1 Introduction↩︎

Contextuality was identified as a necessary and possibly-sufficient condition for universal quantum computing in [1]. The argument of [1] is based on magic state distillation [2] for qudits of odd-prime dimension, and later extended to qudits of arbitrary odd dimensions in [3], and continuous variable systems in [4].¹ The authors of these works showed that qudit states that do not exhibit contextuality with respect to stabilizer measurements have a non-negative discrete Wigner function [5], [6]. The set of such states is known as the Wigner polytope. Because Clifford unitaries and stabilizer measurements are efficiently simulable for states in the Wigner polytope [7], [8], they thus cannot be distilled into pure magic states.

In order to demonstrate that contextuality is not only necessary but also sufficient for qutrit quantum computation, one must demonstrate that a supply of qudits that do exhibit contextuality with respect to stabilizer measurements may be used to achieve universal quantum computation. In the language of magic state distillation, this translates into the question, can any qudit mixed state outside the Wigner polytope be distilled into a pure magic state?

For qudits of odd prime dimension \(p\), the Wigner polytope is a convex polytope with \(p^2\) facets that lives in the \(p^2-1\) dimensional space of qudit density matrices. In [9], it was shown that no finite magic state distillation routine can distill all states that lie outside one of the faces of the Wigner polytope², generalizing the analogous result for qubit states that lie outside the stabilizer polytope [11]. However, the possibility remains that a sequence of magic state distillation routines, based on stabilizer codes of increasing length \(n\) may distill states arbitrarily close to a face of the Wigner polytope. Is there any evidence that such a sequence of magic state distillation routines exist?

The problem simplifies if one focuses on qutrits. There exists a qutrit magic state, first identified by Howard and van Dam [12], sometimes known in the literature in the qutrit strange state \(\mathinner{|{S}\rangle}\) [8], that lies directly above one of the facets of the Wigner polytope³. As discussed in [14], noisy \(\mathinner{|{S}\rangle}\) states can be twirled via Clifford unitaries to lie on a line connecting a pure \(\mathinner{|{S}\rangle}\) state to the maximally mixed state: \[\hat{\rho}(\epsilon)=(1-\epsilon) \mathinner{|{S}\rangle}\mathinner{\langle{S}|} + \epsilon \frac{1}{3},\] with all noise converted to depolarizing noise, parameterized by \(\epsilon\). Any state \(\hat{\rho}(\epsilon)\) for \(\epsilon <3/4\) lies outside the Wigner polytope and exhibits contextuality with respect to stabilizer measurements. We then ask whether or not there exists a family of \(n\)-to-\(1\) magic state distillation routines that distill the strange state with a threshold approaching \(\epsilon = 3/4\) as \(n \to \infty\)? ⁴

At the time [1] was published, and for several years thereafter, while some qutrit and qudit magic state distillation routines had been proposed [10], [15]–[18], no magic state distillation routine that distilled the Howard van Dam strange state was known. It was later discovered that an 11-qutrit CSS code based on the ternary Golay code can distill the \(\mathinner{|{S}\rangle}\) state, with a threshold of \(\epsilon_* = 0.38\) [19]. Do there exist any other qutrit stabilizer codes that distill the strange state? If so, how do their thresholds compare to that of the 11-qutrit Golay code?⁵ In this paper, we carry out a computational search over reasonably small qutrit error-correcting codes to help answer these questions.

To facilitate our search, we prove a simple theorem connecting the performance of a stabilizer code for qudit magic state distillation its complete weight enumerator. For distillation routines for the qutrit strange state, this formula simplifies and depends only on the stabilizer code’s simple weight enumerator.

Our search is limited by practical considerations, as well as the existing classifications of qutrit error-correcting codes in the literature – namely, the classification of qutrit stabilizer states in [21], [22] and a classification of self-orthogonal classical ternary codes available on [23]. We carried out a search over all \([[n,1]]_3\) stabilizer codes with \(n \leq 9\), and a search over all \([[11,1]]_3\) stabilizer codes that can be obtained from a \([[12,0,6]]_3\) stabilizer state. For such codes, we demand transversality of a particular single-qutrit gate (the square of the qutrit Hadamard gate), which allows us to restrict our search to projection onto the trivial syndrome of each stabilizer code. We also searched over all \([[n,1]]_3\) CSS codes for odd \(n \leq 23\) that possess a complete set of transversal single-qudit Clifford gates.

We found that none of the codes we searched with \(n<23\) could distill the \(\mathinner{|{S}\rangle}\) state with better-than-linear⁶ noise suppression, other that the 11-qutrit Golay code of [19]. However, for \(n=23\), we found over \(600\) CSS codes that could distill the strange state with cubic noise suppression – which is approximately \(1/3\) of all the codes we could construct from the ternary self-orthogonal codes listed in [23] – suggesting that for large codes, magic state distillation is somewhat generic. None of these \(23\)-qutrit codes, however, had a threshold that exceeds that of the 11-qutrit Golay code.

Let us point out that the new distillation routines we find here are mainly of theoretical interest. The success probabilities are quite low, and far better yields are obtained via triorthogonal codes [24]–[27] (see [16], [28]–[30] for constructions of qutrit and qudit triorthogonal codes). Nevertheless, the CSS codes we study have a complete set of transversal Clifford gates, and may turn out to be useful for fault-tolerant quantum computation in other settings.

Our paper is organized as follows. In section 2 we briefly review some background material. In section 3 we derive a relation between weight-enumerators and the performance of magic state distillation routines. In section 4 we describe our search space and the results. In section 5 we conclude with some brief discussion.In addition, in appendix 6 we present a useful lemma, and in appendix 7 we study a two qutrit codes, conjectured in [20] to distill the \(\mathinner{|{S}\rangle}\) state, that we find do not distill the \(\mathinner{|{S}\rangle}\) state.

2 Preliminaries↩︎

Here we review some well-done definitions and results concerning stabilizer codes for qudits of dimension \(p\) where \(p\) is any odd prime.

