# Quantum computing thanks to Bianchi groups

Michel Planat

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Michel Planat. Quantum computing thanks to Bianchi groups. Quantum Technology International Conference, Sep 2018, Paris, France. hal-02867890

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# Quantum computing thanks to Bianchi groups

Michel Planat¹,*

¹Université de Bourgogne/Franche Comté, Institut FEMTO-ST UMR 6174, 15 B Avenue des Mont-boucons, F-25044 Besançon, France

**Abstract.** It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group $PSL(2, \mathbb{Z})$ [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups $PSL(2, \mathcal{O}_d)$ (with $\mathcal{O}_d$ the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence $n$-cusped links playing a significant role [4].

## 1 Introduction

A Bianchi group $\Gamma_k = \text{PSL}(2, \mathcal{O}_k) < \text{PSL}(2, \mathbb{C})$ acts as a subset of orientation-preserving isometries of 3-dimensional hyperbolic space $\mathbb{H}_3$ with $\mathcal{O}_k$ the ring of integers of the imaginary quadratic field $\mathcal{I} = \mathbb{Q}(\sqrt{-k})$. The quotient space 3-orbifold $\text{PSL}(2, \mathcal{O}_k)\backslash \mathbb{H}_3$ has a set of cusps in bijection with the class group of $\mathcal{I}$ [5–7]. A torsion-free subgroup $\Gamma_k(l)$ of index $l$ of $\Gamma_k$ is the fundamental group $\pi_1$ of a 3-manifold defined by a link such as the figure-of-eight knot [with $\Gamma_{-3}(12)$], the Whitehead link [with $\Gamma_{-1}(12)$] or the Borromean rings [with $\Gamma_{-1}(24)$]. The fundamental group of the knot complement in the three sphere $S_3$ also called the knot group, or similarly the link group [5, 6], was used to construct appropriate $d$-dimensional fiducial states for universal quantum computing (uqc) [3] thanks to some of their degree $d$ coverings.

In this paper, one starts by upgrading these models of uqc by using other torsion-free subgroups of Bianchi groups and their corresponding 3-manifold such as the Bergé manifold that comes from the Bergé link $L_{6a2}$ [with $\Gamma_{-3}(24)$] or the so-called magic manifold that comes from the link $L_{6a5}$ [with $\Gamma_{-7}(6)$]. The latter link is a small congruence link and belongs to a chain of eight links starting with Thurston’s eight-cusped congruence link [with $\Gamma_{-3}$ and ideal $\langle (5 + \sqrt{-3})/2 \rangle$] [8, 9] and ending with the Whitehead link and then the figure-eight knot.

In Sec. 2, one reminds the permutation based model of universal quantum computing developed by the authors and its relationship to minimal informationally complete POVMs or MICs [1–3]. In Sec. 3, important small index torsion-free subgroups of Bianchi groups and the corresponding substructure of 3-manifolds are derived. In Sec. 4, one specializes in

<footnote>*e-mail: michel.planat@femto-st.fr</footnote>
the connection of the aforementioned Bianchi subgroups to our version of uqc. One features on a remarkable chain of *n*-cusped *n*-links, *n* = 8..1 obtained thanks to (±1, 1)-slope Dehn filling starting with congruence Thurston's link. Their possible role for uqc and the relevant 3-manifolds is discussed.

## 2 Minimal informationally complete POVMs with permutations

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Geometrical structure of low dimensional MICs: (a) the qutrit Hesse SIC, (b) the two-qubit MIC that is the generalized quadrangle of order two $GQ(2, 2)$. The coordinates on each diagram are the *d*-dimensional Pauli operators that act on the fiducial state, as shown.</figcaption></figure>

In our approach [1, 2], minimal informationally complete POVMs (MICs) are derived from appropriate fiducial states under the action of the (generalized) Pauli group. The fiducial states also allow to perform universal quantum computation [10].

