1 Introduction↩︎

Any quantum gate can be written as a product of single-qubit gates and 2-qubit gates [1]. In fact, almost any 2-qubit gate \(V \in \operatorname{U}(4)\) is universal, meaning that every \(n\)-qubit gate can be expressed as a quantum circuit involving only single-qubit gates and copies of \(V\) acting on pairs of qubits [2], [3]. For the sake of comparing different gate decomposition algorithms, it is convenient to fix a particular \(V\); a popular choice is the controlled-NOT (CNOT) gate, shown to be universal in [4].

Once the existence of such quantum circuits has been established, it is natural to ask how large they must be: for instance, what is the minimal \(N\) such that every \(n\)-qubit gate \(V \in \operatorname{U}(2^n)\) can be written as \(L_1 C_1 L_2 C_2 \cdots L_N C_N L_{N+1}\) where the \(C_i\) are CNOTs and the \(L_i\) are in \(\operatorname{U}(2)^{\otimes n}\)? In almost all cases the exact answer is unknown. Shende, Markov, and Bullock [5] showed using a dimension-counting argument that at least \((4^n-3n-1)/4\) CNOTs are required. Many decomposition algorithms have been devised providing upper bounds; the current best seems to be \(\tfrac{22}{48} 4^n - \tfrac{3}{2} 2^n + \tfrac{5}{3}\) CNOTs due to Krol and Al-Ars [6], who also give a comprehensive overview of these algorithms.

The case \(n=2\) is well-understood, at least. Any \(V \in \operatorname{U}(4)\) can be written as the product of at most 3 CNOTs plus single-qubit gates, this bound is sharp, and there are exact algorithms to compute optimal decompositions [7]. A special feature of this case that simplifies analysis is the “magic basis” coordinate change, which conjugates the subgroup \(\operatorname{SU}(2) \otimes \operatorname{SU}(2) \subseteq \operatorname{SU}(4)\) generated by single-qubit gates onto \(\operatorname{SO}(4)\). This allows the technique of Cartan decomposition to be applied directly, which by contrast cannot be done for the subgroup \(\operatorname{SU}(2)^{\otimes n} \subseteq \operatorname{SU}(2^n)\) when \(n > 2\).

In this paper we study the special case of unimodular (\(\det = 1\)) real 3-qubit gates. Wei and Di [8] showed that any \(V \in \operatorname{SO}(8)\) is the product of at most 16 CNOTs plus single-qubit gates. Our main result (Theorem 11) is an explicit quantum circuit realizing any element of \(\operatorname{SO}(8)\) as the product of at most 14 CNOTs plus 35 single-qubit rotations.

Perhaps more interesting than this modest improvement is our technique, which relies on the triality symmetry \(\mathcal{T}: \operatorname{PSO}(8) \to \operatorname{PSO}(8)\). This is a rather exotic map which only appears in association with the \(\mathfrak{so}(8)\) Lie algebra, so we spend some time defining it and exploring its properties. Its utility in the study of 3-qubit gates does not seem to have been observed before; in some ways it behaves as a real 3-qubit version of the magic basis transformation which is so useful in the 2-qubit case. More specifically, it transforms many subgroups defined in terms of tensor products into straightforward block matrix subgroups, which makes matrix factorizations easier to see.

In Section 2 we set up notation and review some Lie algebra and Lie group material, including Cartan decomposition. Section 3 is devoted to the triality map. In Section 4 we discuss an interesting Cartan decomposition of \(\operatorname{PSO}(8)\) involving triality, which we apply in Section 5 to construct our new circuit.

2 Preliminaries↩︎

2.1 Matrix notation↩︎

We use the interval notation \([m,n]\) for the set \(\{m,m{+}1,\ldots,n\}\) when \(m,n\) are integers. As a special case, \([n] = [1,n] = \{1,2,\ldots,n\}\). If \(A\) is an \(m \times n\) matrix and \(I \subseteq [m], J \subseteq [n]\) are subsets, write \(A_{IJ}\) for the \(|I| \times |J|\) submatrix \([A_{ij}]_{i \in I, j \in J}\). For instance, if \(A = \left[ \begin{smallmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{smallmatrix} \right]\) then \(A_{[2], [1,3]} = \left[ \begin{smallmatrix} 1 & 3 \\ 4 & 6 \end{smallmatrix} \right]\).

Given square matrices \(A_1, \ldots, A_k\), write \(A_1 \oplus \cdots \oplus A_k\) for the block-diagonal matrix with \(A_1, \ldots, A_k\) as blocks. For example, \[\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \oplus \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 5 & 6 \\ 0 & 0 & 7 & 8 \end{bmatrix}\] More generally, if \(S_1, \ldots, S_k\) are sets of matrices, then \[S_1 \oplus \cdots \oplus S_k = \{A_1 \oplus \cdots \oplus A_k : A_i \in S_i \text{ for each i}\}.\] We also write \(\operatorname{diag}(x_1,\ldots,x_n)\) for the diagonal matrix with diagonal entries \(x_1,\ldots,x_n\). As a special case that will arise frequently, for each subset \(J \subseteq [n]\) define \(\Delta_{J}\) as the diagonal matrix with \[(\Delta_J)_{ii} = \begin{cases} -1 & \text{if i \in J} \\ 1 & \text{if i \notin J} \end{cases}\] Finally, the operator \(\operatorname{S}\) applied to a matrix group gives the subgroup where \(\det = 1\), and \(\operatorname{P}\) takes the quotient modulo scalar multiplication.

We are especially concerned here with \(\operatorname{PSO}(8)\), the group of real orthogonal matrices of determinant 1, subject to the rule that \(A\) and \(-A\) are considered the same matrix. The reason for making the distinction between \(\operatorname{PSO}(8)\) and \(\operatorname{SO}(8)\) is that the triality map involves an indeterminacy of sign, and so is well-defined as a map \(\operatorname{PSO}(8) \to \operatorname{PSO}(8)\) but not \(\operatorname{SO}(8) \to \operatorname{SO}(8)\). That said, our final circuit decomposition will be perfectly valid in \(\operatorname{SO}(8)\).

2.2 Quantum gate conventions↩︎

Take the Pauli matrices to be \[\sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \qquad \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \qquad \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\] and set \(R_a(\theta) = \exp(-i\theta\sigma_a/2)\) for \(a \in \{x,y,z\}\). These rotations generate \(\operatorname{SU}(2)\) since \(i\sigma_x, i\sigma_y, i\sigma_z\) generate \(\mathfrak{su}(2)\) as a Lie algebra, and it is well-known that any element of \(\operatorname{SU}(2)\) can be written as \(R_{a_1}(\theta_1)R_{a_2}(\theta_2)R_{a_3}(\theta_3)\) where \(a_1 a_2 a_3\) is any word on the alphabet \(\{x,y,z\}\) with no two consecutive letters the same. Our convention is that \(\mathfrak{su}(n)\) consists of the \(n \times n\) skew-Hermitian matrices.

We take \(\mathbb{C}^2\) to have ordered orthonormal basis \(\bra{0}, \bra{1}\), and more generally \((\mathbb{C}^2)^{\otimes n}\) to have basis \(\{\bra{w} : w \in \{0,1\}^n\}\), indexed by the length \(n\) binary words ordered lexicographically. Write \(C_i^j \in \operatorname{U}(2^n)\) for the CNOT gate with control qubit \(i\) and target qubit \(j\), i.e.the linear operator defined by \[\bra{w_1 \cdots w_n} \mapsto \begin{cases} \bra{w_1 \cdots w_n} & \text{if w_i = 0}\\ \bra{w_1 \cdots \tilde{w}_j \cdots w_n} & \text{if w_i = 1} \end{cases}\] where \(\tilde{x} = 1-x\). For instance, with \(n = 2\), \[C_1^2 = \begin{quantikz} & \ctrl{1} & \\ & \targ{} & \end{quantikz} = \left[\begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right] \qquad \text{and} \qquad C_2^1 = \begin{quantikz} & \targ{} & \\ & \ctrl{-1} & \end{quantikz} = \left[\begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{smallmatrix}\right]\]

2.3 Lie group notation and Cartan decomposition↩︎

The following notation will hold for the rest of the paper, unless otherwise noted.

