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arXiv:1904.06920 · Physical Review A

Techniques for fault-tolerant decomposition of a multicontrolled Toffoli gate

Laxmidhar Biswal, Debjyoti Bhattacharjee, Anupam Chattopadhyay, Hafizur Rahaman

Published Dec 19, 2019• 27 citations
arXivHTMLPDF

Abstract

Physical implementation of scalable quantum architectures faces an immense challenge in the form of fragile quantum states. To overcome it, quantum architectures with fault tolerance are desirable. These are achieved currently by using a surface code along with a transversal gate set. This indicates the need for decomposition of universal $n$-qubit multicontrolled Toffoli ($n$-MCT) gates using a transversal gate set. Additionally, the transversal non-Clifford phase gate incurs high latency, which makes it an important factor to consider during decomposition. Additionally, the decomposition of a large MCT gate without ancillas presents an additional hurdle. In this paper, we address both of these issues by introducing the $\mathrm{Clifford}+{Z}_{N}$ gate library. We present an ancilla-free decomposition of MCT gates with linear phase depth and quadratic phase count. Furthermore, we provide a technique for decomposition of MCT gates in unit phase depth using the $\mathrm{Clifford}+{Z}_{N}$ library, albeit at the cost of ancillary lines and exponential phase count.

DOI
10.1103/physreva.100.062326

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Physical implementation of scalable quantum architectures faces an immense challenge in the form of fragile quantum states. To overcome it, quantum architectures with fault tolerance are desirable. These are achieved currently by using a surface code along with a transversal gate set. This indicates the need for decomposition of universal nnn-qubit multicontrolled Toffoli (nnn-MCT) gates using a transversal gate set. Additionally, the transversal non-Clifford phase gate incurs high latency, which makes it an important factor to consider during decomposition. Additionally, the decomposition of a large MCT gate without ancillas presents an additional hurdle. In this paper, we address both of these issues by introducing the Clifford+ZN\mathrm{Clifford}+{Z}_{N}Clifford+ZN​ gate library. We present an ancilla-free decomposition of MCT gates with linear phase depth and quadratic phase count. Furthermore, we provide a technique for decomposition of MCT gates in unit phase depth using the Clifford+ZN\mathrm{Clifford}+{Z}_{N}Clifford+ZN​ library, albeit at the cost of ancillary lines and exponential phase count.
Multi-controlled Toffoli gates are fundamental building blocks in quantum computation, with applications in quantum arithmetic, simulation, and search algorithms. In fault-tolerant architectures, their realization is constrained by the high cost of non-Clifford resources, particularly in terms of T-count and T-depth. Recent advances have demonstrated that the use of ancillary qubits, relative-phase Toffoli gates, and dynamic circuit techniques can substantially reduce this overhead. In this work, we investigate the decomposition of large Toffoli gates using 3- and 4-input relative-phase Toffoli gates in the presence of a single clean ancilla and conditionally clean ancillas. We derive explicit resource bounds for Clifford+T implementations incorporating dynamic-circuit-based uncomputation and measurement-conditioned corrections. Our analysis emphasizes T-depth reduction under fixed CX and T-count overhead, ensuring relevance for near-term devices. We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements. We further validate our theoretical results through experimental evaluation and comparative analysis with existing approaches.
For realizing future fault-tolerant quantum computation, it is very important to reduce so-called the T-depth of a target quantum circuit. Thus, there have been many researches to consider how to decompose Multiple Controlled Toffoli (MCT) gates with few T-depth; these existing methods does not consider the T-depth of each qubit when they decompose one MCT gate. In contrast to existing methods, this paper proposes to consider the T-depth of each qubit when we decompose an MCT gate; we show that we can decrease the T-depth by considering Tdepth of each bit and appropriately selecting ancilla qubits and decomposition methods. In addition, to consider the impact of the decomposition of one MCT gate on the decomposition of subsequent gates, our method utilizes beam search to select a possibly best decomposition for each MCT gate. We confirmed that the proposed method can reduce the T-depth by an average of 17.1% compared to existing methods.
It is considered to be very important to reduce so-called the T-depth of a target quantum circuit for realizing future fault-tolerant quantum computation. Thus, there have been many researches to consider how to decompose Multiple Controlled Toffoli (MCT) gates with few T-depth; these existing methods do not consider the T-depth of each qubit when they decompose one MCT gate. In contrast to existing methods, this paper proposes to consider the T-depth of each qubit when we decompose an MCT gate; we show that we can decrease the T-depth by considering the T-depth of each qubit and appropriately selecting ancilla qubits and decomposition methods. In addition, to consider the impact of the decomposition of one MCT gate on the decomposition of subsequent gates, our method utilizes beam search to select a possibly best decomposition for each MCT gate. We confirmed that the proposed method can reduce the T-depth by an average of 17.1% compared to existing methods.
The decomposition of complex quantum operations into experimentally feasible gate sets has been a central challenge since the early development of quantum computing. The multi-controlled Toffoli (MCT) gate is a key example, with applications across a wide range of quantum algorithms, whose decomposition into smaller gates, however, typically leads to deep circuits. In this work, we introduce a teleportation-based decomposition that implements an MCT gate. This decomposition has unit Toffoli depth, independent of the number of controls, while achieving the lowest Toffoli count among existing approaches for more than five controls at the cost of a linear overhead in ancilla qubits. We also investigate the parameter regimes in which the teleportation-based decomposition is preferable by analyzing the relative quality of distributed entanglement and Toffoli gates. We further demonstrate the advantages of this implementation in circuits that rely on MCT gates, such as the adder operator, quantum read-only memory, quantum neurons, and quantum decision trees.
We present an implementation of multicontrolled () quantum gates that provides significant reductions of cost compared to state-of-the-art methods. The operator applied on the target qubit is a unitary, special unitary, or the Pauli X operator (multicontrolled Toffoli), and requires one clean ancilla, no ancilla, or one dirty ancilla, respectively. We generalize our methods for any number of target qubits, and provide further cost reductions if additional ancilla qubits are available. For each type of gate, we provide implementations for unrestricted (all-to-all) connectivity and for linear-nearest-neighbor (LNN) connectivity. All of the methods use a linear cost of gates from the Clifford+ T (fault-tolerant) set. In the context of LNN architecture, the cost and depth of our circuits scale linearly irrespective of the position of the qubits on which the gate is applied. Our methods directly improve the compilation process of many quantum algorithms, providing optimized circuits. Given the scale of our improvements, for example, the quadratic-to-linear controlled- count for LNN, they will naturally result in a large reduction of errors.
Fault-tolerant quantum computation requires substantial resources for non-Clifford gates. Their count also enters exponentially into the runtime of several classical simulation methods. We present algebraic methods for minimizing the non-Clifford CCZ (Toffoli) count in {CNOT,CCZ} phase circuits using tensor decomposition over F2\mathbb{F}_2F2​. On standard cubic benchmarks, these methods match or improve prior automated Toffoli counts, with most instances optimized in under a minute on a single CPU. With additional computation, the decompositions provide effective initializations for existing T-count methods, improving T counts on several circuits. We also present algebraic methods for CNOT-count optimization at fixed Toffoli count. This gives competitive ancilla-free finite-field multiplication circuits and reduces the CNOT count of XOR AES S-box circuits.