This paper presents novel techniques for the synthesis of reversible networks of Toffoli gates, as well as improvements to previous methods. Gate count and technology oriented cost metrics are used. Our synthesis techniques are independent of the cost metrics. Two new iterative synthesis procedure employing Reed-Muller spectra are introduced and shown to complement earlier synthesis approaches. The template simplification suggested in earlier work is enhanced through introduction of a faster and more efficient template application algorithm, updated (shorter) classification of the templates, and presentation of the new templates of sizes 7 and 9. A novel ``resynthesis'' approach is introduced wherein a sequence of gates is chosen from a network, and the reversible specification it realizes is resynthesized as an independent problem in hopes of reducing the network cost. Empirical results are presented to show that the methods are effective both in terms of the realization of all 3x3 reversible functions and larger reversible benchmark specifications.
The paper introduces dual Toffoli and Peres reversible gates, which operate under disjunctive control, and shows their functionality based on the Barenco et al. quantum model. Both uniform and mixed polarity are considered for the controls. Rewriting rules are presented, which provide a possible reduction of the number of gates and quantum cost of reversible (sub)circuits using standard Toffoli or Peres gates. Finally, a Clifford+T realization of a dual Toffoli and a dual Peres gate is shown, which may be used when mapping reversible circuits to the IBM quantum computers.
The question of finding a lower bound on the number of Toffoli gates in a classical reversible circuit is addressed. A method based on quantum information concepts is proposed. The method involves solely concepts from quantum information - there is no need for an actual physical quantum computer. The method is illustrated on the example of classical Shannon data compression.
The Toffoli gate is an essential logic element, which permits implementation of a reversible processor. It is of relevance both for classical as well as quantum logics. We propose and theoretically study all-optical implementations of three-bit and four-bit Toffoli gates by application of adiabatic population transfer techniques. For a three-bit Toffoli gate we use variants of stimulated Raman adiabatic passage (STIRAP) processes in a $Λ$-type level scheme, driven by two laser pulses at sufficiently large detunings. For the implementation of a four-bit Toffoli gate, we apply reversible adiabat…
Recently, reversible circuit synthesis has been intensively studied. One of the problems that has not been solved for a long time was exact minimization of gate count (GC) in 4-bit circuits. Finally, last year a tool of practical usage for finding optimal gate count Toffoli networks for any 4-variable function was developed. However, not much work has been done yet on exact minimization of quantum cost (QC) in 4-bit circuits. This paper presents an application of the above mentioned tool to reducing QC of 4-bit reversible circuits. It is shown that for benchmarks and for designs taken from rec…
The reversible implementation of classical functions accounts for the bulk of most known quantum algorithms. As a result, a number of reversible circuit constructions over the Clifford+$T$ gate set have been developed in recent years which use both the state and phase spaces, or $X$ and $Z$ bases, to reduce circuit costs beyond what is possible at the strictly classical level. We study and generalize two particular classes of these constructions: relative phase circuits, including Giles and Selinger's multiply-controlled $iX$ gates and Maslov's $4$ qubit Toffoli gate, and measurement-assisted …
Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! {\approx} 2^{44} functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesiz…
Reversible single-target gates are a generalization of Toffoli gates which are a helpful formal representation for the description of synthesis algorithms but are too general for an actual implementation based on some technology. There is an exponential lower bound on the number of Toffoli gates required to implement any reversible function, however, there is also a linear upper bound on the number of single-target gates which can be proven using a constructive proof based on a former presented synthesis algorithm. Since single-target gates can be mapped to a cascade of Toffoli gates, this syn…
Reversible logic has attracted much research interest over the last few decades, especially due to its application in quantum computing. In the construction of reversible gates from basic gates, ancilla bits are commonly used to remove restrictions on the type of gates that a certain set of basic gates generates. With unlimited ancilla bits, many gates (such as Toffoli and Fredkin) become universal reversible gates. However, ancilla bits can be expensive to implement, thus making the problem of minimizing necessary ancilla bits a practical topic. This thesis explores the problem of reversible …
Quantum addition circuits are considered being of two types: 1) Toffolli-adder circuits which use only classical reversible gates (CNOT and Toffoli), and 2) QFT-adder circuits based on the quantum Fourier transformation. We present the first systematic translation of the QFT-addition circuit into a Toffoli-based adder. This result shows that QFT-addition has fundamentally the same fault-tolerance cost (e.g. T-count) as the most cost-efficient Toffoli-adder: instead of using approximate decompositions of the gates from the QFT circuit, it is more efficient to merge gates. In order to achieve th…
We propose the theory of Cayley graphs as a framework to analyse gate counts and quantum costs resulting from reversible circuit synthesis. Several methods have been proposed in the reversible logic synthesis literature by considering different libraries whose gates are associated to the generating sets of certain Cayley graphs. In a Cayley graph, the distance between two vertices corresponds to the optimal circuit size. The lower bound for the diameter of Cayley graphs is also a lower bound for the worst case for any algorithm that uses the corresponding gate library. In this paper, we study …