• \(\mathcal{G}\) is a compact connected Lie group with Lie algebra \(\mathfrak{g}\).

• \(\mathcal{H}_0\) is the connected component of the identity in a Lie group \(\mathcal{H}\).

• \(\theta : \mathcal{G}\to \mathcal{G}\) is an automorphism and an involution, i.e.an invertible function satisfying \(\theta(gh) = \theta(g)\theta(h)\) and \(\theta(\theta(g)) = g\) for all \(g,h \in \mathcal{G}\).

• \(c_g : \mathcal{H}\to \mathcal{H}\) is the conjugation automorphism \(c_g(h) = ghg^{-1}\).

• \(\mathcal{G}^{\pm \theta} = \{g \in \mathcal{G}: \theta(g) = g^{\pm 1}\}\), so \(\mathcal{G}^\theta\) is the subgroup of fixed points of \(\theta\), while \(\mathcal{G}^{-\theta}\) is just a subset.

• \(\mathcal{Z}(S) = \{h \in \mathcal{H}: sh = hs \text{ for all s \in S}\}\) is the centralizer of a subset \(S \subseteq \mathcal{H}\). We write \(\mathcal{Z}(g)\) instead of \(\mathcal{Z}(\{g\})\).

• \(\mathfrak{k}\) and \(\mathfrak{p}\) are the \(1\)- and \((-1)\)-eigenspaces of the induced Lie algebra homomorphism \(d\theta : \mathfrak{g}\to \mathfrak{g}\), so \(\mathfrak{k}\) is a subalgebra and \(\mathfrak{g}= \mathfrak{k}\oplus \mathfrak{p}\).

• \(\mathcal{K}= \exp \mathfrak{k}= (\mathcal{G}^\theta)_0\) is the connected component of the identity in \(\mathcal{G}^\theta\). We will call any closed subgroup arising this way a Cartan subgroup.

• \(\mathfrak{a}\) is a maximal abelian subalgebra of \(\mathfrak{p}\) and \(\mathcal{A}= \exp(\mathfrak{a})\).

• Later we will have more than one involution \(\theta\), in which case we write \(\mathcal{K}_\theta\), \(\mathfrak{a}_\theta\), etc.to be more specific.

We try to consistently use calligraphic capitals for abstract Lie groups, lowercase Fraktur for Lie algebras, capitals for matrices and named matrix groups, Greek letters for automorphisms of groups and Lie algebras, and lowercase \(g,h,k\) etc.for elements of abstract Lie groups. That said, we also use calligraphic capitals for certain distinguished matrices and for the triality automorphism \(\mathcal{T}\).

Besides the decomposition \(\mathcal{G}= \mathcal{K}\mathcal{A}\mathcal{K}\), we will also use a basic corollary of Theorem 1.

How does one compute a Cartan decomposition of \(g \in \mathcal{G}\) in practice? One approach is to compute the Cartan double \(g\theta(g)^{-1}\) of \(g \in \mathcal{G}\). Indeed, if \(g = k_1 a k_2\) with \(k_1, k_2 \in \mathcal{K}\) and \(a \in \mathcal{A}\subseteq \mathcal{G}^{-\theta}\), then \[\label{eq:cartan-double} g\theta(g)^{-1} = (k_1 a k_2)(k_1 a^{-1} k_2)^{-1} = k_1 a^2 k_1^{-1}.\tag{1}\] If \(\mathcal{G}\) is a matrix group (a subgroup of a finite quotient \(\mathcal{G}'\) of \(\operatorname{U}(n)\)), then \(\mathcal{A}\) is a subgroup of a maximal torus in \(\mathcal{G}'\), which in turn must be conjugate to the torus of diagonal matrices [10]. Thus, we can compute \(k_1\) and \(a^2\) as in 1 by diagonalizing \(g\theta(g)^{-1}\) in an appropriate basis. There are finitely many square roots of \(a^2\) in \(\mathcal{A}\) (because the same is true for diagonal matrices), and each possible square root \(a\) can be checked for correctness by seeing whether the putative \(k_2 = a^{-1} k_1^{-1} g\) actually lies in \(\mathcal{K}\). With more work, one can choose a distinguished subset \(\mathcal{A}_\circ \subseteq \mathcal{A}\) such that \(g \in \mathcal{K}a \mathcal{K}\) for a unique \(a \in \mathcal{A}_\circ\), thereby avoiding this last issue; see [9] for the simply connected case.

For instance, in the setting of Example 2 we would find a basis of real eigenvectors for the symmetric unitary matrix \(U\overline{U}^{-1} = UU^T = O_1 D^2 O_1^{-1}\) in order to find \(O_1\) and \(D^2\). The next example covers the Cartan decompositions we will be concerned with.

Note that the sets \(\mathfrak{k}\), \(\mathfrak{p}\), \(\mathcal{K}\) in Theorem 1 are uniquely determined by \(\theta\), while \(\mathfrak{a}\) and \(\mathcal{A}\) are not. It may be convenient to replace \(\mathcal{A}\) in the Cartan decomposition \(\mathcal{G}= \mathcal{K}\mathcal{A}\mathcal{K}\) with a different set \(\mathcal{A}' \subseteq \mathcal{G}\) which is not necessarily a torus. By definition, \(\mathcal{G}= \mathcal{K}\mathcal{A}'\mathcal{K}\) holds if \(\mathcal{A}'\) covers all possible canonical parameters.

As explained above, if \(\mathcal{K}g \mathcal{K}= \mathcal{K}h \mathcal{K}\) then the Cartan doubles \(g\theta(g)^{-1}\) and \(h\theta(h)^{-1}\) are conjugate. If \(\mathcal{G}\) is simply connected, this is a necessary and sufficient condition to have \(\mathcal{K}g \mathcal{K}= \mathcal{K}h \mathcal{K}\), so the conjugacy class of the Cartan double \(g\theta(g)^{-1}\) supplies canonical parameters [9]. For instance, the \(\operatorname{SO}(n)\)-double coset of \(U \in \operatorname{SU}(n)\) is uniquely identified by the eigenvalues of \(UU^T\).

Unfortunately neither \(\operatorname{SO}(n)\) nor \(\operatorname{PSO}(n)\) is simply connected for \(n > 1\). We will not attempt to fix this issue in general, contenting ourselves with definitions that work in our specific cases of interest, which we return to in §2.6.

We end this section with a simple but useful lemma on Lie groups.

2.4 Commuting involutions↩︎

Suppose \(\theta_1, \theta_2 : \mathcal{G}\to \mathcal{G}\) are two commuting involutive automorphisms, i.e. \[\theta_1^2 = \theta_2^2 = \mathop{\mathrm{id}}= \theta_1 \theta_2 \theta_1^{-1} \theta_2^{-1}.\] Apply Cartan decomposition with respect to \(\theta_1\) to \(g \in \mathcal{G}\), writing \(g = k_1 a k_2\) with \(k_1, k_2 \in \mathcal{K}_{\theta_1}\) and \(a_1 \in \mathcal{A}_{\theta_1}\). The commuting property implies \(\theta_2(\mathcal{K}_{\theta_1}) \subseteq \mathcal{K}_{\theta_1}\), so one can Cartan decompose \(k_1, k_2\) using the automorphism \(\theta_2 : \mathcal{K}_{\theta_1} \to \mathcal{K}_{\theta_1}\). The relevant fixed-point subgroup in \(\mathcal{K}_{\theta_1}\) is \((\mathcal{K}_{\theta_1} \cap \mathcal{K}_{\theta_2})_0\), which we abbreviate as \(\mathcal{K}_{\theta_1,\theta_2}\). This gives an expression \[g = k_3 b_1 k_4 a k_5 b_2 k_6.\] with \(k_3,k_4,k_5,k_6 \in \mathcal{K}_{\theta_1,\theta_2}\), and \(b_1, b_2\) in a torus maximal in \((\mathcal{K}_{\theta_1})^{-\theta_2}\).