2.1 The Heisenberg-Weyl displacement group↩︎

The qudit version of the Pauli group is generated by \[\hat{X} = \sum_k \mathinner{|{k+1}\rangle}\mathinner{\langle{k}|}, \quad \hat{Z} = \sum_k \omega^k \mathinner{|{k}\rangle}\mathinner{\langle{k}|},\] and multiplication by \(\omega= e^{2\pi i/p}\), and is known as the Heisenberg-Weyl displacement group. The conventions for Heisenberg Weyl displacement operators we will follow are those in [1], \[\hat{D}(u,v) = \omega^{-2^{-1} u v}\hat{X}^u \hat{Z}^v. \label{Heisenberg-Weyl}\tag{1}\] \(n\)-qudit Heisenberg Weyl-displacement operators are denoted as \[\hat{D}(\vec{u},\vec{v}) = \hat{D}(u_1,v_1) \otimes \hat{D}(u_2,v_2) \otimes \ldots \otimes \hat{D}(u_n,v_n),\] where \(\vec{u}=(u_1, u_2, \ldots, u_n)\) and \(\vec{v}=(v_1, v_2, \ldots, v_n)\).

It is convenient to combine \(\vec{u}\) and \(\vec{v}\) into a symplectic vector \(\chi=(\vec{u},\vec{v})\), and write \(\hat{D}(\chi) =\hat{D}(\vec{u},\vec{v})\). Then \[\hat{D}({\chi})\hat{D}(\chi') = \omega^{2^{-1} [\chi, \chi'] } D_{\chi+\chi'},\] where \[[\chi, \chi']= \vec{u} \cdot \vec{v}'- \vec{u}' \cdot \vec{v}\] is the symplectic inner product (and \(\cdot\) is the usual dot product). For commuting Heisenberg-Weyl operators \([\chi,\chi']\) vanish.

2.2 Qudit stabilizer codes↩︎

An \([[n,k]]_p\) stabilizer code is defined by the simultaneous eigenspace of \(n-k\) independent commuting \(n\)-qudit Heisenberg-Weyl displacement operators, and has dimension \(p^k\). We will assume that each of these operators are of the form given in 1 , with no additional overall phase. As such, to completely specify the stabilizer code, we also have to specify the eigenvalues of each stabilizer, which are of the form \(\omega^a\), where \(a\) can be taken to be an element of \(\mathbb{Z}_p\). In this paper we will restrict our attention to codes defined by the \(1=\omega^0\) eigenspace of each operator, for reasons we explain below. The set of commuting operators \(\{ \hat{D}(\vec{u}_1, \vec{v}_1), \ldots, \hat{D}(\vec{u}_{n-k}, \vec{v}_{n-k}) \}\) can be specified by a symplectic matrix \[H = \left( \begin{array}{c|c} \vec{u}_{1} & \vec{v}_{1} \\ \vec{u}_{2} & \vec{v}_{2} \\ \vdots & \vdots \\ \vec{u}_{n-k} & \vec{v}_{n-k} \end{array}\right),\] such that the symplectic inner product of any two rows vanishes. These operators generate an abelian group of order \(p^{n-k}\), which we refer to as the stabilizer group \(\mathcal{S}\). Because \(\mathcal{S}\) is abelian, there are no overall phases arising from multiplication and \(\mathcal{S}\) consists of all Heisenberg-Weyl Displacement operators of the form \(\hat{D}(\chi)\) where \(\chi\) is in the row-span of \(H\).

A stabilizer code can equivalently be thought of as an additive classical code over \(GF(p^2)\), that is self-orthogonal under the Hermitian inner product [31], [32].

We use the notation \(N(\mathcal{S})\) to denote all \(n\)-qudit Heisenberg-Weyl displacement operators that commute with \(\mathcal{S}\). \(\mathcal{S} \subseteq N(\mathcal{S})\). In general, \(N(\mathcal{S})\) will not form an abelian group, and multiplication of two different elements may give rise to Heisenberg-Weyl operators with overall phases. It is nevertheless useful to define \(N(\mathcal{S})^*\) as \(N(\mathcal{S})\) modulo overall phases (i.e., identify Heisenberg-Weyl operators that differ by an overall phase), and specify \(N(\mathcal{S})^*\) as the row-span of a symplectic matrix. In this way \(N(\mathcal{S})^*\) can be thought of as a classical additive code in \(GF(p^2)\), denoted as \(\mathcal{S}^\perp\), that is the set of all codewords orthogonal to \(\mathcal{S}\) under the Hermitian inner product. We use the notation \(\mathcal{S}^\perp\) and \(N(\mathcal{S})^*\) interchangeably.

Clearly, \(\mathcal{S} \subset \mathcal{S}^\perp\). Any \(L \in \mathcal{S}^\perp\) defines a coset of \(\mathcal{S}\) in \(\mathcal{S}^\perp\) which is of the form⁷ \(\mathcal{S} + L = \{ L + M | M \in \mathcal{S}\}\). All stabilizers in the same coset will commute with each other.

2.3 Weight enumerators↩︎

Let us briefly review the definition of the weight enumerator of a stabilizer code.

Let \(\mathcal{S}\) denote the stabilizer group of an \([[n,k,d]]_3\) stabilizer code. The projector onto the codespace of \(\mathcal{S}\) with trivial syndrome can be written explicitly as, \[\hat{\Pi}^0_{\mathcal{S}} = \frac{1}{p^{n-k}} \sum_{M \in \mathcal{S} } M.\]

Let \(\tilde{ \mathcal{S}}\) be a either a stabilizer code \(\mathcal{S}\), or one of its cosets in \(\mathcal{S}^\perp\). The complete weight enumerator of \(\tilde{\mathcal{S}}\) is a function of \(p^2\) formal variables \(\{y_{ij}\}\), for \(i, j = 0, \ldots, p-1\), defined as follows: \[w(\tilde{\mathcal{S}}; \omega_{ij} ) = \sum_{(\vec{x}|\vec{z}) \in \tilde{ \mathcal{S}}} \prod_{i=1}^n y_{x_i,x_j} \label{complete-weight-enumerator}\tag{2}\]

The simple weight enumerator of \(\tilde{ \mathcal{S}}\) to be a function of two formal variables \(x\) and \(y\), defined as follows: \[w(\tilde{ \mathcal{S}}; x, y) = w(\tilde{ \mathcal{S}}; y_{ij}(x, y) ). \label{simple-weight-enumerator}\tag{3}\] where \[y_{ij}(x,y) = \begin{cases} x & ({i,j})=(0,0) \\ y & ({i,j}) \neq (0,0). \end{cases}\] If we further set \(x=y=1\), then the simple weight enumerator becomes equal to \(|\tilde{ \mathcal{S}}|\). These definitions coincide with the standard definitions of the simple and complete weight enumerators of a classical error-correcting code over the finite field \(GF(p^2)\).