A POVM is a collection of positive semi-definite operators {$E_1, \dots, E_m$} that sum to the identity. In the measurement of a state $\rho$, the *i*-th outcome is obtained with a probability given by the Born rule $p(i) = \text{tr}(\rho E_i)$. For a minimal IC-POVM, one needs $d^2$ one-dimensional projectors $\Pi_i = |\psi_i\rangle\langle\psi_i|$, with $\Pi_i = dE_i$, such that the rank of the Gram matrix with elements $\text{tr}(\Pi_i\Pi_j)$, is precisely $d^2$. A SIC-POVM (the *S* means symmetric) obeys the relation $|\langle\psi_i|\psi_j\rangle|^2 = \text{tr}(\Pi_i\Pi_j) = \frac{d\delta_{ij}+1}{d+1}$, that allows the explicit recovery of the density matrix as in [11, eq. (29)].

New MICs (i.e. whose rank of the Gram matrix is $d^2$) and with Hermitian angles $|\langle\psi_i|\psi_j\rangle|_{i \neq j} \in A = \{a_1, \dots, a_l\}$ have been discovered [2]. A SIC is equiangular with $|A| = 1$ and $a_1 = \frac{1}{\sqrt{d+1}}$. The states encountered are considered to live in a cyclotomic field $\mathbb{F} = \mathbb{Q}[\exp(\frac{2\pi i}{n})]$, with $n = \text{GCD}(d, r)$, the greatest common divisor of $d$ and $r$, for some $r$. The Hermitian angle is defined as $|\langle\psi_i|\psi_j\rangle|_{i \neq j} = \|\langle\psi_i, \psi_j\rangle\|^{\frac{1}{\deg}}$, where $\|\cdot\|$ means the field norm of the pair $(\psi_i, \psi_j)$ in $\mathbb{F}$ and $\deg$ is the degree of the extension $\mathbb{F}$ over the rational field $\mathbb{Q}$ [1].

The fiducial states for SIC-POVMs are quite difficult to derive and seem to follow from algebraic number theory [12]. Except for $d=3$, the MICs derived from permutation groups
are not symmetric and most of them can be recovered thanks to subgroups of index *d* of the modular group Γ [2, Table 1]. For instance, for *d* = 3, the action of a magic state of type (0, 1, ±1) results into the Hesse SIC shown in Fig 1a, arising from the congruence subgroup Γ₀(2) of Γ. For *d* = 4, the action of the two-qubit Pauli group on the magic/fiducial state of type (0, 1, −ω₀, ω₀ − 1) with ω₀ = exp(2iπ/6) results into a MIC whose geometry of triple products of projectors Πᵢ, arising from the the congruence subgroup Γ₀(3) of Γ, turns out to correspond to the commutation graph of Pauli operators, see Fig. 1b and [2, Fig. 2c]. For *d* = 5, the congruence subgroup 5A⁰ of Γ is used to get a MIC whose geometry consists of copies of the Petersen graph (see [2, Fig. 3c]. For *d* = 6, all five congruence subgroups Γ', Γ(2), 3C⁰, Γ₀(4) or Γ₀(5) point out the geometry of Borromean rings (see

The modular group Γ first served as a motivation for investigating the trefoil knot manifold 3₁ in relation to uqc and the corresponding MICs, then the uqc problem was put in the wider frame of Poincaré conjecture, the Thurston’s geometrization conjecture and the related 3-manifolds. E.g. MICs may follow from hyperbolic or Seifert 3-manifolds as shown in Tables 2 to 5 of [3]. A further step is obtained here by restricting our choice of 3-manifolds to low degree coverings of Bianchi subgroups. This is a quite natural procedure to base our uqc models on Bianchi groups Γₖ defined over the ring Oₖ of imaginary quadratic integers [13] since they are a natural generalization of the modular group Γ.

