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Photonic Realization of a Quantum Finite Automaton PHYSICAL REVIEW RESEARCH 2, 013089 (2020) Photonic realization of a quantum finite automaton Carlo Mereghetti ,1 Beatrice Palano ,2 Simone Cialdi ,1,3 Valeria Vento ,1 Matteo G. A. Paris ,1,3 and Stefano Olivares 1,3,* 1Dipartimento di Fisica “Aldo Pontremoli,” Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy 2Dipartimento di Informatica “Giovanni Degli Antoni,” Università degli Studi di Milano, via Celoria 18, 20133 Milano, Italy 3INFN Sezione di Milano, via Celoria 16, 20133 Milano, Italy (Received 23 July 2019; published 28 January 2020) We describe a physical implementation of a quantum finite automaton that recognizes a well-known family of periodic languages. The realization exploits the polarization degree of freedom of single photons and their manipulation through linear optical elements. We use techniques of confidence amplification to reduce the acceptance error probability of the automaton. It is worth remarking that the quantum finite automaton we physically realize is not only interesting per se but it turns out to be a crucial building block in many quantum finite automaton design frameworks theoretically settled in the literature. DOI: 10.1103/PhysRevResearch.2.013089 I. INTRODUCTION has proved to be an extremely complex task, it is also rea- sonable to keep quantum components as small as possible. Quantum computing is a prolific research area, halfway Small-size quantum devices are modeled by quantum finite between physics and computer science [1–5]. Most likely, its automata, a theoretical model for quantum machines with origins may be dated back to the 1970s, when some work on finite memory. quantum information began to appear (see, e.g., Refs. [6,7]). Indeed, in current implementations of quantum computing, In the early 1980s, Feynman suggested that the computational the preparation and initialization of qubits in superposition power of quantum mechanical processes might be beyond that and/or entangled states is often challenging, making worth- of traditional computation models [8]. A similar idea was while the study of quantum computation with restricted mem- put forth by Manin [9]. Almost at the same time, Benioff ory, which requires less demanding resources, as in the case proved that such processes are at least as powerful as Turing of the quantum finite automata. machines [10]. In 1985, Deutsch proposed the notion of a The simplest and most promising from a physical realiza- quantum Turing machine as a physically realizable model for tion viewpoint model of a quantum finite automaton is the a quantum computer [11]. so-called measure-once quantum finite automaton [18–21]. The first impressive result witnessing “quantum power” Such a model also served as a basis for defining several was Shor’s algorithm for integer factorization, which could variants of quantum finite automata introduced and studied in run in polynomial time on a quantum computer [12]. It should plenty of contributions (see, e.g., Refs. [22–28]). Becasue it be stressed that no classical polynomial time factoring algo- is the only model considered in the present paper, from now rithm is currently known. On this fact, the security of many on for the sake of brevity we will simply write “quantum current cryptographic protocols, e.g., Rivest-Shamir-Adleman finite automaton” instead of “measure-once quantum finite (RSA) and Diffie-Hellman, actually relies. Relevant progress automaton.” was made by Grover, who proposed a quantum algorithm for The “hardware” of a (one-way) quantum finite automaton searching an item in an√ unsorted database containing n items, is that of a classical finite automaton. Thus, we have an which runs in time O( n)[13]. input tape scanned by a one-way input head moving one These and other theoretical advances naturally drove much position forward at each move, plus a finite basis state control. attention to efforts on the physical realization of quantum Some basis states are designated as accepting states. At any computational devices (see, e.g., Refs. [14–17]). While we given time during the computation, the state of the quantum can hardly expect to see a full-featured quantum computer in finite automaton is represented by a complex linear combina- the near future, it might be reasonable to envision classical tion of classical basis states, called a superposition. At each computing devices incorporating quantum components. Since step, a unitary transformation associated with the currently the physical realization of quantum computational systems scanned input symbol makes the automaton evolve to the next superposition. Superposition dynamics can transfer the complexity of the problem from a large number of sequential *stefano.olivares@fisica.unimi.it steps to a large number of coherently superposed quantum states. At the end of input processing, the automaton is Published by the American Physical Society under the terms of the observed in its final superposition. This operation makes the Creative Commons Attribution 4.0 International license. Further superposition collapse to a particular (classical) basis state distribution of this work must maintain attribution to the author(s) with a certain probability. The probability that the automaton and the published article’s title, journal citation, and DOI. accepts the input word is given by the probability of observing 2643-1564/2020/2(1)/013089(15) 013089-1 Published by the American Physical Society CARLO MEREGHETTI et al. PHYSICAL REVIEW RESEARCH 2, 013089 (2020) (collapsing into) an accepting basis state. Quantum finite a benchmark upon which to test the descriptional power of automata exhibit both advantages and disadvantages with classical and quantum finite automata, namely the language k respect to their classical (e.g., deterministic or probabilistic) Lm ={a | k ∈ N and k mod m = 0} for any given m > 0. counterparts. Basically, quantum superposition offers some We provide a theoretical definition of a quantum finite au- computational advantages on probabilistic superposition. On tomaton A accepting Lm with isolated cut point and two the other hand, quantum dynamics must be reversible, and this basis states, whereas any classical automaton for Lm requires requirement may impose severe computational limitations to a number of states which grows with m. finite memory devices. As a matter of fact, it is sometimes The photonic implementation of the quantum finite au- impossible to simulate classical finite automata by quantum tomaton A with two basis states is then discussed in Sec. IV. finite automata. In fact, as we will discuss in Sec. II D, isolated There, we start reviewing the standard quantum formalism cut point quantum finite automata recognize a proper subclass used to describe the polarization state of the single photon, of regular languages [18,20,21]. its dynamics, and the link with the formalism used in the Although weaker from a computational power point of previous sections. Then, we explain the working principle of view, quantum finite automata may greatly outperform clas- the photonic implementation of the quantum finite automaton sical ones when descriptional power is at stake. In the realm and propose a discrimination strategy to reduce the acceptance of descriptional complexity [29], models of computation are error probability. Section V describes the experimental appa- compared on the basis of their size. In the case of finite state ratus and reports the results we obtained. Finally, we close the machines, a commonly assumed size measure is the number paper with Sec. VI, where we draw some concluding remarks of finite control states. Most likely, the first contribution and the outlook for our work. explicitly studying the descriptional power of quantum versus classical finite automata is Ref. [30], where an extremely succinct quantum finite automaton is provided, accepting the II. PRELIMINARIES k unary language Lm ={a | k ∈ N and k mod m = 0} for any given m > 0. The construction in Ref. [30]usesasabasic(and A. Formal languages and classical finite automata sole) module a quantum finite automaton A for Lm with 2 ba- Formal language theory studies languages from a math- sis states, whose acceptance reliability is then enhanced within ematical point of view, providing formal tools and methods a suitable modular building framework where traditional com- to analyze language properties. Strictly connected with au- positions (i.e., direct products and sums) of quantum systems tomata theory, the discipline dates back to the 1950s, and are performed. Actually, many (if not all) contributions in the it was originally developed to provide a theoretical basis for literature aiming to design small size quantum finite automata natural language processing. It was soon realized that this the- for several tasks (see, e.g., Refs. [25,31–38]) use the module ory was relevant to the artificial languages (e.g., programming A as a crucial building block. In this sense, the language Lm languages) that had originated in computer science. Since its and the module A turn out to be “paradigmatic” as tools to birth, formal language theory has become established as one build and test size-efficient quantum finite automata. Hence, of the most prominent area in theoretical computer science. Its a physical realization of the module A might be well worth results have huge impacts in numerous fields, including prac- investigating. tical computer science, cryptography and security, discrete In this paper, we put forward a physical implementation of mathematics and combinatorics, graph theory, mathemati- quantum finite automata based on the polarization degree of cal logic, nature-inspired (e.g., quantum, biological, genetic) freedom of single photons and able to recognize a family of computational models, physics, and system theory. periodic languages. More precisely, because of above stressed The reader may find a lot of excellent textbooks where centrality in quantum finite automaton design frameworks, thoughtful presentations of formal language and automata we focus on the physical implementation of the quantum theory and their applications are presented (see, e.g., finite automaton A for the language Lm.
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