Solovay–Kitaev Approximations of Special Orthogonal Matrices

The circuit-gate framework of quantum computing relies on the fact that an arbitrary quantum gate in the form of a unitary matrix of unit determinant can be approximated to a desired accuracy by a fairly short sequence of basic gates, of which the exact bounds are provided by the Solovay–Kitaev theorem. In this work, we show that a version of this theorem is applicable to orthogonal matrices with unit determinant as well, indicating the possibility of using orthogonal matrices for efficient computation. We further develop a version of the Solovay–Kitaev algorithm and discuss the computational experience.


Introduction
A computer program in the context of classical computing is an ordered list of instructions, expressible in terms of elementary operations, readily convertible to the machine language of a classical computer. A quantum program in quantum computing could be described analogously. According to the circuit-gate framework of quantum computing, a quantum algorithm consists of quantum gates acting on quantum states (qubits) where measuring devices are applied at appropriate instances to collapse the wavefunction. Based on the Heisenberg-Born interpretation of quantum mechanics, this circuit-gate framework has achieved significant progress up to date, as the pioneering model of quantum computing. Also, it is proven to be polynomially equivalent to other quantum computational frameworks. Accordingly, a quantum program can be regarded as the application of several unitary matrices, together with measurements at certain instances.
In order to implement unitary operations, basic quantum gates such as Pauli gates, Hadamard gate, and phase gate are available in the circuit-gate framework, in analogy with basic gates in classical computing. It is quite natural to ask how many basic gates are needed to implement an arbitrary unitary operation in a quantum circuit. The remarkable contributions in this regard made independently by Solovay [1] and Kitaev [2] answered this question, resulting in what is known today as the Solovay-Kitaev theorem. This theorem states that it is possible to approximate any 2 × 2 unitary with unit determinant by a product of Oðlog 4 ð1/εÞÞ physically realizable 2 × 2 unitaries (which appear as basic gates) to an arbitrary accuracy ε [3,4]. Recall the other quantum computational frameworks such as quantum walks, quantum Turing machines, and adiabatic computing were proven to be polynomially equivalent to the circuit-gate framework [5][6][7], the Solovay-Kitaev theorem is widely regarded the theoretical proof for the supremacy of quantum computers. In addition, the number of elementary gates needed to implement an arbitrary unitary provides an indicator of the capacity and limitations of quantum computers.
This reveals an interesting aspect of unconventional models of computing. That is, any computational model with similar speed and limitations would be computationally equivalent to quantum computing. If physically realizable, such a model would have the same advantages and limitations as quantum computing. Though little attention has been paid to this subject in the past, several interesting works have investigated the possibility of having such models. A pioneering work was done by Aerts and Czachor in 2007, proposing geometric algebras instead of unitary matrices [8]. The authors named this model cartoon computing and proved its equivalence to quantum computing, demonstrating a simulation of the Deutsch-Jozsa algorithm. A later work investigated entities in cartoon computing equivalent to elementary gates in quantum computing [9]. In 2008, Fernandez and Schneeberger proposed quaternionic computing, in which the possibility of adopting quaternions instead of unitary matrices was proven [10]. In order to show the equivalence to quantum computing, the authors have used the Bernstein-Vazirani theorem in quantum Turing machine framework. In [11], Graydon explored quaternionic quantum processes with respect to standard quantum information theory. Thus, it is an interesting question to ask what other algebraic structures would show similar behaviour, if employed as a computational model.
On the other hand, the progress achieved in three-level quantum systems is noteworthy [12][13][14]. Instead of qubits with states j0i and j1i in standard circuit-gate framework, qutrits having three basis states j0i, j1i, and j2i are used in these systems. Analogous to the single-qubit quantum gates in the form of 2 × 2 matrices in the special (unit determinant) unitary group SUð2Þ, the single-qutrit gates are 3 × 3 special unitaries or the elements in the group SUð3Þ [15][16][17]. Though the realization of SUð3Þ gates has been the topic of interest for several previous works [18][19][20], computational capacity or theoretical bounds on computing of a threelevel quantum system have not been paid the deserved attention. Neither Solovay-Kitaev type approximations were investigated for three-level systems. Nevertheless, a recent work emphasized the significance of the subgroup SOð3Þ of SUð3Þ for qutrit-based quantum computation, showing that any state of a qutrit could be obtained from a oneparameter family of states through the action of SOð3Þ [21]. In this regard, one should not ignore the remarkable relationship between the groups SOð3Þ and SUð2Þ. This motivates us to check whether the Solovay-Kitaev theorem is extendable to SOð3Þ and possible to achieve the quantum speedup in three-level quantum systems, when equipped with orthogonal operators. In addition to that, once this question is resolved, one may know exactly whether the 3 × 3 orthogonal matrices also provide an algebraic structure suitable for efficient computation, such as geometric algebras or quaternions.
In this paper, we show that the question is answered positively. That is, the orthogonal matrices play a role in threelevel quantum systems, equivalent to what the unitaries play in standard quantum circuit framework. More precisely, we show that a version of the Solovay-Kitaev theorem is applicable to 3 × 3 orthogonal matrices with unit determinant. Thus, we indicate the possibility of theoretically replacing the 2 × 2 unitaries in quantum computing by 3 × 3 orthogonals and qubits by qutrits. Using Cornwell's two-to-one homomorphic map from the special unitary group SUð2Þ to special orthogonal group SOð3Þ [22], we prove the possibility of approximating any 3 × 3 orthogonal with unit determinant by a product of Oðlog 4 ð1/εÞÞ elementary 3 × 3 orthogonals of unit determinants to an arbitrary accuracy ε. We further discuss how to find the sequence of appropriate elementary orthogonals, providing a version of Solovay-Kitaev algorithm in SOð3Þ.
The remainder of the paper is organized as follows. In Section 2, our version of the Solovay-Kitaev theorem for SOð3Þ is proven. An approximation scheme for unit-determinant orthogonal matrices in accordance with this theorem and the standard Solovay-Kitaev algorithm is presented in Section 3. Computational experience is discussed in Section 4, and we discuss the implications of our work in Section 5 with several remarks on potential future works.