2.5 Two-qubit magic↩︎

In this subsection we review some useful facts about 2-qubit gates which will also be crucial in our 3-qubit decomposition.

For instance, \[Q = \frac{1}{2}\begin{bmatrix} 1 & 1 & i & i\\ 1 & -1 & i & -i\\ -1 & 1 & i & -i\\ 1 & 1 & -i & -i \end{bmatrix}\] is a magic matrix, as can be verified by checking that \(Q^\dagger (X \otimes I_2) Q\) and \(Q^\dagger (I_2 \otimes X) Q\) are real for \(X \in \{i\sigma_x, i\sigma_y, i\sigma_z\}\).

The existence and use of magic matrices is well-known [11]–[13]. We take a moment to discuss a more conceptual explanation, but this material can safely be skipped.

The basic example is that \(c\) is complex conjugation. Viewing \(c\) as a real-linear map, it has \(1\)- and \((-1)\)-eigenspaces \(V_+\) and \(V_-\), which one thinks of as the “real” and “imaginary” elements of \(\mathbb{C}^n\) with respect to \(c\). If \(A\) is an \(n \times n\) complex matrix commuting with \(c\), then \(AV_+ = V_+\), and if \(B\) is an \(n \times n\) matrix whose columns form an orthonormal basis of \(V_+\), then \(B^\dagger AB\) is a real matrix.

Let \(\iota : \mathbb{C}^2 \to \mathbb{C}^2\) be the conjugate-linear map with \(\iota(\bra{0}) = \bra{1}\) and \(\iota(\bra{1}) = -\bra{0}\). Define \(c = \iota^{\otimes n} : (\mathbb{C}^2)^{\otimes n} \to (\mathbb{C}^2)^{\otimes n}\). Thus, if \(w\) is a binary word and \(\tilde{w}\) denotes its negation, we have \(c(\bra{w}) = (-1)^{\sum w} \bra{\tilde{w}}\). Since \(\sum w + \sum \tilde{w} = n\) for any \(w\), we see that \(c^2 = (-1)^n\). Therefore \(c\) is a real structure on \((\mathbb{C}^2)^{\otimes n}\) if and only if \(n\) is even. Moreover, a matrix of the form \(\left[\begin{smallmatrix} a & -\overline{b}\\ b & \overline{a}\end{smallmatrix}\right]\) commutes with \(\iota\), so \(c\) commutes with all elements of \(\operatorname{SU}(2)^{\otimes n}\). It follows that magic matrices exist for any even \(n\): take the columns to be an orthonormal basis of \(c\)-fixed vectors.

One possible orthogonal basis of \(V_+\) consists of the vectors \[\bra{w} + \bra{c(w)} \qquad \text{and} \qquad i(\bra{w} - \bra{c(w)})\] over binary words \(w\) with \(w_1 = 0\). In the case \(n = 2\), we make a different choice: \[\begin{align} &\tfrac{1}{2}(\bra{00} + \bra{01} - \bra{10} + \bra{11})\\ &\tfrac{1}{2}(\bra{00} - \bra{01} + \bra{10} + \bra{11})\\ &\tfrac{1}{2}(i\bra{00} + i\bra{01} + i\bra{10} - i\bra{11})\\ &\tfrac{1}{2}(i\bra{00} - i\bra{01} - i\bra{10} - i\bra{11}), \end{align}\] This orthonormal basis leads to the matrix 4 . The \(n = 2\) case is special, in fact.

More advanced techniques from representation theory (the Frobenius-Schur indicator) can be used to show that magic matrices do not exist for odd \(n\). In that case the map \(c\) satisfies \(c^2 = -1\), defining a so-called quaternionic structure on \((\mathbb{C}^2)^{\otimes n}\) and yielding matrices \(R\) such that \(R^\dagger \operatorname{SU}(2)^{\otimes n} R\) is contained in the symplectic group \(\operatorname{Sp}(2^{n-1})\) (cf. Definition 8). See [11] for more detail on these constructions.

For the rest of the paper we fix the magic matrix \[\label{eq:magic} \mathcal{Q}= \frac{1}{2}\begin{bmatrix} 1 & 1 & i & i\\ 1 & -1 & i & -i\\ -1 & 1 & i & -i\\ 1 & 1 & -i & -i \end{bmatrix}.\tag{4}\] Also set \(\mathcal{M}= I_2 \otimes \mathcal{Q}\). Standard techniques for two-qubit gates show that \(\mathcal{Q}\) is equivalent to a single CNOT up to multiplication by single-qubit gates: \[\label{eq:magic-circuit} \mathcal{Q}= \begin{quantikz} & \gate{R_x(\pi/2)} & \ctrl{1}& \gate{R_x(-\pi)} & &\\ & \gate{R_z(-\pi/2)} & \targ{} & \gate{R_x(\pi/2)} & \gate{R_z(-\pi/2)} & \end{quantikz}\tag{5}\] Our choice of \(\mathcal{Q}\) is somewhat arbitrary except that it seems to yield nicer formulas (e.g.in Figure 2 later) than some other possible choices. We have not tried to optimize \(\mathcal{Q}\) in any rigorous sense.

The next result is a technical lemma for later use.

2.6 Block matrix subgroups of \(\operatorname{PSO}(n)\)↩︎

Suppose \(\pi = \{\pi_1, \ldots, \pi_k\}\) is a partition of the set \([n]\). That is, \([n]\) is the disjoint union of the sets \(\pi_1, \ldots, \pi_k\), called the blocks of the partition. For instance, \(\{\{1,3,4\},\{2,6\},\{5\}\}\) is a partition of \([6]\) which we will usually abbreviate as \(134|26|5\) (or equivalently \(26|134|5\), etc.). Let \(\operatorname{SO}(\pi)\) be the subgroup of \(\operatorname{SO}(n)\) consisting of block matrices whose blocks occur in positions \(\pi_i \times \pi_i\) for \(i = 1, \ldots, k\), and which all have determinant 1. Following our usual notation, \(\operatorname{PSO}(\pi)\) then denotes the image of \(\operatorname{SO}(\pi)\) in \(\operatorname{PSO}(n)\).

The meet of two partitions \(\pi\) and \(\pi'\) of the same set is \[\pi \cap \pi' = \{b \cap b' : b \in \pi, b' \in \pi', b \cap b' \neq \emptyset\}.\] For instance, \(123|45678 \cap 234|15678 = 1|23|4|5678\).

In general, \(\operatorname{PSO}(\pi) \cap \operatorname{PSO}(\pi')\) may not be connected, e.g.\(\operatorname{PSO}(123|456) \cap \operatorname{PSO}(14|25|36)\) consists of the 4 matrices \(A \oplus A\) where \[A \in \{I_3, \Delta_{\{1\}}, \Delta_{\{2\}}, \Delta_{\{3\}}\}.\] However, since \(123|456 \cap 14|25|36 = 1|2|3|4|5|6\), Lemma 3 correctly predicts that \((\operatorname{PSO}(123|456) \cap \operatorname{PSO}(14|25|36))_0\) is trivial.

If \(\pi = \{\pi_-,\pi_+\}\) has just two blocks, then \(\operatorname{PSO}(\pi)\) is a Cartan subgroup, namely \(\mathcal{K}_{\theta}\) where \(\theta = c_{\Delta_{\pi_-}}\). We can then apply the techniques of Cartan decomposition from §2.3, for which it is useful to have canonical parameters. We start with the slightly simpler case of \(\operatorname{SO}(n)\).

