Clifford+T-based implementation of fault-tolerant quantum circuits over XOR-Majority Graph

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Abstract

To build a scalable quantum computer that is sufficiently large to solve classically intractable problems is one of the biggest engineering quests of current times. But, the fragility of quantum states with short coherence time and vulnerability of quantum components towards external noise becomes an immense challenge to the physical implementation. Thus, the fault-tolerant architecture over the quantum error-correcting code (qecc) such as Surface code in optimal logical level are most desirable and can be achieved transversely using Clifford+T-group. However, the T-gate has high latency. In this paper, we introduce a template matching scheme over the Xor-Majority Graphs to realize a high scalable fault-tolerant quantum circuit. The proposed methodology is validated with multiple quantum benchmarks achieving an improvement of 71.98%, and 78.33% on average Tcount, and Tdepth respectively over state-of-art, and at the same time completely avoiding any ancillary input. When ancillary inputs are permitted, an improvement of 92.77% on Tdepth is obtained.

Introduction

In recent times, quantum computing (QC) is the single biggest breakthrough with great promise to overcome limitations that inherent to classical computers [1], [2], [3]. One of the major challenges towards the implementation of a scalable quantum computer is the presence of external noise, which affects the reliability of the quantum gate operations. To deal with the issue, the fault-tolerant quantum computation with a low execution time is desirable. Fundamentally, the property of fault-tolerance can be achieved by encoding a physical qubit into a logical qubit by using quantum error-correcting code (QECC). In this regard, Surface code is the most promising one and has the highest threshold value i.e. 0.75% per gate [4]. Further, to protect the efficacy of QECC, the error rate should below the threshold limit, and needs transversal operators so as to restrain the diffusion of error within the QECC blocks. Fundamentally CNOT-gate, and all single-qubit primitive quantum gates are default transversal operators [5]. In fact, no such quantum-error correcting code is available which can transversely implement a universal gate set [6]. Currently, the Clifford+T-group is used extensively at a logical level to design fault-tolerant quantum circuit [7], [8], [9]. And more fundamentals on Clifford+T-group is presented in Section 2.

A quantum computer processes the information based on the laws of quantum mechanics in subatomic level. In fact, quantum evolution is unitary in nature which mandates logical reversibility of quantum logic. And for that the reversible logic plays a pivotal role as an intermediary stage while mapping a Boolean logic into quantum logic wherein Multi Controlled Toffoli (MCT) gate provides universality in reversible logic [10], [11], [12]. The quality of mapping from Boolean logic to quantum logic is crucially dependent on the adopted intermediate structure for logic representation. On the quest for a suitable Boolean data structure for a large function, the newly introduced XOR-Majority Graph (XMG) that proposed by Chu et al. [13] could be promising as an intermediary data-structure for the quantum circuit synthesis against the present-state-of-the-approaches [14], [15], due to depth optimization of the synthesis and reduction of node complexity. And, has potential to convert a ripple-carry adder into carry look-ahead adder efficaciously. Therefore, the mapping of XMG data-structure into Clifford+T-based circuit by adhering to the optimization of T-gate and T-cycle could be a suitable approach for the robust fault-tolerant quantum circuit realization of any Boolean function.

Recently, Bandyopadhyay et al. proposed synthesis of improved reversible circuits using AIG- and MIG-based data structures, and evaluated Tcount, Tdepth of equivalent fault-tolerant quantum circuit [16]. In this mapping, the authors studied a comparative analysis over BDD, AIG, MIG, and XMG data structure by means of mapping node-wise functionality into Clifford+T-based structure. Other graph-based quantum circuit synthesis approaches are presented in [15], [17].

A new question crop-up that can the node of XMG, and existing AIG, MIG be implemented by using fault-tolerant templates with low cost (Tcount, Tdepth) in comparing to present-state-of-the-approaches and same can be used in the post-synthesis mapping of XMG towards the realization of the efficient fault-tolerant circuit? From this motivation, we have proposed a template matching scheme for the mapping of XMG data structure [13] into the quantum circuits and also optimized the cost parameter associated with fault-tolerant quantum circuits synthesis.

Here, the key contributions are highlighted as below:

  • A robust Low-cost Clifford+T-based fault-tolerant templates are proposed against node of the XMG data-structure.

  • We propose unit T-depth-based fault-tolerant templates to contain the effect of Decoherence using four ancillary input(s) by reducing Tdepth.

  • We have simulated quantum of XMG-based benchmark function over both kinds of the proposed template(s) and compared the results with present-state-of-the-arts approaches w.r.t. Tcount and Tdepth.

The remainder of this paper is organized as follows: Section 2 provides a detailed background for better apprehension of this paper. In Section 3, our proposed approach is explained. Experimental result and comparative analysis are presented in Section 4 and the work is completed in Section 5.

Section snippets

Background

Here, we discuss some basic fundamentals relating to graph-based Boolean algebra and term, terminologies associated with reversible/quantum logic circuit.

Proposed technique

In this section, we have proposed Clifford+T-based fault-tolerant templates to represent the node functionality of XMG. Then, we apply proposed templates into the XMG data structure using a template matching scheme, in which each node of the XMG replaced by its equivalent templates to find an efficient fault-tolerant quantum circuit for any arbitrary Boolean functions.

Experimental result and analysis

The proposed synthesis approach has been implemented in C++ using XMG and a template-matching scheme (Algorithm 1). Then our proposed synthesis approach is applied over the quantum of benchmark function, and the subsequent results are documented in Table 2, Table 3 separately as design-1 and design-2 based on two above-defined template library TN and TU respectively.

In Table 2, the results have been compared with present-state-of-the-arts [16] over MIG-based fault-tolerant quantum circuit. In

Conclusion

In this paper, we have proposed one template-based mapping approach over XMG in order to achieve fault-tolerant quantum computation. The experimental results validate the very propose of this approach as it reduces the Tcount, Tdepth cost which is desirable for high scalable fault-tolerant quantum computation. Moreover, our design approach is based on matrix identities over Hilbert space (H) which does not violate principle of no-cloning theorem [28].

Literally, quantum states are highly

CRediT authorship contribution statement

Laxmidhar Biswal: Conceptualization, Methodology, Software, Investigation, Visualization, Writing- Original draft preparation. Habibur Rahaman: Software, Validation, Formal analysis, Data curation, Writing - review & editing. Niladri Pratap Maity: Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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