The Standard Error Models for Qubits

The most investigated error model for qubits consists of ``independent, depolarizing errors''. This model has the effect of completely depolarizing each qubit independently with probability $p$ (see Eq. 10). For one qubit, the model is the least biased in the sense that it is symmetric under rotations. As a result, every state of the qubit is equally affected. Independent depolarizing errors are considered to be the quantum analogue of the classical independent bit flip error model.

Depolarizing errors are not typical for physically realized qubits. However, given the ability to control individual qubits, it is possible to enforce the depolarizing model (see below). Consequently, error-correction methods designed to control depolarizing errors apply to all independent error models. Nevertheless, it is worth keeping in mind that given detailed knowledge of the physical errors, a special purpose method is usually better than one designed for depolarizing errors. We therefore begin by showing how one can think about arbitrary error models.

There are several different ways of describing errors affecting a physical system $\mathsf {S}$ of interest. For most situations, in particular if the initial state of $\mathsf {S}$ is pure, errors can be thought of as being the result of coupling to an initially independent environment for some time. Because of this coupling, the effect of error can always be represented by the process of adjoining an environment $\mathsf {E}$ in some initial state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ to the arbitrary state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$ of $\mathsf {S}$, followed by a unitary coupling evolution ${U}^{({\mathsf {E\,S}})}$ acting jointly on $\mathsf {E}$ and $\mathsf {S}$. Symbolically, the process can be written as the map

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\m... ...3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}.$$ (22)

Choosing an arbitrary orthonormal basis consisting of the states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ for the state space of the environment, the process can be rewritten in the form:

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}} {\color{red}\rightarrow} {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mski... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$$

$$= \left(\sum_e\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$$

$$= \sum_e\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbo... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$$

$$= \sum_e \mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mb... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}},$$ (23)

where the last step defines operators ${A_e}^{({\mathsf {S}})}$ acting on $\mathsf {S}$ by ${A_e}^{({\mathsf {S}})}={}^{\scriptscriptstyle\mathsf { E}}\!\mbox{$\langle\hsp... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$. The expression $\sum_e\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hsp... ...rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}{A_e}^{({\mathsf {S}})}$ is called an ``environment labeled operator''. The unitarity condition implies that $\sum_e A_e^\dagger A_e = {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}$ (with system labels omitted). The environment basis $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ need not represent any physically meaningful choice of basis of a real environment. For the purpose of error analysis, the states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ are formal states that ``label'' the error operators $A_e$. One can use an expression of the form shown in Eq. 23 even when the $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ are not normalized or orthogonal, keeping in mind that as a result, the identity implied by the unitarity condition changes.

Note that the state on the right side of Eq. 23 representing the effect of the errors is correlated with the environment. This means that after removing (or ``tracing over'') the environment, the state of $\mathsf {S}$ is usually mixed. Instead of introducing an artificial environment, we can also describe the errors by using the density operator formalism for mixed states. Define $\rho=\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\h... ...ace*{-4.3pt}\langle$}{\psi}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$. The effect of the errors on the density matrix $\rho$ is given by the transformation

$$\rho\;{\color{red}\rightarrow}\; \sum_e A_e \rho A_e^\dagger.$$ (24)

This is the ``operator sum'' formalism [18].

The two ways of writing the effects of errors can be applied to the depolarizing-error model for one qubit. As an environment-labeled operator, depolarization with probability $p$ can be written as

$$\sqrt{1-p}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\ver... ...ace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}\sigma_z \Big),$$ (25)

where we introduced five abstract, orthonormal environment states to label the different events. In this case, one can think of the model as applying no error with probability $1-p$, or completely depolarizing the qubit with probability $p$. The latter event is represented by applying one of ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}},\sigma_x,\sigma_y$ or $\sigma_z$ with equal probability $p/4$. To be able to think of the model as randomly applied Pauli matrices, it is crucial that the environment states labeling the different Pauli matrices be orthogonal. The square roots of the probabilities appear in the operator because in an environment-labeled operator, it is necessary to give quantum amplitudes. Environment labeled operators are useful primarily because of their great flexibility and redundancy.

In the operator sum formalism, depolarization with probability $p$ transforms the input density matrix $\rho$ as

$$\rho \rightarrow (1-p)\rho + {p\over 4}\left( {\mathchoice {\rm 1\mskip-4mu l} {\r... ...l}}+ \sigma_x\rho\sigma_x + \sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z \right)$$

$$= (1-3p/4)\rho + {p\over 4}\left( \sigma_x\rho\sigma_x + \sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z \right).$$ (26)

Because the operator sum formalism has less redundancy, it is easier to tell when two error effects are equivalent.