A MacWilliams identity relates \(w(\mathcal{S};1,z)\) to \(w(\mathcal{S}^\perp; 1,z)\) [32], [33]. This is: \[w(S^\perp;1,z) = \frac{(1+(p^2-1)z)^n}{p^{n-k}} w\left(S;1,\frac{1-z}{1+(p^2-1)z} \right).\] It is conventional to define \(A_{\mathcal{S}}(z)=w(\mathcal{S};1,z)\) and \(B_{\mathcal{S}}(z)=w(\mathcal{S}^\perp; 1,z)\). \(A_{\mathcal{S}}(z)-B_{\mathcal{S}}(z)\) is a polynomial in \(z\) with non-negative coefficients and is the simple weight enumerator of the set of all logical operators for the stabilizer code \(\mathcal{S}\). The lowest power of \(z\) that appears in \(A(z)-B(z)\) is \(z^d\), where \(d\) is the distance of the code.

2.4 Discrete Wigner functions↩︎

A discrete phase space formalism for qudits was formulated in [5], [6], [34], that we briefly review here.

Singe-qudit phase point operators are defined as, \[\hat{A}(0,0) = \frac{1}{p} \sum_{x,z} \hat{D}(u,v), \quad \hat{A}(u,v) = \hat{D}(u,v)\hat{A}(0,0) \hat{D}(u,v)^\dagger.\] These are normalized so that \({\rm{tr~}}\hat{A}(u,v)=1\). Multi-qudit phase point operators are defined as \[\hat{A}(\vec{u},\vec{v})=\bigotimes_i \hat{A}(u_i,v_i),\] and we will use the notation \(\hat{A}(0,0)^{\otimes n} = \hat{A}(0,0) \otimes \hat{A}(0,0) \otimes \ldots \otimes \hat{A}(0,0)\).

Any qudit density matrix \(\hat{\rho}\) can be written as a linear combination of the phase point operators with real, but possibly negative, coefficients. These coefficients define the discrete Wigner function of the qudit state: \[\hat{\rho} = \sum_{u,v} W( \hat{\rho}; u,v)\hat{A}(u,v), \quad W( \hat{\rho}; u, v) = \frac{1}{p} {\rm{tr~}}\hat{\rho} \hat{A}(u,v)\] Note that \({\rm{tr~}}\hat{\rho} = \sum_{u,v} W( \hat{\rho}; u,v) =1\), and the discrete Wigner function defines a quasi-probability distribution.

The Clifford group acts covariantly on the discrete Wigner function, for qudits of odd prime dimension. Heisenberg-Weyl displacement operators act as translations in discrete phase, and a general Clifford unitary can be represented as a symplectic rotation followed by a translation. See [5], [7], [8], [35], and references therein, for more details. This allows one to use the discrete Wigner function to define an efficient classical simulation for Clifford unitaries and stabilizer measurements acting on qudit states with non-negative Wigner functions [7], and develop resource theories of magic for qudits of odd prime dimension – see [8], [36], [37].

3 Distillation and weight enumerators↩︎

In the magic state model of fault-tolerant quantum computing [2], [38], one begins with a quantum computer that can initialize qudits in the computational basis, perform Clifford unitaries, and carry out stabilizer measurements. These operations are conventionally assumed to be noise-free for simplicity. To obtain universal quantum computing we supplement this quantum computer with the ability to initialize ancilla qudits in certain non-stabilizer states known as magic states, which are noisy. A magic state distillation protocol is a way to distill an arbitrarily pure magic state from many noisy magic states using only Clifford unitaries and stabilizer measurements. For the purposes of determining the best attainable threshold [39], any magic state distillation protocol for the qutrit strange state can be thought of as a procedure that projects \(n\) noisy qudits onto the codespace of an \([[n,k]]_p\) stabilizer code – if the projection is successful, one decodes the resulting qudits to obtain \(k\) hopefully-less-noisy magic states.⁸

3.1 A general formulation in terms of complete weight enumerators↩︎

Here we provide a general formulation for qudit magic state distillation in the language complete weight enumerators. These results follow in part from the formulation in [9]. Let us also mention that a similar formulation in terms of signed-weight enumerators for qubit magic state distillation was given in [40].

Let \(\mathcal{S}\) be a group of commuting Heisenberg-Weyl operators acting on \(n\) qudits. (Recall that, each element of \(\mathcal{S}\) is assumed to be of the form given in equation 1 , with no additional overall phase.) The projector onto the joint eigenspace of the code, with eigenvalues \(1=\omega^0\), is given by, \[\hat{\Pi}^0_{\mathcal{S}} = \frac{1}{|\mathcal{S} |} \sum_{M \in \mathcal{S}} M.\]

In magic state distillation, we first project onto the codespace of \(\mathcal{S}\), which we assume for simplicity to be an \([[n,1]]_p\) code, then decode, to obtain a new state \(\rho' = f_{MSD}(\rho)\). We can use the above lemma to compute the Wigner function of \(\rho'\) in terms of the complete weight enumerators of \(\mathcal{S}^\perp\) and its cosets.

