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Quantum Stabilizer Codes Embedding Qubits Into Qudits

We study, by means of the stabilizer formalism, a quantum error correcting code which is alternative to the standard block codes since it embeds a qubit into a qudit. The code exploits the non-commutative geometry of discrete phase space to protect the qubit against both amplitude and phase errors. The performance of such code is evaluated on Weyl channels by means of the entanglement fidelity as function of the error probability. A comparison with standard block codes, like five and seven qubit stabilizer codes, shows its superiority.


I. INTRODUCTION
In order to describe realistic quantum information processes, quantum errors induced by environmental noise must be taken into account. This can be accomplished by introducing the notion of quantum channels, that is to say maps on the set of states of the system that are completely positive and trace preserving [1]. At the same time, one would combat quantum errors to avoid their detrimental effect on quantum information processes. To this end, the method of error correcting codes has been borrowed from classical information theory (for a comprehensive introduction to the quantum theory of error correcting codes we refer to [2]). The underlying idea is to exploit redundancy, that is to encode information in linear subspaces (codes) of the total complex Hilbert space in such a way that errors induced by the interaction with the environment can be detected and corrected.
Usually a logical qubit (a two dimensional complex Hilbert space H 2 ) is encoded into n physical qubits (a 2 n dimensional complex Hilbert space H ⊗n 2 ). This kind of encoding is known as block-encoding H 2 |q logical −→ |q physical ∈ H ⊗n 2 . However, there is also the possibility of embedding a logical qubit into a d-dimensional quantum physical system, i.e. a qudit with complex Hilbert space H d = H ⊗n 2 . We refer to this kind of encoding as embedding or qudit-encoding H 2 |q logical −→ |q physical ∈ H d = H ⊗n 2 , where d = 2 n .
For block-coding schemes a powerful formalism, named stabilizer formalism [3], has been developed describing one of the most important classes of quantum codes, namely the quantum version of linear codes in classical coding theory. The stabilizer formalism can be extended over non binary codes [4] and can be also useful for describing embedding (qudit) codes. Actually, the idea of embedding a qubit into a larger space without resorting to block codes was put forward in [5] where a qubit was encoded into a bosonic mode (infinite dimensional complex Hilbert space) just using the stabilizer formalism. Later on the possibility of qudit encoding was pointed out in Ref. [6] by using the same formalism. There, being in H d , the errors were considered as a generalization of Pauli operators, representing shift errors (X-type) or phase errors (Z-type) or their combination (XZ-type). However, in [6] the proposed code was essentially classical since only Z-type errors were taken into consideration.
In this article we upgrade such a coding scheme to be fully quantum, thus able to correct X-type, Z-type and, XZ-type errors. We then test its effectiveness for d = 18 and d = 50 on a Weyl quantum noisy channel [7] (an error model characterized by errors of X, Z and XZ-types). We allow for the possibility of considering X and Z errors occurring with both symmetric and asymmetric probabilities. Finally, we compare the performance of such qudit coding schemes to those of the conventional five-qubit stabilizer code [ [5,1,3]] [8,9] and seven-qubit stabilizer CSS (Calderbank-Shor-Steane) code [ [7,1,3]] [10,11]. We characterize the performances of these codes by means of the entanglement fidelity [12], rather than the averaged input-output fidelity used in [6]. We show that the use of qudit codes may allow to save space resources while achieving the same performance of block codes.
The layout of the article is as follows. In Section II, we briefly describe block-encoding and then introduce qudit encoding. Special focus is devoted to both Pauli groups of n-qubit vectors and generalized Pauli groups of error operators acting on a qudit. In Section III, the Weyl noisy quantum channel is discussed together with the entanglement fidelity. In Section IV, we study the performance of qudit codes. In Section V, for the sake of comparison, we quantify the performance of relevant block codes. Our final remarks appear in Section VI. In this Section, we briefly recall the block-encoding error correction schemes in terms of the stabilizer formalism. Then, using the same formalism we introduce the qudit-encoding (embedding qubits into qudits) scheme. In both cases we restrict our attention to a single encoded qubit.