### 3 Torsion-free subgroups of Bianchi groups.

<table><caption>**Table 1.** Low index torsion-free subgroups of Bianchi groups Γₖ = PSL(2, Oₖ), *k* = −1, −2, −3, −7, their names in SnapPy in column 4 and their popular name in column 5, when known. The last column shows a connection to Dehn filling of slope (1, 1) of the corresponding manifold. ‘Bold’ manifolds are investigated here in relation to the existence of MICs from their subgroups of finite index.</caption><thead><tr><th>k</th><th>index</th><th>cusps</th><th>names</th><th>other name</th><th>note</th></tr></thead><tbody><tr><td rowspan="4">-1</td><td>12</td><td>2</td><td><b>L5a1</b>, ooct01<sub>00001</sub></td><td rowspan="4">Whitehead link: WL<br>sister of WL<br>Borromean rings</td><td rowspan="4"><i>L6a5</i>(1, 1)</td></tr><tr><td>.</td><td>.</td><td><i>L13n5885</i>, ooct01<sub>00000</sub></td></tr><tr><td>24</td><td>3</td><td><b>L6a4</b>, ooct02<sub>00005</sub></td></tr><tr><td>.</td><td>.</td><td><i>L8n7</i>, ooct02<sub>00001</sub></td></tr><tr><td>.</td><td>.</td><td>.</td><td><i>L10n84</i>, ooct02<sub>00002</sub></td><td></td><td><i>L5a1</i>(1, 1)</td></tr><tr><td rowspan="2">-2</td><td>12</td><td>2</td><td><i>L9a32</i>, 9<sup>2</sup><sub>30</sub></td><td rowspan="2"></td><td rowspan="2"></td></tr><tr><td>.</td><td>.</td><td><i>L9a33</i>, 9<sup>2</sup><sub>24</sub></td></tr><tr><td rowspan="4">-3</td><td>12</td><td>2</td><td><b>K4a1</b>, otet02<sub>00001</sub></td><td rowspan="4">Figure-eight knot<br>sister of <b>K4a1</b><br>Bergé manifold</td><td rowspan="4"><i>L5a1</i>(1, 1)</td></tr><tr><td>.</td><td>1</td><td><i>m003</i>, otet02<sub>00000</sub></td></tr><tr><td>24</td><td>2</td><td><b>L6a2</b>, 6<sup>2</sup><sub>2</sub> otet04<sub>00001</sub></td></tr><tr><td>.</td><td>1</td><td><i>m206</i>, otet04<sub>00002</sub></td></tr><tr><td>.</td><td>.</td><td>.</td><td><i>m207</i>, otet04<sub>00001</sub></td><td></td><td></td></tr><tr><td rowspan="4">-7</td><td>6</td><td>2</td><td><b>L6a1</b>, 6<sup>2</sup><sub>3</sub></td><td rowspan="4">magic manifold</td><td rowspan="4">congruence<br>cyc. 2-cover of <i>L6a5</i></td></tr><tr><td>.</td><td>3</td><td><b>L6a5</b>, 6<sup>3</sup><sub>1</sub></td></tr><tr><td>12</td><td>3</td><td><i>L10n81</i>, 10<sup>3</sup><sub>15</sub></td></tr><tr><td>.</td><td>4</td><td><i>L12n2205</i></td></tr><tr><td>-15</td><td>6</td><td>6</td><td>⟨<sup>1</sup>&frasl;<sub>2</sub>(1 + √-15)⟩</td><td>Baker&amp;Reid [8]</td><td>congruence</td></tr></tbody></table>

Table 1 provides a short list of low index torsion-free subgroups of Bianchi groups Γₖ, *k* ∈ {−1, −2, −3, −7, −15} [13] obtained from the software Magma [14] and their identification as fundamental groups of 3-manifolds obtained from the software SnapPy [15]. In particular,
one recovers the Whitehead link and the Borromean rings ($k = -1$ with index $l = 12$ and 24, respectively) as well as the figure-eight knot ($k = -3$ with index 12) whose relationship to uqc was explored in [3]. Other links of importance in our paper are the link $L6a1$, the Bergé link $L6a2$ and the magic link $L6a5$ whose connections to uqc is summarized in Table 2.

