Chapter 9 - Dynamical Decoupling Controls

Introduction

In the last chapter, it was shown that a symmetry in the system-bath Hamiltonian, if present, could be used to construct states immune to noise. In this chapter we will see that under certain conditions it is possible to reduce errors, create a symmetry, or even remove errors in the evolution of a quantum system. This is done though repeated use of external controls which act on the system. These controls are often called "dynamical decoupling controls" due to their original objective of decoupling the system from the bath. They are quite generally useful controls to consider for the elimination and/or reduction of errors. In this chapter, a simple introduction to dynamical decoupling controls is given and some important concepts discussed.

General Conditions

As stated in Chapter 8 the Hamiltonian describing the evolution of a system and bath which are coupled together can always be written as

	$H=H_{S}\otimes I_{B}+I_{S}\otimes H_{B}+H_{I},\,\!$	(9.1)

where $H_{S}\,\!$ acts only on the system, $H_{B}\,\!$ acts only on the bath, and

$H_{I}=\sum _{\alpha }S_{\alpha }\otimes B_{\alpha },\,\!$

is the interaction Hamiltonian with the $S_{\alpha }\,\!$ acting only on the system and the $B_{\alpha }\,\!$ acting only on the bath.

The idea is to modify the evolution of the system and bath such that the errors are reduced or eliminated using external control Hamiltonians. These controls are called dynamical decoupling controls since they are used to decouple (at least approximately decouple) the system from the bath. Since can be difficult to change states of a bath, indeed one often does not know details of the bath, the controls which are to be used for reducing errors should act on the system. As discussed previously, the errors arise from the system-bath interaction Hamiltonian $H_{I}\,\!$ and, in particular, the system operators $S_{\alpha }\,\!$ are the operators which describe the affect of the coupling on the system. In general the interaction Hamiltonian is time-dependent since the bath operators will change in time. However, for short times we may assume the interaction Hamiltonian is unchanged, or at least approximately constant. This is sometimes called the short-time assumption in dynamical decoupling.

The Magnus Expansion

A fairly good starting point to see how this is done is the so-called Magnus expansion. (See Blanes, et al. and references therein.) The general problem is that a time-dependent operation is to be applied to the Hamiltonian making the Hamiltonian itself time-dependent and one would like to solve the time-dependent Schrodinger equation:

	$i{\frac {\partial }{\partial t}}\left\vert \Psi (t)\right\rangle =H(t)\left\vert \Psi (t)\right\rangle ,\,\!$	(9.2)

which is sometimes written as

	$U^{\prime }(t)=-iH(t)U(t).\,\!$	(9.3)

The question is, what $U(t)\,\!$ will solve this equation? If $U(t)\,\!$ and $H(t)\,\!$ are just numbers, the solution would be

	$U^{\prime }(t)=\exp \left(-i\int _{0}^{t}H(t^{\prime })dt^{\prime }\right).\,\!$	(9.4)

However, when the Schrodinger equation is the equation to be solved, $U(t)\,\!$ and $H(t)\,\!$ are matrices. To be specific, $U(t)\,\!$ is a unitary matrix and $H(t)\,\!$ is a Hermitian matrix. The solution is often written in the form

	$U^{\prime }(t)={\mathcal {T}}\left[\exp \left(-i\int _{0}^{t}H(t^{\prime })dt^{\prime }\right)\right],\,\!$	(9.5)

where ${\mathcal {T}}\,\!$ denotes the time-ordered exponential. In this case, matrices do not commute so that the exponential must be handled with care. Operators must be ordered according to the time where they appear in the operation, and the solution Eq.(9.2) is not the solution to the problem unless $H(t)\,\!$ is a constant matrix.

The solution to this problem is the following,

	$U^{\prime }(T)=\exp \left(-i\Omega (T)T\right),\,\!$	(9.6)

where

	$\Omega (T)=\sum _{k=1}^{\infty }\Omega _{k},\,\!$	(9.7)

and

	${\begin{aligned}\Omega _{1}&={\frac {1}{T}}\int _{0}^{T}H(t_{1})dt_{1},\\\Omega _{2}&=-{\frac {1}{T^{2}}}{\frac {i}{2}}\int _{0}^{T}dt_{1}\int _{0}^{t_{1}}dt_{2}[H(t_{1}),H(t_{2})],\\\Omega _{3}&=-{\frac {1}{T^{3}}}{\frac {1}{6}}\int _{0}^{T}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}([H(t_{1}),[H(t_{2}),H(t_{3})]]+[H(t_{3}),[H(t_{2}),H(t_{1})]])\\&{\mbox{etc.}},\end{aligned}}\,\!$	(9.8)

where $T\,\!$ is some characteristic time scale.