Solovay-Kitaev Theorem in SOð3Þ
2.1. Solovay-Kitaev Theorem. Recall the computational power in the circuit-gate framework is guaranteed by the Solovay-Kitaev theorem; its main focus is on approximating a 2 d × 2 d unitary matrix by basic quantum gates. A set of possible basic gates is referred to as an instruction set in the context of this theorem. Considering single qubit unitary gates, an instruction set G is a finite subset of SUð2Þ such that G contains its own inverse and hGi is dense in SUð2Þ. For example, the set of gates fH, H † , T, T † g makes an instruction set for SUð2Þ, where H and T denote, respectively, Hadamard and phase gates in the circuit-gate framework. The set of all strings that can be made from G without using more than l elements is denoted by G l . Now, the Solovay-Kitaev theorem for a d-qubit system can be stated as follows.
The proof of this theorem is highly constructive, and the algorithmic steps of finding the elements in the instruction set that approximate a given element in SUð2Þ can be found from its proof. A comprehensive version of proof can be found in [4]. An algorithmic version of the theorem with a procedure for finding those elements can be found in [23]. We now explore how a version of this theorem can be adapted to SOð3Þ. Primary motivation for this is the distance relations of the two groups, preserved by a homomorphism from SUð2Þ onto SOð3Þ.

Distance Relations.
The two-to-one homomorphic mapping ρ from SUð2Þ onto SOð3Þ known today as Cornwell's mapping is expressible in several ways [22], from which we adopt the following in [24]. An element U in SUð2Þ is expressible as where α, β ∈ ℂ and jαj 2 + jβj 2 = 1 and its image is given by

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In order to measure the distances, as in the proof of the standard Solovay-Kitaev theorem, we too use the metric induced by the trace norm for consistency. It is customary to use the operator norm in quantum computation according to the matrix formulation of quantum mechanics. However, the standard proof of the Solovay-Kitaev theorem uses trace norm, as it helps to make the proof more comprehensive by incorporating a special property of the trace norm at some point. Since our intention is finding an analogous version in SOð3Þ, it is more appropriate to consider the trace norm for matrices in SOð3Þ as well.
Lemma 2 illustrates how the mapping ρ preserves distances to an order OðεÞ in SOð3Þ with respect to trace norm.
Proof. Due to unitary invariance of the trace norm, it suffices to show that I − ρðUÞ < OðεÞ whenever I − U < ε. We use the fact that any element U in SUð2Þ can be expressed as in equation (1) and the mapping ρ given by equation (2). Then, from which we derive Supposing kI − Uk < ε, it is not difficult to see that the left side of equation (4) is bounded by Oðε 2 Þ as follows. kI Þ. Substituting these in equation (4), kρðIÞ − ρðUÞk 2 < 3ðεðε + ffiffi ffi 2 p ÞÞ 2 + 2ðεðε + 2 ffiffi ffi 2 p ÞÞ 2 . Therefore, kρðIÞ − ρðUÞk < OðεÞ.

