The quantum layout and the mapping of logical to physical qubits are crucial in quantum circuit synthesis for a real quantum computer. Circuits that include large n-bit Toffoli gates (n ≥ 3), such as those designed from cost-expensive gates and hard-to-decompose Exclusive-or Sum of Products (ESOP) expressions, have complications of effective mappings into contemporary quantum layouts, such as the square grid and heavy-hex layouts. These complications are primarily caused by the limited connectivity among the physical qubits in such layouts, leading to the insertion of many additional SWAP gates. This paper introduces a new quantum circuit synthesis methodology by exploring the advantage of a Positive Davio lattice (PDL) as an intermediate representation to create our proposed triangular layout and layout-aware circuits. From these circuits, we introduce and form the SWAT gate, composed of a SWAP gate followed by a 3-bit Toffoli gate. To illustrate the usefulness of our method for existing industrial quantum layouts, we also introduce cost-effective mappings of the resulting circuits onto square grid and heavy-hex layouts without additional SWAP gates. This is done with the help of the SWAT gate. Our research highlights PDLs as an efficient tool for layout-aware quantum circuit synthesis.

A new Boolean-Phase swapping gate is presented with improved quantum generality and cost-effectiveness. Our swapping gate is termed the "p-SWAP gate", where p is the phase difference selected for a set of swapped qubits. The phase p is expressed in radians and -π ≤ p ≤ +π. The generality of p-SWAP gate is demonstrated for Phase applications for selected values of p and for Boolean applications when the value of p is ignored. The cost-effectiveness of p-SWAP gate follows from the lower quantum cost and depth of its final realized (transpiled) quantum circuit, when compared to the standard SWAP gate composed of three Feynman (CNOT) gates. Specifically, our presented p-SWAP gate utilizes only two CNOT gates. The quantum circuit of the p-SWAP gate is visually designed using our previously developed Bloch sphere approach. Experimentally, after transpilation for an IBM quantum computer, the transpiled p-SWAP gate shows an approximate 23

We propose some new uses of toric variety structures in the study of quantum computation for small radices. In particular, we observe the concurrence of the equivalence classes of quantum states under quantum measurement and the orbits of the toric geometric structure of the state space. Visualizations of these state spaces and of certain fundamental unitary transformations in binary and ternary quantum logic and a method to develop new transformations based on these visualization techniques are presented. Transformations discussed included minimal universal sets for permutative ternary quantum circuits. In addition, general structures and synthesis methods based on quantum multiplexers are presented. A general framework for the design of optimal ternary quantum transformations and circuits is additionally presented. Finally, a number of open research areas that are extensions of the work presented herein are given.

Quantum automata can solve certain problems with a smaller state space than classical automata. We developed a quantum finite automaton using ternary rotation quantum gates and the Chrestenson family of ternary quantum gates. The main idea of this paper is to show how to combine rotation ternary quantum circuit-based Quantum Finite Automaton and quantum reversible circuit-based Deterministic Finite Automaton to build a more powerful machine. The combined machine can enable more complex language and pattern recognition. The developed quantum finite automaton and resulting combined machine can be used for robotics applications such as language, gesture, and motion recognition.

The decomposition from the group theory-based methods of Sasao and Saraivanov is extended to design binary quantum cascades, using the quantum rotational gates by the X-axis (CNOT and RX), Y-axis (RY), and Z-axis (controlled-Z) of the Bloch sphere. A class of local transformations is also presented to simplify the final canonical cascade circuits. Our proposed methodology is well suited for quantum layouts, as each single-qubit gate has one target qubit and each double-qubit gate has one control qubit and one target qubit, thereby never creating a graph of triangular connectivity.

A Boolean-Phase swapping gate is introduced for quantum generality and cost-effectiveness, which is termed the "p-SWAP gate", where p is a customizable phase difference for a set of swapped qubits and 0 <= p <= ${\pm \pi}$ radians. The generality of the p-SWAP gate is proposed for quantum Phase oracles requiring a desirable p for a set of swapped qubits, as well as for quantum Boolean oracles when p is ignored. The cost-effectiveness of the p-SWAP gate comes from the lower quantum cost and depth for its final synthesized (transpiled) quantum circuit into a quantum computer, as compared to the standard SWAP gate. In general, the standard SWAP gate is constructed using three Feynman (CNOT) gates, while our p-SWAP gate only utilizes two CNOT gates. In this paper, the desirability of p is geometrically chosen using our proposed Bloch sphere approach, without using any matrices multiplication and unitary representations. After transpilation, the final transpiled p-SWAP gate has approximately 23% quantum cost reduction and 26% depth minimization than those of the final transpiled standard SWAP gate.

This paper extends the decomposition from the group theory based methods of Sasao and Saraivanov to design binary input multivalued output quantum cascades realized with optical NOT, SWAP, and Fredkin Gates. We present this method for 3, 5, and 7valued outputs, but in general it can be used for odd prime-valued outputs. The method can be extended to realize hybrid functions with different valued outputs. A class of local transformations is presented that can simplify the final cascade circuits. Using these simplifying transformations, we present an upper bound on the maximum number of gates in an arbitrary n-variable input and k-valued output function.