The factorization of \(V \in \operatorname{O}(n)\) as \[V = L_1 \begin{bmatrix} \cos D & 0 & -\sin D \\ 0 & I_{n-2p} & 0 \\ \sin D & 0 & \cos D \end{bmatrix} L_2\] with \(L_1,L_2 \in \operatorname{O}(p) \oplus \operatorname{O}(n-p)\) (cf. Example 3) is an instance of the cosine-sine decomposition. A numerical algorithm for computing it can be found in [14]. Using such an algorithm, we can compute matrices \(K_1,K_2 \in \operatorname{SO}(p) \oplus \operatorname{SO}(n-p)\) with \(K_1 V K_2 = U\) whenever \(U,V \in \operatorname{SO}(n)\) have the same \(p\)-canonical parameters (also using the operations in the proof of Proposition 3 to ensure that the blocks of \(K_1\) and \(K_2\) have determinant 1).

Strictly speaking, we are interested in the projective case \(\mathcal{G}= \operatorname{PSO}(n)\) and \(\mathcal{K}= \operatorname{P}(\operatorname{SO}(p) \oplus \operatorname{SO}(n-p))\), in which case a slight modification of Definition 1 is required to get canonical parameters. In fact the difference will never really turn out to matter, but we include the following for completeness.

2.7 Automorphisms and roots↩︎

Constructing a triality symmetry of \(\mathfrak{so}(8)\) will be easier after reviewing some generalities on Lie algebras. For more detailed background, see [10].

An automorphism of a Lie group \(\mathcal{G}\) is an invertible function \(\theta : \mathcal{G}\to \mathcal{G}\) satisfying \(\theta(gh) = \theta(g)\theta(h)\), and an automorphism of a Lie algebra \(\mathfrak{g}\) is an invertible linear function \(\phi : \mathfrak{g}\to \mathfrak{g}\) satisfying \(\phi([X,Y]) = [\phi(X),\phi(Y)]\). In either case, the set of automorphisms forms a group under composition, written \(\mathop{\mathrm{Aut}}(\mathcal{G})\) or \(\mathop{\mathrm{Aut}}(\mathfrak{g})\) as appropriate. Fixing \(g \in \mathcal{G}\), each conjugation function \(c_g(x) = gxg^{-1}\) defines an automorphism of \(\mathcal{G}\), and its derivative \(\mathop{\mathrm{Ad}}_g\) at the identity is an automorphism of \(\mathfrak{g}\). In the case where \(\mathcal{G}\) is a matrix Lie group, \(\mathop{\mathrm{Ad}}_g : \mathfrak{g}\to \mathfrak{g}\) will also simply be conjugation by \(g\). Automorphisms of the type \(c_g\) and \(\mathop{\mathrm{Ad}}_g\) are inner automorphisms, and make up subgroups \(\mathop{\mathrm{Inn}}(\mathcal{G}) \subseteq \mathop{\mathrm{Aut}}(\mathcal{G})\) and \(\mathop{\mathrm{Inn}}(\mathfrak{g}) \subseteq \mathop{\mathrm{Aut}}(\mathfrak{g})\). The outer automorphism group of \(\mathcal{G}\) is the quotient \(\mathop{\mathrm{Out}}(\mathcal{G}) = \mathop{\mathrm{Aut}}(\mathcal{G})/\mathop{\mathrm{Inn}}(\mathcal{G})\), and similarly in the Lie algebra case. A typical example of an non-inner automorphism is complex conjugation acting on \(\operatorname{U}(n)\) or on \(\mathfrak{u}(n)\).

It follows from the classification of compact Lie algebras by their Dynkin diagrams that, with one exception, if \(\mathfrak{g}\) is the Lie algebra of a simple compact group then \(\mathop{\mathrm{Out}}(\mathfrak{g})\) is either trivial or has a single non-identity element \(\phi\) which has \(\phi^2 = \mathop{\mathrm{id}}\). For instance, take \(\phi\) to be complex conjugation in the case \(\mathfrak{g}= \mathfrak{su}(n)\) and conjugation by a reflection in the case \(\mathfrak{g}= \mathfrak{so}(2m)\). The one exception is that \(\mathop{\mathrm{Out}}(\mathfrak{so}(8))\) has order 6, being isomorphic to the symmetric group on 3 elements. An element of order 3 in \(\mathop{\mathrm{Out}}(\mathfrak{so}(8))\) is called a triality map, and we will construct one in §3.1.

Now suppose \(\mathfrak{g}^0\) is the Lie algebra of a compact group \(\mathcal{G}\), and set \(\mathfrak{g}= \mathfrak{g}^0 \otimes \mathbb{C}\). Each \(Z \in \mathfrak{g}^0\) defines a linear operator \(\mathop{\mathrm{ad}}_Z : \mathfrak{g}\to \mathfrak{g}, X \mapsto [Z,X]\), which can be shown¹ to be diagonalizable. Choose a maximal abelian subalgebra \(\mathfrak{h}^0 \subseteq \mathfrak{g}^0\) and set \(\mathfrak{h}= \mathfrak{h}^0 \otimes \mathbb{C}\). Since the operators \(\{\mathop{\mathrm{ad}}_H : H \in \mathfrak{h}\}\) commute, they are simultaneously diagonalizable. This means \(\mathfrak{g}\) breaks up as a direct sum of eigenspaces \(\bigoplus_{\alpha \in \Phi \cup \{0\}} \mathfrak{g}_{\alpha}\) satisfying \[\mathop{\mathrm{ad}}_H(X) = [H,X] = \alpha(H)X \qquad \text{for all H \in \mathfrak{h} and X \in \mathfrak{g}_{\alpha}}.\] Here each eigenvalue \(\alpha\) depends linearly on \(H\), i.e.\(\alpha\) is an element of \(\mathfrak{h}^*\). The zero eigenspace \(\mathfrak{g}_0\) is just \(\mathfrak{h}\) itself. The set of nonzero eigenvalues is a finite subset \(\Phi \subseteq \mathfrak{h}^*\) called the root system of \(\mathfrak{g}\) (with respect to \(\mathfrak{h}\)). Elements of \(\Phi\) are roots and their eigenspaces \(\mathfrak{g}_\alpha\) are root spaces.

Next we describe how an automorphism \(\phi : \mathfrak{g}\to \mathfrak{g}\) is (almost) determined by its action on roots. Assume \(\phi\) maps \(\mathfrak{h}\) to \(\mathfrak{h}\); it follows from [10] that this assumption can always be satisfied without changing the element of \(\mathop{\mathrm{Out}}(\mathfrak{g})\) that \(\phi\) represents. For \(X \in \mathfrak{g}_\alpha\), \[\label{eq:root-action} [H, \phi(X)] = \phi [\phi^{-1}(H), X] = \alpha(\phi^{-1}(H)) \phi(X).\tag{7}\] The linear functional \(H \mapsto \alpha(\phi^{-1}(H))\) is by definition \((\phi^{-1})^*(\alpha)\) where \((\phi^{-1})^*\) is the dual map to \(\phi^{-1}\). In this language, 7 shows that \(\phi\) maps \(\mathfrak{g}_\alpha\) to \(\mathfrak{g}_{(\phi^{-1})^* \alpha}\).

The final key fact is that each root space \(\mathfrak{g}_\alpha\) is 1-dimensional [10]. Thus if we choose a nonzero \(X_\alpha \in \mathfrak{g}_{\alpha}\) for each \(\alpha\) (a root vector), we must have \(\phi(X_{\alpha}) = c_\alpha X_{(\phi^{-1})^* \alpha}\) for some scalars \(c_\alpha\). The conclusion is that, up to these scalars, \(\phi : \mathfrak{g}\to \mathfrak{g}\) is completely determined by its action on \(\mathfrak{h}\). Conversely, if one just starts with the action of \(\phi\) on \(\mathfrak{h}\), the next theorem asserts that the scalars \(c_\alpha\) can be chosen so that the formula \(X_{\alpha} \mapsto c_\alpha X_{(\phi^{-1})^* \alpha}\) actually defines a Lie algebra homomorphism.