Instruction Sets in SOð3Þ.
In the context of single qubit unitary gates, an instruction set G is a finite subset of SUð2Þ such that G contains its own inverse and hGi is dense in SUð2Þ. It is possible to adopt the same definition for instruction sets in SOð3Þ. Interestingly, the image of an instruction set in SUð2Þ under the homomorphism ρ becomes an instruction set in SOð3Þ. Lemmata 3 and 4 prove this claim.

Lemma 3.
Let X and Y be metric spaces, and let A be a dense subset of X. If f : X ⟶ Y is continuous and surjective, then f ðAÞ is dense in Y.
Proof. Let K = f −1 ð f ðAÞÞ. Then, A ⊆ K and since f is continuous, K is closed. This implies A ⊆ K. On the other hand, since A is dense in X, A = X. Thus, K = X, and therefore, f ðAÞ = f ðKÞ = f ðXÞ = Y.

Lemma 4.
If G is an instruction set in SUð2Þ, then hρðGÞi is an instruction set in SOð3Þ.
Proof. Let G be an instruction set in SUð2Þ. Then, ρðGÞ must contain its own inverse and is finite as ρ is a homomorphism. Since ρ is continuous and hGi is dense in SUð2Þ, by Lemma 3 ρðhGiÞ is dense in SOð3Þ. Clearly, since ρ is a homomorphism, we have hρðGÞi = ρðhGiÞ. Therefore, hρðGÞi is an instruction set in SOð3Þ. SOð3Þ. With the results we derived above, it is now possible to establish a version of the Solovay-Kitaev theorem for SOð3Þ.

Theorem 5.
Let G be an instruction set in SUð2Þ. Then, ρðGÞ is an instruction set for SOð3Þ such that for any ε > 0, ρðGÞ l provides an ε-net for SOð3Þ where l = Oðlog 4 ð1/εÞÞ.
Proof. From Lemma 4, ρðGÞ is an instruction set. Let V ∈ S Oð3Þ. Then, there exists some U ∈ SUð2Þ such that V = ρðUÞ. The Solovay-Kitaev theorem guarantees the existence of U 1 ,

Approximations in SOð3Þ
Now we describe how an arbitrary unit-determinant orthogonal matrix can be approximated by ρðGÞ, where G is an instruction set in SUð2Þ. Recall the proof of the Solovay-Kitaev theorem is highly constructive; it provides essential ingredients for finding the sequence of elements from the instruction set approximating the given unitary to a given accuracy ε. As implied by Theorem 5, our algorithmic version for SOð3Þ too is based on the steps in finding those elements as in the proof of the original theorem.
For completion, we first describe the algorithm for finding the approximations in SUð2Þ. We follow the procedure given by Dawson and Nielsen [23] in this regard.

