Quantum codes do not increase fidelity against isotropic errors

Currently the biggest obstacle to the development of quantum computing continues to be control of quantum errors. Since the beginnings of quantum computing in the 90s of the last century one of the main research objectives was to solve this stumbling block. To address the problem, two fundamental tools were developed: quantum error correction codes [1, 2, 3, 4, 5, 6] in combination with fault tolerant quantum computing [7, 8, 9, 10, 11, 12, 13]. These studies culminated in the proof of the quantum threshold theorem, which reads as follows: a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels. However, the proof of this theorem depends on the discretized treatment of quantum errors, inherited from the construction of quantum codes.

We believe that the quantum error model used for the proof of the quantum threshold theorem is not general and that the techniques developed to control quantum errors do not verify the golden rule of error control: correct all small errors exactly. For example, in the case of the coding of a qubit by means of the 5-qubit code [14, 15] it is argued, using error discretization and that this code exactly corrects errors in any of the qubits, that the error probability goes from $p$ to $p^{2}$, once the correction circuit has been applied. But, what is actually happening is that the probability of an error (small with high probability) in all qubits is 1 and that the code cannot correct these simultaneous errors. Then, an error occurs with probability 1 and, once the correction circuit is applied, it becomes undetectable.

Therefore it is necessary to make an analysis of quantum errors regardless of their discretization. The procedure indicated for this is to consider quantum errors as continuous random variables and characterize them by their corresponding density functions. In this article we analyze a specific type of error: isotropic quantum errors. An isotropic error of an $n-$qubit $\Phi$ is one in which the probability of the error $\Psi$ only depends on the distance between the two states, $\|\Psi-\Phi\|$, and not on the direction in which the imprecision $\Psi$ occurs with respect to $\Phi$. They are errors that are easy to analyze due to their central symmetry with respect to $\Phi$.

In work [16] we have studied the ability of an arbitrary quantum code to correct these errors, using the variance as the error measure. If $\Phi$ is the $n-$qubit without error state, $\Psi$ the state resulting from a disturbance modeled by an isotropic quantum error and $\tilde{\Phi}$ the result of applying the quantum code correction circuit, assuming that it does not introduce new errors, the result that we demonstrate in [16] is the following:

where $V(\tilde{\Phi})=E[\|\tilde{\Phi}-\Phi\|^{2}]$ and $V(\Psi)=E[\|\Psi-\Phi\|^{2}]$ are the variances of the corrected state $\tilde{\Phi}$ and the disturbed state $\Psi$ respectively. This means that no quantum code can handle isotropic errors, or even reduce their variance.

Now we are interested in analyzing the ability of quantum codes to increase fidelity against isotropic errors, since the fidelity allows to measure the quantum errors taking into account that the quantum states do not change if they are multiplied by a phase factor, while the variance used in [16] does not take this fact into account.

We represent $n-$qubits as points of the unit real sphere of dimension $2d-1$ being $d=2^{n}$ [17], $S^{2d-1}=\{x\in\mathbb{R}^{2d}\ |\ \|x\|=1\}$, taking coordinates with respect to the computational basis $[|0\rangle,|1\rangle,\dots,|2^{n}-1\rangle]$,

We consider quantum computing errors as random variables with density function defined on $S^{2d-1}$. In [16] it is mentioned that it is easy to relate this representation to the usual representation in quantum computing by density matrices and that the representation through random variables is more accurate.