A. Pauli group of n-qubits and block-encoding A qubit is a two-dimensional quantum system with associated complex Hilbert space H 2 . Let {|0 , |1 } be the canonical basis of this space and consider on it the Pauli operators X = σ z defined in terms of the standard Pauli operators σ x , σ y , σ z realizing the su(2) algebra (throughout the paper i C denotes the imaginary unit of C). They are such that {|0 , |1 } are eigenstates of Z, and Y = XZ. The Pauli operators so defined suffice to describe all possible errors occurring on a single qubit. Together with the identity operator I (with I = I 2×2 ) they form a multiplicative group if we allow them to be multiplied by −1, i.e. {±I, ±X, ±Y, ±Z}. We refer to this group as the Pauli group P H2 . Actually it is a subgroup of the Pauli group realized through the standard sigma Pauli operators and it coincides with the discrete version of the Heisenberg-Weyl group.
For n-qubit errors we can then consider the Pauli group P H ⊗n 2 whose elements result from n-fold direct products (see also [13]) The subscripts on the RHS label the qubits 1,..., n, while j k = 0, 1, 2, 3, label respectively the operators I k , X k , Y k , Z k acting on the k-th qubit. Furthermore λ ∈ {0, 1}.
Since e k (2) = X k Z k , the elements e(λ, j 1 , . . . , j n ) can be rewritten as where a = a 1 ...a n and b = b 1 ...b n are bit strings of length n and, Observe that the order of the group P H ⊗n 2 of n-qubits errors is P H ⊗n 2 = 2 2n+1 . Since the factor ± in front of an error makes no relevant difference on its action, we can actually assume to work with the quotient group P H ⊗n 2 / {±I}, with the major exception being the determination whether elements commute or anti-commute.
The order of the quotient group is P H ⊗n 2 / {±I} = 2 2n . Then, there is a one-to-one correspondence between P H ⊗n 2 / {±I} and the 2n-dimensional binary vector space F 2n 2 whose elements are bit strings of length 2n [14]. A vector v ∈ F 2n 2 is denoted v = (a|b), where a = a 1 ...a n and b = b 1 ...b n are bit strings of length n. Scalars take values in the Galois field F 2 = {0, 1} and vector addition adds components modulo 2. In short, we have the following correspondence e(λ, a, b) ∈ P H ⊗n 2 ↔ v e = (a|b) ∈ F 2n 2 . A quantum stabilizer code C is a vector space C ⊆ H ⊗n 2 stabilized by an Abelian subgroup S ⊆ P H ⊗n 2 , i.e. such that S|ψ = |ψ , ∀|ψ ∈ C.
A quantum stabilizer code C with stabilizer generators g (1) ,..., g (n−1) is a 2-dimensional code space, i.e. a space where to encode a single qubit. The codewords |0 L , |1 L (basis vectors for such code space) can be found as orthogonal eigenvectors (corresponding to the eigenvalue +1) of any of the generators g (j) .
The encoding operation then reads and, H 2 |1 → |1 L ∈ H ⊗n 2 . In view of the correspondence between the Pauli group and the vector space F 2n 2 , let v (j) = a (j) |b (j) be the image of the generators g (j) in F 2n 2 and let introduce the so called parity check matrix (2) . . .
Then, for an error e ∈ P H ⊗n 2 ↔ v e = (a|b) ∈ F 2n 2 , the error syndrome S(e) is given by the bit string [13] S(e) = Hv e = l 1 ...l n−1 , where l j = H T (j) · v e . Errors with non-vanishing error syndrome are detectable. They correspond to operators not in S and not commuting with those in S. That is, a set of error operators E ⊆ P H ⊗n 2 is detectable if E / ∈ Z(S) − S, with Z(S) the centralizer of the subgroup S. Furthermore, the set of error operators E ⊆ P H ⊗n 2 is correctable if the set given by E † E is in turn detectable [15], i.e. E † E / ∈ Z(S) − S. It would be awfully tedious to identify either detectable errors or sets of correctable errors. However, the quantum stabilizer formalism allows to simplify such task [3]. This is a consequence of the fact that by means of such formalism it is sufficient to study the effect of the error operators on the generators of the stabilizer and not on the codewords themselves. Actually the syndrome extraction corresponds to measure the stabilizer generators.
Finally, we denote by [[n, k, d C ]] a quantum stabilizer code C with code parameters n (the length), k (the dimension) and d C (the distance) encoding k-logical qubits into n-physical qubits and correcting d C −1 2 -qubit errors ( x denotes the largest integer less than x).

