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arXiv:2010.00255 · IEEE Transactions on Quantum Engineering

Efficient Construction of a Control Modular Adder on a Carry-Lookahead Adder Using Relative-Phase Toffoli Gates

Kento Oonishi, Tomoki Tanaka, Shumpei Uno, Takahiko Satoh, Rodney Van Meter, Noboru Kunihiro

Published Dec 17, 2021• 15 citations
arXivHTMLPDF

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 minimizingKQ (where K is the number of logical qubits required by the algorithm, and Q is the elementary gate step), defined by the product of the number of qubits and the depth of the circuit. In this article, 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 with 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 especially 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 (n / \sqrt{\log n})$$T$gates simultaneously. In NISQ,cnotgates are the major error source. We propose a new control modular adder that uses only 35% of the number ofcnotgates of the original. Moreover, we show that the$\text{KQ}_{\text{CX}}$(the product of the number of qubits andcnot-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.

DOI
10.1109/tqe.2021.3136195

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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 minimizingKQ (where K is the number of logical qubits required by the algorithm, and Q is the elementary gate step), defined by the product of the number of qubits and the depth of the circuit. In this article, 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 with Van Meter and Itoh’s, based on a carry-lookahead adder. In FTQ,TTTgates incur heavy cost due to distillation, which fabricates ancilla for runningTTTgates with high accuracy but consumes a lot of especially prepared ancilla qubits and a lot of time. Thus, we must reduce the number ofTTTgates. We propose a new control modular adder that uses only 20% of the number ofTTTgates of the original. Moreover, when we take distillation into consideration, we find that we minimizeKQT\text{KQ}_{T}KQT​(the product of the number of qubits andTTT-depth) by running\Theta (n / \sqrt{\log n})$$Tgates simultaneously. In NISQ,cnotgates are the major error source. We propose a new control modular adder that uses only 35% of the number ofcnotgates of the original. Moreover, we show that theKQCX\text{KQ}_{\text{CX}}KQCX​(the product of the number of qubits andcnot-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.
Quantum Computing is making significant advancements toward creating machines capable of implementing quantum algorithms in various fields, such as quantum cryptography, quantum image processing, and optimization. The development of quantum arithmetic circuits for modulo addition is vital for implementing these quantum algorithms. While it is ideal to use quantum circuits based on fault-tolerant gates to overcome noise and decoherence errors, the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers cannot handle the additional computational cost associated with fault-tolerant designs. Our research aims to minimize circuit depth, which can reduce noise and facilitate the implementation of quantum modulo addition circuits on NISQ machines. This work presents quantum carry-lookahead modulo (2n - 1) adder (QCLMA), which is designed to receive two n-bit numbers and perform their addition with an O(log n) depth. Compared to existing work of O(n) depth, our proposed QCLMA reduces the depth and helps increase the noise fidelity. In order to increase error resilience, we also focus on creating a tree structure based Carry path, unlike the chain based Carry path of the current work. We run experiments on Quantum Computer IBM Cairo to evaluate the performance of the proposed QCLMA against the existing work and define Quantum State Fidelity Ratio (QSFR) to quantify the closeness of the correct output to the top output. When compared against existing work, the proposed QCLMA achieves a 47.21% increase in QSFR for 4-qubit modulo addition showcasing its superior noise fidelity.
Progress in quantum hardware design is progressing toward machines of sufficient size to begin realizing quantum algorithms in disciplines such as encryption and physics. Quantum circuits for addition are crucial to realize many quantum algorithms on these machines. Ideally, quantum circuits based on fault-tolerant gates and error-correcting codes should be used as they tolerant environmental noise. However, current machines called Noisy Intermediate Scale Quantum (NISQ) machines cannot support the overhead associated with faulttolerant design. In response, low depth circuits such as quantum carry lookahead adders (QCLA)s have caught the attention of researchers. The risk for noise errors and decoherence increase as the number of gate layers (or depth) in the circuit increases. This work presents an out-of-place QCLA based on Clifford+T gates. The QCLAs optimized for T gate count and make use of a novel uncomputation gate to save T gates. We base our QCLAs on Clifford+T gates because they can eventually be made faulttolerant with error-correcting codes once quantum hardware that can support fault-tolerant designs becomes available. We focus on T gate cost as the T gate is significantly more costly to make faulttolerant than the other Clifford+T gates. The proposed QCLAs are compared and shown to be superior to existing works in terms of T-count and therefore the total number of quantum gates. Finally, we illustrate the application of the proposed QCLAs in quantum image processing by presenting quantum circuits for bilinear interpolation.
