Quantum Automata Research Articles

Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits

We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as tln⁡t, saturating the bound established by Huang \cite{Huang_2020}. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.

Entanglement structure in the volume-law phase of hybrid quantum automaton circuits

We study entanglement fluctuations and quantum error correction in the weakly monitored volume-law phase of quantum automaton circuits subject to repeated local measurements. We numerically observe that the entanglement entropy exhibits strong fluctuation with the exponent close to the ``growth exponent'' of the Kardar-Parisi-Zhang (KPZ) universality class, the same as other local random circuits studied previously. We also investigate the dynamically generated quantum error correction code in the purification process and show that this model has different contiguous code distances for two types of errors that exhibit similar sublinear power-law scaling. We give an interpretation of these results by mapping them to various quantities in a classical particle model. We demonstrate that the subleading correction term of the entanglement entropy and the sublinear power-law scaling of the contiguous code distance in the volume-law phase are both the emergent phenomena of the hybrid random dynamics. Finally, we show that this classical particle dynamics itself has a type of error correction ability and can dynamically generate a classical linear code.

Distinction between transport and Rényi entropy growth in kinetically constrained models

Conservation laws and the associated hydrodynamic modes have important consequences on the growth of higher R\'enyi entropies in isolated quantum systems. It has been shown in various random unitary circuits and Hamiltonian systems that the dynamics of the R\'enyi entropies in the presence of a U(1) symmetry obey ${S}^{(n\ensuremath{\ge}2)}(t)\ensuremath{\propto}{t}^{1/z}$, where $z$ is identified as the dynamical exponent characterizing transport of the conserved charges. Here, however, we demonstrate that this simple identification may not hold in certain quantum systems with kinetic constraints. In particular, we study two types of U(1)-symmetric quantum automaton circuits with XNOR (exclusive NOR) and Fredkin constraints, respectively. We find numerically that while spin transport in both models is subdiffusive, the second R\'enyi entropy grows diffusively in the XNOR model, and superdiffusively in the Fredkin model. For systems with the XNOR constraint, this distinction arises since the spin correlation function can be attributed to an emergent tracer dynamics of tagged particles, whereas the R\'enyi entropies are constrained by collective transport of the particles. Our results suggest that care must be taken when relating transport and entanglement entropy dynamics in generic quantum systems with conservation laws.

Quantum simulations of hydrodynamics via the Madelung transformation

Developing numerical methods to simulate efficiently nonlinear fluid dynamics on universal quantum computers is a challenging problem. In this paper, a generalization of the Madelung transform is defined to solve quantum relativistic charged fluid equations interacting with external electromagnetic forces via the Dirac equation. The Dirac equation is discretized into discrete-time quantum walks which can be efficiently implemented on universal quantum computers. A variant of this algorithm is proposed to implement simulations using current noisy intermediate scale quantum (NISQ) devices in the case of homogeneous external forces. High resolution (up to $N={2}^{17}$ grid points) numerical simulations of relativistic and nonrelativistic hydrodynamical shocks on current IBM NISQs are performed with this algorithm. This paper demonstrates that fluid dynamics can be simulated on NISQs, and opens the door to simulating other fluids, including plasmas, with more general quantum walks and quantum automata.

Clifford algebra from quantum automata and unitary Wilson fermions

We introduce a spacetime discretization of the Dirac equation that has the form of a quantum automaton and that is invariant upon changing the representation of the Clifford algebra, like the Dirac equation itself. Our derivation follows Dirac's original one: We required that the square of the discrete Dirac scheme be what we define as an acceptable discretization of the Klein-Gordon equation. In contrast to standard lattice gauge theory in discrete time, in which unitarity needs to be proven, we show that the quantum automaton delivers naturally unitary Wilson fermions for any choice of Wilson's parameter.

Hidden quasiconservation laws in fracton hydrodynamics.