B. Generalized Pauli group and qudit-encoding
A qudit is a d-dimensional quantum system with associated complex Hilbert space H d . On this space we can introduce a generalized version of the Pauli operators X and Z considered in the previous Subsection. They can be defined through their action on the canonical basis {|k } k∈Z d of H d [16], where "⊕" denotes addition of integers modulo d and ω is a primitive dth root of unity (ω d = 1). The X and Z operators so defined are unitary (X † = X −1 and Z † = Z −1 ), but not Hermitian, and satisfy X d = Z d = I (with I = I d×d ) together with the commutation relations It is then possible to consider the Pauli group P H d consisting of all operators e of the form e (l, n, m) = ω l X n Z m , where l, n, m ∈ Z d . Similarly to the previous Subsection, for errors on a qudit we may refer to the quotient group In addition to the reasons stated in the introductory Section about passing from block to embedding codes, another motivation is to understand whether or not finite dimensional versions of the shift-resistant quantum codes of Ref. [5] are effective. Hence, following Ref. [5], we consider d = 2r 1 r 2 , and introduce a code C stabilized by the Abelian subgroup S ⊂ P H d generated by X 2r1 , Z 2r2 . The codewords (basis for C) are eigenstates of Z 2r2 and X 2r1 with eigenvalue one. Hence they only contain |k s with values of k that are multiples of r 1 and that are invariant under a shift by 2r 1 . They read The encoding operation becomes in this case If the states (5), (6) undergo an amplitude shift, the value of k modulo r 1 is determined by measuring the stabilizer generator Z 2r2 , and the shift can be corrected by adjusting k to the nearest multiple of r 1 .
The codewords in the basis of X's eigenstates can be found by observing that the eigenstates of X and Z operators are connected by the Fourier transform where X |i = ω i |i . Then, it turns out that The codewords (7), (8), have the same form of (5), (6), but with r 1 and r 2 interchanged. Hence, if they undergo a phase shift, the value of k modulo r 2 is determined by measuring the stabilizer generator X 2r1 , and the shift can be corrected by adjusting k to the nearest multiple of r 2 .
To understand what is the set of correctable errors according to the condition E † E / ∈ Z(S) − S we have with the help of (4) The phases on the right hand sides are non-trivial only if Therefore, the code C can correct all shifts corresponding to these conditions. They amount to r 1 r 2 = d/2 and this is also the number of possible error syndromes. Finally, notice that other families of qudit codes could be constructed by generalizing the Pauli operators in a different way, e.g. by making them Hermitian.

III. THE WEYL QUANTUM CHANNEL AS ERROR MODEL
In this Section, we first discuss the Weyl quantum channel for qudit states and for qubit states, and then the entanglement fidelity as quantifier of codes performances.

A. General form of the Weyl quantum channel
Consider a completely positive trace preserving map (CPT map or quantum channel) Λ (d) , Weyl channel is a bistochastic quantum channel of the following form [7], where ρ ∈ S (H d ) and π(n, m) is an arbitrary probability distribution, thus respecting 0 ≤ π(n, m) ≤ 1, where "⊕", as specified earlier, denotes the addition of integers modulo d. They also satisfy the (Weyl) commutation relations (4), i.e.
where 0 ≤ n, n , m, m ≤ d − 1. Notice that U n, m may be rewritten as U n, m = U n, 0 U 0, m . Furthermore, U n,0 ≡ X n and U 0,m ≡ Z m . Therefore in what follows we will consider the d-dimensional Weyl channel acting as For d = 2 we have the most general channel acting on a qubit For instance, if we take π(0, 0) Weyl becomes the standard symmetric qubit depolarizing channel.
To justify the choice of Weyl's error model, we point out that the quantum codes employed here are designed to error-correct arbitrary quantum errors such as X-errors, Z-errors and combinations of the two (Y -errors). Since the Kraus decomposition of the Weyl channel is defined in terms of powers of these aforementioned error operators, it certainly constitutes a natural test-bed where quantifying the performance of the selected qudit codes. Such a test-bed turns also out to be very general, with the possibility of encompassing physically relevant scenarios.
Hereafter, when we consider the two-dimensional Weyl channel, we mean to take into consideration the following channel parametrization with, In this way we have Then, we can also consider the possibility of having asymmetric X and Z errors' probabilities [17][18][19][20][21][22]. In such a case the probabilities π (n, m) are defined as follows, with π X and π Z not identical. In particular, we shall consider with π Z (1) def = p and κ ∈ [0, 1/p] to guarantee that κp is a valid probability value. Notice that for κ = 1 we recover the symmetric case.
Finally, when considering block codes on n qubits, the error map simply becomes Λ (2)⊗n Weyl acting on ρ ∈ S(H ⊗n 2 ).