A new methodology is introduced to solve classical Boolean problems as Hamiltonians, using the quantum approximate optimization algorithm (QAOA). This methodology is termed the “Boolean-Hamiltonians Transform for QAOA” (BHT-QAOA). Because a great deal of research and studies are mainly focused on solving combinatorial optimization problems using QAOA, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for Boolean problems, by transforming such problems from Boolean oracles (in different structures) into Phase oracles, and then into the Hamiltonians of QAOA. From such a transformation, we noticed that the total utilized numbers of qubits and quantum gates are dramatically minimized for the generated Hamiltonians of QAOA. In this article, arbitrary Boolean problems are examined by successfully solving them with our BHT-QAOA, using different structures based on various logic synthesis methods, an IBM quantum computer, and a classical optimization minimizer. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, and machine learning, just to name a few, in the hybrid classical-quantum domain.

The Grover controlled-diffuser (CUs) for quantum Boolean oracles (Uω) is introduced as a new approach for Grover’s algorithm [1-3], to search for all solutions for arbitrary logical structures of such oracles, since the standard Grover diffuser (Us) is not able to find all correct solutions for some logical structures of Uω. This protocol constructs the quantum circuit of the CUs operator [4] of Grover's algorithm, which relies on the states of the output qubit (as the reflection of Boolean decisions from a Uω) without relying on the conventional phase kickback mechanism. The CUs operator successfully searches for all correct solutions for all Uω regardless of their different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few.

Conventional presentations of quantum algorithms overwhelmingly rely on mathematical formalism, posing an unnecessary barrier to conceptual understanding. The growing influence of quantum computing across diverse domains necessitates more accessible education on the subject. To engage a broader audience in quantum computing, we introduce a new approach for teaching quantum algorithms: drawing parallels to classical pseudocode. This approach, which we call “QuCode,” is demonstrated through its application to several black box quantum algorithms: Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm.

The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. In quantum computing, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool. On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY, XZ, and/or YZ planes, which are the 2D circular top and side views of the Bloch sphere. Collectively, the Bloch sphere and its planes are termed the Bloch sphere approach (BSA). With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native "basis" gates ( , , , and ). Our designed generic and cost-effective quantum gates and libraries are listed as follows, where 2 n 5 qubits. Quantum libraries: GALA-n [3, 6] and CALA-n [4, 7], which have become part of the IBM Qiskit ecosystem [8] n-bit Toffoli gate [2-4] n-bit Boolean gates (AND, NAND, OR, NOR, implication, and inhibition) [3, 4] n-bit controlled- ( ) and controlled- ( ) gates [3, 4] n-bit Fredkin gate [3, 4] n-bit Miller gate [3, 4] Boolean-Phase SWAP gate: p-SWAP [4, 5] Because the quantum operations of all IBM native gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, the XY-plane of the Bloch sphere is utilized here for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using other projectional planes of the Bloch sphere, i.e., the XZ and YZ planes, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as a generic and open geometrical framework for prospective quantum computing research for building interesting and innovative quantum gates and circuits.

The Grover controlled-diffuser (CUs) for quantum Boolean oracles (Uω) is introduced as a new approach for Grover’s algorithm [1-3], to search for all solutions for arbitrary logical structures of such oracles, since the standard Grover diffuser (Us) [1-3] is not able to find all correct solutions for some logical structures of Uω. This protocol constructs the quantum circuit of the CUs operator [4] of Grover's algorithm, which relies on the states of the output qubit (as the reflection of Boolean decisions from a Uω) without relying on the conventional phase kickback mechanism. The CUs operator successfully searches for all correct solutions for all Uω regardless of their different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few.

This study is concerned with the task of robotic piece picking, which has been a persistent challenge in the realm of robotic manipulation. Despite notable progress in this fields, this domain still presents numerous hurdles. This paper presents a robotic grasping detection algorithm that finds sufficiently stable grasping poses for a densely stocked cuboid-shaped packages using the depth image of the picking scene. Using depth sensing and computer vision algorithms, the proposed method computes a highly accurate grasping region and pose for the object of interest. The method achieves 95.17 % success rate on the DPB (Densely Packed Boxes) dataset. The proposed algorithm improves the success rate of the grasping synthesis methods that optimizes the distance to the geometric centroid of an object surface as a stable grasping point by 37.68%.

A controlled-diffusion operator for Boolean oracles is designed as a new approach for Grover’s algorithm to search for solutions for arbitrary logical structures of such oracles, since the Grover diffusion operator is not able to find correct solutions for some logical structures of Boolean oracles. We also show that the Phase oracles do not work sometimes correctly using the Grover diffusion operator. Our proposed controlled-diffusion operator relies on the states of output qubit, as the reflection of Boolean decisions from a Boolean oracle without relying on the phase kickback. We prove that on many examples of Boolean and Phase oracles the Grover diffusion operator is not working correctly. The oracles in these examples are constructed using different structures of POS, SOP, ESOP, CSP-SAT, and XOR-SAT. Our mathematical models and experiments prove that the proposed controlled-diffusion operator successfully searches for all solutions for all Boolean oracles regardless of their different logical structures.