Choose any representatives of the logical Pauli operators \(\bar{X}\) and \(\bar{Z}\), to define logical Heisenberg-Weyl displacement operators \(\bar{D}(x,z)\) and logical phase point operators \(\bar{A}(x,z)\). The Wigner function of \(\hat{\rho}'\) is given by, \[W(\hat{\rho}'; x,z) = \frac{1}{\nu} {\rm{tr~}}\left( p^{-1} \bar{A}(x,z) \hat{\Pi}^0_{\mathcal{S}} \hat{\rho}^{\otimes n} \right).\] Let, \[p^{-1} \bar{A}(0,0) \hat{\Pi}^0_{\mathcal{S}} = p^{-2}\sum_{u,v} \bar{D}(u,v) \hat{\Pi}^0_{\mathcal{S}} = \frac{1}{p^{n+1}} \sum_{(\vec{u}|\vec{v}) \in \mathcal{S}^\perp} \hat{D}(\vec{u},\vec{v}) \equiv \hat{\Pi}^0_{\mathcal{S}^\perp}.\] Although \(\hat{\Pi}^0_{\mathcal{S}^\perp}\) is not a projector onto the codespace of a stabilizer code, in Appendix 6, we also show that, \[W(\hat{\rho}'; 0,0) = \frac{1}{\nu} {\rm{tr~}}\left( \hat{\Pi}^0_{\mathcal{S}^\perp} \hat{\rho}^{\otimes n} \right) = \frac{1}{\nu} {\rm{tr~}}\left( \hat{\Pi}^0_{\mathcal{S}^\perp} \hat{\rho}^{\otimes n} \right) = \frac{1}{\nu} w( \mathcal{S}; \{ W(\hat{\rho},i,j) \}). \label{msd-1}\tag{4}\] In general, \[W(\hat{\rho}';u,v) = \frac{1}{\nu} w( \mathcal{S}+\bar{D}(u,v); \{ W(\hat{\rho};i,j) \}). \label{msd-2}\tag{5}\] As a cross-check, note that \(\sum_{u,v} w( \mathcal{S}+\bar{D}(u,v); \{ y_{ij} \}) = w(\mathcal{S}^\perp; \{y_{ij}\}),\) so the resulting Wigner function is correctly normalized.

The problem of computing the output state of a general qudit magic state distillation routine defined by a stabilizer \(\mathcal{S}\) is thus reduced to computing the complete weight enumerators of \(\mathcal{S}\), and its cosets in \(S^\perp\).

3.2 The strange state and simple-weight enumerators↩︎

We now restrict our attention to the special case where \(\hat{\rho}\) a twirled qutrit strange state \(\mathinner{|{S}\rangle}\). For this particular magic state, we will show that the complete weight enumerators in the previous subsection can be replaced by simple weight enumerators.

The qutrit strange state is given by \[\mathinner{|{S}\rangle}= \frac{1}{\sqrt{2}} \left (\mathinner{|{1}\rangle}- \mathinner{|{2}\rangle} \right).\] After twirling by symplectic rotations, as described in [14], all noise is converted to depolarizing noise, and noisy \(\mathinner{|{S}\rangle}\) states are described by the density matrix: \[\hat{\rho}_S(\epsilon)=(1-\epsilon) \mathinner{|{S}\rangle}\mathinner{\langle{S}|} + \epsilon \frac{1}{3}. \label{mixed}\tag{6}\] This corresponds to the discrete Wigner function: \[W(\hat{\rho}_\epsilon; u,v) = \begin{cases} x & (u,v)=(0,0) \\ y & (u,v) \neq (0,0) \end{cases} = \begin{pmatrix} y & y & y \\ y & y & y \\ x & y & y \end{pmatrix}, \label{wigner-function}\tag{7}\] where the parameters \(x\) and \(y\) satisfy \(x+8y=1\), and are related to \(\epsilon\) via: \[\begin{align} 3x & = & -1+4\frac{\epsilon}{3} \tag{8} \\ 6y & = & 1-\frac{\epsilon}{3} \tag{9}. \end{align}\]

Applying Theorem 1 to \(\hat{\rho}_S(\epsilon)\), we find the following lemma.

Let us assume we distilling using a stabilizer code that has a complete set of transversal Clifford gates. The output state \(\hat{\rho}'=f_{MSD}(\hat{\rho})\) of such a distillation protocol will also be of the form in equation 7 , with parameter \(\epsilon'\). (Alternatively, if the stabilizer code does not have a complete set of Clifford gates, one could still twirl the output state by symplectic rotations as well to bring it into this form.) Let \(W(\hat{\rho}';0,0)=x'\) and \(W(\hat{\rho}';i,j)=y'\) for \((i,j) \neq (0,0)\). Then using equation 4 , we find, \[\begin{align} x' & = & \frac{w(\mathcal{S}; x,y)}{w(\mathcal{S}^\perp;x,y)} \\ y' & = & \frac{1}{8} \frac{w(\mathcal{S}^\perp;x,y)-w(\mathcal{S};x,y)}{w(\mathcal{S}^\perp;x,y)}. \end{align}\] The formal variables \(x\) and \(y\) used to define the simple weight enumerator in equation 3 are now reinterpreted as entries of the discrete Wigner function in equation 7 .

Let us rewrite this expression in terms of the noise parameter \(\epsilon\). We find, \[\epsilon' = 3\frac{3A(z(\epsilon))+B(z(\epsilon))}{4B(z(\epsilon)}. \label{msd-formula}\tag{10}\] where, \[z(\epsilon) = \frac{3-\epsilon}{8 \epsilon-6}, \label{zeps}\tag{11}\] \(A(z)=w(\mathcal{S}; 1,z)\) and \(B(Z)=w(\mathcal{S}^\perp; 1,z)\).

The error suppression of the magic state distillation routine is determined by the largest power of \(\epsilon\) that divides \(33A(z(\epsilon))+B(z(\epsilon))\). By contrast, the distance of the error-correcting code is given by the largest power of \(z\) that divides \(B(z)-A(z)\).