In the remainder of this section, we discuss how one can use active intervention to simplify the error model. To realize this simplification, we intentionally randomize the qubit so that the environment cannot distinguish between the different ``axes'' defined by the Pauli spin matrices. Here is a simple randomization that actively converts an arbitrary error model for a qubit into one that consists of randomly applying Pauli operators according to some distribution. The distribution is not necessarily uniform so the new error model is not yet depolarizing. Before the errors act, apply a random Pauli operator $\sigma_u$ ($u=0,x,y,z$, $\sigma_0={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}$). After the errors act apply the inverse of that operator, $\sigma_u^{-1} = \sigma_u$; then ``forget'' which operator was applied. This randomization method is called ``twirling'' [19]. To understand twirling, we use environment labeled operators to demonstrate some of the techniques useful in this context. The sequence of actions implementing twirling can be written as follows (omitting labels for $\mathsf {S}$):

$$\begin{array}[b]{rcll} \mbox{$\vert\hspace*{-3pt}\vert\hsp... ...as used by absorbing its memory in $\mathsf {E}$.} \end{array}$$ (27)

The system $\mathsf {C}$ that was artificially introduced to carry the memory of $u$ may be a classical memory because there is no need for coherence between different $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{u}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {C}}}}$.

To determine the equivalent random Pauli operator error model, it is necessary to rewrite the total effect of the procedure using an environment labeled sum involving orthogonal environment states and Pauli operators. To do so, express $A_e$ as a sum of the Pauli operators, $A_e=\sum_v\alpha_{ev}\sigma_v$, using the fact that the $\sigma_v$ are a linear basis for the space of one-qubit operators. Recall the fact that $\sigma_u$ anticommutes with $\sigma_v$ if $0\not=u\not=v\not=0$. Thus $\sigma_u\sigma_v\sigma_u=(-1)^{\langle v,u\rangle}\sigma_v$, where $\langle v,u\rangle = 1$ if $0\not=u\not=v\not=0$ and $\langle v,u\rangle = 0$ otherwise. We can now rewrite the last expression of Eq. 27 as follows:

$$\sum_{eu} \mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{eu... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$} = \sum_{eu} \mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{eu... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$$

$$= \sum_v\left(\sum_{eu}{1\over 2}\alpha_{ev}(-1)^{\langle v,u\rangl... ...pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}.$$ (28)

It can be checked that the states ${1\over 2}\sum_{u}(-1)^{\langle v,u\rangle}\mbox{$\vert\hspace*{-3pt}\vert\hspa... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {EC}}}}$ are orthonormal for different $e$ and $v$. As a result the states $\sum_{eu}{1\over 2}\alpha_{ev}(-1)^{\langle v,u\rangle}\mbox{$\vert\hspace*{-3p... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {EC}}}}$ are orthogonal for different $v$ and have probability (square norm) given by $p_v=\sum_e\vert\alpha_{ev}\vert^2$. Introducing $\sqrt{p_v}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\tilde v}\mbox{$... ...angle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {EC}}}}$, we can write the sum of Eq. 28 as

$$\sum_v\left(\sum_{eu}{1\over 2}\alpha_{ev}(-1)^{\langle v,u\... ...mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$},$$ (29)

showing that the twirled error model behaves like randomly applied Pauli matrices with $\sigma_v$ applied with probability $p_v$. It is a recommended exercise to reproduce the above argument using the operator sum formalism.

To obtain the standard depolarizing error model with equal probabilities for the Pauli matrices, it is necessary to strengthen the randomization procedure by applying a random member $U$ of the group generated by the $90^\circ$ rotations around the $x$, $y$ and $z$ axes before the error and then undoing $U$ by applying $U^{-1}$.

Randomization can be used to transform any one-qubit error model into the depolarizing error model. This explains why the depolarizing model is so useful for analyzing error correction techniques in situations in which errors act independently on different qubits. However, in many physical situations, the independence assumptions are not satisfied. For example, errors from common internal couplings between qubits are generally pairwise correlated to first order. In addition, the operations required to manipulate the qubits and to control the encoded information act on pairs at a time, which tends to spread even single qubit errors. Still, in all these cases, the primary error processes are local. This means that there usually exists an environment labeled sum expression for the total error process in which the amplitudes associated with errors acting simultaneously at $k$ locations in time and space decrease exponentially with $k$. In such cases, error-correction methods that handle all or most errors involving sufficiently few qubits are still applicable.