We define the variance of a random variable $X$ as the mean of the quadratic deviation from the mean value $\mu$ of $X$, $V(X)=E[\|X-\mu\|^{2}]$. In our case, since the random variable $X$ represents a quantum computing error, the mean value of $X$ is the $n-$qubit $\Phi$ resulting from an errorless computation. Without loss of generality, we will assume that the mean value of every quantum computing error will always be $\Phi=|0\rangle$. To achieve this, it suffices to move $\Phi$ into $|0\rangle$ through a unitary transformation. Therefore, using the pure quantum states given by Formula (1), the variance of $X$ will be:

Obviously the variance satisfies $V(X)\in[0,4]$. In [18] the variance of the sum of two independent errors on $S^{2d-1}$ is presented for the first time. It is proved for isotropic errors and it is conjectured in general that:

Considering the representation of errors through random variables, the definition of fidelity is very simple:

Then, the problem we want to address is the following: Let $\Phi_{0}$ be an $m-$qubit and $\Phi$ the corresponding $n-$qubit encoded by an $(n,m)-$quantum code $\mathcal{C}$. Suppose that the coded state $\Phi$ is changed by error, becoming the state $\Psi$. Now, to fix the error we apply the code correction circuit, obtaining the final state $\tilde{\Phi}$. While $\Phi$ is a pure state, $\Psi$ and $\tilde{\Phi}$ are random variables (mixed states).

We also want to study the possibility of not using quantum codes. In this case, we suppose that the initial state $\Phi_{0}$ is changed by error, becoming the state $\Psi_{0}$. State $\Psi_{0}$ is also a random variable. Then our goal is to compare the fidelities of $\Psi$, $\tilde{\Phi}$ and $\Psi_{0}$:

In order to compare the fidelities we will assume that the corrector circuit of $\mathcal{C}$ does not introduce new errors and it does not increase the execution time. In other words, we are going to estimate the theoretical capacity of the code to correct quantum computing errors.

In the case of isotropic errors we shall prove that:

This result leads us to the conclusion that the best option to optimize fidelity against isotropic errors is not to use quantum codes. This result goes in the same direction as that obtained in [16], which indicates that quantum codes do not reduce the variance against isotropic errors.

However, the most widely used model of errors in quantum computing is qubit independent errors. The study of this type of quantum error is much more complex than that of isotropic errors, because it does not have the same symmetry. Despite this technical difficulty, we have proved in [19] that the $5-$qubit code [14, 15] is not able to reduce the variance against qubit independent errors. This result, together with those obtained in [16] and in this article, clearly reveals the difficulty of the quantum error control challenge and clearly shows that the continuous nature of quantum errors cannot be ignored.

There are many works related to the control of quantum computing errors, in addition to those already mentioned above. General studies and surveys on the subject [20, 21, 22, 23, 24, 25, 26, 27], about the quantum computation threshold theorem [28, 29, 30, 31], quantum error correction codes [32, 33, 34, 35], concatenated quantum error correction codes [36, 37] and articles related to topological quantum codes [38, 39]. Lately, quantum computing error control has focused on both coherent errors [40, 41] and cross-talk errors [42, 43]. Finally, we cannot forget the hardest error to control in quantum computing, the quantum decoherence [44]. As we have commented above, these quantum computing errors can be analyzed in the framework of random variables that has been set in [16, 18]. In the conclusions we analyze in more detail the characteristics of the different types of error from the point of view of their control and in view of the result obtained in this paper.

The outline of the article is as follows: in section 2 we study the fidelity of the quantum stages $\Psi$, $\Psi_{0}$ and $\tilde{\Phi}$; in section 3 we prove the relationship between them given by Formula (5); finally, in section 4 we analyze the conclusions that can be obtained from the proved result.

Associated with the $(n,m)-$quantum code $\mathcal{C}$, the following parameters are defined: $d=2^{n}$ is the dimension of $\mathcal{C}$, $d^{\,\prime}=2^{m}$ and $d^{\,\prime\prime}$ is the number of discrete errors that $\mathcal{C}$ corrects.