3.2.1 Example: the \([[11,1,5]]_3\) Golay code↩︎

To illustrate the above formalism, let us apply it to the 11-qutrit Golay code of [19]. The 11-qutrit Golay code is an \([[11,1,5]]_3\) CSS-code formed using two copies of the (self-dual) classical ternary Golay code.

The weight enumerator of the 11-qutrit Golay code is computed (e.g., via Magma [41]) to be: \[A(z) = 1+528 z^6+7920 z^8+11000 z^9+23760 z^{10}+15840 z^{11}.\] Using the MacWilliams identity, we find, \[B(z) = 1+528 z^5+528 z^6+15840 z^7+40920 z^8+129800 z^9+198000 z^{10}+145824 z^{11}.\] Substitute these results into equation 10 , to obtain \[\begin{align} \small \epsilon' & = \frac{3 \left(48336 z^{11}+67320 z^{10}+40700 z^9+16170 z^8+3960 z^7+528 z^6+132 z^5+1\right)}{145824 z^{11}+198000 z^{10}+129800 z^9+40920 z^8+15840 z^7+528 z^6+528 z^5+1} \\ & = \left(3021 \epsilon ^{11}-24816 \epsilon ^{10}+92180 \epsilon ^9-203280 \epsilon ^8+292710 \epsilon ^7-283536 \epsilon ^6+181764 \epsilon ^5-71280 \epsilon ^4+13365 \epsilon ^3\right)/ \\ & \phantom{=} \large(990 \epsilon ^{11}-7920 \epsilon ^{10}+27500 \epsilon ^9-50490 \epsilon ^8+37620 \epsilon ^7+47256 \epsilon ^6-172656 \epsilon ^5+243540 \epsilon ^4-204930 \epsilon ^3 \\ & +106920 \epsilon ^2-32076 \epsilon +4374\large) \\ & \approx 55 \epsilon^3/18 + O(\epsilon^4). \label{golayeq} \end{align}\tag{12}\]

The threshold of the code, \(\epsilon_*\), is the critical value of \(\epsilon\) such that, \(\epsilon<\epsilon_*\) implies \(\epsilon'<\epsilon\). Using 12 , we find \[\begin{align} \epsilon_* & =\frac{1}{45} \left(-262 \sqrt[3]{\frac{2}{405 \sqrt{109}-2981}}+2^{2/3} \sqrt[3]{405 \sqrt{109}-2981}+31\right) \\ & \approx 0.387. \end{align}\] Interestingly, \(z_*=z(\epsilon_*)\) satisfies a simple cubic equation, \(11 z_*^3+12 z_*^2+3 z_*+1=0\).

3.3 Conditions for magic state distillation↩︎

There are two conditions that a stabilizer code must satisfy for it to qualify as a magic state distillation routine for the strange state. We first require that the probability of successful projection to be nonzero in the limit \(\epsilon \to 0\), which translates into the requirement \[B\left( z( \epsilon=0) \right) =B(-1/2) \neq 0. \label{condition-1}\tag{13}\] We also require that the noise suppression be better than linear.

Assuming equation 13 is satisfied, the noise-suppression exponent, \(\delta\), of the magic state distillation routine, \(\epsilon' = \theta(\epsilon^\delta)\), is determined by the smallest power of \(\epsilon\) that divides \(3A(z(\epsilon))+B(z(\epsilon))\). Let us write \[3A(z(\epsilon)) + B(z(\epsilon)) = C_0 + C_1 \epsilon + C_2 \epsilon^2 + \ldots\] Generically, we expect \(C_0\) and \(C_1\) will be non-zero. The necessary and sufficient condition for \(C_0\) and \(C_1\) to vanish are, \[\begin{align} \left( 3A(z(\epsilon)) + B(z(\epsilon)) \right)|_{\epsilon=0} & = 0 \\ \frac{d}{d\epsilon}|_{\epsilon=0} \left( 3A(z(\epsilon)) + B(z(\epsilon)) \right) & = 0. \end{align}\] Translated into \(z\), these conditions become \[\begin{align} 3A(-1/2)+B(-1/2) & = & 0, \tag{14} \\ 3A'(-1/2) + B'(-1/2) & = & 0 . \tag{15} \end{align}\] As a check, observe that, for the weight enumerators of the 11-qutrit Golay code, \(A(-1/2)=2187/64\), \(B(-1/2)=6561/64\), \(A'(-1/2)=-8019/16\) and \(B'(-1/2)=24057/16\), these conditions are satisfied.

The MacWilliams identity simplifies at \(z=-1/2\) to: \[B(-1/2)= 3(-1)^n A(-1/2).\] For \(n\) odd, 14 is therefore automatically satisfied. We can also ask, when do we get cubic noise suppression? If condition in equation 15 is satisfied, the condition for \(C_2\) to vanish is that \[3 A''(-1/2)+B''(-1/2)=0. \label{three}\tag{16}\] Taking derivatives of the MacWilliams identity, we find that, \[\begin{align} B'(-1/2) & = & (-1)^{n+1} \left(3 A'(-1/2)+8n A(-1/2) \right), \tag{17}\\ B''(-1/2) & = & \frac{1}{3} (-1)^n \left(9 A''(-1/2)+48 (n-1) A'(-1/2)+64(n^2-n)A(-1/2)\right). \tag{18} \end{align}\] When \(n\) is odd, then the MacWilliams identities above along with 15 imply that condition 16 is satisfied. Thus, cubic noise suppression is guaranteed for any odd-length stabilizer code whose weight enumerator satisfies the two conditions, 13 and 15 .

4 Search for distillation routines↩︎

With the above results in place, a computational search for distillation routines for the qutrit strange state is straightforward. For each stabilizer code \(\mathcal{S}\) in our search space, we compute the simple weight enumerator \(A(z)\), and then \(B(z)\) using the MacWilliams identity. We then check if conditions 13 and 15 are satisfied.

4.1 Equivalence of distillation routines↩︎

When are two stabilizer codes equivalent for the purposes of magic state distillation?

Conventionally, one considers two quantum error-correcting codes to be equivalent if they differ by permutations of coordinates and by local Clifford operations. However, two stabilizer codes which differ by local Clifford transformations will in general give rise to different magic state distillation protocols.

