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

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^\prime_{r^\prime s^\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. (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho\,\!} can be viewed as a vector. Simply rearrange the elements of the matrix into a column vector. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!} . Now, we recall that the density matrix is required to be, Hermitian, positive semi-definite, and have trace one. Respectively, we write

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (\ref{hermiticity}), (\ref{dmatpositivity}), and (\ref{normalization}), 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 (\ref{As1}), 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\eta(\alpha)}\,\!} can be absorbed into the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{(\alpha)}\,\!} and the map can be written as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.