The BHT-QAOA is a hybrid classical-quantum algorithm that solves arbitrary classical Boolean problems as Hamiltonians in the quantum domain, using the quantum approximate optimization algorithm (QAOA) [1]. The BHT-QAOA stands for the "Boolean-Hamiltonians Transform for QAOA" [2]. Research and studies are mainly focused on solving combinatorial optimization problems using QAOA, e.g., the MaxCut problem [1]. However, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for classical Boolean problems, as expressed in the following steps and demonstrated in the figure below. Design classical Boolean problems into quantum Boolean oracles in different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few. Convert these quantum Boolean oracles into quantum Phase oracles. Transform these quantum Phase oracles into the Hamiltonians (HC and HM) of QAOA. Please observe that, from the aforementioned steps, the total utilized numbers of qubits and quantum gates are significantly reduced for the final generated Hamiltonians (HC and HM) of QAOA. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, machine learning, just to name a few, in the hybrid classical-quantum domain.

A cost-effective n-bit Toffoli gate is proposed to be realized (or transpiled) based on the layouts (linear, T-like, and I-like) and the number of n physical qubits for IBM quantum computers. This proposed gate is termed the "layout-aware n-bit Toffoli gate". The layout-aware n-bit Toffoli gate is designed using the visual approach of the Bloch sphere, from the visual representations of the rotational quantum operations for IBM native gates. In this paper, we also proposed a new formula for the quantum cost, which calculates the total number of native gates, the crossing connections, and the depth of the final transpiled quantum circuit. This formula is termed the "transpilation quantum cost". After transpilation, our proposed layout-aware n-bit Toffoli gate always has a much lower transpilation quantum cost than that of the conventional n-bit Toffoli gate, where 3 <= n <= 7 qubits, for different IBM quantum computers.

A generic Boolean-phase SWAP gate is introduced for quantum cost-effectiveness. This gate is termed the "p-SWAP", and p is a customizable phase difference between the swapped qubits, where 0 <= p <= ±π radians. The cost-effectiveness of p-SWAP gate comes from a lower quantum cost for its final transpiled quantum circuit into a real quantum computer. The p-SWAP gate only utilizes two Feynman (CNOT) gates, as compared to the standard SWAP gate constructed from three CNOT gates. The quantum circuit of p-SWAP gate is geometrically designed using our Bloch sphere approach. The generality of p-SWAP gate is proposed for Phase oracles requiring a desirable p for a combination set of swapped qubits, as well as for Boolean oracles requiring a cost-effective SWAP gate when p is ignored. In this paper, after transpilation (synthesization) into a real quantum computer, it was concluded that the transpiled quantum circuit of p-SWAP gate has a lower quantum cost than that of the standard SWAP gate.

The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. For that, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool. On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY-plane, which is the 2D circular top-view of the Bloch sphere. Collectively, the Bloch sphere and its XY-plane are termed the Bloch sphere approach (BSA). With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native gates ( , , , and ), as follows. Quantum libraries (GALA-n [3, 6] and CALA-n [4, 7]), which have become part of the IBM Qiskit ecosystem [8] n-bit Toffoli gate [2-4] n-bit Boolean gates (AND, NAND, OR, NOR, implication, and inhibition) [3, 4] n-bit controlled- ( ) and controlled- ( ) gates [3, 4] n-bit Fredkin gate [3, 4] n-bit Miller gate [3, 4] Boolean-Phase SWAP gate (p-SWAP) [4, 5] Because the quantum operations of all IBM native "basis" gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, we propose the XY-plane of the Bloch sphere for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using different projectional planes of the Bloch sphere, e.g., the XZ-plane and the YZ-plane, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as an open geometrical framework for prospective quantum computing research.

We introduce a new quantum layout-aware approach to realize cost-effective n-bit gates using the Bloch sphere, for 2 ≤ n ≤ 5 qubits. These n-bit gates are entirely constructed from the Clifford+T gates, in the approach of selecting sequences of rotations visualized on the Bloch sphere. This Bloch sphere approach ensures to match the quantum layout for synthesizing (transpiling) these n-bit gates into an IBM quantum computer. Various standard n-bit gates (Toffoli, Fredkin, etc.) and their operational equivalent of our proposed n-bit gates are examined and evaluated, in the context of the final quantum costs, as the final counts of generated IBM native gates. In this paper, we demonstrate that all our n-bit gates always have lower quantum costs than those of standard n-bit gates after transpilation. Hence, our Bloch sphere approach can be used to build a quantum library of various cost-effective n-bit gates for different layouts of IBM quantum computers.