This observation increases the size of our search space substantially – to search for all magic state distillation routines associated with a given stabilizer code \(\mathcal{S}\), we need to search over all orbits of \(\mathcal{S}\) under local Clifford transformations, and all possible eigenvalues of the stabilizers. For each \([[n,1]]_3\) code, this could increase the search space⁹ by a factor as large as, \(|SL(2,\mathbb{Z}_3)|^n 3^{n-1}=24^n 3^{n-1}\).

It is possible, however, to substantially reduce this search space size, by demanding that the both the stabilizer codes we study and the magic states we wish to distill possess certain symmetries.

Let the magic state we wish to distill be invariant under a subgroup \(G\) of the single-qudit Clifford group. It is natural to require¹⁰ our noisy input states first undergo a twirling procedure (as described in detail for qutrits in [14]) so that they are also invariant under \(G\). Let \(C\) be a single-qudit Clifford-unitary, if a twirled noisy magic state \(\hat{\rho}_M\), satisfies \(C\hat{\rho}_M (C)^\dagger=\hat{\rho}_M\), then two stabilizer codes which differ via local Clifford transformations belonging to the subgroup of the Clifford group generated by by \(C\) are equivalent for distillation of the \(\hat{\rho}_M\) magic state. The qutrit magic state \(\mathinner{|{S}\rangle}\) is a simultaneous eigenvector of all symplectic rotations, as emphasized in [14]. Assuming noisy input magic states are twirled by applying a random symplectic rotation, two stabilizer codes are equivalent for magic state distillation if and only if they are related to each other by local symplectic transformations. This reduction in the size of search-space is only possible for the qutrit strange state, by virtue of its exceptional symmetry – and is not expected to be true for any other qutrit or qudit magic state.

It is also natural to require that some, or all, of the unitaries in \(G\) be transversal gates for the stabilizer code used for distillation. This is not a sufficient condition for distillation, but it is true for all magic state distillation routines studied in the literature that we are aware of, other than triorthogonal codes (which possess a transversal non-Clifford gate) [24].

In this paper, we are interested in distilling the qutrit magic state \(\mathinner{|{S}\rangle}\). The symmetry group of \(\mathinner{|{S}\rangle}\), is the set of all symplectic rotations. Therefore, it is natural to demand that our code possess a complete set of transversal Clifford gates. However, we can instead impose a slightly less restrictive requirement, and demand only that \(\hat{H}^2\) be a transversal gate, where \(\hat{H}\) denotes the single-qutrit Hadamard gate. Notice that, acting with \((\hat{H}^2)^{\otimes n}\) on a stabilizer code corresponds to multiplying each row of the symplectic matrix defining the code by \(-1\), leaving the eigenvalues unchanged. Alternatively, its action can be thought of as changing the eigenvalues \(\omega^a\) of any stabilizer to \(\omega^{-a}\). Demanding that \((H^2)^{\otimes n}\) commute with the codespace is therefore equivalent to demanding that the eigenvalues of all stabilizers in the code must be \(+1=\omega^0\). This condition allows us to restrict our search to projection onto the trivial syndrome.

Our search space thus consists of two classes of codes: a search over all small quantum error correcting codes, and a search over CSS codes with a complete set of transversal Clifford gates. For both classes, we restrict attention to projection onto the trivial syndrome, due to our requirement that \((\hat{H}^2)^{\otimes n}\) be transversal.

4.2 Small quantum error-correcting codes↩︎

Any \(n\)-qudit stabilizer state can be thought of as a \([[n,0,d]]_3\) stabilizer code, and any \([n-1,1]_3\) stabilizer code can be obtained by shortening an \([[n,0]]_3\) stabilizer code. Using the graph-state formalism, [21], [22] classified all \([[n,0,d]]_3\) stabilizer states up to local Clifford transformations for \(n\leq 10\), and all¹¹ \(12\)-qutrit stabilizer states of maximum distance¹² 6.

For each \([[n,0]]_3\) code, we computed all inequivalent ways of shortening it to obtain an \([[n-1,1]]_3\) stabilizer code. For each resulting \([[n-1,1]]_3\) code, we computed the weight enumerator \(A(z)\) and the performance for magic state distillation, as described in the previous section. We thus searched over all \([[n,1,d]]_3\) codes for \(m \leq 9\), and all \([[11,1,d]]\) codes that can be obtained from shortening a \([[12,0,6]]_3\) stabilizer state.

Remarkably the only code we found that could distill the qutrit strange state with better-than-linear noise-suppression was the 11-qutrit Golay code. There were also a few \([[9,1]]_3\) and \([11,1]]_3\) codes that distilled the strange state with linear noise suppression, (i.e., \(\epsilon' = \alpha \epsilon\) with \(\alpha<1\)), all with lower thresholds than the 11-qutrit Golay code.

4.3 CSS codes↩︎

A maximal self-orthogonal classical ternary code has parameters \([n, \lfloor \frac{n}{2} \rfloor]_3\). For \(n\) odd, therefore, using the CSS construction [43], [44], two copies of a self-orthogonal ternary code of odd length can be combined to form a \([[n, 1, d]]_3\) qutrit code, whose transversal gate set is the entire single-qutrit Clifford group. Explicitly, if \(G_c\) is the generator matrix of the self-orthogonal classical ternary code, the quantum CSS code is given by symplectic matrix: \[H = \left( \begin{array}{c|c} G_c & 0 \\ 0 & G_c \end{array}\right).\] Any such CSS code is a natural candidate for a magic state distillation routine for the strange state.

Classical maximal self-orthogonal ternary codes up to size \(n=23\) have been classified in [45]–[50], and are conveniently available on a website maintained by Harada and Munemasa [23].

We computed the weight enumerators of all CSS-codes constructed this way from the codes given in [23]. We found that no indecomposable¹³ 13, 15, 17 or 19-qutrit CSS codes were able to distill the strange state.

