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, the Schrodinger equation was discussed as the way in which to describe quantum systems' evolution. The evolution described by Schrodinger's Equation is a the evolution of a system which has been isolated from everything else, i.e., it describes a 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 Sudarshan, Mathews, and Rau in 1961 (SMR) and was later taken up by 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 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.

	$\rho ^{\prime }=A\rho ,\,\!$	(8.1)

or more explicitly

	$\rho _{r^{\prime }s^{\prime }}^{\prime }=A_{r^{\prime }s^{\prime },rs}\rho _{rs}.\,\!$	(8.2)

One way to think of this is a linear mapping from one vector to another by a matrix. ($\rho \,\!$ can be viewed as a vector. Simply rearrange the elements of the matrix into a column vector. $A\,\!$ 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 $A\,\!$. Now, we recall that the density matrix is required to be, Hermitian, positive semi-definite, and have trace one. Respectively, we write

	$\rho =\rho ^{\dagger },\,\!$	(8.3)

	$\rho \geq 0,\,\!$	(8.4)

and

	${\mbox{Tr}}\rho =1.\,\!$	(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. (8.3), (8.4), and (8.5), the mapping $A\,\!$ (which can be considered a matrix) has the following properties:

	$A_{sr,s^{\prime }r^{\prime }}=(A_{rs,r^{\prime }s^{\prime }})^{*},\,\!$	(8.6)

	$x_{r}^{*}x_{s}A_{sr,s^{\prime }r^{\prime }}y_{s^{\prime }}y_{r^{\prime }}\geq 0,\,\!$	(8.7)

and

	$A_{rr,s^{\prime }r^{\prime }}=\delta _{s^{\prime }r^{\prime }}.\,\!$	(8.8)

We could also introduce a new matrix $B\,\!$ which is related to $A\,\!$ by relabeling, by

	$B_{rr^{\prime },ss^{\prime }}\equiv A_{sr,s^{\prime }r^{\prime }},\,\!$	(8.9)

with the following properties:

	$B_{rr^{\prime },ss^{\prime }}=(B_{ss^{\prime },rr^{\prime }})^{*},\,\!$	(8.10)

	$z_{rr^{\prime }}^{*}B_{rr^{\prime },ss^{\prime }}z_{ss^{\prime }}\geq 0,\,\!$	(8.11)

and

	$B_{rr^{\prime },rs^{\prime }}=\delta _{r^{\prime }s^{\prime }}.\,\!$	(8.12)

Now the important point to note is that $B\,\!$ can be considered a Hermitian matrix, and as such, it is diagonalizable. Letting $\eta (\alpha )\,\!$ be its eigenvalues and $C_{(\alpha )}\,\!$ the corresponding eigenvectors, we see that the mapping (8.2), can be written as

	$\rho ^{\prime }=\sum _{\alpha }\eta (\alpha )C_{(\alpha )}^{}\rho C_{(\alpha )}^{\dagger }.\,\!$	(8.13)

If all of the $\eta (\alpha )\,\!$ are positive, then a factor of ${\sqrt {\eta (\alpha )}}\,\!$ can be absorbed into the $C_{(\alpha )}\,\!$ and the map can be written as

	$\rho ^{\prime }=\sum _{\alpha }C_{(\alpha )}^{}\rho C_{(\alpha )}^{\dagger }.\,\!$	(8.14)

This is what is sometimes called the Operator-Sum representation or Kraus decomposition, originally given in SMR.

Physics Behind the Noise and Completely Positive Maps

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.

The dynamics of an open quantum system coupled to a bath is formally obtained from the evolution $U=\exp(-iHt)\,\!$ under the combined system-bath Hamiltonian

	$H=H_{S}\otimes I+I\otimes H_{B}+\sum _{\gamma }S_{\gamma }\otimes B_{\gamma },\,\!$	(8.15)

where $H_{S}\,\!$ is the Hamiltonian for the system alone, $H_{B}\,\!$ is the Hamiltonian for the bath alone, and the $S_{\gamma }\,\!$ and $B_{\gamma }\,\!$ 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 $t=0\,\!$,