B. Entanglement Fidelity
Entanglement fidelity is a reliable performance measure of the efficiency of quantum error correcting codes [23]. Suppose a two dimensional code C is such that C ⊂ H with dim C H = N (here H can be either H ⊗n 2 or H d , hence N is either 2 n or d). Then consider errors taking place through a CPT map Λ : S (H) → S (H) (which can be either Λ (2)⊗n or Λ (d) ) written, in terms of the Kraus decomposition, as To recover the errors by means of the code C a recovery operation must be applied according to the syndrome extraction. Suppose it is described by a CPT map R : If the code is effective, we expect that the map resulting from the composition of Λ and R restricted to the subspace C, namely [R • Λ] | C , will be close to the identity map id C on C. In order to evaluate this closeness we can consider a state ρ = tr C |ψ ψ| written in terms of a purification |ψ ∈ C ⊗ C and see how well entanglement (between C and the reference system identical to C) is preserved by means of This is the entanglement fidelity [12] for the map In terms of the Kraus error operators, F can be rewritten as [24] F ρ, Finally, choosing a purification described by a maximally entangled unit vector |ψ ∈ C ⊗ C for the mixed state ρ = 1 dim C C I C , we obtain This is the expression we will use in the following.

IV. QUDIT CODES FOR WEYL ERRORS
In this Section, we analyze in details how the qudit codes devised in Sec. II work on the Weyl channel for d = 18 and d = 50 and determine their performance by means of the entanglement fidelity.
Our main motivation to use the qudit codes with d = 18 and d = 50 is that they represent the lowest-dimensional perfect qudit systems where a two-dimensional quantum systems (a qubit) can be encoded and protected against arbitrary shift errors of the form X n Z m by one and two units, respectively. Furthermore, the physical dimensionality of these codes is chosen so to be as much as possible comparable with the physical dimensionality of code-spaces characterizing well-known standard stabilizer error correction schemes capable of correcting arbitrary single-qubit errors and, possibly, few two-qubits errors. For this reason, the five [8,9] and seven-qubit [10,11] quantum stabilizer codes seem to be a convenient choice. In particular, recalling that a stabilizer code is perfect if all the eigenvalues of the generators constitute valid syndromes for correcting an error, it turns out that both the five and the qudit code with d = 18 are perfect and require minimal quantum resources for their task.
A. The d = 18 qudit code Encoding. The encoding operation is characterized by, where the codewords |0 L and |1 L are defined according to (5), (6) as, respectively. The stabilizer group S of this code is generated by the two error operators X 6 and Z 6 . Here r 1 = r 2 = 3.
A simple calculation shows that provided we restrict our focus to the two-dimensional code space C (d=18) ⊂ H 18 , the following identities hold, and, Therefore the total number of error operators to be considered amounts to be 36 rather than 18 2 . Then, the Weyl channel (11) may be rewritten as, where we have relabeled the error operators as follows, In the symmetric case, the probabilities π (n, m) are defined like in Eq. (13), where now Correctability. According to (9) with r 1 = r 2 = 3 the set of correctable errors A (d=18) correctable is given by the following 9 errors, where, Recovery Operators. Following the recipe provided in [25], it turns out that the two 9-dimensional orthogonal subspaces V 0 L and V 1 L of H 18 generated by the action of A (d=18) correctable on |0 L and |1 L are given by, with k ∈ I (d=18) def = {0, 1, 2, 6, 7, 8, 12, 13, 14}. The coefficients √ π k denote the errors amplitudes where, for instance, √ π 8 def = π (1, − 1). Notice that v i L k |v j L k = δ kk δ ij with k, k ∈ I (d=18) and i, j ∈ {0, 1}. Therefore, it follows that V 0 L ⊕ V 1 L = H 18 . The recovery superoperator R ↔ {R k } with k ∈ I (d=18) is defined by means of [25], In the case under investigation, the entanglement fidelity (18) reads, After a simple calculation, F (d=18) (p) becomes, We stress that this error correction scheme is effective as long as the failure probability 1 − F (d=18) (p) is strictly smaller than the error probability p. This implies that the d = 18 -dimensional qudit code is effective only when 0 ≤ p 0.24. Furthermore, we point out that in its range of effectiveness, F (d=18) (p) in (20) is a monotonic decreasing function of p.

