Abstract
Control modular addition is a core arithmetic function, and we must consider the computational cost for actual quantum computers to realize efficient implementation. To achieve a low computational cost in a control modular adder, we focus on minimizing KQ, defined by the product of the number of qubits and the depth of the circuit. In this paper, we construct an efficient control modular adder with small KQ by using relative-phase Toffoli gates in two major types of quantum computers: Fault-Tolerant Quantum Computers (FTQ) on the Logical layer and Noisy Intermediate-Scale Quantum Computers (NISQ). We give a more efficient construction compared to Van Meter and Itoh's, based on a carry-lookahead adder. In FTQ, $T$ gates incur heavy cost due to distillation, which fabricates ancilla for running $T$ gates with high accuracy but consumes a lot of specially prepared ancilla qubits and a lot of time. Thus, we must reduce the number of $T$ gates. We propose a new control modular adder that uses only 20% of the number of $T$ gates of the original. Moreover, when we take distillation into consideration, we find that we minimize $\text{KQ}_{T}$ (the product of the number of qubits and $T$-depth) by running $\Theta\left(n / \sqrt{\log n} \right)$ $T$ gates simultaneously. In NISQ, CNOT gates are the major error source. We propose a new control modular adder that uses only 35% of the number of CNOT gates of the original. Moreover, we show that the $\text{KQ}_{\text{CX}}$ (the product of the number of qubits and CNOT-depth) of our circuit is 38% of the original. Thus, we realize an efficient control modular adder, improving prospects for the efficient execution of arithmetic in quantum computers.
Highlights
Functional but imperfect quantum computers have emerged, called Noisy Intermediate-Scale Quantum Computers (NISQ) [2], with machines from IBM [3], [4], Google [5], Rigetti [6], IonQ [7], and Honeywell [8] all accessible via the web.Many researchers have constructed simple quantum circuits for NISQ machines
OUR CONTRIBUTION In this study, we propose a method for optimizing a control modular adder based on a carry-lookahead adder for both Fault-Tolerant Quantum Computers (FTQ) and NISQ
We propose a form of the control modular adder reducing CNOT gates
Summary
Functional but imperfect quantum computers have emerged, called Noisy Intermediate-Scale Quantum Computers (NISQ) [2], with machines from IBM [3], [4], Google [5], Rigetti [6], IonQ [7], and Honeywell [8] all accessible via the web. In Shor’s algorithm, the control modular exponentiation step dominates the total cost, leading many researchers to study its construction [31]–[41]. The operation of (3) requires an addition by the result of multiplication This addition can be decomposed as follows, where a and b are non-negative integers less than N , and k is a n-bit number expressed as k = Thapliyal et al.’s construction reduces KQ in FTQ, but similar optimization can be applied to NISQ by Maslov’s relative-phase Toffoli gates [49]. B. OUR CONTRIBUTION In this study, we propose a method for optimizing a control modular adder based on a carry-lookahead adder for both FTQ and NISQ. In second-level optimization, we minimize KQ for FTQ or NISQ by using relative-phase Toffoli gates (Section IV). W√e show that we achieve smallest KQT when we run Θ n/ log n T gates simultaneously
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