3 Triality↩︎

In this section we construct triality automorphisms for \(\mathfrak{so}(8)\) and \(\operatorname{PSO}(8)\). One possible construction—indeed, Cartan’s original construction—proceeds via the algebra of octonions [15]. We give a different description relying on quaternion arithmetic, which we extracted from an article of Baez [16]; see also [17]. The reader who is just interested in a quick explicit description of these maps for purposes of calculation can skip to the end of §3.1 (or §3.4) and §3.2 respectively.

3.1 Triality for \(\mathfrak{so}(8)\)↩︎

As discussed in the previous section, constructing a triality automorphism mostly amounts to describing its action on roots, so we start with an explicit description of the root system and root vectors for \(\mathfrak{so}(2m)\). Let \(f_{ji}\) be the matrix with \(1\) in position \((j,i)\) and \(-1\) in position \((i,j)\) and \(0\)’s elsewhere. Let \(H_i = f_{2i,2i-1}\) for \(i = 1, \ldots, m\) and \(\mathfrak{h}^0 = \operatorname{span}_{\mathbb{R}} \{H_1, \ldots,H_m\}\). Then the \({2m \choose 2}\) matrices \(\{f_{ji} : 1 \leq i < j \leq m\}\) form a basis of \(\mathfrak{so}(2m)\), and \(\mathfrak{h}^0\) is a maximal abelian subalgebra. For instance, in the case \(m = 4\), \[a_1 H_1 + a_2 H_2 + a_3 H_3 + a_4 H_4 = \left[\begin{smallmatrix} 0 & -a_1 & 0 & 0 & 0 & 0 & 0 & 0\\ a_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -a_2 & 0 & 0 & 0 & 0\\ 0 & 0 & a_2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -a_3 & 0 & 0\\ 0 & 0 & 0 & 0 & a_3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -a_4\\ 0 & 0 & 0 & 0 & 0 & 0 & a_4 & 0 \end{smallmatrix}\right]\] and \(\mathfrak{h}^0\) is the set of all such matrices for \(a_1,a_2,a_3,a_4 \in \mathbb{R}\).

Write \(\mathfrak{so}(n,\mathbb{C}) := \mathfrak{so}(n) \otimes \mathbb{C}\) and \(\mathfrak{h}:= \mathfrak{h}^0 \otimes \mathbb{C}\). Note that \(\mathfrak{so}(n,\mathbb{C})\) is the space of complex skew-symmetric (not skew-Hermitian!) matrices. Let \(e_1, \ldots, e_m \in \mathfrak{h}^*\) be the dual basis to \(-iH_1,\ldots,-iH_m\), so \(e_j(H_k) = i\delta_{jk}\). The root system of \(\mathfrak{so}(2m,\mathbb{C})\) then consists of the \(2m(m-1)\) linear functionals \(\{\pm e_p \pm e_q : p \neq q\}\). For root vectors we take

• \(X_{e_p-e_q} \in \mathfrak{so}(2m,\mathbb{C})_{e_p-e_q}\) the skew-symmetric matrix with \(\tfrac{1}{2}\left[ \begin{smallmatrix} 1 & i \\ -i & 1 \end{smallmatrix}\right]\) in entries \([2p-1,2p] \times [2q-1,2q]\) and \(0\)’s elsewhere above the diagonal,

• \(X_{e_p+e_q} \in \mathfrak{so}(2m,\mathbb{C})_{e_p+e_q}\) the skew-symmetric matrix with \(\tfrac{1}{2}\left[ \begin{smallmatrix} 1 & -i \\ -i & -1 \end{smallmatrix}\right]\) in entries \([2p-1,2p] \times [2q-1,2q]\) and \(0\)’s elsewhere above the diagonal,

• \(X_{-\alpha} = \overline{X_{\alpha}} \in \mathfrak{so}(2m,\mathbb{C})_{\alpha}\) if \(\alpha = e_p\pm e_q\),

where we assume \(p < q\) in all cases.

For convenience, identify \(\mathfrak{h}^0 \subseteq \mathfrak{so}(2m)\) with \(\mathbb{R}^m\) by taking \(H_1, \ldots, H_m\) to be the standard basis, and likewise identify \(i(\mathfrak{h}^0)^* = \operatorname{span}_{\mathbb{R}} \{e_1,\ldots,e_m\}\) with \(\mathbb{R}^m\) by taking \(e_1, \ldots, e_m\) as the standard basis. For instance, \((1,0,0,-1)\) represents either \(H_1-H_4\) or \(e_1-e_4\) as determined by context.

Let \[D_m = \{\pm e_i \pm e_j : i,j \in [m], i \neq j\} \subseteq \mathbb{R}^m\] be the root system of \(\mathfrak{so}(2m,\mathbb{C})\) for \(m > 1\). The root lattice \(\mathbb{Z}D_m\) is the set of integer linear combinations of elements of \(D_m\), and its dual lattice is \[(\mathbb{Z}D_m)^* = \{x \in \mathbb{Z}^m : x \cdot v \in \mathbb{Z}\text{ for all v \in \mathbb{Z}D_m} \}.\]

In words: \(\mathbb{Z}D_m\) is the set of integer vectors with even sum, while \((\mathbb{Z}D_m)^*\) consists of the vectors \((x_1, \ldots, x_m)\) with the \(x_i\) either being all integers or all half-integers.

We now turn to the case \(m = 4\) in particular. Identify \((x_1,x_2,x_3,x_4) \in \mathbb{R}^4\) with the quaternion \(x_1 + x_2 i + x_3 j + x_4 k\). The only knowledge we require of the algebra of quaternions \(\mathbb{H}= \{x_1 + x_2 i + x_3 j + x_4 k : x_1,x_2,x_3,x_4 \in \mathbb{R}\}\) is the basic multiplication rules \[i^2 = j^2 = k^2 = -1, \qquad ij = -ji = k,\quad jk = -kj = i,\quad ki = -ik = j,\] the associativity of the multiplication, and the fact that the norm \[|x_1+x_2 i+x_3 j + x_4 k| = \sqrt{x_1^2+x_2^2+x_3^2+x_4^2}\] satisfies \(|q_1 q_2| = |q_1||q_2|\).

The miracle that occurs after identifying \(\mathbb{R}^4\) with \(\mathbb{H}\) is that \((\mathbb{Z}D_4)^*\) is not just a lattice but a subring of \(\mathbb{H}\), i.e.it is closed under multiplication.

Proposition 5(b) identifies \((\mathbb{Z}D_4)^*\) with the set of Hurwitz quaternions. Hurwitz proposed them as a notion of “integral” quaternions, as they enjoy some number-theoretic benefits over the more obvious set of quaternions with integer coefficients. It is a good exercise to check that that the product of two Hurwitz quaternions must be a third.

Because of the subring property, the linear map \(m_q(x) = qx\) defined by each \(q \in (\mathbb{Z}D_4)^*\) maps \((\mathbb{Z}D_4)^*\) to itself. Therefore we also get a dual map \(m_q^* : \mathbb{Z}D_4 \to \mathbb{Z}D_4\). In order to lead to an automorphism of \(\mathfrak{so}(8)\), this map must preserve the root system \(D_4\), which consists of those integer vectors in \(\mathbb{Z}D_4\) with length \(\sqrt{2}\). Hence \(m_q^*\) should be length-preserving, which by the property \(|qx| = |q||x|\) happens exactly if \(|q| = 1\). There are \(24\) norm 1 Hurwitz quaternions: \[\pm 1, \pm i, \pm j, \pm k, \qquad \text{and} \qquad \frac{\pm 1 \pm i \pm j \pm k}{2}.\] Set \(\omega = \tfrac{-1+i+j+k}{2}\). One checks that \(\omega^3 = 1\), so \((m_{\omega}^*)^3 = \mathop{\mathrm{id}}\). We have now constructed an order 3 operator \(m_{\omega}^* \otimes \mathbb{C}\) on \(\mathbb{Z}D_4 \otimes \mathbb{C}\simeq \mathfrak{h}^*\) preserving the root system \(D_4\), which extends to an order 3 automorphism of \(\mathfrak{so}(8,\mathbb{C})\) by Theorem 4. The latter is the desired triality map.