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The algorithm in SUð2Þ can be expressed in pseudocode as follows.
The algorithm is a function which takes two inputs: U is an arbitrary element in SUð2Þ which we desire to approximate by G, and n a nonnegative integer which controls the accuracy of the approximation. This function returns sequence elements from an instruction set G in SUð2Þ which approximates U to an accuracy of ε n , a strictly decreasing function of n. The Solovay-Kitaev algorithm is recursive and the recursion terminates when n = 0.
In this step, we find an ε 0 approximation to U. To find such an approximation, we have to assure that we have constructed an ε 0 -net: a set containing elements from hGi such that for any unitary matrix we can find an ε 0 approximation from it. Since ε 0 is a constant and hGi is dense in SUð2Þ, we can build a gate net by enumerating and sorting a large number of elements from G l 0 for sufficiently large but fixed positive integer l 0 and creating a search algorithm to find the closed approximation. If n ≠ 0, then we find an ε n−1 approximation to U: IfΔ is an ε n approximation to Δ = UU † n−1 , then by the unitary invariance of the norm, Thus, finding an ε n approximation to Δ with ε n < ε n−1 allows us to find an improved approximation (i.e., ε n < ε n−1 ) to U. To find such an approximation, first we decompose Δ = UVW † V † , where U, V are unitaries with kI − Vk, kI − Uk < k 1 ffiffiffiffiffiffiffiffi ε n−1 p , where k 1 is positive constant: This decomposition is known as the balance group commutator. To find such a decomposition, we use the fact that any arbitrary unitary can be represented as a rotation in the Bloch sphere. If Δ is a rotation by an angle θ about some axis n on the Bloch sphere, consider α satisfying Then, ifṼ is a rotation by α about the x axis andW is a rotation by α about the y axis, on the bloch sphere, then N = VWṼ †W † is conjugate to Δ (i.e., Δ = SNS † ) for some unitary S. Since N and Δ are unitary matrices, they are diagonalizable; moreover, they have the same eigenvalues. Thus, by diagonalizing N and Δ, we find a diagonal matrix D and two unitary matrices S Δ and S N such that Now, letting S = S Δ S † N , we have V = SṼS † and W = SWS † satisfying Also, for sufficiently small ε n−1 , V and W satisfy for some positive constant k 1 . Now, we find ε n−1 approximations to both V and W: By replacing Δ by ε n−1 and δ by k 1 ffiffiffiffiffiffiffiffi ε n−1 p in Lemma 6, the group commutator of V n−1 and W n−1 turns out to be a k 2 ε 3/2 n−1 approximation to Δ for some positive constant k 2 . Now, if ð1/k 2 2 Þ < ε n−1 , then k 2 ε 3/2 n−1 < ε n−1 . Hence, ε n = k 2 ε 3/2 n−1 provides an improved approximation for U. Accordingly, the value of ε 0 is determined by this constant k 2 ; i.e., for this construction to guarantee that ε 0 > ε 1 > ⋯, the value of ε 0 must be strictly less than 1/k 2 2 (i.e., ε 0 < ð1/k 2 2 Þ). This algorithm concludes by returning the sequences of elements in G that approximate the group commutator as well as U n−1 .

Solovay-Kitaev Algorithm in SOð3Þ.
In light of the algorithmic steps described, now it is possible to provide the algorithmic version for SOð3Þ as follows.
This algorithm is a function which takes two inputs: S: an arbitrary element in SOð3Þ which we intend to approximate and n: a nonnegative integer which controls the accuracy of the approximation. This function returns a sequence of elements from an instruction set ρðGÞ ⊂ SOð3Þ, where G is an instruction set in SUð2Þ, which approximates S Advances in Mathematical Physics to an accuracy of ε n , where ε n is a decreasing function of n, i.e, ε n ⟶ 0 as n ⟶ ∞.
In this step, we find U∈SUð2Þ such that ρðUÞ = S, where ρ is the homomorphic mapping from SUð2Þ to S Oð3Þ, so that we can find a Solovay-Kitaev approximation for U in SUð2Þ (SKð:, :Þ is the Solovay-Kitaev function in SUð2Þ): Supposing that for a given depth m the SK function approximates any unitary matrix V ∈ SUð2Þ to accuracy k m , we find a k n approximationŨ to U. Next, we find ρðŨÞ: By Lemma 2,S turns out to be an ε n = ck n approximation for S, for some positive constant c. SinceŨ is a sequence of elements from G, there are U 1 , U 2 , ⋯U n ∈ G such thatŨ = U 1 U 2 ⋯ U n . Then,S = ρðŨÞ = ρðU 1 U 2 ⋯ U n Þ = ρðU 1 ÞρðU 2 Þ ⋯ ρðU n Þ, because ρ is a homomorphism. Thus,S is a sequence of elements from ρðGÞ which approximates S to an accuracy ε n . For a given depth m, k m is approximation error associated with the Solovay-Kitaev approximations in SUð2Þ. Therefore, we can ensure that k 0 > k 1 > ⋯, which implies ε 0 > ε 2 > ⋯. Finally, this function returns a sequence of instructions from ρðGÞ which approximates S to an accuracy of k n .

