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.
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.
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.
The Clifford+ T gate set is commonly used to perform universal quantum computation. In such setup the T gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of T gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial T -count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of T gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of π /4 Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last T gate of the circuit.
In quantum computing, computational tasks are represented by quantum circuits. These circuits are composed of gates whose physical realization comes at a cost. Typically, gates from the so-called Clifford group are considered cheap, while non Clifford gates are considered expensive. Consequently, non-Clifford operations are often seen as a resource whose use should be minimized. In this thesis, following recent work by Beverland and others, we study lower bounds for the number of non-Clifford gates in quantum circuits. We focus on lower bounds that can be derived from monotones, which are real-valued functions of quantum states that are non-increasing under Clifford operations. We first provide a detailed presentation of two recently introduced monotones: the stabilizer nullity and the dyadic monotone. We then discuss how these monotones can be used to give lower bounds for the non-Clifford resources for two important quantum operations: the multiply-controlled Pauli Z gate and the modular adder.
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.
Optimal Hadamard gate count for Clifford+T synthesis of Pauli rotations sequences
The Clifford+T gate set is commonly used to perform universal quantum computation. In such setup the T gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of T gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial T-count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of T gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of π/4 Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last T gate of the circuit.