Let us make this construction more explicit. Recall that we are identifying

• the basis \(H_1, H_2,H_3,H_4\) of \(\mathfrak{h}^0\)

• the basis \(e_1,e_2,e_3,e_4\) of \(i(\mathfrak{h}^0)^*\)

• the basis \(1,i,j,k\) of \(\mathbb{H}\).

Since \[\omega i = \frac{-1-i+j-k}{2}, \qquad \omega j = \frac{-1-i-j+k}{2}, \qquad \omega k = \frac{-1+i-j-k}{2},\] the linear map \(T : \mathfrak{h}^0 \to \mathfrak{h}^0, H \mapsto \omega H\) has matrix \[\label{eq:triality-matrix-h} \frac{1}{2} \begin{bmatrix} -1 & -1 & -1 & -1\\ 1 & -1 & -1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 1 & -1 \end{bmatrix}.\tag{8}\] Note that this matrix is orthogonal, so under our identification of \(\mathfrak{h}^0\) and \(i(\mathfrak{h}^0)^*\), the same matrix represents the map \((T^{-1})^*\). Accordingly, we write the action of both \(T\) and \((T^{-1})^*\) as just \(x \mapsto \omega x\). As per Theorem 4, we can extend \(T\) to an automorphism of \(\mathfrak{so}(8,\mathbb{C})\) by sending \(X_\alpha \mapsto c_\alpha X_{\omega \alpha}\) for appropriate scalars \(c_\alpha\). We will not discuss a conceptual way to think about these scalars, but the required automorphism equations \(\tau([X_{\alpha},X_{\beta}]) = [\tau(X_{\alpha}), \tau(X_{\beta})]\) for all \(\alpha,\beta \in \Phi\) lead to a system of equations for the \(c_{\alpha}\), using the fact that \([X_{\alpha}, X_{\beta}] \in \mathfrak{g}_{\alpha+\beta}\). This yields the following explicit definition.

What we really want is an automorphism of the real subalgebra \(\mathfrak{so}(8) \subseteq \mathfrak{so}(8,\mathbb{C})\). By construction, \(\overline{X_\alpha} = X_{-\alpha}\) for all \(\alpha\), so the real subalgebra \(\mathfrak{so}(8)\) is spanned by \(\mathfrak{h}^0\) together with all vectors \(X_{\alpha}+X_{-\alpha}\) and \(i(X_{\alpha}-X_{-\alpha})\). Evidently \(\tau\) preserves the span of such vectors, so it restricts to an automorphism of \(\mathfrak{so}(8)\) as desired. See Example 10 below for an example calculation with \(\tau\).

Even more explicitly, consider the matrix of \(\tau\) with respect to the basis \(\{f_{ji} : 1 < i < j \leq 8\}\) of \(\mathfrak{so}(8)\), ordered with the pairs \((j,i)\) in lex order. Reading the entries of this \(28 \times 28\) matrix left to right across rows, starting with row 1, gives the string

where \(-\) indicates \(-1/2\), \(+\) indicates \(1/2\), and each digit \(d\) is short for a string of \(d\) \(0\)’s (so 66 means a string of 12 zeros).

Several arbitrary choices went into our construction of \(\tau\), and one could equally well call any \(\alpha \circ \tau \circ \alpha^{-1}\) for \(\alpha \in \mathop{\mathrm{Aut}}(\mathfrak{so}(8))\) a triality map. Empirically, Definition 6 seems to lead to nicer formulas later than some other possible choices, but we have not tried to make this idea precise.

3.2 Triality for \(\operatorname{PSO}(8)\)↩︎

We would like to lift \(\tau\) to an automorphism \(\mathcal{T}\) acting on real 3-qubit gates by setting \(\mathcal{T}(\exp(X)) = \exp \tau(X)\). Unfortunately this simply does not work if we view \(X\) as an element of \(\operatorname{SO}(8)\). For instance, since \(\exp \left[ \begin{smallmatrix} 0 & -x \\ x & 0 \end{smallmatrix} \right] = \left[ \begin{smallmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{smallmatrix} \right]\), we would have \(\mathcal{T}(\exp(2\pi H_1)) = \mathcal{T}(I) = I\) but \[\exp(\tau(2\pi H_1)) = \exp(\pi(-H_1+H_2+H_3+H_4)) = -I.\] Instead we use \(\operatorname{PSO}(8)\).

Since \(\operatorname{PSO}(8)\) has trivial center, Lemma 4 assures us this is well-defined.

Since the matrices \(f_{ji}\) generate \(\mathfrak{so}(8)\), the corresponding 1-parameter families of rotations \(\exp(\theta f_{ji})\) generate \(\operatorname{PSO}(8)\). These are called Givens rotations. One can check that \(\tau(f_{ji})^2 = -1/4\) for all \(i,j\), so \(\exp(\theta f_{ji}) = \cos(\theta_{ji}/2) + 2\sin(\theta_{ji}/2) \tau(f_{ji})\). These facts lead to the following algorithm computing \(\mathcal{T}(V)\) for an arbitrary \(V \in \operatorname{PSO}(8)\), amounting to repeated applications of Example 10.

(a) Write \(V\) as a product of Givens rotations \(\exp(\theta_{ji} f_{ji})\) in some order.

(b) Compute each \(\tau(f_{ji})\), e.g.using the fact that each \(f_{ji}\) is a linear combination of the 4 root vectors \(X_{\pm e_i \pm e_j}\).

(c) Replace each \(\exp(\theta_{ji} f_{ji})\) in the decomposition from (a) by \[\exp(\theta_{ji} \tau(f_{ji})) = \cos(\theta_{ji}/2) + 2\sin(\theta_{ji}/2) \tau(f_{ji}).\] The resulting product is \(\mathcal{T}(V)\).

3.3 Some correspondences under triality↩︎

We now work out how \(\mathcal{T}\) acts on some interesting subgroups \(\mathcal{G}\) of \(\operatorname{PSO}(8)\). Let \(\mu : \operatorname{SU}(8) \to \operatorname{SU}(8)\) be conjugation by \(\mathcal{M}^\dagger\), so \(\mu(U) = \mathcal{M}^\dagger U \mathcal{M}\) and \(\mu(\operatorname{SU}(2)^{\otimes 3}) = \operatorname{SU}(2) \otimes \operatorname{SO}(4)\). It will be convenient to record correspondences between elements \(V \in \operatorname{PSU}(8)\) and \(\mathcal{T}(\mu(V))\). Of course, this only makes sense if \(\mu(V)\) happens to be real, so more accurately we view \(\mu\) as a map \(\operatorname{PSO}(8) \to \mathcal{M}\operatorname{PSO}(8)\mathcal{M}^\dagger\). The reason for the use of \(\mu\) is explained in §4.

Figure 2 shows the results of applying \(\mathcal{T}\circ \mu\) to various one-parameter subgroups \(\exp(\mathbb{R}X) = \{\exp(tX) : t \in \mathbb{R}\}\). Since these amount to just computing the one element \(\tau(\mu(X))\), we omit the verifications.

The entries in Figure 2 can be used to work out what happens to larger subgroups. For instance:

An interesting feature of the triality map is that it realizes all of the exceptional isomorphisms between the classical matrix Lie algebras of types \(A_n, B_n, C_n, D_n\), in various ways. In this way, it is similar to the magic basis change \(\mu\), which realizes the isomorphism \(\operatorname{SO}(4) \simeq (\operatorname{SU}(2) \times \operatorname{SU}(2))/\{\pm 1\}\). For instance:

• \(A_1 \simeq B_1\): Proposition 6(a-c) show \(\operatorname{PSU}(2) \simeq \operatorname{PSO}(3)\).

• \(D_2 \simeq A_1 \oplus A_1\): One can check \(\mathcal{T}^{-1}\operatorname{PSO}(125|378|4|6) = \operatorname{P}(I_2 \otimes \operatorname{SO}(4))\), so \(\operatorname{PSO}(4) \simeq \operatorname{PSO}(3) \times \operatorname{PSO}(3)\).