Asymmetric errors
By repeating the steps of the previous Subsection using Eqs. (16) and (17) we get Eqs. (20) and (21) become, to the leading order in p with p 1, It results that the presence of asymmetric Weyl errors with κ < 1 increases the performance of the correction scheme. This can be understood by noticing that as soon as κ → 0 the noise model becomes classical-like and the errors become of a single type, namely Z type, hence more easy to correct (this limiting case is similar to that investigated in Ref. [6]). On the contrary, for κ > 1 the performance of the code is lowered by errors asymmetries.

B. The d = 50 qudit code
The encoding operation in this case is characterized by, where the codewords |0 L and |1 L are defined according to (5), (6) as, respectively. The stabilizer group S of this code is generated by the two error operators X 10 and Z 10 . Here r 1 = r 2 = 5. Indeed, it can be shown that provided we restrict our focus to the two-dimensional code space C (d=50) ⊂ H 50 , the set of all (50 2 ) non-normalized errors can be reduced to 100 operators. Following the same line of reasoning of the previous Subsection, it is possible to find the recovery superoperator. After some algebraic calculations, we arrive at the following expression for the entanglement fidelity We emphasize that this error correction scheme is effective as long as the failure probability 1 − F (d=50) (p) is strictly smaller than the error probability p. This implies that the d = 50 -dimensional qudit code is effective only when 0 ≤ p 0.43. This p-range of effectiveness is larger than that of the d = 18-dimensional qudit code. Furthermore, comparing the p-expansions of (20) and (22) to the leading orders for p 1, it follows that From the above equation, it follows that the d = 50-dimensional qudit code outperforms the d = 18-dimensional qudit code in the p-range where both error correction schemes are effective as expected. Moreover, F (d=50) (p) in (22) is a monotonic decreasing function of the error probability parameter belonging in its range of effectiveness.
By taking probabilities as in Eqs. (16) and (17) we obtain for the d = 50 qudit code Eqs. (22) and (23) become, to the leading order in p with p 1, Also in this case, for κ < 1 the presence of asymmetric Weyl errors increases the performance of the correction scheme, while for κ > 1 the performance of the code is lowered. Furthermore, comparing (21) and (23) we get, to the leading order in p with p 1, That is, the d = 50 qudit code outperforms the d = 18 one for any κ ∈ [0, 1/p].