We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden quasiconservation laws, in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. First, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of harmonic multipole charges within linear response; we calculate the decay or growth of these charges due to dangerously irrelevant nonlinearities. Second, we consider a model with xy and x^{2}-y^{2} quadrupole conservation, in addition to dipole conservation, which is described by isotropic fourth-order subdiffusion, yet has dangerously irrelevant sixth-order corrections necessary to relax the harmonic multipole charges. We confirm our predictions for the anomalously slow decay of the harmonic conserved charges in each setting by using numerical simulations, both of the nonlinear hydrodynamic differential equations, and in quantum automaton circuits on a square lattice.

Two-way and one-way quantum and classical automata with advice for online minimization problems

We consider online algorithms. Typically, the model is investigated with respect to the competitive ratio. In this paper, we explore two-way automata and one-way automata as models for online algorithms. We focus on quantum and classical online algorithms. We show that there are problems that can be solved more efficiently by two-way automata with quantum and classical states than by classical two-way automata in the case of sublogarithmic memory (sublinear size), even if classical automata get advice bits. Additionally, we show that there are problems that can be solved more efficiently by one-way quantum automata than by classical one-way automata in the case of sublogarithmic memory (resp., sublinear size) and in the case of logarithmic memory (resp., linear size) even if classical automata get advice bits.

Measurement-induced criticality in Z2 -symmetric quantum automaton circuits

We study entanglement dynamics in hybrid $\mathbb{Z}_2$-symmetric quantum automaton circuits subject to local composite measurements. We show that there exists an entanglement phase transition from a volume law phase to a critical phase by varying the measurement rate $p$. By analyzing the underlying classical bit string dynamics, we demonstrate that the critical point belongs to parity-conserving universality class. We further show that the critical phase with $p>p_c$ is related to the diffusion-annihilation process and is protected by the $\mathbb{Z}_2$-symmetric measurement. We give an interpretation of the entanglement entropy in terms of a two-species particle model and identify the coefficient in front of the critical logarithmic entanglement scaling as the local persistent coefficient. The critical behavior observed at $p\geq p_c$ and the associated dynamical exponents are also confirmed in the purification dynamics.

Multifractality in nonunitary random dynamics

We explore the multifractality of the steady state wave function in non-unitary random quantum dynamics in one dimension. We focus on two classes of random systems: the hybrid Clifford circuit model and the non-unitary free fermion dynamics. In the hybrid Clifford model, we map the measurement driven transition to an Anderson localization transition in an effective graph space by using properties of the stabilizer state. We show that the volume law phase with nonzero measurement rate is non-ergodic in the graph space and exhibits weak multifractal behavior. We apply the same method to the hybrid Clifford quantum automaton circuit and obtain similar multifractality in the volume law phase. For the non-unitary random free fermion system with a critical steady state, we compute the moments of the probability distribution of the single particle wave function and demonstrate that it is also weakly multifractal and has strong variations in real space.

Multipole conservation laws and subdiffusion in any dimension.

Subdiffusion is a generic feature of chaotic many-body dynamics with multipole conservation laws and subsystem symmetries. We numerically study this subdiffusive dynamics, using quantum automaton random unitary circuits, in a broad range of models including one-dimensional models with dipole and quadrupole conservation, two-dimensional models with dipole conservation, and two-dimensional models with subsystem symmetry on the triangular lattice. Our results are in complete agreement with recent hydrodynamic predictions for such theories.