$\rho (t=0)=\rho _{S}\otimes \rho _{B}.\,\!$

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 partial trace to get

	$\rho _{S}(t)=\mathrm {Tr} _{B}[{U}\left(\rho _{S}(0)\otimes \rho _{B}(0)\right){U}^{\dagger }],\,\!$	(8.16)

where $\rho _{S}(0)\,\!$ is the initial density matrix of the (open) system, and $\rho _{B}(0)\,\!$ is the initial density matrix of the bath. If we take the bath to be in a state $\rho _{B}=\sum _{i}\mu _{i}|i\rangle \langle i|\,\!$, then we can write Eq. (8.16) as

	${\begin{aligned}\rho _{S}^{\prime }(t)&=\sum _{i,j}\langle j|(U_{tot}(t)\rho _{S}(0)\otimes \mu _{i}|i\rangle \langle i|U_{tot}^{\dagger }(t)|j\rangle \\&=\sum _{i,j}\mu _{i}\langle j|U_{tot}(t)|i\rangle \rho _{S}(0)\langle i|U_{tot}^{\dagger }(t)|j\rangle .\end{aligned}}$	(8.17)

Now we may define a new index $\alpha \equiv (i,j)\,\!$ and also we may define

	$A_{\alpha }(t)\equiv {\sqrt {\mu _{i}}}\;\langle j|U_{tot}(t)|i\rangle ,\,\!$	(8.18)

and in an analogous way $A_{\alpha }^{\dagger }\,\!$ so that the operator-sum representation is given by

	$\rho _{S}^{\prime }(t)=\sum _{\alpha }A_{\alpha }(t)\rho _{S}A_{\alpha }^{\dagger }(t).\,\!$	(8.19)

Note the relation to Eq.(8.14). In some places the definition (8.18) will be used without the factor ${\sqrt {\mu _{i}}}\,\!$. This will be of little consequence, but should be noted.

The equation (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 SMR and the map was called, and rather appropriately, a dynamical map. The $A\,\!$s are often called 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,

$\sum _{\alpha }A_{\alpha }^{\dagger }A_{\alpha }=I_{S}.\,\!$

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) SMR, Kraus, Schumacher:

	${\begin{aligned}{\mathcal {E}}_{t}(\rho (0))&\equiv \rho (t)\\&=\!\sum _{\mu \nu }\!A_{\mu \nu }(t)\rho (0)A_{\mu \nu }^{\dagger }(t)\!=\!\sum _{\alpha ,\beta }\!\chi _{\alpha \beta }(t)K_{\alpha }\rho (0)K_{\beta }^{\dagger }.\;\;\;\;\;\;\end{aligned}}$	(8.20)

The operators $A_{\mu \nu }(t)={\sqrt {\lambda _{\nu }}}\langle \mu |U(t)|\nu \rangle \,\!$ where $U=\exp(-iHt/\hbar )\,\!$ and the initial bath density matrix is written as $\rho _{B}(0)=\sum _{\nu }\lambda _{\nu }|\nu \rangle \langle \nu |\,\!$ Lidar, et al.. They satisfy the normalization condition: $\sum _{\mu }A_{\mu \nu }^{\dagger }A_{\mu \nu }=I_{S}\,\!$. The matrix $\chi _{\alpha \beta }(t)=\sum _{i}b_{i,\alpha }b_{i,\beta }^{\ast }\,\!$ is a time-dependent, hermitian coefficient matrix defined by a transformation of the Kraus operators to a fixed operator basis $K_{\alpha }\,\!$: $A_{\mu \nu }(t)=\sum _{\alpha }b_{\mu \nu ;\alpha }(t)K_{\alpha }\,\!$.

We can express our $A_{\alpha }\,\!$ in terms of a fixed basis of operators. The $A_{\alpha }\,\!$ are functions of time (since $U_{tot}\,\!$ is) and are thus not fixed operators. We can expand these $A\,\!$'s in terms of a complete fixed chosen basis $\{K_{m}\}\,\!$:

	$A_{\alpha }=\sum _{m}b_{\alpha ,m}(t)K_{m},\,\!$	(8.21)

so that the time-dependence is in the $b\,\!$'s. Now we would like to see how Eq. (8.21) transforms under a change of basis, that is, when the $K\,\!$'s are different. We will do this to first order in time. We should note that the very general form of the Hamiltonian (Eq. (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 ${\sqrt {\mu _{i}}}\,\!$)