A. The Five-Qubit Stabilizer Code
Encoding. The [ [5,1,3]] code is the smallest single-error correcting quantum code. Of all quantum codes that encode 1 qubit and correct all single-qubit errors, the [ [5,1,3]] is the most efficient, saturating the quantum Hamming bound. It encodes 1 qubit in 5 qubits. The cardinality of its stabilizer group S is |S| = 2 5−1 = 16 and the 5 − 1 = 4 group generators are given by [13], The distance of the code is d C = 3 and therefore the weight of the smallest error A † l A k that cannot be detected by the code is 3. Finally, we recall that it is a non-degenerate code since the smallest weight for elements of S (other than identity) is 4 and therefore it is greater than the distance d = 3. The encoding operation for the [[5, 1, 3]] code is characterized by, 2 and, H 2 |1 → |1 L ∈ H ⊗5 2 , where the codewords |0 L and |1 L are defined as [13], and, respectively. Then, the action of the channel Λ (2)⊗5 on ρ ∈ S H ⊗5 2 can be written as, where, according to the notation of Section II, A k ∝ X(a 1 a 2 a 3 a 4 a 5 )Z (b 1 b 2 b 3 b 4 b 5 ).
Correctability. The sixteen weight zero and one quantum error operators in (24) are given by, where the coefficientsp l with l = 0,..., 15 can be easily deduced from the above distribution of errors and Eq.(13), p 0 = (1 − p) 10 ,p 1 = p(1 − p) 9 , . . . ,p 6 = p 2 (1 − p) 8 , . . . ,p 15 = p(1 − p) 9 . It is straightforward, though tedious, to check that, for the above given errors of weight zero and one we have: where S A † l A k is the error syndrome of the error operator A † l A k defined according to (2) as, The quantity H [ [5,1,3]] is the parity check matrix for the five-qubit code while v A † l A k is the vector in the 10-dimensional binary vector space F 10 2 corresponding to the error operator A † l A k . Hence, the set of correctable error operators is given by, where the cardinality of A defining the channel in (24) equals 2 10 .
Recovery Operators. All weight zero and one error operators satisfy the error correction conditions [15], where P C is the orthogonal projector operator (P C = P 2 C and, P C = P † C ) on the code space C ([ [5,1,3]]) defined as, The two sixteen-dimensional orthogonal subspaces V 0 L and V 1 L of H ⊗5 2 generated by the action of A correctable on |0 L and |1 L are given by, with k = 0, 1,..., 15. Notice that v i L l |v j L l = δ ll δ ij with l, l ∈ {0, 1,..., 15} and i, j ∈ {0, 1}. Therefore, it follows The recovery superoperator R ↔ {R l } with l = 1 ,...,16 is defined by means of [25], Finally, the composition of this recovery operation R with the map Λ (2)⊗5 (ρ) in (24) yields, Entanglement Fidelity. Here we want to describe the action of R•Λ (2)⊗5 in (25) restricted to the code subspace C [ [5,1,3]] . Note that the recovery operators can be expressed as, with l ∈ {0,..., 15}. Recalling that in this case A l = A † l , it turns out that, We now need to compute the 2 × 2 matrix representation [R l A k ] |C of each R l A k with l = 0,..., 15 and k = 0,..., 2 10 − 1 where, For l, k = 0,.., 15, we note that [R l+1 A k ] |C becomes, while for any pair (l, k) with l, = 0,..., 15 and k > 15, it follows that, We conclude that the only matrices [R l A k ] |C with non-vanishing trace are given by, with s = 1,.., 16. Therefore, the entanglement fidelity (18) can be written as and results, We remark that this error correction scheme is effective as long as the failure probability 1 − F [[5,1,3]] (p) is strictly smaller than the error probability p. This implies that the five-qubit code is effective only when 0 ≤ p 2.9 × 10 −2 .
Finally, we emphasize that this block-encoding scheme is less efficient than the previously-mentioned qudit-encoding schemes as it appears in Figure 1.

Asymmetric errors
By taking probabilities as in Eqs. (16) and (17) we obtain for the five-qubit code We stress that unlike the finding uncovered in [17], asymmetries in the considered Weyl noisy channel do affect the performance of the five-qubit code quantified in terms of the entanglement fidelity. This difference is ultimately a consequence of the fact that while in [17] it is assumed error probabilities p X , p Y and p Z all linear in the error probability p (although weighted with different coefficients α X , α Y and α Z with α X + α Y + α Z = 1), here we assumed p Y quadratic in p while keeping both p X and p Z linear in p. From Eqs. (26) and (27) we obtain, to the leading order in p with p 1, Hence, also in this case, the presence of asymmetric errors increases the performance of the correction scheme for κ < 1, while for κ > 1 the performance of the code is lowered.

