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Kirchhoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers

Vadim V. Kostrykin, Robert Schrader

Published Jan 1, 2000• 158 citations
OpenAlex

Abstract

In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary $n\times n$ matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.

Related papers

Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers

V. Kostrykin, R. Schrader

In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S-matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.

arXiv:quant-ph/9910053•Fortschritte der Physik•Aug 1, 2000•110 citations

Kirchhoff's rule for quantum wires

V Kostrykin, R Schrader

We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S -matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S -matrix of a graph is factorizable in terms of the S -matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.

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Kirchhoff's rule for quantum wires

Vadim V. Kostrykin, R. Schrader, Technische Univ. Berlin (Germany). Sonderforschungsbereich 288 -Differentialgeometrie und Quantenphysik

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms. (orig.)

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Abstract In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case, where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary conditions for quantum graphs, building on previous works on transparent boundary conditions for the stationary Schrödinger equation on a line. Physically relevant constraints that make the vertex transparent under boundary conditions in the form of (weight) continuity and Kirchhoff rules are derived using two approaches, where transparent boundary conditions for the time-independent Schrödinger equation are found. The latter is derived by extending the transparent boundary condition concept to the time-independent Schrödinger equation on driven quantum graphs. For the constant potential, these constraints are also obtained using the scattering approach. We also discuss how the eigenvalues and eigenfunctions of a quantum graph are influenced not only by its topology, but also by the shape/type of a potential when an external field is involved.

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We consider an inverse problem for a finite graph $ (X, E) $ where we are given a subset of vertices $ B\subset X $ and the distances $ d_{(X, E)}(b_1, b_2) $ between all pairs of vertices $ b_1, b_2\in B $. The distance between vertices $ x_1, x_2\in X $ is defined as the minimal number of edges in paths connecting the vertices. This problem can be regarded as a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We develop quantum computing methods to find a solution to the problem, and show that the solution is unique under certain conditions. We prove the following uniqueness result: when $ (X, E) $ is a tree and $ B $ is the set of leaves of the tree, the graph $ (X, E) $ is uniquely determined in the class of all connected graphs having a fixed number of vertices. We present a quantum algorithm which, under arbitrary conditions, produces the graph $ (X, E) $, or one of those, which has the given number of vertices and the required distances between vertices in $ B $. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $ O(|X|^2) $ qubits, the same order as the number of elements in the adjacency matrix of $ (X, E) $, and it has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in the theory of computation, and show that a slight modification of the inverse problem above is NP-complete: all NP-problems can be reduced to a discrete inverse problem that we consider.

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Scattering theory and quantum walks

Mark S. Hillery, Edgar Feldman

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each. We are interested in walks that proceed from one half line, through the graph, to the other. The particle propagates freely on the half lines but is scattered at each vertex in the original graph. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph. An example is presented. Classical random walks on graphs can be used to construct algorithms that solve 2-SAT, graph connectivity problems, and for finding satsifying assignments for Boolean functions. A hope is that recently defined quantum walks will prove similarly useful in the development of quantum algorithms. In fact, it has recently been shown that it is possible to use a quantum walk to perform a search on the hypercube faster than can be done classically [1]. In this problem

•Dec 8, 2003•0 citations
In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n×nn\times nn×n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.
In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S-matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.
We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S -matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S -matrix of a graph is factorizable in terms of the S -matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.
In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms. (orig.)
Abstract In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case, where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary conditions for quantum graphs, building on previous works on transparent boundary conditions for the stationary Schrödinger equation on a line. Physically relevant constraints that make the vertex transparent under boundary conditions in the form of (weight) continuity and Kirchhoff rules are derived using two approaches, where transparent boundary conditions for the time-independent Schrödinger equation are found. The latter is derived by extending the transparent boundary condition concept to the time-independent Schrödinger equation on driven quantum graphs. For the constant potential, these constraints are also obtained using the scattering approach. We also discuss how the eigenvalues and eigenfunctions of a quantum graph are influenced not only by its topology, but also by the shape/type of a potential when an external field is involved.
We consider an inverse problem for a finite graph $ (X, E) $ where we are given a subset of vertices $ B\subset X $ and the distances $ d_{(X, E)}(b_1, b_2) $ between all pairs of vertices $ b_1, b_2\in B $. The distance between vertices $ x_1, x_2\in X $ is defined as the minimal number of edges in paths connecting the vertices. This problem can be regarded as a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We develop quantum computing methods to find a solution to the problem, and show that the solution is unique under certain conditions. We prove the following uniqueness result: when $ (X, E) $ is a tree and $ B $ is the set of leaves of the tree, the graph $ (X, E) $ is uniquely determined in the class of all connected graphs having a fixed number of vertices. We present a quantum algorithm which, under arbitrary conditions, produces the graph $ (X, E) $, or one of those, which has the given number of vertices and the required distances between vertices in $ B $. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $ O(|X|^2) $ qubits, the same order as the number of elements in the adjacency matrix of $ (X, E) $, and it has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in the theory of computation, and show that a slight modification of the inverse problem above is NP-complete: all NP-problems can be reduced to a discrete inverse problem that we consider.
We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each. We are interested in walks that proceed from one half line, through the graph, to the other. The particle propagates freely on the half lines but is scattered at each vertex in the original graph. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph. An example is presented. Classical random walks on graphs can be used to construct algorithms that solve 2-SAT, graph connectivity problems, and for finding satsifying assignments for Boolean functions. A hope is that recently defined quantum walks will prove similarly useful in the development of quantum algorithms. In fact, it has recently been shown that it is possible to use a quantum walk to perform a search on the hypercube faster than can be done classically [1]. In this problem