First we are going to study how we can compare the fidelity of the quantum states $\Psi$ and $\tilde{\Phi}$, which are $n-$qubits encoded with the quantum code $\mathcal{C}$, and the fidelity of the state $\Psi_{0}$, which is an unencoded $m-$qubit state. The working scheme in these two scenarios is illustrated in Figure 1. We assume that the $\mathcal{C}$ correction circuit, which is applied after each quantum gate in the coded algorithm, does not introduce new errors and is ideally applied for a time $t=0$. In this way we study the theoretical capacity of $\mathcal{C}$ to control isotropic errors, that is, its capacity to increase the fidelity of the final state $\tilde{\Phi}$ with respect to $\Psi$, and we can compare it with the fidelity of the final state $\Psi_{0}$ in the scheme without the quantum code $\mathcal{C}$.

We analyze the isotropic error as a decoherence error over a unit of time, which corresponds to the time it takes to apply a quantum gate in the coded algorithm. To compare it with the uncoded algorithm we have to bear in mind that the unit of time in this case will be at most the $n-$th part of the unit of time in the coded algorithm. To relate the probability distributions in both cases we use the following equality of variances:

where $E$ is the decoherence error during a unit of time in the coded algorithm and $E_{1}$, $E_{2}$, …$E_{n}$ are independent decoherence errors corresponding to a unit of time in the uncoded algorithm. Using the following generalization of Formula (3) demonstrated in [18]:

where $v_{u}$ is the variance of each of the independent errors, we obtain the following relation of $v_{u}$ with the variance $v_{c}$ of the error $E$:

In the case of the normal probability distribution defined in [16, 18], with the following density function:

where the parameter $\sigma$ belongs to the interval $[0,1)$, the above variances have a very simple expression and are independent of the dimension: $v_{c}=2(1-\sigma_{c})$ and $v_{u}=2(1-\sigma_{u})$. The relationship between them given in Formula (7) translates into a very simple relationship between the corresponding sigma parameters:

From now on we are going to follow the scheme proposed in [16] to calculate the variances of states $\Psi$ and $\tilde{\Phi}$, but to calculate the fidelities of these states and of state $\Psi_{0}$.

The results obtained in the previous section allow us to easily prove the following theorem.

To compare the fidelities of states $\Psi_{0}$ and $\tilde{\Phi}$ we need to use Theorems 2 and 4. However, we must establish a previous result in order to establish the relationship between these states.

The previous lemma allows us to obtain the main result of this article.

If the isotropic distributions of $\Psi$ and $\Psi_{0}$ are normal the condition given in Formula (18) for Theorems 4 and 6 is not necessary. Indeed, Corollaries 1, 2 and 3 clearly imply that:

On the other hand, the condition given by Formula (18) for Theorems 4 and 6 is a sufficient condition. However, it is not necessary because it has been obtained by underestimating the fidelity of $\Psi_{0}$ and overestimating that of $\tilde{\Phi}$. It is verified for very general isotropic distributions, such as for density functions $f(\theta_{0})$ that satisfy the following:

Figure 2 shows the curves of $F^{2}(\Psi_{0})$, $F^{2}(\tilde{\Phi})$ and $F^{2}(\Psi)$ for normal isotropic distributions and $n=5$ ($d=32$), in the extreme cases $d^{\prime}=16$ ($d^{\prime\prime}=2$) and $d^{\prime}=2$ ($d^{\prime\prime}=16$).

The conclusion of the study carried out in this article, in view of the results summarized in Formula (22), is that the best option to obtain the highest fidelity against isotropic errors is not to use quantum codes. On the other hand, the improvement of the fidelity of $\tilde{\Phi}$ versus that of $\Psi$ seems to be closely related to the dimension of the subspaces to which these states belong: $d^{\prime}$ for $\tilde{\Phi}$ versus $d$ for $\Psi$. See Theorems 1 and 3 and Corollaries 1 and 3.