B. The Seven-Qubit Stabilizer Code
The [ [7,1,3]]-CSS code encodes 1 qubit in 7 qubits. The cardinality of its stabilizer group S is |S| = 2 7−1 = 64 and the set of 7 − 1 = 6 group generators is given by [13], The distance of the code is d C = 3 and therefore the weight of the smallest error A † l A k that cannot be detected by the code is 3. Finally, we recall that it is a non-degenerate code since the smallest weight for elements of S (other than identity) is 4 and therefore it is greater than the distance d C = 3. The encoding for the [[7, 1, 3]] code is given by [13], and, Following Observe that the seven-qubit code is effective for 0 ≤ p 2.6 × 10 −2 . Comparing the p-expansions of (26) and (28) to the leading orders for p 1, it follows that and, in addition, the p-range of applicability of the five-qubit code is larger than that of the seven-qubit code. Thus, for the symmetric Weyl channel considered, the five-qubit code outperforms the seven-qubit code. However, we shall see that this ordering does not hold when considering asymmetric scenarios.

Asymmetric errors
In the asymmetric scenario of Eqs. (16) and (17) From Eqs. (28) and (29) it follows, to the leading order in p with p 1, Once again, for κ < 1 it results that the presence of asymmetric errors increases the performance of the correction scheme, while for κ > 1 the performance of the code is lowered. Furthermore, by comparing (27) and (29) it follows, to the leading order in p with p 1, Thus, with respect to the noise model discussed in Ref. [17], we conclude that here the comparison between five-qubit code and seven-qubit code is slightly more involved.

VI. FINAL REMARKS
In this article, we discussed how to protect a qubit embedded into a qudit from both amplitude and phase errors occurring in the discrete phase space. A code has been devised using stabilizer formalism and its performances compared with those of common block codes for a general Weyl noisy quantum channel allowing symmetric and asymmetric error probabilities.
Specifically we have considered the d = 18 and d = 50 qudit stabilizer codes together with five and the CSS (Calderbank-Steane-Shor) seven-qubit quantum stabilizer codes. The performances of these codes were quantified by means of the entanglement fidelity as function of the error probability.
We uncovered that qudit codes have an enormously wider (by approximately an order of magnitude) range of applicability in the error probability. Furthermore, already the d = 18 qudit code outperforms the five and sevenqubit block codes for symmetric errors (see Fig.1). Our theoretical analysis leads to the conclusion that the qudit codes with d = 18 and d = 50 outperform the common five and CSS seven-qubit stabilizer codes. This in principle allows one to save space resources (since d = 18 < dim C H ⊗5 2 = 32 << dim C H ⊗7 2 = 128), however one should also account for the difficulties in implementing qudit systems, an issue that seems to be nontrivial and not quite settled yet. For an overview of the experiments performed for producing quantum optical qudits, we refer to [26].
The performance of qudit code are also robust against asymmetries in errors' probabilities. In fact, restricting our analysis to κ > 1.1, it results that the d = 18 qudit code outperforms the seven-qubit code until strong asymmetries come into play, as can be seen by comparing (21)  Comparative results for the various codes performances in presence of asymmetries are graphically represented in Figure 2. The different uncovered behaviors in the four error correcting schemes employed in this article can be ascribed to the fact that the errors in P H ⊗5 2 (or P H ⊗7 2 ) are fundamentally different from those in P H18 and P H50 . Finally, it could be interesting to consider the presence of correlations between X and Z errors in the qudit code. These can be introduced as follow π (n, m) def = (1 − µ) π (n) π (m) + µδ n, m π (m) , where µ with 0 ≤ µ ≤ 1 represent the degree of correlation. Following the very same line of reasoning provided in Section IV, it can be shown that for instance the entanglement fidelity becomes, ≈ 2p 2 ≥ 0, that is memory effects lead to better performances. The reason is that in the limit of very strong correlations µ → 1, only one type of error (namely Y = XZ) takes place. As such, this case shows similarities with the case of asymmetric errors with κ → 0.
In conclusion, we are strongly motivated by our investigation to believe that encoding a qubit into a qudit can be a useful approach in quantum coding. We are aware of the difficulties in realizing and controlling qudit systems even of low dimensionality, however we have witnessed a lot of progress along this direction recently. Quantum optical qudits can be generated by means of experimental schemes based upon interferometric set-ups, orbital angular momentum entanglement and, biphoton polarization [26]. For instance, in the interferometric scheme employed in [27], high symmetry and maximally entangled qutrits are realized with a fidelity up to 0.985 as the superposition state of the three possible paths of a single photon in a three-arms interferometer. Therefore, the realization of the discussed qudit codes seems not futureless.