Compared with the idea of universal quantum computation, a direct synthesis of a multiqubit logic gate can greatly improve the efficiency of quantum information processing tasks. Here we propose an efficient scheme to implement a three-qubit controlled-not (Toffoli) gate of neutral atoms based on unconventional Rydberg pumping. By adjusting the strengths of Rabi frequencies of driving fields, the Toffoli gate can be achieved within one step, which is also insensitive to the fluctuation of the Rydberg-Rydberg interaction. Considering different atom alignments, we can obtain a high-fidelity Toff…
The quantum Toffoli gate allows universal reversible classical computation. It is also an important primitive in many quantum circuits and quantum error correction schemes. Here we demonstrate the realization of a Toffoli gate with three superconducting transmon qubits coupled to a microwave resonator. By exploiting the third energy level of the transmon qubit, the number of elementary gates needed for the implementation of the Toffoli gate, as well as the total gate time can be reduced significantly in comparison to theoretical proposals using two-level systems only. We characterize the perfo…
Universal quantum gates and quantum error correction~(QEC) lie in the heart of quantum information science. Large-scale quantum computing depends on a universal set of quantum gates, in which some gates may be easily carried out, while others are hard with a certain physical system. There is a unique three-qubit quantum gate called the Deutsch gate~[$\mathbb{D}(θ)$], from which alone a circuit can be constructed so that any feasible quantum computing is attainable. As far as we know, however, $\mathbb{D}(θ)$ has not been demonstrated. Here we design an easily realizable $\mathbb{D}(θ)$ by usin…
Most of the work on implementing arithmetic on a quantum computer has borrowed from results in classical reversible computing (e.g. [VBE95], [BBF02], [DKR04]). These quantum networks are inherently classical, as they can be implemented with only the Toffoli gate. Draper [D00] has proposed an inherently "quantum" network for addition based on the quantum Fourier transform. His approach has the advantage that it requires no carry qubits (the previous approaches required O(n) carry qubits). The network in [D00] uses quantum rotation gates, which must either be implemented with exponential precisi…
A breadth-first search method for determining optimal 3-line circuits composed of quantum NOT, CNOT, controlled-V and controlled-V+ (NCV) gates is introduced. Results are presented for simple gate count and for technology motivated cost metrics. The optimal NCV circuits are also compared to NCV circuits derived from optimal NOT, CNOT and Toffoli (NCT) gate circuits. The work presented here provides basic results and motivation for continued study of the direct synthesis of NCV circuits, and establishes relations between function realizations in different circuit cost metrics.
The basic principles and results of Conservative Logic introduced by Fredkin and Toffoli on the basis of a seminal paper of Landauer are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental conn…
A single-shot Toffoli, or controlled-controlled-NOT, gate is desirable for classical and quantum information processing. The Toffoli gate alone is universal for reversible computing and, accompanied by the Hadamard gate, forms a universal gate set for quantum computing. The Toffoli gate is also a key ingredient for (non-topological) quantum error correction. Currently Toffoli gates are achieved by decomposing into sequentially implemented single- and two-qubit gates, which requires much longer times and yields lower overall fidelities compared to a single-shot implementation. We develop a quan…
We report optimal and asymptotically optimal reversible circuits composed of NOT, CNOT, and Toffoli (NCT) gates, keeping the count by the subsets of the gate types used. This study fine tunes the circuit complexity figures for the realization of reversible functions via reversible NCT circuits. An important consequence is a result on the limitation of the use of the $T$-count quantum circuit metric popular in applications.
Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16!~2^44 functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present an algorithm that synthesizes an optimal circuit for any 4-bit reversible specification. We employ several techniques to make the probl…
The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its CNOT-optimality. We study three-qubit circuits which contain less than six CNOT gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator. …
The reversible circuit synthesis problem can be reduced to permutation group. This allows Schreier-Sims Algorithm for the strong generating set-finding problem to be used to find tight bounds on the synthesis of 3-bit reversible circuits using the NFT library. The tight bounds include the maximum and minimum length of 3-bit reversible circuits, the maximum and minimum cost of 3-bit reversible circuits. The analysis shows better results than that found in the literature for the lower bound of the cost. The analysis also shows that there are 1960 universal reversible sub-libraries from the main …
This paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits. The results we established are twofold. The first shows that ternary Swap, ternary Not and ternary Toffoli gates are universal for the realization of arbitrary $n\times n$ ternary quantum switching networks without ancilla bits. The second result proves that all $n\times n$ quantum ternary networks can be generated by Not, Controlled-Not, Multiply-Two, and Toffoli gates. Our approach is constructive. key words: ternary quantum logic synthesis, quantum circuit op…
Homogenous Boolean function is an essential part of any cryptographic system. The ability to construct an optimized reversible circuits for homogeneous Boolean functions might arise the possibility of building cryptographic system on novel computing paradigms such as quantum computers. This paper shows a factorization algorithm to synthesize such circuits.