According to [16], a $d$-fold covering of the fundamental group $\pi_1(M)$ of a 3-manifold $M$ is uniquely determined by the conjugacy class of a subgroup of index $d$ of $\pi_1$. To recognize the $d$-covering from the conjugacy class and conversely, one makes use of the cardinality signature of subgroups of $\pi_1(M)$ denoted $\eta_d(\pi_1(M))$, $d = 1..d_{max}$ and one identifies $\eta_d(\pi_1(M))$ both in a particular $d$-covering of $M$ (with SnapPy) and in the representative of a particular conjugacy class of subgroups of $\pi_1(M)$ (with Magma).

# 4 A Bianchi factory for quantum computing

First one provides a reminder about the concept of Dehn filling that will be useful in Sect. 4.2. Let us start with a Lens space $L(p, q)$ that is 3-manifold obtained by gluing the boundaries of two solid tori together, so that the meridian of the first solid torus goes to a $(p, q)$-curve on the second solid torus [where a $(p, q)$-curve wraps around the longitude $p$ times and around the meridian $q$ times]. Then we generalize this concept to a knot exterior, i.e. the complement of an open solid torus knotted like the knot. One glues a solid torus so that its meridian curve goes to a $(p, q)$-curve on the torus boundary of the knot exterior, an operation called Dehn surgery [6, p. 275],[7, p 259],[17]. According to Lickorish's theorem, every closed, orientable, connected 3-manifold is obtained by performing Dehn surgery on a link in the 3-sphere.

One shows below how the Bianchi subgroups are related to uqc and the related MICs and how useful is Dehn filling in our context.

## 4.1 Universal quantum computing from the Bianchi factory

In this subsection, one specializes on uqc based on the qutrit Hesse SIC (shown in Fig. 1a) and on the two-qubit geometry of the generalized quadrangle $GQ(2, 2)$ (shown in Fig. 1b) obtained with the 'magic' link $L6a5$. The qutrit uqc follows from a link called 'L14n63788' with Poincaré polyhedron shown in Fig. 2a and the two-qubit uqc follows from a 3-manifold with Poincaré polyhedron shown in Fig 2b. The former case corresponds to the 3-fold irregular covering the fundamental group $\pi_1(L6a5)$ whose 3-manifold has first homology $\mathbb{Z}^5$, 5 cusps and volume $\approx 16.00$. The latter case corresponds to the 4-fold irregular covering whose 3-manifold has first homology $\mathbb{Z}/2 \oplus \mathbb{Z}^4$, 4 cusps and volume $\approx 21.34$.

For the other links investigated in Table 2 one proceeds in the same way.

## 4.2 Congruence links in the Bianchi factory

As announced earlier, there is an interesting sequence of links starting at the Thurston's link and ending at the figure-eight knot that one obtains by applying (±1, 1)-slope Dehn fillings. The Dehn fillings of last five links called $M_i$, $i = 5..1$ were studied in [18]. Observe that $M_3$ is the magic link $L6a4$, $M_4$ is the 4-link $L8n7$ and $M_5$ is the 5-link $L10n113$ of Table 2. The full sequence is