• \(A_3 \simeq D_3\): One can check that \(\mathcal{T}^{-1}\operatorname{PSO}(123578|4|6)\) is the group \[\left\{ \left[\begin{smallmatrix} \operatorname{Re}(U) & -\operatorname{Im}(U) \\ \operatorname{Im}(U) & \operatorname{Re}(U) \end{smallmatrix} \right] : U \in \operatorname{SU}(4) \right\},\] so \(\operatorname{SU}(4) / \{\pm I\} \simeq \operatorname{PSO}(6)\).

• \(C_2 \simeq B_2\): \(\operatorname{P}(\operatorname{Sp}(2)) \simeq \operatorname{PSO}(5)\) follows from Lemma 7 in the next section, where the symplectic groups \(\operatorname{Sp}(n)\) are also defined.

3.4 Pauli operators and triality↩︎

In §3.1 we used the basis of skew-symmetric unit matrices \(\{f_{ji} : 1 \leq i < j \leq 8\}\) of \(\mathfrak{so}(8)\). It turns out that, up to scalars, the triality map identifies these matrices with another natural basis of \(\mathfrak{so}(8)\), the basis of Pauli operators.

We will use a word \(abc\cdots\) on the alphabet \(\{X,Y,Z,I\}\) as an abbreviation for the tensor product \(i\sigma_a \otimes \sigma_b \otimes \sigma_c \otimes \cdots\), where \(\sigma_I\) means the \(2 \times 2\) identity \(I_2\). For instance, \[YXI = i\sigma_y \otimes \sigma_x \otimes I_2 = \left[ \begin{smallmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \end{smallmatrix} \right]\] Each \(\sigma_a\) is Hermitian, so such a tensor product is skew-Hermitian. Indeed, the \(4^n\) \(n\)-qubit Pauli operators \(W \in \{I,X,Y,Z\}^n\) form a basis of \(\mathfrak{su}(2^n)\). Since \(\sigma_x, \sigma_z, I_2\) are real while \(\sigma_y\) is pure imaginary, a Pauli operator \(W\) is real iff it has an odd number of \(Y\)’s. One can calculate using the binomial theorem that the number of such operators is \({2^n \choose 2} = \dim \mathfrak{so}(2^n)\), so they form a basis of \(\mathfrak{so}(2^n)\).

The following array shows the correspondence between Pauli operators and skew-symmetric unit matrices \(f_{ji}\), where a Pauli operator \(\pm W\) in position \((j,i)\) means that \(\tau(W) = \pm 2f_{ji}\): \[\label{eq:pauli-correspondence} \text{\texttt{ \begin{tabular}{cccccccc} · & · & · & · & · & · & · & · \\ +IIY & · & · & · & · & · & · & ·\\ +YXZ & -YXX & · & · & · & · & · & ·\\ +XYX & +XYZ & -ZZY & · & · & · & · & ·\\ -IYZ & +IYX & +YZI & -XIY & · & · & · & ·\\ -ZYX & -ZYZ & -XZY & -YII & -ZIY & · & · & ·\\ -YZZ & +YZX & -IYI & +ZXY & -YXI & +XXY & · & ·\\ -YIX & -YIZ & +IXY & +ZYI & -YYY & +XYI & -IZY & · \end{tabular}}}\tag{9}\] For instance, \(\tau(i\sigma_y \otimes \sigma_x \otimes \sigma_x) = -2f_{32}\). Alternatively, one could view a Pauli operator \(W\) as an element of \(\operatorname{PSO}(8)\), in which case 9 says that \[\mathcal{T}(W) = \mathcal{T}(\exp(\pi W/2)) = \exp(\pm \pi f_{ji}) = J_{\{i,j\}}\] if \(W\) occurs in entry \((j,i)\).

The significance of this is that it makes it easy to find the Cartan decomposition \(\mathfrak{k}\oplus \mathfrak{p}\) with respect to an involution of the form \(c' = \tau^{-1} \circ c \circ \tau\): \(\mathfrak{k}\) is the span of the Pauli operators fixed by \(c'\), while \(\mathfrak{p}\) is spanned by those negated by \(c'\).

4 Real circuit elements↩︎

It is a priori unclear when a long product of CNOTs and single-qubit gates actually multiplies to a real matrix. This is one advantage of using the magic matrix \(\mathcal{M}\): we know at least that any element of \(\mathcal{M}^\dagger (R_y(\mathbb{R}) \otimes \operatorname{SU}(2) \otimes \operatorname{SU}(2))\mathcal{M}\) is guaranteed to be real. In this section we discuss some other basic circuit elements that will be useful in constructing real 3-qubit gates. These considerations also lead us toward to an interesting subgroup of \(\operatorname{PSO}(8)\) that turns out to be key to our Cartan decomposition in §5 later.

For instance, \(\operatorname{Sp}(1) = \operatorname{SU}(2)\). The symplectic group \(\operatorname{Sp}(n)\) is a compact simply connected group of dimension \(n(2n+1)\). Setting \(\Omega = I_n \otimes i\sigma_y = \left[ \begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix} \right]^{\oplus n}\), an alternative phrasing of Definition 8 is that \(\operatorname{Sp}(n) = \{U \in \operatorname{U}(2n) : \Omega \overline{U} \Omega^T = U\}\). This reveals \(\operatorname{Sp}(n)\) as a Cartan subgroup of \(\operatorname{U}(2n)\).

As an aside, the algebra of quaternions \(\mathbb{H}\) is isomorphic to the algebra of \(2 \times 2\) matrices of the form \(\left[ \begin{smallmatrix} w & -\bar{z} \\ z & \bar{w} \end{smallmatrix} \right]\) by identifying this matrix with \(w+jz\). Making this replacement for each \(2 \times 2\) block reveals \(\operatorname{Sp}(n)\) as the group of \(n \times n\) unitary quaternionic matrices. We will not use this perspective, but it has the nice feature of unifying the classical compact matrix groups \(\operatorname{O}(n)\), \(\operatorname{U}(n)\), and \(\operatorname{Sp}(n)\) as being the unitary matrices whose entries come from the three possible associative real division algebras \(\mathbb{R}\), \(\mathbb{C}\), and \(\mathbb{H}\).

Combining Lemmas 5 and 6 gives:

The discussion above identifies \(\mu\operatorname{P}(\operatorname{Sp}(2) \otimes \operatorname{SU}(2))\) as a subgroup of \(\operatorname{PSO}(8)\) determined by relatively simple circuit elements. The second key fact about it is that it is a Cartan subgroup, which we leverage in the next section to derive our new circuit for \(\operatorname{PSO}(8)\).

Given the utility of the slightly mysterious subgroup \(\mathcal{K}_\chi = \mu \operatorname{P}(\operatorname{Sp}(2) \otimes \operatorname{SU}(2))\), it would be useful to have other descriptions; here is one possibility.

We can also describe the Lie algebra of \(\mathcal{K}_\chi = \mu \operatorname{P}(\operatorname{Sp}(2) \otimes \operatorname{SU}(2))\).

One might wonder whether this construction can be simplified. One issue is that, unlike more familiar involutions like complex conjugation or conjugation by a fixed matrix, \(\chi\) is not a linear function. That is, the entries of \(\chi(V)\) do not depend linearly on the entries of \(V\). For instance, \[\chi\left( \left[ \begin{smallmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{smallmatrix} \right] \oplus I_6 \right) = \left[\begin{smallmatrix} c & s \\ -s & c \end{smallmatrix} \right] \oplus \left[\begin{smallmatrix} c & -s \\ s & c \end{smallmatrix} \right] \oplus \left[\begin{smallmatrix} c & -s \\ s & c \end{smallmatrix} \right] \oplus \left[\begin{smallmatrix} c & -s \\ s & c \end{smallmatrix} \right]\] where \(c = \cos(t/2)\) and \(s = \sin(t/2)\).