It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, i.e. ignoring the ancilla bits, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate s…
It is well known that most of the frequently used reversible logic gates (e.g., NOT, CNOT, SWAP, Toffoli, Fredkin) are self-inverse and are represented by square matrices that are unitary and Hermitian. However, with a simple minded argument, it is established that the most of the allowed reversible gates are non-self-inverse (unitary but non-Hermitian) in nature. It is also shown that the % of non-Hermitian gates increases with the dimension. For example, 58.33% of the 2-bit gates, 98.10% of the 3-bit gates and 99.99% of the 4-bit gates are non-Hermitian. As classical reversible gates are ess…
In this paper, a new non-search based synthesis algorithm for reversible circuits is proposed. Compared with the widely used search-based methods, our algorithm is guarantied to produce a result and can lead to a solution with much fewer steps. To evaluate the proposed method, several circuits taken from the literature are used. The experimental results corroborate the expected findings.
Many synthesis approaches for reversible and quantum logic have been proposed so far. However, most of them generate circuits with respect to simple metrics, i.e. gate count or quantum cost. On the other hand, to physically realize reversible and quantum hardware, additional constraints exist. In this paper, we describe cost metrics beyond gate count and quantum cost that should be considered while synthesizing reversible and quantum logic for the respective target technologies. We show that the evaluation of a synthesis approach may differ if additional costs are applied. In addition, a new c…
Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and stu…
We propose a linear-size synthesis of the multi-controlled Toffoli gate on qudits with at most one borrowed ancilla. This one ancilla can even be saved when the qudit dimension is odd. Our synthesis leads to improvements in various quantum algorithms implemented on qudits. In particular, we obtain (i) a linear-size and one-clean-ancilla synthesis of multi-controlled qudit gates; (ii) an optimal-size and one-clean-ancilla synthesis of unitaries on qudits; (iii) a near-optimal-size and ancilla-free/one-borrowed-ancilla implementation of classical reversible functions as qudit gates.
Efficient constructions for quantum logic are essential since quantum computation is experimentally challenging. This thesis develops quantum logic synthesis as a paradigm for reducing the resource overhead in fault-tolerant quantum computing. The model for error correction considered here is the surface code. After developing the theory behind general logic synthesis, the resource costs of magic-state distillation for the $T = \exp(i π(I-Z)/8)$ gate are quantitatively analyzed. The resource costs for a relatively new protocol distilling multi-qubit Fourier states are calculated for the first …
Portland State University（波特兰州立大学）；Intel Labs, Intel Corporation（英特尔实验室，英特尔公司）
We use machine learning techniques to design a 50 ns three-qubit flux-tunable controlled-controlled-phase gate with fidelity of >99.99% for nearest-neighbor coupled transmons in circuit quantum electrodynamics architectures. We explain our gate design procedure where we enforce realistic constraints, and analyze the new gate's robustness under decoherence, distortion, and random noise. Our controlled-controlled-phase gate in combination with two single-qubit gates realizes a Toffoli gate which is widely used in quantum circuits, logic synthesis, quantum error correction, and quantum games.