	$A_{\alpha }\approx \delta _{ij}+(-i\Delta t)\sum _{\gamma }S_{\gamma }\langle i|B_{\gamma }|j\rangle .\,\!$	(8.22)

Since the $S_{\gamma }\,\!$ are Hermitian, they can be expanded in a complete basis for the algebra of the $d\times d\,\!$ unitary matrices, plus the identity if the system is $d\,\!$-dimensional. Therefore, under a unitary transformation of the operators $K\,\!$ or for that matter $A\,\!$, the operators transform as (see the note above equ.(\ref{bsnKs}))

	$K_{m}^{\prime }=UK_{m}U^{\dagger }=\sum _{n}Ua_{mn}\lambda _{n}U^{\dagger },\,\!$	(8.23)

where the set $\{\lambda _{n}\}\,\!$ are a complete set of basis elements for the algebra of $d\,\!$-dimensional unitary matrices, with $n\,\!$ ranging from zero to $d^{2}-1\,\!$ with $\lambda _{0}\equiv I\,\!$. This transformation can be written as

	${\begin{aligned}A_{\alpha }^{\prime }&=\sum _{m}b_{\alpha m}K_{m}^{\prime }\\&=\sum _{m,n,p}b_{\alpha m}a_{mn}R_{np}\lambda _{p},\end{aligned}}$	(8.24)

where

	$U\lambda _{n}U^{\dagger }=\sum _{p}R_{np}\lambda _{p}\,\!$	(8.25)

defines an action by the adjoint representation of the group. Alternatively, we can redefine the $b_{\alpha m}\,\!$ such that

	${\tilde {b}}_{\alpha n}=b_{\alpha m}a_{mn},\,\!$	(8.26)

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. (8.21) and (8.22) we can identify $b\,\!$s and $K\,\!$s as Lidar, et al.

	$K_{\gamma }=S_{\gamma },\;\;\;\;\;\;\;b_{\gamma ,m}=(-i\Delta t)\langle i|B_{\gamma }|j\rangle .\,\!$	(8.27)

Unitary Degree of Freedom in the OSR

The operator sum decomposition is not unique. There is a freedom in choosing the operators in Eq. (8.14) (or (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.

Let us consider an operator-sum decomposition, of the form given in Eq. (8.19) which represents some quantum process whereby one quantum system, a bath, interacts with another and is then averaged away (traced out). Now consider another given by a linear combination of the operators $A_{\alpha }\,\!$,

	$D_{\alpha }=\sum _{\beta }u_{\beta \alpha }A_{\alpha }.\,\!$	(8.28)

Now, let us construct the operator-sum representation for the set of operators $\{D_{\beta }\}\,\!$. This is

	$\sum _{\beta }D_{\beta }\rho _{S}D_{\beta }^{\dagger }=\sum _{\alpha ,\gamma }u_{\beta \alpha }A_{\alpha }\rho _{S}u_{\beta \gamma }^{*}A_{\gamma }^{\dagger }.\,\!$	(8.29)

Rewriting this as

	$\sum _{\beta }D_{\beta }\rho D_{\beta }^{\dagger }=\sum _{\alpha ,\gamma }(u_{\gamma \beta })^{\dagger }u_{\beta \alpha }A_{\alpha }\rho A_{\gamma }^{\dagger },\,\!$	(8.30)

where $(u_{\gamma \beta })^{\dagger }=u_{\beta \gamma }^{*}\,\!$, (i.e. the $(u_{\gamma \beta })\,\!$ are elements of a matrix $U\,\!$) we see that if $(u_{\gamma \beta })^{\dagger }u_{\beta \alpha }=\delta _{\gamma \alpha }\,\!$, then

	$\sum _{\beta }D_{\beta }\rho _{S}D_{\beta }^{\dagger }=\rho _{S}^{\prime }.\,\!$	(8.31)

Therefore the two different sets of operators $\{A_{\alpha }\}\,\!$ and $\{D_{\beta }\}\,\!$ give rise to the same open-system evolution of $\rho _{S}\,\!$ if they are related by a unitary transformation of the form of $U\,\!$. Note that if the two sets do not have the same number of elements, then we may either append zeroes (as is done in Nielsen and Chuang's book) or let $U\,\!$ be a ''right unitary matrix'', which means the matrix $U^{\dagger }U=I\,\!$, but the matrix need not be square or have $UU^{\dagger }=I\,\!$.