 
<math>u_{kj}A_{ij} = \sqrt{\mu_j}\langle i| U_{tot}^\dagger(t) |j\rangle.\,\!</math>.  However, this clearly makes no difference in the result since we are tracing over the bath.  One could just as well include this unitary transformation which changes the basis of the bath in <math>U_{tot}\,\!</math>.  


===Examples===

In this section several examples are given which are quite important to quantum error correction and quantum computing.  These are all examples of actions on single qubit density operators which, not only provide very relevant examples, but are also quite simple.  

====Example 1: Bit-flip====

Bit-flip errors have a direct analogy in classical computation.  If a <math>\left\vert 1\right\rangle\,\!</math> is input, but, due to noise a  <math>\left\vert 0\right\rangle\,\!</math> is output (or vice versa) then a bit-flip error has occurred.  If a quantum bit-flip error occurs with some probability <math>p\,\!</math>, we may express this as
{{Equation|<math>
\rho^\prime = (1-p)\rho + p \sigma_x \rho \sigma_x,
\,\!</math>|8.34}} 
where <math>\sigma_x\,\!</math> is the  <math>X\,\!</math> [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli Matrix]], and <math>\rho \,\!</math> can be expressed as in Eq.[[Chapter 3 - Physics of Quantum Information#eq3.35|(3.35)]].  This has a clear interpretation.  There is a probability <math>p\,\!</math> that there is a bit flip, and there is a probability <math>(1-p)\,\!</math> that nothing happens to the density operator.  The operators of the operator-sum representation of this map can be taken to be <math>\sqrt{p}I\,\!</math> and <math>\sqrt{(1-p)} \sigma_x\,\!</math>.  This is a very important type of error and therefore will be discussed in connection with quantum error prevention methods.  

====Example 2: Phase-flip====

Phase-flip errors do not have a direct analogue in classical computation.  In this case a <math>\left\vert 1\right\rangle\,\!</math> will acquire a (-1) sign change due to some noise, but a  <math>\left\vert 0\right\rangle\,\!</math> is unaffected.  If a quantum phase-flip error occurs with some probability <math>p\,\!</math>, we may express the phase flip error as
{{Equation|<math>
\rho^\prime = (1-p)\rho + p \sigma_z \rho \sigma_z,
\,\!</math>|8.35}} 
where <math>\sigma_z\,\!</math> is the  <math>Z\,\!</math> [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli Matrix]], and <math>\rho \,\!</math> can be expressed as in Eq.[[Chapter 3 - Physics of Quantum Information#eq3.35|(3.35)]].  Here there is a probability <math>p\,\!</math> that a phase-flip occurs, and there is a probability <math>(1-p)\,\!</math> that nothing happens to the density operator.  The operators of the operator-sum representation of this map can be taken to be <math>\sqrt{p}I\,\!</math> and <math>\sqrt{(1-p)} \sigma_z\,\!</math>.  This is another very important type of error.  

====Example 3: Bit-flip and Phase-flip====

Suppose that both errors occur.  If both errors occur with some probability <math>p\,\!</math>, we may express the error as
{{Equation|<math>
\rho^\prime = (1-p)\rho + p \sigma_y \rho \sigma_y,
\,\!</math>|8.36}} 
where <math>\sigma_y=i\sigma_z\sigma_x\,\!</math>, is the  <math>Y\,\!</math> [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli Matrix]]. As before, <math>\rho \,\!</math> can be expressed as in Eq.[[Chapter 3 - Physics of Quantum Information#eq3.35|(3.35)]].  Here there is a probability <math>p\,\!</math> that a <math>Y \,\!</math> occurs, and there is a probability <math>(1-p)\,\!</math> that nothing happens to the density operator.  The operators of the operator-sum representation of this map can be taken to be <math>\sqrt{p}I\,\!</math> and <math>\sqrt{(1-p)} \sigma_y\,\!</math>.   Note that <math>\sigma_y^\dagger = (i\sigma_z\sigma_x)^\dagger = -i\sigma_x\sigma_z\,\!</math>.


It is important to note that the [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli Matrices]] form a basis for all of the possible errors on a qubit as discussed in Section [[Chapter 2 - Qubits and Collections of Qubits#Two-State Example: Bloch Sphere|3.5.4]].

====Example 4: Depolarizing Error====

A depolarizing error is an error which is symmetric in the three possible errors.  For example, suppose that each of the three possible types of errors on a qubit all occur with equal probability.  Let this probability be <math> p/3 \,\!</math>.  The depolarizing operation is then
{{Equation|<math>
\rho^\prime = (1-p)\rho + (p/3) \sigma_x \rho \sigma_x + (p/3) \sigma_y \rho \sigma_y + (p/3) \sigma_z \rho \sigma_z,
\,\!</math>|8.37}} 
Using Eq.[[Chapter 3 - Physics of Quantum Information#eq3.35|(3.35)]], this can also be written as
{{Equation|<math>
\rho^\prime = \frac{1}{2}(I + (1-4p/3) n_x \sigma_x + (1-4p/3) n_y\sigma_y  + (1-4p/3) n_z\sigma_z) = \frac{1}{2}(I +(1-4p/3)\vec{n}\cdot\vec{\sigma}).
\,\!</math>|8.38}}
Thus the depolarizing error has quite interesting and simple properties since it can be seen as a uniform shrinking of the polarization vector <math>\vec{n}\,\!</math>.

===Notes===

Errors are the obstacle to building a quantum computing device.  These errors are caused by noise in quantum systems.  Noise in quantum systems is also known as decoherence although the original meaning of decoherence referred (roughly) to the loss of off-diagonal terms of the density matrix.  (An analogy of such loss is the <math>Z\,\!</math> error example above.)  When a set of entangled states experiences decoherence, entanglement is lost.  Since entanglement is believed to be the source of advantages of quantum computing and a great deal of other quantum information processing, this is a serious concern.  

In the following chapters, several methods designed to reduce or eliminate noise in quantum systems will be discussed.