Quantum walks in spaces with applied potentials

Eur. Phys. J. Plus (2023) 138:312 https://doi.org/10.1140/epjp/s13360-023-03921-6 Regular Article Quantum walks in spaces with applied potentials Georgios D. Varsamis1, Ioannis G. Karafyllidis1,2,a , Georgios Ch. Sirakoulis1 1 Department of Electrical and Computer Engineering, Democritus University of Thrace, 67100 Xanthi, Greece 2 National Centre for Scientific Research Demokritos, 15342 Athens, Greece Received: 19 September 2022 / Accepted: 22 March 2023 © The Author(s) 2023 Abstract Discrete quantum walks are a universal model of quantum computation equivalent to the quantum circuit model and can be mapped onto quantum circuits and executed using quantum computers. Quantum walks can model and simulate many physical systems and several quantum algorithms are based on them. Discrete quantum walks have been extensively studied, but quantum walks that evolve in spaces in which potentials are applied received little or no attention. Here, we formulate the discrete quantum walk model in one and two-dimensional spaces in which potentials are applied. In this formulation the quantum walker carries a “charge” affected by the potentials and the walk evolution is driven by both constant and time-varying potentials. We reproduce the tunneling through a barrier phenomenon and study the quantum walk evolution in one and two-dimensional spaces with various potential distributions. We demonstrate that our formulation can serve as a basis for applied quantum computing by studying maze running and the motion of vehicles in urban spaces. In these spaces curbs and buildings are modeled as impenetrable potential barriers and traffic lights as time-varying potential barriers. Quantum walks in spaces with applied potentials may open the way for the development of novel quantum algorithms in which inputs are introduced as potential profiles. 1 Introduction Quantum walks are proven to be a universal model of quantum computation, equivalent to the quantum circuit (gate) model [1–5]. Quantum walks can directly be mapped onto quantum circuits and executed by quantum computers [6–8] and can also be implemented in various platforms [9]. Furthermore, quantum walks can be combined with quantum cellular automata to perform efficient quantum computations [10–12]. Among others, quantum walks have been used in quantum search [13, 14], in data safety [15], in Hash functions generation [16], in quantum encryption [17], in quantum transport, in graphene [18, 19] and in the simulation of bosonic and fermionic quantum systems [20]. In all above quantum walk applications, the quantum walk evolves freely in one and two-dimensional lattices and graphs, i.e., in spaces with no applied potentials. The specificities of each problem enter the quantum computation through the structure of lattices and graphs. Areas in which the quantum walker should not enter are defined using broken links. In this paper, we extent the quantum walk model of quantum computation by introducing potentials in the spaces where the quantum walk evolves. The quantum walker carries a “charge” which is affected by the applied potential and, therefore, the potential profiles can be used to enter the inputs of the specific quantum computation in addition to the structure of lattices and graphs. The output of the quantum computation is the probability distribution of the possible locations of the quantum walker, or, in specific cases, the final location of the quantum walker. The fact that the quantum walker carries a “charge” along with the fact that the probability amplitudes of its location change with time, allows us to use quantum phase gates to introduce the potential effect on the evolution of the quantum walk. In the next section, we present the formulation of the quantum walk in spaces with potentials. In the third section, we present the mapping of the model onto quantum circuits and simulate the function of the circuit modules using Qiskit and IBM’s quantum computer. In the fourth section we study the evolution of quantum walks in one-dimensional spaces with applied potentials, with an emphasis on the reproduction of the tunneling through a potential barrier phenomenon. In the next section, we study the quantum walk evolution in two-dimensional spaces with applied potentials. In the sixth section we apply our model to develop a basis for mazerunning and routing quantum algorithms and we introduce time-varying potentials and study the possibility of using quantum walks to simulate the motion of vehicles in urban spaces. Immovable obstacles such as curbs and buildings are modeled as impenetrable potential barriers. Traffic lights are modeled as time-varying potential barriers. Red traffic lights are modeled as potential barriers with near-zero transmission coefficients and green lights as zero applied potentials. Finally, we present our conclusions. We believe a e-mail: (corresponding author) 0123456789().: V,-vol 123 312 Page 2 of 10 Eur. Phys. J. Plus (2023) 138:312 that quantum walks in spaces with applied potentials can be used as a basis to develop novel quantum algorithms and also novel hybrid quantum/classical algorithms. 2 Model formulation We formulate our model using the one-dimensional discrete quantum walk. Its extension to more dimensions is straightforward. The quantum walk evolves in a one-dimensional discrete lattice and the associated quantum computational basis is |n, where n is a positive integer representing the lattice sites. The basis states |n label the possible positions of the quantum walker and span the corresponding Hilbert space, H P , which is called the position space [1, 2]. To determine the direction of the quantum walker motion, a two-state quantum system, called a coin, is used. The coin basis states are labeled as |0 and |1 and span a two-dimensional Hilbert space, H C , called the coin space [1, 2]. The quantum walk evolves in the Hilbert space, H QW , which comprises H P and H C and is given by their tensor product: H QW  H P ⊗ HC (1) Next, we define the unitary operators that act on the basis states and drive the quantum walk. The first operator acts on the coin  The coin operator is represented by a 2X2 unitary matrix: basis states and is called the coin operator, C.     a11 a12 (2) C a21 a22  on the coin basis states is given by: The action of C      a11 a12 1 a11  C|0    a11 |0 + a21 |1 a21 a22 0 a21 and      a12 a11 a12 0    a12 |0 + a22 |1 C|1  a21 a22 1 a22 (3) (4) The second operator is called the shift operator,  S, and acts on the position basis states.  S shifts the quantum walker to the left or right according to the current coin state. Suppose that at time step t the quantum walker is located at lattice sitei with probability amplitude ai,t . The coin is tossed at time step t + 1 and if its state is |0 the action of shift operator moves the walker to lattice site i-1 and the probability amplitude becomes ai,t+1 :  S|0ai, (...truncated)