In 2004, Cuccaro et al found a quantum-quantum adder with O(n)O(n)O(n) gate cost and O(1)O(1)O(1) ancilla qubits. Since then, it's been an open question whether classical-quantum adders can achieve the same asymptotic complexity. These costs are particularly relevant to modular arithmetic circuits, which often offset by the classically known modulus. In this paper, I construct an adder that uses 3 clean ancillae and 4n±O(1)4n \pm O(1)4n±O(1) Toffoli gates to add a classical offset into a quantum register. I also present an adder with a Toffoli cost of 3n±O(1)3n \pm O(1)3n±O(1) that uses 2 clean ancillae and n−2n-2n−2 dirty ancillae. I further show that applying the presented adders conditioned on a control qubit requires no additional workspace or Toffolis.
Progress in quantum hardware design is progressing toward machines of sufficient size to begin realizing quantum algorithms in disciplines such as encryption and physics. Quantum circuits for addition are crucial to realize many quantum algorithms on these machines. Ideally, quantum circuits based on fault-tolerant gates and error-correcting codes should be used as they tolerant environmental noise. However, current machines called Noisy Intermediate Scale Quantum (NISQ) machines cannot support the overhead associated with fault-tolerant design. In response, low depth circuits such as quantum carry lookahead adders (QCLA)s have caught the attention of researchers. The risk for noise errors and decoherence increase as the number of gate layers (or depth) in the circuit increases. This work presents an out-of-place QCLA based on Clifford+T gates. The QCLAs optimized for T gate count and make use of a novel uncomputation gate to save T gates. We base our QCLAs on Clifford+T gates because they can eventually be made fault-tolerant with error-correcting codes once quantum hardware that can support fault-tolerant designs becomes available. We focus on T gate cost as the T gate is significantly more costly to make fault-tolerant than the other Clifford+T gates. The proposed QCLAs are compared and shown to be superior to existing works in terms of T-count and therefore the total number of quantum gates. Finally, we illustrate the application of the proposed QCLAs in quantum image processing by presenting quantum circuits for bilinear interpolation.
The Toffoli gate is a fundamental building block for quantum arithmetic and reversible logic, yet its efficient realization remains a major challenge in both near-term and fault-tolerant quantum architectures. Recent advances in dynamic quantum circuit capabilities, including mid-circuit measurement and classical feedforward, provide new opportunities for reducing the resource overhead of non-Clifford operations. In this work, we propose a set of dynamic decomposition strategies for multi-controlled Toffoli gates that exploit adaptive circuit execution and ancilla-assisted constructions. Our methods systematically reduce entangling-gate count, T-count, and T-depth compared with conventional static decompositions, while preserving fault-tolerance guarantees. Through analytical cost models and experimental evaluation, we demonstrate that relative-phase primitives and measurement-conditioned corrections enable scalable implementations with improved depth and resource efficiency.
We provide evidence that commonly held intuitions when designing quantum circuits can be misleading. In particular, we show that 1) reducing the T-count can increase the total depth; 2) it may be beneficial to trade controlled NOTs for measurements in noisy intermediate-scale quantum (NISQ) circuits; 2) measurement-based uncomputation of relative phase Toffoli ancillae can make up to 30% of a circuit’s depth; and 4) area and volume cost metrics can misreport the resource analysis. Our findings assume that qubits are and will remain a very scarce resource. The results are applicable for both NISQ and quantum error-corrected protected circuits. Our method uses multiple ways of decomposing Toffoli gates into Clifford+T gates. We illustrate our method on addition and multiplication circuits using ripple-carry. As a byproduct result, we show systematically that for a practically significant range of circuit widths, ripple-carry addition circuits are more resource-efficient than the carry-lookahead addition ones. The methods and circuits were implemented in the open-source QUANTIFY software.