In this article we have analyzed the ability of quantum codes to increase fidelity of quantum states affected by isotropic decoherence errors. The results obtained, despite being those expected for this type of quantum errors, are not good from the point of view of controlling errors in quantum computing. The ability of quantum codes to reduce errors does not compensate the multiplication of the number of gates that they require. This fact implies that the best option against isotropic errors is not to use quantum codes. This result is similar to that obtained in [16]: quantum codes do not reduce the variance of isotropic errors; and in [19]: the $5-$qubit quantum code do not reduce the variance of qubit independent errors. The last result is more worrying since it negatively affects the standard model of error in quantum computing. For this reason, it would be important to study the behavior of fidelity in this case.

These results indicate that continuous errors must be taken into account, since it is not possible to ensure that the golden rule of error control “correct all small errors exactly” is fulfilled. Therefore, the study of the stochastic model of quantum errors, focused on discrete errors, must be extended to continuous errors.

For future research, we believe that the continuous quantum computing error model should be further developed. The results on the ability of quantum codes to increase the fidelity or to reduce the variance of quantum errors should be extended to other types of error. It is also important to develop models of the behavior of quantum errors in highly entangled quantum systems. We need to know better the behavior of errors in this type of systems so important for quantum computing. Finally, all these approaches should allow a reformulation of fault-tolerant quantum computing for continuous errors.

The values of the integrals that have been used throughout the article are included in this Appendix.

$\displaystyle\int_{0}^{\pi}\sin^{k}(\theta)d\theta=\left\{\begin{array}[]{l}\displaystyle 2\,\frac{(k-1)!!}{k!!}\quad\mbox{}\quad k=1,\,3,\,5,\dots\\ \\ \displaystyle\pi\,\frac{(k-1)!!}{k!!}\quad\mbox{}\quad k=2,\,4,\,6,\dots\end{array}\right.$

$\displaystyle\int_{0}^{\pi}\frac{\sin^{2d-2}(\theta_{0})}{(1+\sigma^{2}-2\sigma\cos(\theta_{0}))^{d}}d\theta_{0}=\frac{(2d-3)!!}{(2d-2)!!}\frac{\pi}{(1-\sigma^{2})}\quad\mbox{}\quad d=1,\,2,\,3,\,\,\dots$

$\displaystyle\int_{0}^{\pi}\frac{\cos(\theta_{0})\sin^{2d-2}(\theta_{0})}{(1+\sigma^{2}-2\sigma\cos(\theta_{0}))^{d}}d\theta_{0}=\frac{(2d-3)!!}{(2d-2)!!}\frac{\sigma}{(1-\sigma^{2})}\pi\quad\mbox{}\quad d=1,\,2,\,3,\,\,\dots$

$\displaystyle\int_{0}^{\pi}\frac{\sin^{2d}(\theta_{0})}{(1+\sigma^{2}-2\sigma\cos(\theta_{0}))^{d}}d\theta_{0}=\frac{(2d-1)!!}{(2d)!!}\pi\quad\mbox{}\quad d=0,\,1,\,2,\,\,\dots$

Starting from the first integral, the surface of a unit sphere of arbitrary even $(2d)$ or odd $(2d-1)$ dimension can be calculated.

$\begin{array}[]{ccl}|{\cal S}_{2d}|&=&\displaystyle\int_{0}^{\pi}\cdots\int_{0}^{\pi}\int_{0}^{2\pi}\sin^{2d-1}(\theta_{0})\,\cdots\,\sin^{1}(\theta_{2d-2})\ d\theta_{0}\,\cdots\,d\theta_{2d-2}d\theta_{2d-1}\\ \\ &=&\displaystyle 2\frac{(2d-2)!!}{(2d-1)!!}\ \frac{(2d-3)!!}{(2d-2)!!}\pi\ 2\frac{(2d-4)!!}{(2d-3)!!}\ \cdots\ \frac{(2-1)!!}{2!!}\pi\ 2\frac{(1-1)!!}{1!!}\ 2\pi\\ \\ &=&\displaystyle\frac{2(2\pi)^{d}}{(2d-1)!!}\\ \end{array}$