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arXiv:2102.11380 · Globecom

Decomposition of Clifford Gates

Tefjol Pllaha, Kalle Volanto, Olav Tirkkonen

Published Dec 1, 2021• 7 citations
arXivHTMLPDF

Abstract

In fault-tolerant quantum computation and quan-tum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. This information is encoded by the support (Pllaha et al., 2020) of the given circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a minimal product of Clifford transvections. The algorithm can be directly used for computing the support of any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.

DOI
10.1109/globecom46510.2021.9685501

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In fault-tolerant quantum computation and quan-tum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. This information is encoded by the support (Pllaha et al., 2020) of the given circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a minimal product of Clifford transvections. The algorithm can be directly used for computing the support of any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.
Approximating a given unitary operator into a discrete set of gates is of great importance in quantum computation. This problem can be seen as a source quantization problem where the unitary operator is considered as the source. Elements of the Clifford hierarchy can be implemented fault-tolerantly using quantum gate teleportation. This motivates the development of algorithms that quantize unitary operators into Clifford hierarchy elements. In this paper, we present a fast algorithm for approximating arbitrary n-qubit unitaries with Clifford matrices. The algorithm relies on the Clifford transvection decomposition of the Clifford group. The algorithm greedily applies Clifford transvections taking the input unitary closer to the identity matrix. To make the algorithm fast, we take advantage of the Pauli basis representation for unitary matrices to parallelize matrix distance computations. The complexity of the proposed algorithm is O(n 4n). Based on simulation results, the proposed algorithm outperforms the state of the art method.
Decomposing unitary operators into Clifford gates and non-Clifford gates is an extremely important problem in quantum computing, as each extra quantum gate in a quantum circuit makes it more prone to errors. The first objective in this problem is usually to minimize the number of non-Clifford gates. The second objective, and the objective of this thesis, is to minimize the number of 2-qubit gates in the Clifford operators that are left. Most algorithms from the literature for decomposing Clifford operators into Clifford gates involve using a decomposition such as the Bruhat decomposition to first split the input operator into multiple separate operators. These operators are then individually decomposed into Clifford gates. A problem with this approach is that the minimum number of 2-qubit gates needed to implement all the separate operators can be greater than what is needed to implement the input Clifford operator in the first place. This thesis tackles this problem by developing an algorithm for decomposing Clifford operators into Clifford gates without splitting the input operator into multiple smaller operators first. To help with the decomposition steps, this thesis defines a set of operators called 2-qubit symplectic transvections. This results in an algorithm that performs well for small to medium sized Clifford operators. This thesis assumes that the order of the qubits does not affect the implementation cost for quantum gates. This means that swap gates can be implemented for free by relabeling the qubits. It is also assumed that 1-qubit gates have negligible implementation cost compared to 2-qubit gates.
Quantum error correction is essential for achieving practical quantum computing but has a significant computational overhead. Among fault-tolerant (FT) gate operations, non-Clifford gates, such as T, are particularly expensive due to their reliance on magic state distillation. These costly T gates appear frequently in FT circuits as many quantum algorithms require arbitrary single-qubit rotations, such as Rx and Rz gates, which must be decomposed into a sequence of T and Clifford gates. In many quantum circuits, Rx and Rz gates can be fused to form a single U3 unitary. However, existing synthesis methods, such as gridsynth, rely on indirect decompositions, requiring separate Rz decompositions that result in a threefold increase in T count.
Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in CN×Nas a 2m×2m binary sym-plectic matrix, where N=2m. We show that for an [[ m, m-k ]] stabilizer code every logical Clifford operator has 2k(k+1)/2symplectic solutions, and we enumerate them efficiently using symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary symplectic matrices. For a given operator, our assembly of all of its physical realizations enables optimization over them with respect to a suitable metric. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the well-known [[ 6,4,2 ]] code. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a.
Quantum gates are usually described through matrix and tensor-product formalisms that often obscure their geometric structure. In this work, we formulate the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework. We show that the Pauli group is naturally identified with the group of blades up to a global phase, thereby providing a geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical behavior suggests unexpectedly compact decompositions for Clifford operators. Finally, we show that Clifford+T universality admits a natural geometric interpretation through π/8-rotors within this framework.
The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second ‘levels’, respectively. Non-Clifford gates from the third level or higher, such as the T gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined 26 years ago by Gottesman–Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.