In this paper we present a method for minimizing reversible quantum circuits using the Quantum Operator Form (QOF); a new representation of quantum circuit and of quantum-realized reversible circuits based on the CNOT, CV and CV$^\dagger$ quantum gates. The proposed form is a quantum extension to the well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the usage of different quantum gates. Therefore QOF permits minimization of quantum circuits by using properties of different gates than only the multi-control Toffoli gates. We introduce a set of minimization rules and a pseud…
In this paper we consider circuit synthesis for n-wire linear reversible circuits using the C-NOT gate library. These circuits are an important class of reversible circuits with applications to quantum computation. Previous algorithms, based on Gaussian elimination and LU-decomposition, yield circuits with O(n^2) gates in the worst-case. However, an information theoretic bound suggests that it may be possible to reduce this to as few as O(n^2/log n) gates. We present an algorithm that is optimal up to a multiplicative constant, as well as Theta(log n) times faster than previous methods. While …
Reversible logic has applications in various research areas including low-power design and quantum computation. In this paper, a rule-based optimization approach for reversible circuits is proposed which uses both negative and positive control Toffoli gates during the optimization. To this end, a set of rules for removing NOT gates and optimizing sub-circuits with common-target gates are proposed. To evaluate the proposed approach, the best-reported synthesized circuits and the results of a recent synthesis algorithm which uses both negative and positive controls are used. Our experiments reve…
The Toffoli gate serving as a basic building block for reversible quantum computation, has manifested its great potentials in improving the error-tolerant rate in quantum communication. While current route to the creation of Toffoli gate requires implementing sequential single- and two-qubit gates, limited by longer operation time and lower average fidelity. We develop a new theoretical protocol to construct a universal $(n+1)$-qubit Toffoli gate sphere based on the Rydberg blockade mechanism, by constraining the behavior of one central target atom with $n$ surrounding control atoms. Its merit…
A rotation-based synthesis framework for reversible logic is proposed. We develop a canonical representation based on binary decision diagrams and introduce operators to manipulate the developed representation model. Furthermore, a recursive functional bi-decomposition approach is proposed to automatically synthesize a given function. While Boolean reversible logic is particularly addressed, our framework constructs intermediate quantum states that may be in superposition, hence we combine techniques from reversible Boolean logic and quantum computation. The proposed approach results in quadra…
Reversible circuits for SR flip flop, JK flip flop, D flip flop, T flip flop, Master Slave D flip flop and Master Slave JK flip flop have been provided with three different logical approaches. All the circuits have been optimized with the help of existing local optimization algorithms (e.g. template matching, moving rule and deletion rule) and the optimized sequential circuits have been compared with the earlier proposals for the same. It has been shown that the present proposals have lower gate complexities and lower number of garbage bits compared to the earlier proposals. It has also been s…
Distributed quantum computation requires to apply quantum remote gates on separate nodes or subsystems of network. On the other hand, Toffoli gate is a universal and well-known quantum gate. It is frequently used in synthesis of quantum circuits. In this paper, a general protocol for implementing a remote n-qubit controlled-U gate is presented with minimum required resources. Then, the proposed method is applied for implementing a Toffoli gate in bipartite and tripartite systems. This method also is optimal when group of the qubits belong to one part, section or subsystem of network. Because i…
Many universal reversible libraries that contain more than one gate type have been proposed in the literature. Practical implementation of reversible circuits is much easier if a single gate type is used in the circuit construction. This paper proposes a reversible n-bit gate that is universal for reversible circuits synthesis. The proposed gate is extendable according to the size of the circuit. The paper shows that the size of the synthesized circuits using the proposed gate is comparable with the size of the synthesized circuits using the hybrid reversible libraries for 3-in/out reversible …
Linear Nearest Neighbor (LNN) synthesis in reversible circuits has emerged as an important issue in terms of technological implementation for quantum computation. The objective is to obtain a LNN architecture with minimum gate cost. As achieving optimal synthesis is a hard problem, heuristic methods have been proposed in recent literature. In this work we present a graph partitioning based approach for LNN synthesis with reduction in circuit cost. In particular, the number of SWAP gates required to convert a given gate-level quantum circuit to its equivalent LNN configuration is minimized. Our…
Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search…
The synthesis of reversible functions has been an intensively studied research area in the last decade. Since almost all proposed approaches rely on representations of exponential size (such as truth tables and permutations), they cannot be applied efficiently to reversible functions with more than 15 variables. In this paper, we propose an ancilla-free synthesis approach based on Young subgroups using symbolic function representations that can efficiently be implemented with binary decision diagrams (BDDs). As a result, the algorithm not only allows to synthesize large reversible functions wi…
In this paper, simultaneous reduction of circuit depth and synthesis cost of reversible circuits in quantum technologies with limited interaction is addressed. We developed a cycle-based synthesis algorithm which uses negative controls and limited distance between gate lines. To improve circuit depth, a new parallel structure is introduced in which before synthesis a set of disjoint cycles are extracted from the input specification and distributed into some subsets. The cycles of each subset are synthesized independently on different sets of ancillae. Accordingly, each disjoint set can be synt…
We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known deco…
Reversible computing has emerged as a possible low cost alternative to conventional computing in terms of speed, power consumption and computing capability. In order to achieve reliable circuits in reversible computing, provision for fault tolerance is necessary. A number of fault models, fault tolerant techniques (such as parity-preserving) and testing approaches have proposed in literature. This dissertation exploits parity-preserving characteristics of two reversible gates which provide low cost parity-preserving based fault tolerance. In order to extend online testability of reversible cir…
Atomic-scale logic and the minimization of heating (dissipation) are both very high on the agenda for future computation hardware. An approach to achieve these would be to replace networks of transistors directly by classical reversible logic gates built from the coherent dynamics of a few interacting atoms. As superpositions are unnecessary before and after each such gate (inputs and outputs are bits), the dephasing time only needs to exceed a single gate operation time, while fault tolerance should be achieved with low overhead, by classical coding. Such gates could thus be a spin-off of qua…