Somewhat surprisingly, however, we found a total of 646 inequivalent indecomposable 23-qutrit CSS codes¹⁴ that were able to distill the strange state with cubic noise suppression. A complete list of the \(646\) classical ternary codes that gave rise to these codes is included in MAGMA format [41] as supplemental data along with this arXiv submission.

There are a total of 1928 indecomposable maximally-self-dual \([23,11]_3\) codes, so the probability that a randomly chosen code will give rise to a quantum CSS code that distills the strange state is \(0.335\), which is very close to \(1/3\). The probability that a randomly chosen maximally self-orthogonal \([11,5]_3\) code gives rise to a CSS code that distills the strange state is also \(1/3\). This seems to suggest that quantum error correcting codes that distill the qutrit strange state are actually quite common.

The thresholds that arise from these codes range from \(0.063\) to \(0.318\), and are plotted in a histogram in Figure 1. None of these thresholds exceed that of the ternary Golay code.

The code with the highest threshold was a \([[23,1,5]]_3\) code. It is formed from two copies of the classical \([23, 11, 6]_3\) code whose generator matrix is: \[G_C = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 2 & 2 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 2 & 2 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & 1 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 2 & 1 & 2 & 2 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 0 & 1 \end{pmatrix}. \label{23-code}\tag{19}\] The quantum CSS code has weight enumerator \[\begin{align} A(z) & = 2079023616 z^{23}+6035662080 z^{22}+8258226816 z^{21}+7208904960 z^{20}+4504066560 z^{19}+2182781824 z^{18} \\ &+ 790797312 z^{17}+252077184 z^{16}+52015680 z^{15}+14590080 z^{14}+2083104 z^{13}+628800 z^{12}+121824 z^{11} \\ & +58320 z^{10}+16120 z^9+4608 z^8+720 z^6+1. \end{align}\] It gives rise to a distillation performance \[\epsilon' \approx \frac{73 \epsilon ^3}{18}+O\left(\epsilon ^4\right),\] plotted in Figure 2.

The probability of successful projection onto the codespace for these codes is very low. We refer to the probability for 23 pure copies of the strange state to successfully project onto the codespace as the success probability of the code. The success probabilities range from \(1/5159780352 \approx 1.9 \times 10^{-10}\) to \({1}/{35831808} \approx 2.8 \times 10^{-8}\). The highest probability of successful distillation is attained for the 23-qutrit code in equation 19 is \({1}/{35831808}\) – three other codes have the same success probability and very similar thresholds. It appears that the success probability is correlated with the threshold. A plot of success probability versus threshold for the 646 codes that distill the strange state is shown in Figure 3.

5 Discussion↩︎

Magic state distillation [2], [38] is a somewhat mysterious approach to fault-tolerant quantum computing. [2] identified two magic states for qubits, which they denote as \(\mathinner{|{H}\rangle}\) and \(\mathinner{|{T}\rangle}\). For the qubit \(\mathinner{|{H}\rangle}\) state of [2] and its qudit generalizations defined in [16], [28] – magic state distillation is better understood, and systematic approaches to construct distillation routines with low overhead have been proposed in the literature via the formalism of triothogonal codes. However, for the \(\mathinner{|{T}\rangle}\) state of [2] few distillation routines are known [51], and the mechanism behind distillation remains unclear. The qutrit analogue of the \(\mathinner{|{T}\rangle}\) state, which lies directly above a face of the stabilizer polytope, is the strange state \(\mathinner{|{S}\rangle}\) [12], [14] that we study here. The study of distillation of the strange state may in fact be more tractable than that of the qubit \(\mathinner{|{T}\rangle}\) state, thanks to the simple relation between distillation performance and simple enumerators valid only for the qutrit strange state. We hope that, by enlarging the landscape of codes known to distill this state, will lead, in the future, to a better understanding of magic state distillation.

In this work, we carried out a computational search and found that over \(600\) \([[n,1,d]]_3\) codes that distill the qutrit strange state, with cubic noise suppression. This seems to suggest that the ability to distill the qutrit strange state is a generic feature, at least of codes constructed to have a complete set of transversal Clifford gates. While none of these codes have thresholds that exceed that of the 11-qutrit Golay code, their existence does provide some hope that there may exist larger codes with thresholds better than the 11-qutrit Golay code. These results also do provide some support for the belief that a sequence of distillation routines of increasing size whose threshold approaches the limit set by contextuality exists, as required to prove the conjecture that contextuality is sufficient for qutrit quantum computation [1].

Acknowledgements↩︎

SP thanks Prof. P.S. Satsangi for inspiration and guidance. We also thank Mark Howard and Amolak Kalra for comments on a draft of this manuscript. We acknowledge the support of MeitY QCAL, Amazon Braket and DST-SERB grant (CRG/2021/009137).

Appendices↩︎

6 Projecting phase point operators onto a stabilizer code↩︎

In this appendix we prove a lemma which is needed for the proof of Theorem 1 in the main text.

Let \(\mathcal{S}^\perp\) be the dual of \(\mathcal{S}.\) \(|\mathcal{S}^\perp|=p^{n+k}\). Let us also point out that, if we define, \[\hat{\Pi}^0_{\mathcal{S}^\perp} \equiv \sum_{u,v} \bar{D}(u,v) \hat{\Pi}^0_{\mathcal{S}} = \frac{1}{p^{n+k}} \sum_{(\vec{u}|\vec{v}) \in \mathcal{S}^\perp} \hat{D}(\vec{u},\vec{v}),\] then one can also show \[\begin{align} {\rm{tr~}}\left( \hat{\Pi}^0_{\mathcal{S}^\perp} \hat{A}(\vec{u},\vec{v}) \right) & = & \begin{cases} 0 & (\vec{u}|\vec{v}) \notin \mathcal{S} \\ 1 & (\vec{u}|\vec{v}) \in \mathcal{S} \end{cases}. \end{align}\]

7 Some codes that do not distill the strange state↩︎

It was recently [20] proposed that two qutrit stabilizer codes – a \([[13,1]]_3\) and a \([[29,1]]_3\) CSS code – distill the strange state, and estimated their thresholds to be of \(0.425\) and \(\approx 0.7\), respectively. We computed the distillation performance of these codes exactly, via their simple weight enumerators, and found that, contrary to the estimates in [20], these two codes do not, in fact, distill the strange state.