$$ \begin{aligned} \text{Thurston's link : } & \frac{\text{``}(5 + \sqrt{-3})\text{''}}{2} \xrightarrow{\text{``(1,1)''}} \text{7-link } L14n64180 \xrightarrow{\text{``(1,1)''}} \text{``}2 + 0\sqrt{-1}\text{''} : \text{6-link } L12n2256 \\ & \xrightarrow{\text{``(1,1)''}} \text{``}2 + 0\sqrt{-3}\text{''} : \text{5-link } L10n113 \xrightarrow{\text{``(1,1)''}} \text{4-link } L8n7 \xrightarrow{\text{``(1,1,1)''}} \text{``}(1 + \sqrt{-7})\text{''} : \text{magic 3-link } L6a5 \\ & \xrightarrow{\text{``(1,1)''}} \text{Whitehead link } L5a1 \xrightarrow{\text{``(1,1)''}} \text{figure-eight knot } K4a1 \xrightarrow{\text{``(1,1)''}} \text{Poincaré homology sphere.} \end{aligned} \tag{1} $$
<table><caption>**Table 2.** A few 3-manifolds (in column 5) occurring as a model of uqc (in column 6). When not identified in SnapPy the cardinality structure $η_d(M)$ of finite index subgroups of the 3-manifold $M$ is made explicit. The third column identifies the first homology type, with 1 meaning the ring of integers $\mathbb{Z}$. The 3-manifolds occurring as a model of uqc with source the trefoil knot $K3a1=3$, the figure-eight knot $K4a1$ and the Whitehead link $L5a1$ have been investigated in [3] and thus are not included in the table.</caption><thead><tr><th>source</th><th>index</th><th>cusps</th><th>homology</th><th>names or η<sub>d</sub>(M)</th><th>uqc</th></tr></thead><tbody><tr><td rowspan="2">L6a1</td><td>3</td><td>3</td><td>irr: <sup>1</sup>&frasl;<sub>2</sub> + <sup>1</sup>&frasl;<sub>2</sub></td><td>L12n2181</td><td>Hesse SIC</td></tr><tr><td>4</td><td>5</td><td>irr: 1<sup>+5</sup></td><td>L14n63905</td><td>2QB MIC</td></tr><tr><td rowspan="4">Bergé L6a2</td><td>4</td><td>4</td><td>irr: 1<sup>+4</sup></td><td>otet16<sub>00025</sub></td><td>2QB MIC</td></tr><tr><td>5</td><td>4</td><td>irr: 1<sup>+4</sup></td><td>otet20<sub>00543</sub>; otet20<sub>00041</sub>; otet20<sub>00549</sub></td><td>5-dit MIC</td></tr><tr><td>5</td><td>2</td><td>irr: <sup>1</sup>&frasl;<sub>2</sub> + 1<sup>+4</sup></td><td>otet20<sub>00574</sub></td><td>.</td></tr><tr><td>5</td><td>2</td><td>irr: <sup>1</sup>&frasl;<sub>3</sub> + 1<sup>+2</sup></td><td>otet20<sub>00665</sub></td><td>.</td></tr><tr><td rowspan="2">magic L6a5</td><td>3</td><td>5</td><td>irr: 1<sup>+5</sup></td><td>L14n63788</td><td>Hesse SIC</td></tr><tr><td>4</td><td>4</td><td>irr: <sup>1</sup>&frasl;<sub>2</sub> + 1<sup>+4</sup></td><td>{31, 174, 4324, 82357, ...}</td><td>2QB MIC</td></tr><tr><td rowspan="4">Borr. L6a4</td><td>3</td><td>4</td><td>irr: 1<sup>+4</sup></td><td>ooc06<sub>00466</sub></td><td>Hesse SIC</td></tr><tr><td>3</td><td>4</td><td>irr: 1<sup>+5</sup></td><td>ooc06<sub>00398</sub></td><td>.</td></tr><tr><td>4</td><td>4</td><td>irr: <sup>1</sup>&frasl;<sub>2</sub> + 1<sup>+4</sup></td><td>{63, 300, 10747, ...}</td><td>2QB MIC</td></tr><tr><td>4</td><td>6</td><td>irr: <sup>1</sup>&frasl;<sub>2</sub> + 1<sup>+6</sup></td><td>{127, 2871, 478956, ...}</td><td>.</td></tr><tr><td>4-link L8n7</td><td>3</td><td>6</td><td>irr: 1<sup>+6</sup></td><td>ooc06<sub>00340</sub></td><td>Hesse SIC</td></tr><tr><td>5-link L10n113</td><td>3</td><td>8</td><td>irr: 1<sup>+8</sup></td><td>ocube06<sub>03005</sub>; ocube06<sub>03322</sub></td><td>Hesse SIC</td></tr><tr><td>6-link L12n2256</td><td>3</td><td>9</td><td>irr: 1<sup>+9</sup></td><td>{511, 20122, ...}</td><td>Hesse SIC</td></tr></tbody></table>

and is pictured in Fig. 3. Applying (1, 1)-slope Dehn filling on the figure-eight knot, the sequence terminates to the spherical manifold $M_0$ that is the Poincaré homology sphere (also known as Poincaré dodecahedral space) [5, 6].