Perhaps \(\mathcal{K}_{\chi}\) can be described as the fixed-point set of an entirely different involution \(\phi\)? If this were the case, then since \(d\phi\) is an automorphism, it would be an isometry with respect to the Killing form on \(\mathfrak{so}(8)\). Its \((-1)\)-eigenspace is therefore the orthogonal complement \(\mathfrak{k}_{\chi}^\perp\) of the \(1\)-eigenspace \(\mathfrak{k}_\chi\). By the same logic applied to \(d\chi\), \(\mathfrak{k}_{\chi}^\perp\) is simply \(\mathfrak{p}\). Therefore \(d\phi\) and \(d\chi\) have the same eigenvalues and eigenspaces, i.e.they are the same linear map. Since \(d\phi\) determines \(\phi\) by the formula \(\phi(\exp(X)) = \exp(d\phi(X))\), this means \(\phi = \chi\).

5 Cartan decompositions for \(\operatorname{PSO}(8)\)↩︎

As with many other circuit decompositions in quantum computing [6], [8], [11], [18]–[20], ours relies on iterated Cartan decomposition. Here is a more specific outline, which the methods of [8] also follow.

1. Choose commuting involutive automorphisms \(\theta_1, \theta_2\) of \(\operatorname{PSO}(8)\), and sets

   • \(\mathcal{A}\subseteq \operatorname{PSO}(8)\) with \(\mathcal{K}_{\theta_1}\mathcal{A}\mathcal{K}_{\theta_1} = \operatorname{PSO}(8)\)

   • \(\mathcal{B}\subseteq \operatorname{PSO}(8)\) with \(\mathcal{K}_{\theta_1,\theta_2}\mathcal{B}\mathcal{K}_{\theta_1,\theta_2} = \mathcal{K}_{\theta_1}\).

2. Given \(V \in \operatorname{PSO}(8)\), use two Cartan decompositions as in §2.4 to write \[\label{eq:double-decomp} V = K_1 B_1 K_2 A K_3 B_2 K_4\tag{13}\] where \(A \in \mathcal{A}\) and \(B_1,B_2 \in \mathcal{B}\) and \(K_1,\ldots,K_4 \in \mathcal{K}_{\theta_1,\theta_2}\).

3. Find explicit decompositions for \(\mathcal{M}\) and for elements of \(\mu^{-1}(\mathcal{A})\) and \(\mu^{-1}(\mathcal{B})\) and \(\mu^{-1}(\mathcal{K}_{\theta_1,\theta_2})\) in terms of CNOTs and single-qubit gates.

This works because we can rewrite 13 as \[\begin{align} V &= \mathcal{M}^\dagger \mu^{-1}(V) \mathcal{M}= \mathcal{M}^\dagger \mu^{-1}(K_1 B_1 K_2 A K_3 B_2 K_4) \mathcal{M}\\ &= \mathcal{M}^\dagger \mu^{-1}(K_1)\mu^{-1}(B_1)\cdots \mu^{-1}(K_4)\mathcal{M}. \end{align}\] As described in more detail in §4, the point of applying \(\mu^{-1}\) and then its inverse here is to make it easier to see which circuit elements will lead to a real matrix.

There is a sense in which \(\operatorname{PSO}(8)\) is a rich setting for Cartan decomposition techniques: among all connected simple compact Lie groups, only \(\operatorname{PSO}(8)\) and its simply connected cover \(\operatorname{Spin}(8)\) have more than one distinct outer automorphism of order 2.

For the rest of the paper, let \(\psi_1, \psi_2 : \operatorname{PSO}(8) \to \operatorname{PSO}(8)\) denote conjugation by \(\Delta_{\{1,2,5\}}\) and \(\Delta_{\{6,7\}}\), and \(\chi_i = \mathcal{T}^{-1} \circ \psi_i \circ \mathcal{T}\). Note that \(\psi_1, \chi_1\) were called just \(\psi,\chi\) in §4. A simpler description of \(\chi_2\) is possible: \[\chi_2(V) = \mathcal{T}^{-1}(\Delta_{\{6,7\}} \mathcal{T}(V)\Delta_{\{6,7\}}) = \mathcal{T}^{-1}(\Delta_{\{6,7\}})V \mathcal{T}^{-1}(\Delta_{\{6,7\}}),\] so \(\chi_2\) is conjugation by \(\mathcal{T}^{-1}(\Delta_{\{6,7\}}) = i\sigma_x \otimes \sigma_x \otimes \sigma_y\). No such simplification is possible for \(\chi_1\), since \(\Delta_{\{1,2,5\}} \notin \operatorname{PSO}(8)\); as observed in §4, \(\chi_1\) is not even linear.

Because of Proposition 10, our method amounts to applying triality, using two Cartan decompositions coming from subgroups conjugate to \[\operatorname{PSO}(8) \supseteq \operatorname{P}(\operatorname{SO}(3) \times \operatorname{SO}(5)) \supseteq \operatorname{P}(\operatorname{SO}(3) \times \operatorname{SO}(2) \times \operatorname{SO}(3)),\] and applying inverse triality. The next lemma provides explicit quantum circuits for each factor that will appear in these decompositions.

Actually, we have already proven (b) and (c): together they amount to the Cartan decomposition of \(\operatorname{Sp}(2)\) used in the proof of Lemma 6. We give a second proof below using what we have learned about triality. It is also worth noting that the set \(\mathcal{A}\) in (a) is not a torus or even a group, so (a) is not expressing the “standard” Cartan decomposition as in Theorem 1. Rather, \(\mathcal{A}\) was chosen to cover all possible canonical parameters with respect to \(\mathcal{K}_{\chi_1}\) while still having a short circuit description.

Write the circuit in (a) as the product of gates appearing in
Figure [-@fig:fig:triality], i.e.
$$\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/psaqnzom.png}\label{mbuqrdxv}\end{figure} \, \raisebox{8mm}{\text{and}} \,
\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/lodtvxhb.png}\label{uqlyanbj}\end{figure} \,\raisebox{8mm}{\text{and}} \,
\raisebox{-2mm}{\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/cjvkfexi.png}\label{jianmrtw}\end{figure}} \,\raisebox{8mm}{\text{and etc.}} \,$$ {#eq: sublabel=eq:mbuqrdxv,eq:uqlyanbj,eq:jianmrtw} 
Consulting Figure [-@fig:fig:triality] to apply
$\mathcal{T}\circ \mu$, we see $\mathcal{T}(\mathcal{A})$ is the set
of matrices $$\begin{align}
\begin{bmatrix}
-\cos \alpha_1 & 0 & 0 & 0 & 0 & 0 & \sin \alpha_1 & 0\\
0 & \cos \alpha_2 & 0 & -\sin \alpha_2 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & \sin \alpha_3 & -\cos \alpha_3 & 0 & 0\\
0 & 0 & 0 & 0 & -\cos \alpha_3 & -\sin \alpha_3 & 0 & 0\\
\sin \alpha_1 & 0 & 0 & 0 & 0 & 0 & \cos \alpha_1 & 0\\
0 & -\sin \alpha_2 & 0 & -\cos \alpha_2 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}
\end{align}$$ The submatrix
$\operatorname{diag}(-\cos \alpha_1, \cos \alpha_2, -\cos \alpha_3)$
in rows and columns $\{1,2,5\}$ obviously has the desired property.

By assembling the circuits derived in Lemma 8, we arrive at our new decomposition for real 3-qubit gates.

In this decomposition there are 14 CNOT gates total and 35 single-qubit rotations. Also, there are \(6+4+6 = 16\) free angles above and 4 free elements of \(\operatorname{SU}(2)\), for a total of \(16 + 4\cdot 3 = 28\) free parameters. Since this is also the dimension of \(\operatorname{SO}(8)\), the circuit is optimal in the sense that none of the parameters can be eliminated. Of course, this does not rule out the existence of shorter circuits.

Acknowledgements↩︎

I’m grateful to Jim van Meter for useful comments and for helping me navigate the literature in this area.

References↩︎