The \([[13,1 ,4]]_3\) CSS-code of [20] generated from two copies of the \([13,6,3]_3\) maximal self-orthogonal ternary code with generator matrix is, in row-reduced form, \[M_{13}= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 1 & 2 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 2 & 2\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 1 & 2 & 2\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 2 & 0 & 1 & 1 & 0 \end{pmatrix}.\] The weight enumerator of the \([[13,1 ,4]]_3\) code is \[A(z) = 1+8 z^3+600 z^6+720 z^7+4320 z^8+18320 z^9+61200 z^{10}+151200 z^{11}+178144 z^{12}+116928 z^{13}.\] This gives rise to a relation \[\tiny \epsilon' = \frac{19683-157464 \epsilon+612360 \epsilon^2-1516320 \epsilon^3+2708235 \epsilon^4-3744144 \epsilon^5+4143960 \epsilon^6-3674592 \epsilon^7+2543865 \epsilon^8-1312920 \epsilon^9+471264 \epsilon^{10}-104384 \epsilon^{11}+10713 \epsilon^{12}}{2 \left(6561-52488 \epsilon+204120 \epsilon^2-495720 \epsilon^3+844425 \epsilon^4-1078704 \epsilon^5+1074888 \epsilon^6-849744 \epsilon^7+529395 \epsilon^8-251400 \epsilon^9+85152 \epsilon^{10}-18184 \epsilon^{11}+1827 \epsilon^{12}\right)},\] plotted in Figure 4. As \(\epsilon \to 0\), we see that \(\epsilon' \to 3/2\). The reason for this is that the pure state \(\mathinner{|{S}\rangle}^{\otimes 13}\) is orthogonal to the codespace of the stabilizer code, so the code is completely unsuitable for magic state distillation. This can be seen from the fact that \(B(-1/2)=0\). This feature was shared by \(6\) out of the \(7\) possible \(13\)-qutrit codes that arise from our construction using the classical ternary codes listed in [23].

The \([[29,1,7]]_3\) CSS code has weight enumerator \[\begin{align} A(z) = & 1+40 z^6+4280 z^9+96 z^{10}+2832 z^{11}+196584 z^{12}+198768 z^{13}+1773408 z^{14}+15542368 z^{15} \\ & +91797024 z^{16} +565547232 z^{17}+3037545272 z^{18}+13979050848 z^{19}+55970778960 z^{20}+192507694176 z^{21} \\ & +559711606992 z^{22}+1361197350960 z^{23}+2723501140720 z^{24}+4358977591776 z^{25} \\ & +5363568387600 z^{26}+4767481212256 z^{27}+2724627154368 z^{28}+751557878400 z^{29}. \end{align}\] (Computing the above weight enumerator on the machines we had access to took approximately one day of computation time.)

The resulting relation between \(\epsilon'\) and \(\epsilon\) is \(\epsilon' = \frac{1937 \epsilon}{224}+O\left({\epsilon}^2\right)\), and is shown in Figure 5 below. The threshold for distillation is zero.

Therefore, at present, the \(11\)-qutrit Golay code has the highest threshold for distillation of the strange state, which is still quite far from the threshold set by contextuality [1]. We have not yet carried out a search over 29-qutrit codes for distillation, as, to our knowledge a complete list of \([29, 14]_3\) self-orthogonal ternary codes is not yet available.

References↩︎

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1. There are certain subtleties associated with state-independent contextuality for qubits and qudits of even dimension – for simplicity, we focus exclusively on qudits of odd-prime dimension in this paper.↩︎

2. Some distillation routines that distill qutrit states up to one of the hyperedges of the Wigner polytope were found in [10].↩︎

3. See [13] for another application of the strange state.↩︎

4. An additional consideration is the increase in overhead cost of distillation as \(n \to \infty\).↩︎

5. Recently, [20] reported the existence of 13-qutrit and 29-qutrit codes that could distill the qutrit strange state with high thresholds. However, we attempted to reproduce these results and found that neither of the codes reported in [20] are able to distill the qutrit strange state. See Appendix 7.↩︎

6. We also found a few 9 and 11 qutrit codes that could distill the state with linear noise suppression.↩︎

7. In this expression, addition means addition as symplectic vectors (or equivelently, vectors in \(GF(p^2)^n\)), or multiplication as Heisenberg-Weyl displacement operators.↩︎

8. Sometimes, i.e., in the case of triorthogonal codes, error-correction is also possible prior to decoding, which can result in a substantial reduction in overhead. However, this is not possible for distillation routines for the qutrit strange state.↩︎

9. Any local Clifford operator can be written as a product of \(V_{\hat{F}} \hat{D}(\chi)\) where \(V_{\hat{F}}\) is a symplectic rotation. Acting on a stabilizer code with a transversal Heisenberg-Weyl displacement operator can only change the eigenvalues of the stabilizers. Symplectic rotations leave the eigenvalues unchanged but transform the stabilizers by a product of local \(SL(2,\mathbb{Z}_p)\) transformations. Therefore, in principle, a search over magic state distillation routines seems to require a search over all local \(SL(2,\mathbb{Z}_3)\) orbits of each \([[n,1]]_3\) stabilizer codes, with all possible allowed eigenvalues for each of the stabilizers.↩︎

10. One, in principle, can also consider magic state distillation without twirling the input states – see [42].↩︎

11. Danielsen has also classified all optimal 11-qutrit stabilizer states, but we could not obtain them.↩︎

12. The distance of a stabilizer state is the minimum weight of any non-trivial element of its stabilizer group.↩︎

13. A code is said to be indecomposable if its generator matrix cannot be written as the direct sum of two smaller generator matrices.↩︎

14. The 646 inequivalent CSS codes gave rise to 263 different simple weight enumerators.↩︎