Quantum State Complexity in Computationally Tractable Quantum Circuits

Characterizing the quantum complexity of local random quantum circuits is a very deep problem with implications to the seemingly disparate fields of quantum information theory, quantum many-body physics and high energy physics. While our theoretical understanding of these systems has progressed in recent years, numerical approaches for studying these models remains severely limited. In this paper, we discuss a special class of numerically tractable quantum circuits, known as quantum automaton circuits, which may be particularly well suited for this task. These are circuits which preserve the computational basis, yet can produce highly entangled output wave functions. Using ideas from quantum complexity theory, especially those concerning unitary designs, we argue that automaton wave functions have high quantum state complexity. We look at a wide variety of metrics, including measurements of the output bit-string distribution and characterization of the generalized entanglement properties of the quantum state, and find that automaton wave functions closely approximate the behavior of fully Haar random states. In addition to this, we identify the generalized out-of-time ordered 2k-point correlation functions as a particularly useful probe of complexity in automaton circuits. Using these correlators, we are able to numerically study the growth of complexity well beyond the scrambling time for very large systems. As a result, we are able to present evidence of a linear growth of design complexity in local quantum circuits, consistent with conjectures from quantum information theory.

Yuri Manin on Biomolecules as Quantum Computers

In 1980, Russian mathematician Yuri Manin published Computable and Uncomputable. On pages 14 and 15 of his introduction, Manin suggests that “Molecular biology furnishes examples of the behavior of natural (not engineered by humans) systems which we have to describe in terms initially devised for discrete automata.” Manin then describes the remarkable energy efficiency of naturally occurring biomolecular processes such as DNA replication. He proposes modeling such behaviors in terms of unitary rotations in a finite-dimensional Hilbert space. The decomposition of such systems then corresponds to the tensor product decomposition of the state space, that is, to quantum entanglement. Manin’s initial focus on biological molecules as examples of highly energy-efficient quantum automata is unique among quantum computing’s founding figures since both he and other early leaders quickly moved to the then-new and exciting concept of von Neumann automata. The von Neumann formalism reinterpreted molecular quantum computing in terms of qubits, which made it possible to imagine the power of quantum computing as not much more than a superposition of virtual binary computers. This paper provides the original excerpt of Manin’s molecular computing argument. A useful analytical feature of Manin’s pre-von-Neumann model of quantum computation is its openness to new formalisms that avoid accidentally making classical physics dominant over the quantum world by expressing quantum states only in terms of concepts such as automata that assume extreme classical precision and complexity.

Measurement-induced phase transitions in quantum automaton circuits

We study the entanglement dynamics in a generic quantum automaton circuit subjected to projective measurements. We design an efficient algorithm which not only allows us to perform large scale simulation for the R\'enyi entropy but also provides a physical picture for the entanglement dynamics, which can be interpreted in terms of a classical bit-string model which belongs to the directed percolation universality class. We study the purification dynamics of a state formed by EPR pairs, and the growth of entanglement starting from a product state. In both cases, we verify numerically that the dynamics is in the universality class of classical directed percolation.

RTL Design of Efficient High-Speed Adders using Quantum-Dot Cellular Automatation

This article presents the design of high-speed adders using Quantum cellular automata. Quantum Automata is an efficient evaluation platform than CMOS in nanotechnology. Power, surface and performance play a virtual role in nanotechnology. Quantum Automata is an emerging technology in nanotechnology. QCA provides more speed, less power consumption and large scale integration in VLSI. By using Quantum-Dot cellular automata, we can reduce the power of leaks. Generally, for transferring the data and store data, electric fields are used in CMOS technology but in QCA, storing the data and transfer the data done by electronic polarization. This article presents different combinational logic circuits depend on QCA technology. The proposed modified CSLA (carry select adder) offer the best results of delay compared to the Ripple carry adder (RCA).

Experimental demonstration of quantum finite automaton

In quantum information science, a major task is to find the quantum models that can outperform their classical counterparts. Automaton is a fundamental computing model that has wide applications in many fields. It has been shown that the quantum version of automaton can solve certain problem using a much smaller state space compared to the classical automaton. Here we report an experimental demonstration of an optical quantum automaton, which is used to solve the promise problems of determining whether the length of an input string can be divided by a prime number P with no remainder or with a remainder of R. Our quantum automaton can solve such problem using a state space with only three orthonormal states, whereas the classical automaton needs no less than P states. Our results demonstrate the quantum benefits of a quantum automaton over its classical counterpart and paves the way for implementing quantum automaton for more complicated and practical applications.