A First-Order Theory

To show how this theory of dynamical decoupling controls could work in an ideal case, let us consider a simple example. Suppose that the external controls (decoupling controls) are so strong that the Hamiltonian evolution can be neglected during the time the external controls are turned on. Due to their strength, we will also assume that they can be implemented in a very short time and that there are $N\,\!$ different controls to be used. We will first use a given control $U_{n}\,\!$ and then its inverse. Between the controls the system evolves for a short time $\Delta t\,\!$. After all control pulses have been implemented, the effective evolution of the system will be

	$U_{eff}(T)=\prod _{n=0}^{N-1}U_{n}^{-1}U(\Delta t)U_{n},\,\!$	(9.9)

where

	$U(\Delta t)=\exp(-iH\Delta t),\,\!$	(9.10)

$T=N\Delta t,\,\!$ and $H\,\!$ is free evolution given by Eq.(9.1) above. Furthermore, suppose that the time $\Delta t\,\!$ is small so that

	$U(\Delta t)\approx I-iH\Delta t.\,\!$	(9.11)

Now suppose that we let $U_{eff}=\exp(-iH_{eff}T)\,\!$. Inserting Eq.(9.11) into Eq.(9.9) and keeping only first order terms in the product gives

	$H_{eff}={\frac {1}{N}}\sum _{n=0}^{N-1}U_{n}^{-1}HU_{n}.\,\!$	(9.12)

This is a simple expression for the effective Hamiltonian evolution of a system undergoing a series of dynamical decoupling controls. Note that the assumptions are that the operations are strong (since the free evolution is neglected during the control pulses) and fast (since we assume that the Hamiltonian $H\,\!$ of Eq.(9.1) is constant during the entire time $T\,\!$ of this cycle of control pulses). Due to these strong and fast assumptions, these are often referred to as "bang-bang" controls.

It is important to note that these controls are rather unrealistic. That is, these criteria are never met completely. However, they are met approximately in some systems, most notably in nuclear magnetic resonance experiments where the so-called average Hamiltonian theory originated. More realistic pulses can be, and have been, explored for use in actual physical systems where they have been shown to reduce noise very effectively. This has been done by generalizing the theory beyond the first-order limit and without the assumption that the pulses are extremely strong.

The Single Qubit Case

The simplest case involves the elimination of an error on a single qubit. There are several types of errors that can degrade a qubit state as discussed in Section 6.4. There is a bit-flip, phase-flip, or both. In this section the first-order approximation is used to show how to eliminate first phase errors and then arbitrary errors on an arbitrary qubit state.

Phase Errors

Let us suppose that the Hamiltonian for the free evolution contains only an interaction part which induces a phase error,

	$H_{i}=\sigma _{z}\otimes B,\,\!$	(9.13)

where $B\,\!$ is the bath operator. This interaction Hamiltonian will couple the system to the bath and thus case errors. The factor $\sigma _{z}\,\!$ indicates that it is a phase error. Using the first-order theory, the objective is to find a series of pulses which will effectively decouple the system from the bath. In this case it can be done with only one decoupling pulse, $U_{X}(\pi )\,\!$. This will be denoted $U_{1}=U_{X}(\pi )\,\!$ and the identity (doing nothing) will be denoted $U_{0}=I\,\!$. The effective Hamiltonian is

	$H_{eff}={\frac {1}{2}}\sum _{n=0}^{1}U_{n}^{-1}H_{i}U_{n}={\frac {1}{2}}(U_{0}^{\dagger }H_{i}U_{0}+U_{1}^{\dagger }H_{i}U_{1}).\,\!$	(9.14)

A rotation about the x-axis by an angle $\pi \,\!$ will rotate the Pauli matrix $\sigma _{z}\,\!$ to $-\sigma _{z}\,\!$. (See Section C.5, in particular Section C.5.1.) This is because $U_{1}=U_{X}(\pi )=\sigma _{x}\,\!$. After the pulse sequence $U_{0},U_{1}\,\!$, the system is decoupled from the bath because the effective Hamiltonian is zero! There is no more interaction between the system and bath! That is

	$H_{eff}={\frac {1}{2}}(\sigma _{z}\otimes B+\sigma _{x}\sigma _{z}\sigma _{x}\otimes B=\sigma _{z}\otimes B-\sigma _{z}\otimes B=0.\,\!$	(9.15)

Thus the noise has been removed from the system.

Arbitrary Single Qubit Errors

	$\,\!$	(9.)