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Quantum Information and Computation

Realization of Deterministic Quantum Circuits for Non-Deterministic or Incompletely Specified Quantum State Machines

Manjith Kumar, Marek A. Perkowski

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract In classical logic design, there are machine learning methods based on converting a set of input-output traces to non-deterministic automata that are then converted to deterministic automata and synthesized using logic gates. This approach has not yet been extended to quantum automata. In this paper, we present a method to convert a set of input-output traces to a non-deterministic automaton, which is then converted to an incompletely specified multi-output Boolean function. The existing logic synthesis approaches for designing quantum circuits are insufficient to handle incompletely specified functions. So, we present a novel algorithm to synthesize logic functions with don’t cares using permutative quantum gates. The original MMD (D.M.Miller, D. Maslov, and G.W.Dueck) algorithm minimizes only completely specified reversible functions using cascades of reversible gates. In this paper, this algorithm is modified to allow for the inclusion of don’t cares within the given function’s truth table (reversible or irreversible). The distinguishing property of our presented algorithm QAS, is that it does not add any ancilla qubits if it is not necessary. This algorithm solves the problem of synthesizing deterministic and non-deterministic quantum state machines, both completely specified and incompletely specified.

DOI
10.2478/qic-2025-0038

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Abstract In classical logic design, there are machine learning methods based on converting a set of input-output traces to non-deterministic automata that are then converted to deterministic automata and synthesized using logic gates. This approach has not yet been extended to quantum automata. In this paper, we present a method to convert a set of input-output traces to a non-deterministic automaton, which is then converted to an incompletely specified multi-output Boolean function. The existing logic synthesis approaches for designing quantum circuits are insufficient to handle incompletely specified functions. So, we present a novel algorithm to synthesize logic functions with don’t cares using permutative quantum gates. The original MMD (D.M.Miller, D. Maslov, and G.W.Dueck) algorithm minimizes only completely specified reversible functions using cascades of reversible gates. In this paper, this algorithm is modified to allow for the inclusion of don’t cares within the given function’s truth table (reversible or irreversible). The distinguishing property of our presented algorithm QAS, is that it does not add any ancilla qubits if it is not necessary. This algorithm solves the problem of synthesizing deterministic and non-deterministic quantum state machines, both completely specified and incompletely specified.
Compact realizations of reversible logic functions are of interest in the design of quantum computers. In this paper we present an exact synthesis algorithm, based on Boolean satisfiability (SAT), that finds the minimal elementary quantum gate realization for a given reversible function. Since these gates work in terms of qubits, a multi-valued encoding is proposed. Don't care conditions appear naturally in many reversible functions. Constant inputs are often required when a function is embedded into a reversible one. The proposed algorithm takes full advantage of don't care conditions and automatically sets the constant inputs to their optimal values. The effectiveness of the algorithm is shown on a set of benchmark functions.
This paper proposes an approach to optimally synthesize quantum circuits by symbolic reachability analysis, where the primary inputs and outputs are basis binary and the internal signals can be nonbinary in a multiple-valued domain. The authors present an optimal synthesis method to minimize quantum cost and some speedup methods with nonoptimal quantum cost. The methods here are applicable to small reversible functions. Unlike previous works that use permutative reversible gates, a lower level library that includes nonpermutative quantum gates is used here. The proposed approach obtains the minimum cost quantum circuits for Miller gate, half adder, and full adder, which are better than previous results. This cost is minimum for any circuit using the set of quantum gates in this paper, where the control qubit of 2-qubit gates is always basis binary. In addition, the minimum quantum cost in the same manner for Fredkin, Peres, and Toffoli gates is proven. The method can also find the best conversion from an irreversible function to a reversible circuit as a byproduct of the generality of its formulation, thus synthesizing in principle arbitrary multi-output Boolean functions with quantum gate library. This paper constitutes the first successful experience of applying formal methods and satisfiability to quantum logic synthesis.
Reversible logic finds many applications, especially in the area of quantum computing. A completely specified n-input, n-output Boolean function is called reversible if it maps each input assignment to a unique output assignment and vice versa. Logic synthesis for reversible functions differs substantially from traditional logic synthesis and is currently an active area of research. The authors present an algorithm and tool for the synthesis of reversible functions. The algorithm uses the positive-polarity Reed-Muller expansion of a reversible function to synthesize the function as a network of Toffoli gates. At each stage, candidate factors, which represent subexpressions common between the Reed-Muller expansions of multiple outputs, are explored in the order of their attractiveness. The algorithm utilizes a priority-based search tree, and heuristics are used to rapidly prune the search space. The synthesis algorithm currently targets the generalized n-bit Toffoli gate library. However, other algorithms exist that can convert an n-bit Toffoli gate into a cascade of smaller Toffoli gates. Experimental results indicate that the authors' algorithm quickly synthesizes circuits when tested on the set of all reversible functions of three variables. Furthermore, it is able to quickly synthesize all four-variable and most five-variable reversible functions that were in the test suite. The authors also present results for some benchmark functions widely discussed in literature and some new benchmarks that the authors have developed. The algorithm is shown to synthesize many, but not all, randomly generated reversible functions of as many as 16 variables with a maximum gate count of 25
Abstract — This paper investigates a universal approach of synthesizing arbitrary ternary logic circuits in quantum computation based on the truth table technology. It takes into account of the relationship of classical logic and quantum logic circuits. By adding inputs with constant value and garbage outputs, the classical non-reversible logic can be transformed into reversible logic. Combined with group theory, it provides an algorithm using the ternary Swap gate, ternary NOT gate and ternary Toffoli gate library. Simultaneously, the main result shows that the numbers of qutrits we use are minimal compared to other methods. We also illustrate with two examples to test our approach. Index Terms — ternary quantum computation, reversible logic, circuit synthesis, group theory I.
The design of state machines for quantum algorithms remains a largely unexplored topic. The abstract models for quantum computation proposed by Feynman [1], Deutsch [2] etc have been related to the theory of Quantum Turing Machines (QTM) and the work was later expanded by Bernstein and Vazirani [3] to lay the foundations of quantum complexity theory. However, no papers deal with the following problem: “how to realize the quantum state machine in hardware as an operational quantum circuit that the internal states of which can be measured and used as inputs in a system of such machines”. This is an equivalent to the fundamental problem of “structural synthesis of FSM”. This paper creates models of quantum state machines that are useful from an engineering perspective. The objective of the paper is to propose models that can be used for circuit-level implementation. We restrict our study to binary logic realized in quantum, the so-called binary permutative quantum circuits; the “permutative machines”. The presented synthesis procedure is analogous to classical logic circuits synthesis and is a necessary step in the entire design procedure of quantum automata. Thus state minimization, state assignment and excitation function realization aspects are also of interest and some preliminary experimental results are presented.
The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.