Unary probabilistic and quantum automata on promise problems

We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error unary QFAs are more powerful than bounded-error unary PFAs, and, contrary to the binary language case, the computational power of Las-Vegas QFAs and bounded-error PFAs is equivalent to the computational power of deterministic finite automata (DFAs). Then, we present a new family of unary promise problems defined with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentially more succinct than DFAs.

Design of binary subtractor using actin quantum cellular automata

Actin is a biological protein that provides support to the cellular structure and plays a crucial role in cytoskeletal and intra-cellular signalling events. Logic circuits can be designed with actin filaments with the help of actin quantum automata. The authors use a rule (4,27) to implement some novel designs of logic subtractor circuits on this automata to achieve the difference in two binary bits. Logic design of both half and full binary subtractors is proposed in this study. Actin-based quantum cellular automata can be used in different combinations of input to get optimised results from the circuits. The authors focus on consolidating the designs inside single automata block to generate output in a less number of timesteps and less overheads. The designs are simulated with Simulink and this way output is verified for these different design approaches. Reliability and fault-tolerance check is another interesting part of this study. To get a better idea of the optimisation achieved, the authors have also presented a comparative study between the proposed designs in terms of circuit size and efficiency. With all these parameters involved, this study explores opportunities for future implementation of unconventional computing in nano-scale and cost-effective bio-molecular networks.

XOR transform with actin quantum automata for better security against cryptanalysis

XOR is a basic cryptographic transform, used as a one way function inside of complex cryptosystems. In this paper we propose first of a kind design of the XOR transform with a universal logic designing paradigm. The system offers enhanced security to the transform by propagating randomly generated information along with the XOR output over the unsecured channel. This additional security is a inherent attribute of the actin automata and is exploited in this paper. The system can be enacted on a Partitioned one-Dimensional Quantum Cellular Automata. A simulation of the proposed system is illustrated with Simulink tool in MATLAB. Based on the work done by Adamatzky et al. we observe that the actin automata is capable of operating on quantum input bits and can generate appropriate results when required. In this paper, the actin automata is simulated with classical bits instead of quantum bits for designing the transform. We have also tried to establish the proposed system by means of low power consumption verified with NI Multisim and comparison with cryptographic complexity with traditional XOR transform. Additionally, we have analyzed the system for enhanced cryptanalytic complexity against various attacks.

Promise problems solved by quantum and classical finite automata

The concept of promise problems was introduced and started to be systematically explored by Even, Selman, Yacobi, Goldreich, and other scholars. It has been argued that promise problems should be seen as partial decision problems and as such that they are more fundamental than decision problems and formal languages that used to be considered as the basic ones for complexity theory. The main purpose of this paper is to explore the promise problems accepted by classical, quantum and also semi-quantum finite automata. More specifically, we first introduce two acceptance modes of promise problems, recognizability and solvability, and explore their basic properties. Afterwards, we show several results concerning descriptional complexity on promise problems. In particular, we prove: (1) there is a promise problem that can be recognized exactly by measure-once one-way quantum finite automata (MO-1QFA), but no deterministic finite automata (DFA) can recognize it; (2) there is a promise problem that can be solved with error probability ϵ≤1/3 by one-way finite automaton with quantum and classical states (1QCFA), but no one-way probability finite automaton (PFA) can solve it with error probability ϵ≤1/3; and especially, (3) there are promise problems A(p) with size p that can be solved with any error probability by MO-1QFA with only two quantum basis states, but they can not be solved exactly by any MO-1QFA with two quantum basis states; in contrast, the minimal PFA solving A(p)with any error probability (usually smaller than 1/2) has p states. Finally, we mention a number of problems related to promise for further study.

Decision Problems on Unary Probabilistic and Quantum Automata

It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.