Many of the links of the sequence correspond to 'congruence' manifolds [19]. A Bianchi subgroup $\Gamma_B$ of $PSL(2, \mathbb{O}_d)$ is called a 'congruence' subgroup if there exists an ideal $I \subset \mathbb{O}_d$ so that $\Gamma_B$ contains the 'principal congruence group' $\Gamma(I) = \ker\{PSL(2, \mathbb{O}_d) \to PSL(2, \mathbb{O}_d/I)\}$. Details about these links are available on-line at [20]. Of course congruence Bianchi subgroups generalize congruence subgroups that played a leading role in our approach of uqc [2].

The eight links under scrutiny are pictured in Fig. 3. Four of them are congruence links and are identified by a pair $(-k, I)$ meaning that they act on $\mathbb{Q}(\sqrt{-k})$ with the ideal $I$: the Thurston's link $(-3, (5 + \sqrt{-3})/2)$, the 6-link $L12n2256 : (-1, (2 + 0\sqrt{-1}))$, the 5-link $L10n113 : (-3, (2 + 0\sqrt{-3}))$ and the magic link $L6a5 : (-1, (1 + \sqrt{-7})/2)$. The link $L10n113$ is also $otet10_{00027}$ and the link $L12n2256$ is $ooc04_{00042}$.

The 3-manifolds that we could identify as connected to MICs are in Table 2. This becomes a cumbersome task for links larger than the 6-link. For the link $L8n7$ the cardinality sequence is $η_d = \{63, 794, 23753, 280162, ...\}$, for the link $L10n113$ it is $η_d = \{31, 176, 1987, 7628, 11682, ...\}$ and for the 6-link it is $η_d = \{63, 580, 12243, 94274, ...\}$. For the latter link, SnapPy is able to build the Dirichlet domain (not shown) of symmetry group $D_6$, corresponding to the Hesse SIC while it is not the case for the Hesse SICs attached to the 5-link.

We expect that the full sequence should have a meaning in the context of uqc.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** (a) The 3-manifold L14n63788 corresponding to the 3-fold covering of L6a5 relevant to qutrit universal quantum computing, (b) The 3-manifold corresponding to the 4-fold covering of L6a5 relevant to two-qubit universal quantum computing.</figcaption></figure>

**Conclusion**

It is not yet known which vista a practical quantum computer will follow. The authors of [3] developed a view of universal quantum computing based on 3-manifolds. In a nutshell, there exists a connection between the Poincaré conjecture -it states that every simply connected closed 3-manifold is homeomorphic to the 3-sphere- and the Bloch sphere $S_3$ that houses the qubits. According to this approach, the dressing of qubits in $S_3$ leads to 3-manifolds (that have been investigated in much details by W. P. Thurston and led a proof of the Poincaré conjecture) many of them corresponding to MICs (minimal informationally complete POVMs) and the related uqc (universal quantum computing). In [2], uqc, was based on the modular group $\Gamma$ that corresponds to the trefoil knot manifold in [3]. In the present paper, the 3-manifolds under investigation derive from subgroups of Bianchi groups, a generalization of $\Gamma$. There seems to exist a nice connection between some congruence subgroups of Bianchi groups through the chain (1) and quantum computing that the author started to investigate. At a more practical level, 3-manifolds can also be seen as (three-dimensional) quasiparticles beyond anyons [22].
<figure><img src="image_4.png" /><figcaption>**Figure 3.** The consecutive links from (a) to (h) obtained from (1,1) Dehn filling at a single cusp [except for (-1, 1) filling at (e)]. The links (a), (c), (d), (f) are congruence. The sequence (d) to (h) already appears in [18].</figcaption></figure>

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