Quantum Circuit Designs of Integer Division Optimizing T-Count and T-Depth

Abstract

Quantum circuits for mathematical functions such as division are necessary to use quantum computers for scientific computing. Quantum circuits based on Clifford+T gates can easily be made fault-tolerant but the T gate is very costly to implement. The small number of qubits available in existing quantum computers adds another constraint on quantum circuits. As a result, reducing T-count and qubit cost have become important optimization goals. The design of quantum circuits for integer division has caught the attention of researchers and designs have been proposed in the literature. However, these designs suffer from excessive T gate and qubit costs. Many of these designs also produce significant garbage output resulting in additional qubit and T gate costs to eliminate these outputs. In this work, we propose two quantum integer division circuits. The first proposed quantum integer division circuit is based on the restoring division algorithm and the second proposed design implements the non-restoring division algorithm. Both proposed designs are optimized in terms of T-count, T-depth and qubits. Both proposed quantum circuit designs are based on (i) a quantum subtractor, (ii) a quantum adder-subtractor circuit, and (iii) a novel quantum conditional addition circuit. Our proposed restoring division circuit achieves average T-count savings from 79.03% to 91.69% compared to the existing works. Our proposed non-restoring division circuit achieves average Tcount savings from 49.75% to 90.37% compared to the existing works. Further, both our proposed designs have linear T-depth.

Quantum Circuit Design of Toom 3-Way Multiplication

“…Note that in the case of quantum, the computation is not as straightforward and the resource counts can be higher since we need to deal with quantum-related property of the circuit, such as reversibility, which requires performing uncomputation to clear up garbage outputs, i.e., outputs that are neither one of the primary inputs nor a useful output but must exist in the quantum circuit to preserve reversibility [22]. Hence, resource consumption in quantum computation tends to be higher than the classical counterparts.…”

“…In contrast, analyzing its cost in the quantum case can provide insights and open the possibility to the higher Toom-k multiplication. Additionally, despite the view that division circuit is expensive, there have been several attempts to achieve relatively efficient quantum dividers, either for binary integer [20][21][22] or floating point arithmetic case [23]. Furthermore, the nontrivial division operation inherent to Toom 3-way (and higher order) multiplication is not a general one; instead, it uses exact integer division, which we later found to enable some further optimization to the circuit.…”

Acongestion‐awaremixed integer linear programming model for placement and scheduling of quantum circuits with atwo‐levelheuristic solution approach

“…Wu et al 20 presented a linear “Turing‐machine‐like” architecture with a multi‐laser control “head”, where a linear chain of ions moves back and forth under the laser head. Some studies 21‐25 have focused on the fault‐tolerant implementations for quantum circuits and designed fault‐tolerant circuits for quantum operations.…”

Quantum circuit design for objective function maximization in gate-model quantum computers

“…The aim of the T-count optimization is to reduce the number of T-gates without substantially increasing the number of qubits. The method also applied for quantum circuit designs of integer division [44]. In the optimization takes into consideration both the T-count and T-depth, since T-depth is also an important performance measure to reduce the implementation costs.…”