Computational Experience
One challenge encountered in the computation is that ρ −1 fails to exist as the map is not one to one. This however was overcome using the fact that for given A ∈ SOð3Þ it is possible to find U ∈ SUð2Þ such that ρðUÞ = A, for which the following construction was used. Any element in SOð3Þ can be represented by a real number θ, the angle of rotation, and a rotation axis a = ða x , a y , a z Þ, which is a 3-dimensional unit vector, denoted by R a ðθÞ. The corresponding matrix R a ðθÞ can be expressed explicitly by where For a given rotational matrix R a ðθÞ, define One can verify that U ∈ SUð2Þ and ρðUÞ = R a ðθÞ. Therefore, for an element A ∈ SOð3Þ, in order to find an element U ∈ SUð2Þ such that ρðUÞ = A, under this construction, we need to find a unit vector a ∈ ℝ 3 and a real number θ such that R a ðθÞ = A.
Let A ∈ SOð3Þ, and suppose A is the corresponding rotational matrix of R a ðθÞ (i.e., A = R a ðθÞ). If v is any vector parallel to a, then it must satisfy Av = v, because the rotation of v around the axis of rotation must result in a. Since A ∈ SOð3Þ, we can always find an eigenvalue A which is equal to 1, from which it immediately follows that a is an eigenvector which corresponds to the eigenvalue 1. So by diagonalizing A, we find the unit vector a ′ which is parallel to each other a. Now, since both a and a ′ are unit vectors, we must have a = a ′ or a′ = a. By equation (18), the trace of the matrix A reduces to TrðAÞ = 1 + 2 cos θ, which immediately results in θ = cos −1 ððTrðAÞ − 1Þ/2Þ: Now, by defining α ∈ ℝ such that jαj = jθj and choosing the right sign for α to match the rotational axis a′ (i.e., aθ = a′α), we get R a ′ðθÞ = R a ðαÞ = A.
Accordingly, we implemented our algorithm in SOð3Þ to find the Solovay-Kitaev approximates to several special unitary matrices. The computational experiment was conducted in accordance with the algorithmic steps mentioned and the bound was obeyed as in Theorem 5. We implemented with different instruction sets and the implementation with the instruction set fS 1 , Advances in Mathematical Physics resulted in much shorter length for a given ε than the others. It would be an interesting future work to identify any classes or subgroups of matrices that can be approximated best by each instruction set, perhaps with a comparison of different instruction sets.

Discussion
Unconventional computing with different algebraic structures had been the topic of interest for a few previous works for which the primary motivation was quantum computing. Based on the fact that the circuit-gate framework of quantum computing relies on the Solovay-Kitaev theorem, we investigated the possibility of deriving a version of this theorem for SOð3Þ on a three-level quantum system, indicating the potential of using orthogonal matrices for efficient computation.
Three-level quantum systems and relevant operators had already been a topic of interest. In analogy with standard circuit-gate framework, it was customary to use the elements in the unitary group SUð3Þ as the operators in these systems. Despite the recent experimental achievements, theoretical bounds, capacity, and other related questions on three-level quantum systems were seldom explored. With our version of the Solovay-Kitaev theorem, it is now known that efficient computation is possible with the orthogonal subgroup SOð3Þ of SUð3Þ. This is a noticeable distinction when compared with the subgroup SOð2Þ of SUð2Þ. Being an Abelian group, it is impossible to perform Solovay-Kitaev type approximations on SOð2Þ. Thus, an instruction set in standard quantum computation enforces the inclusion of T (the phase gate), T 2 , or T † 2 , the nonorthogonal gates. However, the fault-tolerant implementation of the phase gate T is much more complicated than the orthogonal gates [25]. Therefore, quantum speedup only using orthogonals is beyond feasibility in standard circuit-gate framework, though desired. In contrast to this, as our results indicate, quantum speedup with orthogonals is theoretically feasible in a three-level quantum system.
It is worthwhile to consider our version of the Solovay-Kitaev theorem in the context quantum compilation [26,27] in which the conversion of a nonfault-tolerant circuit into a fault tolerant one is investigated. A recent paper introduced several efficient methods for quantum compilation using physical machine descriptions, which included one method based on the Solovay-Kitaev approximations [28]. Although compilation and optimization of three-level quantum circuits have been the subject of a few other works [19,20,29], none is based on the Solovay-Kitaev theorem. It would be an interesting future task to investigate whether efficient compilations are possible for a three-level system with orthogonals. This would be possible by constructing a Solovay-Kitaevbased compilation method analogous to the one in [28], for which our algorithm in Section 3.1 would be helpful.
Our aim was particularly on exploring the approximation power of special orthogonal matrices. Therefore, we confined our study to a particular form of instruction sets in SOð3Þ; that is, the images of instruction sets in SUð2Þ. A closer inspection reveals that an arbitrary instruction set in SOð3Þ behaves similarly, resulting in the same length for a given accuracy. Therefore, it is an immediate consequence of Theorem 5 that slightly different versions of the Solovay-Kitaev theorem and algorithm for SOð3Þ can be established. However, the applicability of the Solovay-Kitaev theorem to other Lie groups than SOð3Þ still remains a nontrivial and theoretically interesting topic, which has not been investigated in literature. It would be a potential future task to see if the theorem is extendable to those groups.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.