The Ising Machine as a Physical Non-Deterministic Computing Device: Formal Proof, Polynomial-Time Equivalence for NP, and Cryptographic Implications at Scale
We present a rigorous, self-contained proof that an idealised Ising Machine (IM)-defined axiomatically from the experimentally observed properties of coherent Ising machines (CIMs), quantum annealers, and memristive/optical Ising processors-satisfies every clause of the canonical definition of a non-deterministic computing device in the sense of Turing (1936), Rabin and Scott (1959), and Cook (1971). Concretely, we show that (i) The physical ground-state search performed by an N-spin Ising Machine is not a deterministic computation: the machine simultaneously and physically explores an exponential number of spin configurations, and the branch that is "selected" (the ground state) is not the output of any single deterministic trajectory. (ii) There exists a polynomial-time simulation of any non-deterministic Turing machine (NDTM) by a suitably scaled Ising Machine, and conversely, every Ising Machine computation can be simulated by an NDTM in polynomial time. Hence the Ising Machine decides exactly the class NP in polynomial physical time. (iii) Because NP is, by definition, the class of languages decidable in polynomial time on a non-deterministic machine, a scalable Ising Machine would solve every NP problem in polynomial time, rendering all cryptographic schemes whose security relies on NP-hard or presumed-intractable problems (RSA, ECC, lattice-based, subset-sum-based, code-based, etc.) insecure. The argument is structured axiomatically: we first isolate the minimal physical axioms an Ising Machine must satisfy, then construct a formal mapping to the NDTM model, and finally derive the complexity-theoretic and cryptographic consequences. We discuss the crucial role of the scalability assumption and the engineering gap that currently prevents the full realisation of the threat.
We present a rigorous, self-contained proof that an idealised Ising Machine (IM)-defined axiomatically from the experimentally observed properties of coherent Ising machines (CIMs), quantum annealers, and memristive/optical Ising processors-satisfies every clause of the canonical definition of a non-deterministic computing device in the sense of Turing (1936), Rabin and Scott (1959), and Cook (1971). Concretely, we show that (i) The physical ground-state search performed by an N-spin Ising Machine is not a deterministic computation: the machine simultaneously and physically explores an exponential number of spin configurations, and the branch that is "selected" (the ground state) is not the output of any single deterministic trajectory. (ii) There exists a polynomial-time simulation of any non-deterministic Turing machine (NDTM) by a suitably scaled Ising Machine, and conversely, every Ising Machine computation can be simulated by an NDTM in polynomial time. Hence the Ising Machine decides exactly the class NP in polynomial physical time. (iii) Because NP is, by definition, the class of languages decidable in polynomial time on a non-deterministic machine, a scalable Ising Machine would solve every NP problem in polynomial time, rendering all cryptographic schemes whose security relies on NP-hard or presumed-intractable problems (RSA, ECC, lattice-based, subset-sum-based, code-based, etc.) insecure. The argument is structured axiomatically: we first isolate the minimal physical axioms an Ising Machine must satisfy, then construct a formal mapping to the NDTM model, and finally derive the complexity-theoretic and cryptographic consequences. We discuss the crucial role of the scalability assumption and the engineering gap that currently prevents the full realisation of the threat.
The central problem in computer science is the conjecture that two complexity classes, P (polynomial time) and NP (nondeterministic polynomial time-roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: P not equal NP. As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all #P problems, a computationally intractable class, in polynomial time. Specifically, Witten [Witten, E. (1989) Commun. Math. Phys. 121, 351-391] has identified expectation values in a certain SU(2)-field theory with values of the Jones polynomial [Jones, V. (1985) Bull. Am. Math. Soc. 12, 103-111] that are #P-hard [Jaeger, F., Vertigen, D. & Welsh, D. (1990) Math. Proc. Comb. Philos. Soc. 108, 35-53]. This suggests that some physical system whose effective Lagrangian contains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving NP or even #P-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.
In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch's model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(logT) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class $\BPP$. The class $\BQP$ of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
Ising machines are hardware solvers which aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because it is possible to formulate any problem in the complexity class NP as an Ising problem with only polynomial overhead. A scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications for a wide variety of optimization problems. In this review, we survey the current status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors, and digital hardware accelerators; dynamical-systems solvers implemented with optics and electronics; and superconducting-circuit quantum annealers. We compare and contrast their performance using standard metrics such as the ground-state success probability and time-to-solution, give their scaling relations with problem size, and discuss their strengths and weaknesses.
Abstract In this chapter, we introduce the definitions of automata Turing machines. We present different flavors of Turing machines including deterministic, probabilistic, and multi-tape machines. We present several complexity classes, i.e. collections of problems that can be solved by a machine with similar resources, including the P and NP classes. We present the Church-Turing thesis, which states that if a problem can be solved by a physical machine, then it can also be solved by any deterministic Turing machine. In the last part of the chapter, we introduce quantum Turing machines and the quantum circuit model. We study the complexity class BQP. This is the class of problems that can be solved by a quantum computer with polynomial resources and a small error probability. Finally, we introduce QMA, the quantum version of the complexity class NP. We present the k-LOCAL problem, and we show that this problem is in QMA
Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
In this paper, we introduce two formal means by which physical adversarial actions and features can be modeled in cryptography and security: The concepts of a “physical Turing machine (PhTM or φ-TM)” and of a “technology” on which the PhTM operates. We show by two examples how these concepts can be applied: Firstly, we sketch their use in formalizing physical adversarial computations (quantum computation [4], optical techniques [26, 15], etc.) in classical cryptpography, which an adversary might carry out to attack complexity-based schemes. Secondly, we work out in more detail the application of PhTMs in the formal treatment of physical unclonable functions and physical cryptography in general, in which disordered, unclonable physical objects are used for cryptographic purposes. PhTMs allow a rigid formal expression of the required properties of these objects (such as their physical unclonability), and enable us to lead formal reductionist proofs in this field. The hybrid nature of PhTMs thereby allows us to combine physical with computational assumptions in the proof. As an example, we lead a formal proof of a physical scheme that combines a classical digital signature with an unclonable, unique object in order to “label” or “tag” valuable objects securely and in a forgery-proof manner. We stress that PhTMs as introduced in this paper cannot directly and straightforwardly answer the question which physical tasks are eventually feasible and infeasible in our universe. But such an expectation would be unreasonably high; recall that classical Turing machines also do not allow to draw a simple line between feasible and infeasible computations, as the NP vs. P issue shows. Rather, they provide us with a formal backbone in which relevant physical security features can be expressed and security proofs can be led. Apart from the applications sketched in this paper, many other uses of PhTMs lie at hand, for example in defining security against side channels or invasive attacks, or in the development of a “physical” structural complexity theory, which are left to future work.